goodwin a.r.h., sengers j.v., peters c.j. (eds.) applied thermodynamics of fluids (rsc, 2010)(isbn...

Applied Thermodynamics of Fluids

Applied Thermodynamics ofFluids

Edited by

A. R. H. GoodwinSchlumberger Technology Corporation, Sugar Land, TX, USA

J. V. SengersInstitute for Physical Science and Technology, University of Maryland, CollegePark, MD, USA

C. J. PetersDepartment of Process & Energy, Delft University of Technology, Delft,The Netherlands and Department of Chemical Engineering,The Petroleum Institute, Abu Dhabi, United Arab Emirates

International Union of Pure and Applied ChemistryPhysical and BioPhysical Chemistry Division

IUPAC Associated OrganizationInternational Association of Chemical Thermodynamics

ISBN: 978-1-84755-806-0

A catalogue record for this book is available from the British Library

r International Union of Pure and Applied Chemistry 2010

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Preface

Monographs concerned with the field of chemical thermodynamics were initi-ated by Commission 1.2 of the International Union of Pure and AppliedChemistry (IUPAC) and summarize the state of knowledge with regard toexperimental and theoretical methods in thermodynamics and thermo-chemistry. These texts have appeared in two series, the first1,2 reportingmethods in thermochemistry. The present volume is the eighth in the secondseries3–9 concerned with methods of measuring the thermophysical propertiesof substances and is now published under the auspices of both IUPAC andthe International Association of Chemical Thermodynamics (IACT) that isan affiliate of IUPAC. The first volume of the second series was concernedwith calorimetry of non-reacting systems,3 while the third volume5 continuedthe theme of non-reacting systems with measurements of the transport prop-erties characteristic of the relaxation of a fluid from a non-equilbrium state.The fourth monograph was concerned with the calorimetry of reactingfluids6 and also provided updates to the first series.1,2 The sixth and seventhvolumes8,9 were concerned with the measurement of the thermodynamicproperties of single and multiple phases respectively. They were an update ofthe second volume4 and are focused on measurements of a broader class ofthermodynamic properties and state variables over a wide range of tempera-ture and pressure including techniques with industrial applications for chemi-cally non-reacting systems. These are also of interest to the data evaluatorwho needs to assess the reliability of experimental data obtained with specifictechniques.The fifth volume7 presented the theoretical basis for equations of state of

both fluids and fluid mixtures along with practical uses of each equation typeand has been independently described as important for engineers and physicalchemists.10 Nevertheless, some subject matter of importance to the practitionerwas omitted from reference 7, including equations of state for chemicallyreacting and non-equilibrium fluids, and others have undergone significant


Edited by A. R. H. Goodwin, J. V. Sengers and C. J. Petersr International Union of Pure and Applied Chemistry 2010

Published by the Royal Society of Chemistry, www.rsc.org

v

developments that now deserve more detailed explanations than providedhitherto.In view of both developments in the field and the desire to provide the

content omitted in the previous volume led to a request made to the Board ofIACT to initiate an update of Volume V that was overwhelmingly endorsed andbecame IUPAC project 2008-014-1-100 in 2008. The principal remit providedfor this volume, as it was for Volumes VI and VII, is to serve as a guide to thescientist or technician who use equations of state for fluids. It is complementaryto, rather than a replacement for Volume V, and concentrates on the appli-cation of theory, includes the material omitted previously and greater attentionis given to those topics that have significant enhancements. The success of thefifth volume7 implied this book should also be comprehensive and mustnecessarily include matters of fundamental importance.With this broad remit, the Editors suggested a revision of the title to Applied

Thermodynamics of Fluids to reflect a greater emphasis on the application oftheory and proceeded to address the needs of practitioners within academia,government and industry by assembling an international team of distinguishedexperts to provide each chapter.This volume compliments other recent publications associated with IUPAC

and IACT that have covered a range of diverse issues reporting applications ofsolubility data,11 to the topical issue of alternate sources of energy,12 heatcapacities of liquids and vapours and the application of chemical thermo-dynamics to other matters of current industrial and scientific research includingseparation technology, biology, medicine and petroleum in one13 of elevenmonographs of an IUPAC series entitled Chemistry for the 21st Century.14

Anthony R. H. GoodwinChairman, International Association of Chemical Thermodynamics and

Associate Member, Physical and BioPhysical Division (I)of the International Union of Pure and Applied Chemistry

References

1. Experimental Thermochemistry, F. D. Rossini, ed., For IUPAC, Inter-science, New York, 1956.

2. Experimental Thermochemistry, Volume II, H. A. Skinner, ed., for IUPAC,Interscience, New York, 1962.

3. Experimental Thermodynamics, Volume I, Calorimetry of Non-ReactingSystems, J. P. McCullough and D. W. Scott, eds, for IUPAC, Butter-worths, London, 1968.

4. Experimental Thermodynamics, Volume II, Experimental Thermodynamicsof Non-Reacting Fluids, B. Le Neindre and B. Vodar, eds, For IUPAC,Butterworths, London, 1975.

vi Preface

5. Experimental Thermodynamics, Volume III, Measurement of the TransportProperties of Fluids, W. A. Wakeham, A. Nagashima and J. V. Sengers,eds., For IUPAC, Blackwell Scientific Publications, Oxford, 1991.

6. Experimental Thermodynamics, Volume IV, Solution Calorimetry, K. N.Marsh and P. A. G. O’Hare, eds, For IUPAC, Blackwell Scientific Pub-lications, Oxford, 1994.

7. Experimental Thermodynamics, Volume V, Equations of State for Fluids andFluid Mixtures, Parts I and II, J. V. Sengers, R. F. Kayser, C. J. Peters andH. J. White, Jr., eds, For IUPAC, Elsevier, Amsterdam, 2000.

8. Experimental Thermodynamics, Volume VI, Measurement of the Thermo-dynamic Properties of Single Phases, A. R. H. Goodwin, K. N. Marsh andW. A. Wakeham, eds, for IUPAC, Elsevier, Amsterdam, 2003.

9. Experimental Thermodynamics, Volume VII, Measurement of the Thermo-dynamic Properties of Multiple Phases, R. D. Weir and T. W. de Loos, eds,For IUPAC, Elsevier, Amsterdam, 2005.

10. L. S. Garcia-Colin and F. J. Uribe, J. Stat. Phys., 2002, 106, 403-404; 2003,112, 885.

11. Developments and Applications of Solubility T. M. Letcher, ed., for theIUPAC, Royal Society of Chemistry, Cambridge, 2007.

12. Future Energy: Improved, Sustainable and Clean Options for our Planet,T. M. Letcher, ed., for IUPAC, Elsevier, Amsterdam, 2008.

13. Chemical Thermodynamics, T. M. Letcher, ed., for IUPAC, BlackwellScientific Publications, Oxford, 2000.

14. C. L. Watkins, J. Chem. Educ., 2000, 77, 973.

viiPreface

Contents

List of Contributors xix

Experimental Thermodynamics Series xxi

Acknowledgments xxiii

Chapter 1 Introduction 1

Anthony R. H. Goodwin, Jan V. Sengers and Cornelis

J. PetersReferences 2

Chapter 2 Fundamental Considerations 5

Angel Martın Martınez and Cor J. Peters

2.1 Introduction 52.2 Basic Thermodynamics 5

2.2.1 Homogeneous Functions 82.2.2 Thermodynamic Properties from Differentia-

tion of Fundamental Equations 92.3 Deviation Functions 11

2.3.1 Residual Functions 122.3.2 Evaluation of Residual Functions 13

2.4 Mixing and Departure Functions 132.4.1 Departure Functions with Temperature, Mo-

lar Volume and Composition as the Inde-pendent Variables 14

2.4.2 Departure Functions with Temperature,Pressure and Composition as the IndependentVariables 16

2.5 Mixing and Excess Functions 172.6 Partial Molar Properties 192.7 Fugacity and Fugacity Coefficients 20


Edited by A. R. H. Goodwin, J. V. Sengers and C. J. Peters

r International Union of Pure and Applied Chemistry 2010


ix

2.8 Activity Coefficients 222.9 The Phase Rule 24

2.10 Equilibrium Conditions 252.10.1 Phase Equilibria 252.10.2 Chemical Equilibria 27

2.11 Stability and the Critical State 282.11.1 Densities and Fields 282.11.2 Stability 282.11.3 Critical State 29

References 32

Chapter 3 The Virial Equation of State 33

J. P. Martin Trusler

3.1 Introduction 333.1.1 Temperature Dependence of the Virial

Coefficients 343.1.2 Composition Dependence of the Virial

Coefficients 353.1.3 Convergence of the Virial Series 363.1.4 The Pressure Series 37

3.2 Theoretical Background 383.2.1 Virial Coefficients of Hard-Core-Square-Well

Molecules 393.3 Thermodynamic Properties of Gases 40

3.3.1 Perfect-gas and Residual Properties 403.3.2 Helmholtz Energy and Gibbs Energy 413.3.3 Perfect-Gas Properties 413.3.4 Residual Properties 44

3.4 Estimation of Second and Third Virial Coefficients 473.4.1 Application of Intermolecular Potential-energy

Functions 473.4.2 Corresponding-states Methods 48

References 51

Chapter 4 Cubic and Generalized van der Waals Equations of State 53

Ioannis G. Economou

4.1 Introduction 534.2 Cubic Equation of State Formulation 54

4.2.1 The van der Waals Equation of State (1873) 544.2.2 The Redlich and Kwong Equation of State

(1949) 564.2.3 The Soave, Redlich and Kwong Equation of

State (1972) 564.2.4 The Peng and Robinson Equation of State (1976) 57

x Contents

4.2.5 The Patel and Teja (PT) Equation of State(1982) 58

4.2.6 The a Parameter 584.2.7 Volume Translation 594.2.8 The Elliott, Suresh and Donohue (ESD)

Equation of State (1990) 604.2.9 Higher-Order Equations of State Rooted to

the Cubic Equations of State 614.2.10 Extension of Cubic Equations of State to

Mixtures 624.3 Applications 64

4.3.1 Pure Components 644.3.2 Oil and Gas Industry – Hydrocarbons and

Petroleum Fractions 654.3.3 Chemical Industry – Polar and Hydrogen

Bonding Fluids 684.3.4 Polymers 744.3.5 Transport Properties 77

4.4 Conclusions 81References 81

Chapter 5 Mixing and Combining Rules 84

Anthony R. H. Goodwin and Stanley I. Sandler

5.1 Introduction 845.2 The Virial Equation of State 855.3 Cubic Equations of State 87

5.3.1 Mixing Rules 885.3.2 Combining Rules 925.3.3 Non-Quadratic Mixing and Combining Rules 975.3.4 Mixing Rules that Combine an Equation of

State with an Activity-Coefficient Model 1005.4 Multi-Parameter Equations of State 111

5.4.1 Benedict, Webb, and Rubin Equation of State 1115.4.2 Generalization with the Acentric Factor 1145.4.3 Helmholtz-Function Equations of State 118

5.5 Mixing Rules for Hard Spheres and Association 1215.5.1 Mixing and Combining Rules for SAFT 1235.5.2 Cubic Plus Association Equation of State 125

References 127

Chapter 6 The Corresponding-States Principle 135

James F. Ely

6.1 Introduction 1356.2 Theoretical Considerations 139

xiContents

6.3 Determination of Shape Factors 1426.3.1 Other Reference Fluids 1446.3.2 Exact Shape Factors 1466.3.3 Shape Factors from Generalized Equations of

State 1546.4 Mixtures 156

6.4.1 van der Waals One-Fluid Theory 1586.4.2 Mixture Corresponding-States Relations 160

6.5 Applications of Corresponding-States Theory 1626.5.1 Extended Corresponding-States for Natural

Gas Systems 1646.5.2 Extended Lee-Kesler 1656.5.3 Generalized Crossover Cubic Equation of

State 1656.6 Conclusions 166References 166

Chapter 7 Thermodynamics of Fluids at Meso and Nano Scales 172

Mikhail A. Anisimov and Christopher E. Bertrand

7.1 Introduction 1727.2 Thermodynamic Approach to Meso-Heterogeneous

Systems 1747.2.1 Equilibrium Fluctuations 1747.2.2 Local Helmholtz Energy 176

7.3 Applications of Meso-Thermodynamics 1797.3.1 Van der Waals Theory of a Smooth Interface 1797.3.2 Polymer Chain in a Dilute Solution 1827.3.3 Building a Nanoparticle Through Self Assembly 1847.3.4 Modulated Fluid Phases 187

7.4 Meso-Thermodynamics of Criticality 1897.4.1 Critical Fluctuations 1897.4.2 Scaling Relations 1927.4.3 Near-Critical Interface 1937.4.4 Divergence of Tolman’s Length 195

7.5 Competition of Meso-Scales 1967.5.1 Crossover to Tricriticality in Polymer

Solutions 1967.5.2 Tolman’s Length in Polymer Solutions 2007.5.3 Finite-size Scaling 202

7.6 Non-Equilibrium Meso-Thermodynamics of Fluid

Phase Separation 2057.6.1 Relaxation of Fluctuations 2067.6.2 Critical Slowing Down 2077.6.3 Homogeneous Nucleation 208

xii Contents

7.6.4 Spinodal Decomposition 2087.7 Conclusion 209References 210

Chapter 8 SAFT Associating Fluids and Fluid Mixtures 215

Clare McCabe and Amparo Galindo

8.1 Introduction 2158.2 Statistical Mechanical Theories of Association and

Wertheim’s Theory 2168.3 SAFT Equations of State 222

8.3.1 SAFT-HS and SAFT-HR 2248.3.2 Soft-SAFT 2258.3.3 SAFT-VR 2268.3.4 PC-SAFT 2268.3.5 Summary 227

8.4 Extensions of the SAFT Approach 2288.4.1 Modelling the Critical Region 2288.4.2 Polar Fluids 2348.4.3 Ion-Containing Fluids 2408.4.4 Modelling Inhomogeneous Fluids 2478.4.5 Dense Phases: Liquid Crystals and Solids 249

8.5 Parameter Estimation: Towards more Predictive

Approaches 2498.5.1 Pure-component Parameter Estimation 2498.5.2 Use of Quantum Mechanics in SAFT

Equations of State 2518.5.3 Unlike Binary Intermolecular Parameters 252

8.6 SAFT Group-Contribution Approaches 2538.6.1 Homonuclear Group-Contribution Models in

SAFT 2558.6.2 Heteronuclear Group Contribution Models in

SAFT 2568.7 Concluding Remarks 260References 260

Chapter 9 Polydisperse Fluids 280

Dieter Browarzik

9.1 Introduction 2809.2 Influence of Polydispersity on the Liquid+Liquid

Equilibrium of a Polymer Solution 2819.3 Approaches to Polydispersity 283

9.3.1 The Pseudo-component Method 2839.3.2 Continuous Thermodynamics 285

xiiiContents

9.4 Application to Real Systems 2999.4.1 Polymer Systems 2999.4.2 Petroleum Fluids, Asphaltenes, Waxes and

Other Applications 3069.5 Conclusions 313References 314

Chapter 10 Thermodynamic Behaviour of Fluids near Critical Points 321

Hassan Behnejad, Jan V. Sengers and Mikhail A. Anisimov

10.1 Introduction 32110.2 General Theory of Critical Behaviour 322

10.2.1 Scaling Fields, Critical Exponents, andCritical Amplitudes 322

10.2.2 Parametric Equation of State 32510.3. One-Component Fluids 328

10.3.1 Simple Scaling 32810.3.2 Revised Scaling 33210.3.3 Complete Scaling 33310.3.4 Vapour-Liquid Equilibrium 33510.3.5 Symmetric Corrections to Scaling 338

10.4 Binary Fluid Mixtures 33910.4.1 Isomorphic Critical Behaviour of Mixtures 33910.4.2 Incompressible Liquid Mixtures 34010.4.3 Weakly Compressible Liquid Mixtures 34410.4.4 Compressible Fluid Mixtures 34710.4.5 Dilute Solutions 347

10.5 Crossover Critical Behaviour 34910.5.1 Crossover from Ising-like to Mean-Field

Critical Behaviour 34910.5.2 Effective Critical Exponents 35110.5.3 Global Crossover Behaviour of Fluids 354

10.6 Discussion 359Acknowledgements 359References 359

Chapter 11 Phase Behaviour of Ionic Liquid Systems 368

Maaike C. Kron and Cor J. Peters

11.1 Introduction 36811.2 Phase Behaviour of Binary Ionic Liquid Systems 369

11.2.1 Phase Behaviour of (Ionic Liquid+GasMixtures) 369

11.2.2 Phase Behaviour of (Ionic Liquid+Water) 37211.2.3 Phase Behaviour of (Ionic Liquid+Organic) 373

xiv Contents

11.3 Phase Behaviour of Ternary Ionic Liquid Systems 37411.3.1 Phase Behaviour of (Ionic Liquid+Carbon

Dioxide+Organic) 37411.3.2 Phase Behaviour of (Ionic Liquid+

Aliphatic+Aromatic) 37611.3.3 Phase Behaviour of (Ionic Liquid+

Water+Alcohol) 37711.3.4 Phase Behaviour of Ionic Liquid Systems

with Azeotropic Organic Mixtures 37811.4 Modeling of the Phase Behaviour of Ionic Liquid

Systems 37911.4.1 Molecular Simulations 37911.4.2 Excess Gibbs-energy Methods 38011.4.3 Equation of State Modeling 38111.4.4 Quantum Chemical Methods 383

References 383

Chapter 12 Multi-parameter Equations of State for Pure Fluids and

Mixtures 394

Eric W. Lemmon and Roland Span

12.1 Introduction 39412.2 The Development of a Thermodynamic Property

Formulation 39512.3 Fitting an Equation of State to Experimental Data 397

12.3.1 Recent Nonlinear Fitting Methods 40212.4 Pressure-Explicit Equations of State 404

12.4.1 Cubic Equations 40412.4.2 The Benedict-Webb-Rubin Equation of State 40412.4.3 The Bender Equation of State 40512.4.4 The Jacobsen-Stewart Equation of State 40612.4.5 Thermodynamic Properties from Pressure-

Explicit Equations of State 40612.5 Fundamental Equations 408

12.5.1 The Equation of Keenan, Keyes, Hill, andMoore 409

12.5.2 The Equations of Haar, Gallagher, and Kell 40912.5.3 The Equation of Schmidt and Wagner 41112.5.4 Reference Equations of Wagner 41112.5.5 Technical Equations of Span and of Lemmon 41212.5.6 Recent Equations of State 41812.5.7 Thermodynamic Properties from Helmholtz

Energy Equations of State 41912.6 Comparisons of Property Formulations 42012.7 Recommended Multi-Parameter Equations of State 42312.8 Equations of State for Mixtures 424

xvContents

12.8.1 Extended Corresponding States Methods 42512.8.2 Mixture Properties from Helmholtz Energy

Equations of State 42612.9 Software for Calculating Thermodynamic Properties 428References 428

Chapter 13 Equations of State in Chemical Reacting Systems 433

Selva Pereda, Esteban Brignole and Susana Bottini

13.1 Introduction 43313.2 The Chemical Equilibrium Problem 43413.3 Reactions under Near-Critical Conditions 43613.4 Modelling Reacting Systems with Group

Contribution Equations of State 43913.4.1 Group Contribution with Association

Equation of State (GCA-EoS) 44013.5 Phase Equilibrium Engineering of Supercritical

Gas-Liquid Reactors 44613.5.1 Solvent Selection 44613.5.2 Boundaries of Feasible Operating

Regions 45013.6 Concluding Remarks 454References 455

Chapter 14 Applied Non-Equilibrium Thermodynamics 460

Signe Kjelstrup and Dick Bedeaux

14.1 Introduction 46014.1.1 A Systematic Thermodynamic Theory for

Transport 46114.1.2 On the Validity of the Assumption of Local

Equilibrium 46414.1.3 Concluding remarks 465

14.2 Fluxes and Forces from the Second Law of

Thermodynamics 46614.2.1 Continuous phases 46714.2.2 Maxwell-Stefan Equations 47214.2.3 Discontinuous Systems 47414.2.4 Concluding Remarks 481

14.3 Chemical Reactions 48114.3.1 Thermal Diffusion in a Reacting System 48114.3.2 Mesoscopic Description Along the

Reaction Coordinate 48414.3.3 Heterogeneous Catalysis 48614.3.4 Concluding Remarks 487

xvi Contents

14.4 The Path of Energy-Efficient Operation 48814.4.1 An Optimisation Procedure 48814.4.2 Optimal Heat Exchange 48914.4.3 The Highway Hypothesis for a Chemical

Reactor 49114.4.4 Energy-Efficient Production of Hydrogen

Gas 49314.4 Conclusions 494References 496

Subject Index 499

xviiContents

List of Contributors

M. A. Anisimov (USA)H. Behnejad (Iran)D. Bedeaux (Norway)C. E. Bertrand (USA)S. Bottini (Argentina)E. Brignole (Argentina)D. Browarzik (Germany)I. G. Economou (Greece)J. F. Ely (USA)A. Galindo (UK)A. R. H. Goodwin (USA)

S. Kjelstrup (Norway)M. C. Kroon (The Netherlands)E. W. Lemmon (USA)A. M. Martınez (Spain)C. McCabe (USA)S. Pereda (Argentina)C. J. Peters (The Netherlands)S. I. Sandler (USA)J. V. Sengers (USA)R. Span (Germany)J. P. Martin Trusler (UK)

xix

Experimental ThermodynamicsSeries

Calorimetry of Non-Reacting Systems

Experimental Thermodynamics, Volume IEdited by J. P. McCullough and D. W. ScottButterworths: London. 1968.

Experimental Thermodynamics of Non-Reacting Fluids

Experimental Thermodynamics, Volume IIEdited by B. Le Neindre and B. VodarButterworths, London, 1975

Measurement of the Transport Properties of Fluids

Experimental Thermodynamics, Volume IIIEdited by W. A. Wakeham, A. Nagashima, and J. V. SengersBlackwell Scientific Publications, Oxford, 1991

Solution Calorimetry

Experimental Thermodynamics, Volume IVEdited by K. N. Marsh and P. A. G. O’HareBlackwell Scientific Publications, Oxford, 1994

Equations of State for Fluids and Fluid Mixtures

Experimental Thermodynamics, Volume VEdited by J. V. Sengers, R. F. Kayser, C. J. Peters, and H. J. White, Jr.Elsevier, Amsterdam, 2000

Measurement of the Thermodynamic Properties of Single Phases

Experimental Thermodynamics, Volume VIEdited by A. R. H. Goodwin, K. N. Marsh, W. A. WakehamElsevier, Amsterdam, 2003

xxi

Measurement of the Thermodynamic Properties of Multiple Phases

Experimental Thermodynamics, Volume VIIEdited by R. D. Weir and T. W. de LoosElsevier, Amsterdam, 2005


Edited by A. R. H. Goodwin, J. V. Sengers, C. J. PetersRSC, Cambridge, 2010

xxii Experimental Thermodynamics Series

Acknowledgments

We are indebted to the authors and grateful to past and present members ofthe IACT for their unwaving support for this project.Some of the illustrations that appear in this volume have been published

elsewhere. The present authors, editors and publishers are grateful to all thoseconcerned in the original publications for permission to use their Figures again.Some of the Figures have been edited for consistency of presentation.

xxiii

CHAPTER 1

Introduction

ANTHONY R. H. GOODWIN,a JAN V. SENGERSb ANDCORNELIS J. PETERSc, d

a Schlumberger Technology Corporation, Sugar Land, TX, USA; bUniversityof Maryland, Institute for Physical Science and Technology, College Park,MD, USA; cDelft University of Technology, Delft, The Netherlands;d Chemical Engineering Program, Petroleum Institute, Abu Dhabi, UnitedArab Emirates

In the series Experimental Thermodynamics1–7 the first and only volume con-cerned with equations of state for fluids and fluid mixtures was Volume V.5

This volume, which was published during 2000 in two parts,5 provided descrip-tions of equations of state, over an entire range of approaches and range ofvariables for their wide use in science, engineering and industry. Reference 5included methods required to develop equations of state, including their theo-retical bases and practical uses along with their strengths and limitations. Fur-thermore, the volume contained not only equations of state for simple fluidsand fluid mixtures, but also for and more important classes of complex fluidsand mixtures. In particular, the volume included associating fluids, ionic fluids,poly-disperse systems, polymers, and micelle-forming and other self-organizingsystems. However, some subject matter of importance to the practitioner wasomitted including equations of state specifically for chemically reacting fluidsand methods applicable to non-equilibrium thermodynamics.Since the year 2000, there have been over 15,000 publications in the academic

scientific and engineering archival literature that in some form or other areconcerned with equations of state. In addition, equations of state for chemicallyreacting and non-equilibrium fluids have received additional theoretical and




1

practical attention and now deserve more detailed explanations than hithertoprovided.Both developments in the field and the desire to provide the content omitted

in the previous volume motivated this text entitled Applied Thermodynamics ofFluids. The change in title also reflects a greater emphasis that has been placedon the application of theory, without recourse to derivations of the constitutiveequations, while retaining the fundamental aspects. The volume includesthermodynamics at the nano and meso scale in Chapter 7, chemically reactingsystems in Chapter 13 and the application of non-equilibrium thermodynamicsin Chapter 14. This volume is intended to address the needs of practitionerswithin academia, government and industry. However, chapters from reference5 regarding self-assembled systems and analytical solvable integral equationshave been omitted in this work and as have in both volumes the use of com-puter simulations for the calculation of thermodynamic properties. The latterwould deserve an in-depth coverage of its own and because of size limitationsalong with a recent special issue of Fluid Phase Equilibria8–14 and other publica-tions15–17 reporting the Industrial Fluid Property Simulation Challenges, it wasdecided not to include the topic in the present volume.Some chapters from Volume V5 have been revised and updated and are

included here because of their fundamental importance to the topic: theseappear in this text as Chapter 2, regarding Fundamental Considerations that isessential to determine the validity of any method adopted, Chapter 3, entitledVirial Equation of State, Chapter 4 concerned with Cubic and Generalized vander Waals Equations, Chapter 5, Mixing and Combining Rules, Chapter 6 onCorresponding States. Significantly greater prominence has been is given toStatistical Associating Fluid Theory (SAFT) in Chapter 8, while Chapter 9provides and update on Poly-disperse fluids, Chapter 10 is concerned with themore general topic of Critical Behaviour, Chapter 11 reports on Ionic Fluids andChapter 12 on the Multi-parameter Equations of State.Applied Thermodynamics is published under the auspices of the Physical and

BioPhysical Division (I) of the International Union of Pure and Applied Chemis-try as a project proposed by the International Association of Chemical Thermo-dynamics (IACT) in its capacity as an organization affiliated with IUPAC.Consequently, throughout the text we have adopted the quantities, units andsymbols of physical chemistry defined by IUPAC in the text commonly knownas the Green Book.18 We have also adopted the ISO guidelines for theexpression of uncertainty19 and vocabulary in metrology.20 Values of thefundamental constants and atomic masses of the elements have been obtainedfrom references 21 and 22, respectively.

References

1. Experimental Thermodynamics, Volume I, Calorimetry of Non-ReactingSystems, J. P. McCullough and D. W. Scott, eds., for IUPAC, Butter-worths, London, 1968.

2 Chapter 1

2. Experimental Thermodynamics, Volume II, Experimental Thermodynamicsof Non-Reacting Fluids, B. Le Neindre and B. Vodar, eds., For IUPAC,Butterworths, London, 1975.

3. Experimental Thermodynamics, Volume III, Measurement of the TransportProperties of Fluids, W. A. Wakeham, A. Nagashima and J. V. Sengers,eds., For IUPAC, Blackwell Scientific Publications, Oxford, 1991.

4. Experimental Thermodynamics, Volume IV, Solution Calorimetry, K. N.Marsh and P. A. G. O’Hare, eds., For IUPAC, Blackwell ScientificPublications, Oxford, 1994.

5. Experimental Thermodynamics, Volume V, Equations of State for Fluidsand Fluid Mixtures, Parts I and II, J. V. Sengers, R.F. Kayser, C. J.Peters and H. J. White, Jr., eds., For IUPAC, Elsevier, Amsterdam,2000.

6. Experimental Thermodynamics, Volume VI, Measurement of the Thermo-dynamic Properties of Single Phases, A. R. H. Goodwin, K. N. Marsh andW. A. Wakeham, eds., for IUPAC, Elsevier, Amsterdam, 2003.

7. Experimental Thermodynamics, Volume VII, Measurement of the Thermo-dynamic Properties of Multiple Phases, R. D. Weir and T. W. de Loos, eds.,For IUPAC, Elsevier, Amsterdam, 2005.

8. R. D. Mountain, Fluid Phase Equilib., 2008, 274, 1.9. F. H. Case, J. Brennan, A. Chaka, K. D. Dobbs, D. G. Friend, P. A.

Gordon, J. D. Moore, R. D. Mountain, J. D. Olson, R. B. Ross,M. Schiller, V. K. Shen and E. A. Stahlberg, Fluid Phase Equilib., 2008,274, 2–9.

10. J. D. Olson and L. C. Wilson, Fluid Phase Equilib., 2008, 274, 10–15.11. B. Eckl, J. Vrabec and H. Hasse, Fluid Phase Equilib., 2008, 274, 16–26.12. T. J. Muller, S. Roy, W. Zhao, A. Maaß and D. Reith, Fluid Phase Equilib.,

2008, 274, 27–35.13. X. Li, L. Zhao, T. Cheng, L. Liu and H. Sun, Fluid Phase Equilib., 2008,

274, 36–43.14. M. H. Ketko, J. Rafferty, J. I. Siepmann and J. J. Potoff, Fluid Phase

Equilib., 2008, 274, 44–49.15. F. Case, A. Chaka, D. G. Friend, D. Frurip, J. Golab, R. Johnson,

J. Moore, R. D. Mountain, J. Olson, M. Schiller and J. Storer, Fluid PhaseEquilib., 2004, 217, 1–10.

16. F. Case, A. Chaka, D. G. Friend, D. Frurip, J. Golab, P. Gordon,R. Johnson, P. Kolar, J. Moore, R. D. Mountain, J. Olson, R. Ross andM. Schiller, Fluid Phase Equilib., 2005, 236, 1–14.

17. F. H. Case, J. Brennan, A. Chaka, K. D. Dobbs, D. G. Friend, D. Frurip,P. A. Gordon, J. Moore, R. D. Mountain, J. Olson, R. B. Ross, M. Schillerand V. K. Shen, Fluid Phase Equilib., 2007, 260, 153–163.

18. M. Quack, J. Stohner, H. L. Strauss, M. Takami, A. J. Thor, E. R. Cohen,T. Cvitas, J. G. Frey, B. Holstrom, K. Kuchitsu, R. Marquardt, I. Millsand F. Pavese, Quantities, Units and Symbols in Physical Chemistry, RSCPublishing, Cambridge 2007.

3Introduction

19. Guide to the Expression of Uncertainty in Measurement, InternationalStandards Organization, Geneva, Switzerland, 1995.

20. International Vocabulary of Basic and General Terms in Metrology, Inter-national Standards Organization, Geneva, Switzerland, 1993.

21. P. J. Mohr, B. N. Taylor and D. B. Newell, J. Phys. Chem. Ref. Data, 2008,37, 1187–1284.

22. M. E. Wieser, Pure Appl. Chem., 2006, 78, 2051–2066.

4 Chapter 1

CHAPTER 2

Fundamental Considerations

ANGEL MARTIN MARTINEZa AND COR J. PETERSb, c

aDepartment of Chemical Engineering and Environmental Technology,Faculty of Science, University of Valladolid, Prado de la Magdalenas/n 47011, Valladolid, Spain; bDelft University of Technology, Faculty ofMechanical, Maritime and Materials Engineering, Department of Processand Energy, Laboratory of Process Equipment, Leeghwaterstraat 44,2628 CA Delft, The Netherlands; c The Petroleum Institute, Department ofChemical Engineering, Bu Hasa Building, Room 2203, P.O. Box 2533, AbuDhabi, United Arab Emirates

2.1 Introduction

This chapter provides a thermodynamic toolbox and contains most of theimportant basic relations that are used in other chapters. The scope is restrictedalmost exclusively to the second law of thermodynamics and its consequence,but the treatment is still intended to be exemplary rather than definitive. Newresults are not presented as befits a discussion of fundamentals which arenecessarily invariant with time.

2.2 Basic Thermodynamics

The state of a system may be described in terms of a small number of variables.For a phase in the absence of any external field, the second law of thermo-dynamics may be written as

dU ¼ TdS � pdV þXi

midni ð2:1Þ




5

This equation shows that the change dU in the energy U may be described interms of simultaneous changes dS in the entropy S, dV in the volume V, and dniin the amount of substance ni of the i components. It is often convenient toeliminate the size of the phase by writing eq 2.1 in terms of intensive variables.For example, division by the total amount n

n ¼Xi

ni; ð2:2Þ

gives

dUm ¼ TdSm � pdVm þXi

midxi ð2:3Þ

where the subscript m denotes a molar quantity and the mole fraction ofcomponent i is defined by

xi ¼ ni=N; ð2:4Þ

Equation 2.1, or eq 2.3 for intensive variables, is the fundamental expression ofthe second law of thermodynamics. However, entropy, in particular, is not avery convenient experimental variable and, consequently, alternative formshave been derived from the fundamental eq 2.1. Introduction of the followingcharacteristic functions:

H ¼ U þ pV ; ð2:5Þ

A ¼ U � TS; ð2:6Þ

and

G ¼ U þ pV � TS ¼ H � TS ¼ Aþ pV : ð2:7Þ

For enthalpy, Helmholtz and Gibbs functions and use of Legendre transfor-mations1,2 with the fundamental eq 2.1 gives the following alternative forms ofthe second law

dH ¼ TdS þ VdpþXi

midni; ð2:8Þ

dA ¼ �SdT þ pdV þXi

midni; ð2:9Þ

and

dG ¼ �SdT þ VdpþXi

midni ð2:10Þ

Another modification of eq 2.1, frequently used in statistical mechanics, is

d pVð Þ ¼ SdT þ pdV þXi

nidmi; ð2:11Þ

6 Chapter 2

or, as the Gibbs-Duhem equation, in the study of phase equilibria

0 ¼ SdT � VdpþXi

nidmi: ð2:12Þ

From eq 2.1 and eqs 2.8 to 2.10 we find the following:

mi ¼@U

@ni

� �S;V ;nji

¼ @H

@ni

� �S;p;nj

¼ @A

@ni

� �T ;V ;nj

¼ @G

@ni

� �T ;p;nj

; ð2:13Þ

where the subscript nj means that the amount of substance nj of all the com-ponents are constant except for component i. The quantity mi is the chemicalpotential of species i. In terms of intensive variables these equations are

dHm ¼ TdSm þ VmdpþXi

midxi; ð2:14Þ

dAm ¼ �SmdT � pdVm þXi

midxi; ð2:15Þ

dGm ¼ �SmdT þ VmdpþXi

midxi; ð2:16Þ

d pVmð Þ ¼ SmdT þ pdVm þXi

nidmi; ð2:17Þ

and

0 ¼ SmdT � VmdpþXi

xidmi: ð2:18Þ

For a system of constant composition, eq 2.1 and eqs 2.8 to 2.10 reduce to

dU ¼ TdS � pdV ; ð2:19Þ

dH ¼ TdS þ Vdp; ð2:20Þ

dA ¼ �SdT � pdV ; ð2:21Þ

and

dG ¼ �SdT þ Vdp: ð2:22Þ

A large number of thermodynamic relations may be derived from the aboveequations by conventional manipulations. Table 2.1 summarizes the mostfrequently used equations.

7Fundamental Considerations

2.2.1 Homogeneous Functions

A homogeneous function F of the first order in any number of the variables x,y, z,. . .is defined by:

F lx; ly; lz; � � �ð Þ ¼ lF x; y; z; � � �ð Þ; ð2:23Þ

where l is an arbitrary number. If each independent variable is made l timeslarger, the function F increases l times. For large enough systems, all extensivethermodynamic functions are homogeneous and of the first order in amount ofsubstance ni at fixed temperature and pressure. For homogeneous functions ofthe first order, Euler’s theorem on homogeneous functions applies:

lF x; y; z; � � �ð Þ ¼ x@F

@x

� �y;z;:::

þy @F

@y

� �x;z;:::

þz @F

@z

� �x;y;:::

þ � � � : ð2:24Þ

Table 2.1 Frequently used thermodynamic relationships with general validity.

dU ¼ TdS � pdV dH ¼ TdS þ VdpdA ¼ �SdT � pdV dG ¼ �SdT þ Vdp

CV ¼@U

@T

� �V

CP ¼@H

@T

� �p

@S

@T

� �V

¼ CV

T

@S

@T

� �p

¼ Cp

T

@CV

V

� �T

¼ T@2p

@T2

� �V

@Cp

@p

� �T

¼ �T @2V

@T2

� �p

CV ¼ T@p

@T

� �V

@V

@T

� �p

¼ �T @V

@T

� �2

p

@p

@V

� �T

kS ¼ �1

V

@p

@V

� �S

kT ¼ �1

V

@V

@p

� �T

kT � kS ¼1

V

@V

@T

� �p

@T

@p

� �S

¼ 1

V

@V

@T

� �2

p

@T

@S

� �p

Maxwell:@T

@V

� �S

¼ � @p

@S

� �V

@S

@V

� �T

¼ @p

@T

� �V

@S

@p

� �T

¼ � @V

@T

� �p

Helmholtz:@U

@V

� �T

¼ T@p

@T

� �V

�p @H

@p

� �T

¼ V � T@V

@T

� �p

Gibbs-Helmholtz:@ A=Tð Þ@T

� �V

¼ � U

T2

@ G=Tð Þ@T

� �p

¼ � H

T2

8 Chapter 2

Equation 2.24 relates the value of the function to the values of its derivatives.For l¼ 1, eq 2.24 reduces to:

F x; y; z; � � �ð Þ ¼ x@F

@x

� �y;z;:::

þy @F

@y

� �x;z;:::

þz @F

@z

� �x;y;:::

þ � � � : ð2:25Þ

By applying Euler’s theorem to the various characteristic functions U¼U(S, V,n1, n2, � � �), H¼H(S, p, n1, n2,� � �), A¼A(V, T, n1, n2, � � �), G¼G(p, T, n1, n2,� � �), respectively, the following expressions result:

U ¼ TS � pV þXi

nimi; ð2:26Þ

H ¼ TS þXi

nimi; ð2:27Þ

A ¼ �pV þXi

nimi; ð2:28Þ

and

G ¼Xi

nidmi: ð2:29Þ

For further details on Euler’s theorem see references 1 to 3.

2.2.2 Thermodynamic Properties from Differentiation of

Fundamental Equations

The quantities U¼U(S, V, n1, n2, � � �, xi), H¼H(S, p, n1, n2, � � �, xi), A¼A(V,T, n1, n2, � � �, xi), and G¼G(p, T, n1, n2, � � �, xi) are examples of thermodynamicpotentials from which all properties of a system can be obtained without theneed for integration. For example, eq 2.10 gives directly the heat capacity atconstant pressure

Cp ¼ T@S

@T

� �p;n

¼ �T @2G

@T2

� �p;n

; ð2:30Þ

and the isothermal compressibility

kT ¼ �1

V

@V

@p

� �T ;n

¼ � @2G

@p2

� �T ;n

,@G

@p

� �T ;n

: ð2:31Þ

Avoiding integration is often advantageous in theoretical applications ofthermodynamics because there are no constants of integration. On the otherhand, very often the derivatives needed involve variables that are difficult tomeasure experimentally. Even with the Gibbs function surface, which is closelylinked to the convenient experimental variables of temperature, pressure and


Table

2.2

Thermodynamic

properties

expressed

inderivatives

ofthecharacteristicfunctions.

U(S,V)

H(p,S)

A(T,V)

G(T

,p)

p�

@U @V�� S

�@A @V�� S

V@H @p�� S

@G @p�� T

T@U @S�� V

@H @S�� p

S�

@A @T�� V

�@G @T�� p

UH�p@H @p�� S

A�T

@A @T�� V

G�p@G @p�� T

�T

@G @T�� p

HU�V

@U @V�� S

A�T

@A @T�� V

�V

@A @V�� S

G�T

@G @T�� p

AU�S

@U @S�� V

H�S

@H @S�� p

�p@H @p�� S

G�p@G @p�� T

GU�V

@U @V�� S

�S

@U @S�� V

H�S

@H @S�� p

A�V

@A @V�� T

Cp

@H=@S

ðÞ p

@2H=dS2

ðÞ p

�T

@2G

@T2

�� p

CV

@U=dS

ðÞ V

@2U=@S2

ðÞ V

�T

@2A

@T2

�� V

k TV

@2A

@V2

�� T

�� 1

�@G=@p2

�� T

@G=@p

ðÞ T

k sV

@2U

@V2

�� S

�� 1

�@2H=@p2

�� S

@H=@p

ðÞ S

10 Chapter 2

composition, differences in the Gibbs function can be studied only at equili-brium. At constant composition the characteristic functions U¼U(S, V, n1, n2,� � �, xi),H¼H(S, p, n1, n2, � � �, xi), A¼A(V, T, n1, n2, � � �, xi), and G¼G(p, T, n1,n2, � � �, xi) reduce to U¼U(S, V),H¼H(S, p), A¼A(V, T) and G¼G(p, T). Bydifferentiation of these functions and use of eqs 2.5 and 2.19 to 2.21, the valueof any thermodynamic property can be expressed in terms of the derivatives ofeach characteristic function. Table 2.2 summarizes the most relevant results.Further details can be found elsewhere.3

2.3 Deviation Functions

Almost all definitions of molar properties for mixtures lack an unambiguousdefinition in the sense that they can be related directly to measurable properties.Therefore, it is common practice to compare an actual mixture property with itscorresponding value obtained from an arbitrary model, for instance, an equa-tion of state. This approach leads to the introduction of deviation functions. Fora general mixture molar property Mm, the deviation function is defined by:4

MDm ¼Mm �Mm calcð Þ: ð2:32Þ

An important aspect in this definition is the choice of the independent variables.Many analytical equations of state are expressions explicit in pressure: that is,temperature, molar volume (or density) and composition x¼ x1, x2, � � �, xi arethe natural independent variables. Therefore, eq 2.32 can be rewritten into:

MD T ;V ; nð Þ ¼MðT ;V ; nÞ �Mðcalc;T ;V ; nÞ; ð2:33Þ

where n denotes the amounts n1, n2, � � �, ni. The value of M obtained from themodel is evaluated at the same values of the independent variables as used for themixture property. Alternatively, temperature, pressure and composition may bea suitable choice as independent variables for deviation functions, for example:

MDðT ; p; nÞ ¼MðT ; p; nÞ �Mðcalc;T ; p; nÞ; ð2:34Þ

In this case the value of M obtained from the model is evaluated at the samevalues of T, p and n as used for the actual mixture property. Both approachesare interrelated as follows:4

MDðT ;V ; nÞ ¼MDðT ; p; nÞ þZppr

@M calcð Þ@p

� �T ;n

dp: ð2:35Þ

In this equation, pr is the reference pressure at which the molar volume of themixture obtained from the model is equal to the molar volume of the actual


mixture at the same temperature and composition as the mixture. A necessaryfeature of the model is that such a pr exists.

2.3.1 Residual Functions

As pointed out in the previous section, the calculation of deviation functionsrequires a choice of an appropriate model. If the model system is chosen to bean ideal gas mixture, which is an obvious choice for fluid mixtures, then thedeviation functions are called residual functions. With temperature, volumeand composition as independent variables eq 2.33 becomes:

MDðT ;V ; nÞ ¼MRðT ;V ; nÞ ¼MðT ;V ; nÞ �MpgðT ;V ; nÞ; ð2:36Þ

or with T and p as independent variables

MDðT ; p; nÞ ¼MRðT ; p; nÞ ¼MðT ; p; nÞ �MpgðT ; p; nÞ; ð2:37Þ

which is a particular form of eq 2.34. The two sets of residual functions arerelated by eq 2.35 in the form:

MRðT ;V ; nÞ ¼MRðT ; p; nÞ þZppr

@Mpg

@p

� �T ;n

dp; ð2:38Þ

where the reference pressure pr¼RT/Vm is sufficiently low for the thermo-dynamic property M of the real fluid to have the ideal-gas value.Some thermodynamic properties (U, H, CV and Cp) of an ideal gas are

independent of pressure, while others like S, A and G are not. Consequently,from eq 2.37, it can be easily seen that the following equations hold:

UR T ;V ; nð Þ ¼ UR T ; p; nð Þ; ð2:39Þ

HR T ;V ; nð Þ ¼ HR T ; p; nð Þ; ð2:40Þ

CRV T ;V ; nð Þ ¼ CR

V T ; p; nð Þ; ð2:41Þ

CRp T ;V ; nð Þ ¼ CR

p T ; p; nð Þ; ð2:42Þ

SR T ;V ; nð Þ ¼ SR T ; p; nð Þ þ R lnZ; ð2:43Þ

ARðT ;V ; nÞ ¼ ARðT ; p; nÞ þ RT lnZ; ð2:44Þ

and

GRðT ;V ; nÞ ¼ GRðT ; p; nÞ þ RT lnZ: ð2:45Þ

where Z¼ pV/nRT and is known as the compression or compressibility factor.

12 Chapter 2

2.3.2 Evaluation of Residual Functions

With temperature, molar volume and composition as the independent vari-ables, general expressions for the residual functions of thermodynamic prop-erties are readily obtained from eq 2.36. Abbott and Nass4 have given adefinitive list of expressions for various properties as residual functions andthese are summarised in Table 2.3.For temperature, pressure and compositions as the independent variables,

Abbott and Nass4 also evaluated general expressions for the residual functions.Table 2.4 summarizes the results. For further details see references 2 and 4.

2.4 Mixing and Departure Functions

In terms of the independent variables, temperature, pressure and composition,departure functions compare the value of a general thermodynamic propertyM(T, p, n) with the corresponding property in the ideal-gas state and at areference pressure pr, that is M

pg(T, pr, n). According to the ideal gas law, thereference pressure pr is related to the reference volume Vr¼ nRT/pr. Similarly,as was the case for the residual functions, the independent variables tempera-ture, molar volume and composition can also be used to define departure

Table 2.3 Residual functions with volume or density as an independentvariable (r¼ n/V and Z¼ pV/nRT).

MR(T, V, n) Residual function

UR¼�nRT2

Zr0

@Z

@T

� �r;n

drr

HR¼�nRT2

Zr0

@Z

@T

� �r;nþnRT Z � 1ð Þ

SR¼�nR

Zr0

T@Z

@T

� �r;nþZ � 1

( )drr

AR¼nRT

Zr0

Z � 1ð Þ drr

GR¼nRT

Zr0

Z � 1ð Þ drrþ nRT Z � 1ð Þ

CVR¼

�RTZr0

T@2Z

@T2

� �r;nþ2 @Z

@T

� �r;n

( )drr

CPR¼

CRV � Rþ R Z þ T

@Z

@T

� �r;n

( )2

Z þ r@Z

@r

� �T ;n

( )�1pR¼ rRT Z � 1ð Þ


functions. Based on these independent variables, the general thermodynamicM(T, V, n) is compared with the corresponding ideal-gas propertyMpg(T, V, n)

2.4.1 Departure Functions with Temperature, Molar Volume and

Composition as the Independent Variables

The following equality can be derived for the departure function of a generalthermodynamic M(T, V, n):

M T ;V ; nð Þ �Mpg T ;V ; nð Þ ¼ZVN

@M

@V

� �T ;n

� @Mpg

@V

� �T ;n

( )dV

þZVVr

@Mpg

@V

� �T ;n

dV :

ð2:46Þ

Applying conventional thermodynamic manipulations on eq 2.46, the follow-ing relations can be obtained:

U �Upg ¼ZVN

T@p

@T

� �V ;n

�p !

dV ; ð2:47Þ

H �Hpg ¼ZVN

T@p

@T

� �V ;n

�p( )

dV þ nRT Z � 1ð Þ; ð2:48Þ

S � Spg ¼ZVN

@p

@T

� �V ;n

� nR

V

( )dV þ nR ln

V

Vr

� �; ð2:49Þ

A� Apg ¼ZVN

p� nRT

V

� �dV � nRT ln

V

Vr

� �; ð2:50Þ

and

G� Gpg ¼ �ZVN

p� nRT

V

� �dV � nRT ln

V

Vr

� �þ nRT Z � 1ð Þ: ð2:51Þ

If the reference volume Vr is replaced by the actual volume V of the system,then the corresponding residual functions are recovered (and are listed inTable 2.3). The thermodynamic properties for the reference state are defined bythe following relations:

Upg ¼Xi

niUpgi ¼

Xi

niHpgi � nRT

Xi

ni; ð2:52Þ

14 Chapter 2

Spg ¼Xi

niSpgi � R

Xi

ni ln ni; ð2:53Þ

Hpg ¼Xi

niHpgi ; ð2:54Þ

Apg ¼Xi

niAi þ RTXi

ni ln ni ¼Xi

niHpgi

� TXi

niSpgi � RT

Xi

ni þ RTXi

ni ln ni; ð2:55Þ

and

Gpg ¼Xi

niGpgi þ RT

Xi

ni ln ni ¼Xi

niHpgi � T

Xi

niSpgi þ RT

Xi

ni ln ni:

ð2:56ÞFrom eqs 2.52 through 2.56, the mixture properties for ideal gases are obtainedas follows:

DmixU ¼ Upg �Xi

niUpgi ¼ 0; ð2:57Þ

DmixH ¼ Hpg �Xi

niHpgi ¼ 0; ð2:58Þ

DmixS ¼ Spg �Xi

niSpgi ¼ �R

Xi

ni ln ni; ð2:59Þ

DmixA ¼ Apg �Xi

niApgi ¼ RT

Xi

ni ln ni; ð2:60Þ

and

DmixG ¼ Gpg �Xi

niGpgi ¼ RT

Xi

ni ln ni: ð2:61Þ

Departure functions are conveniently evaluated from eq 2.50 as the generatingfunction. The following calculation procedure is the appropriate route tofollow:

1. Equation 2.46 gives (A–Apg);

2: S � Spgð Þ ¼ � @ A� Apgð Þ@T

� �V ;n

; ð2:62Þ

3: ðU �UpgÞ ¼ ðA� ApgÞ þ TðS � SpgÞ; ð2:63Þ

4: ðH �HpgÞ ¼ ðA� ApgÞ þ TðS � SpgÞ þ nRTðZ � 1Þ; ð2:64Þ

and,

5: ðG� GpgÞ ¼ ðA� ApgÞ þ nRTðZ � 1Þ: ð2:65Þ


2.4.2 Departure Functions with Temperature, Pressure and

Composition as the Independent Variables

In this case, the general thermodynamic property is M(T, p, x) and the fol-lowing equality for the departure function can be derived:

M T ; p; nð Þ �Mid T ; pr; nð Þ

¼Zp0

@M

@p

� �T ;n

� @Mpg

@p

� �T ;n

( )dpþ

Zpr0

@Mpg

@p

� �T ;n

dp:ð2:66Þ

The following relations can be obtained from eq 2.62:

U �Upg ¼Zp0

V � T@V

@T

� �p;n

( )dp� nRT Z � 1ð Þ; ð2:67Þ

H �Hpg ¼Zp0

V � T@V

@T

� �p;n

( )dp; ð2:68Þ

S � Spg ¼Zp0

nR

p� T

@V

@T

� �p;n

( )dp� nR ln

p

pr

� �; ð2:69Þ

A� Apg ¼Zp0

V � nRT

p

� �dp� nRT Z � 1ð Þ þ nRT ln

p

pr

� �; ð2:70Þ

and

G� Gpg ¼Zp0

V � nRT

p

� �dpþ nRT ln

p

pr

� �: ð2:71Þ

If the reference pressure pr is replaced by the actual pressure p of the system,then the corresponding residual functions are recovered and listed in Table 2.4.Again, the thermodynamic properties for the reference state can be obtained

from eqs 2.52 to 2.56. In order to obtain the departure functions for this set ofindependent variables, the following procedure is recommended:

1. Equation 2.71 gives (G – Gpg);

2: S � Spgð Þ ¼ � @ G� Gpgð Þ@T

� �p;n

; ð2:72Þ

3: U �Upgð Þ ¼ G� Gpgð Þ þ T S � Spgð Þ � nRT Z � 1ð Þ; ð2:73Þ

16 Chapter 2

4: H �Hpgð Þ ¼ G� Gpgð Þ þ T S � Spgð Þ; ð2:74Þ

and

5: A� Apgð Þ ¼ G� Gpgð Þ � nRT Z � 1ð Þ: ð2:75Þ

Further details on departure functions can be found in reference 5.

2.5 Mixing and Excess Functions

Deviation functions for mixtures are concerned mainly with variation incomposition rather than pressure or density. Consequently, it is convenient touse molar quantities. Molar excess functions are defined by:

MEm ¼MD

m ¼Mm �Midm : ð2:76Þ

where Midm is the molar value for the ideal mixture at the same temperature and

pressure. If, for example,Mid¼Vid, then the molar volume of the ideal mixture(denoted by the superscript id) is given by:

V idm ¼

Xi

xiV�i : ð2:77Þ

Table 2.4 Residual functions with pressure as the independent variable (r¼ n/Vand Z¼ pV/nRT).

MR(T, V, n) Residual function

UR¼�nRT2

Zp0

@Z

@T

� �p;n

dp

p� nRT Z � 1ð Þ

HR¼�nRT2

Zp0

@Z

@T

� �p;n

dp

p

SR¼�nR

Zp0

T@Z

@T

� �p;n

þZ � 1

( )dp

p

AR¼nRT

Zp0

Z � 1ð Þ dpp� nRT Z � 1ð Þ

GR¼nRT

Zp0

Z � 1ð Þ dpp

CVR¼

�RTZp0

T@2Z

@T2

� �p;n

þ2 @Z

@T

� �p;n

( )dp

p

CPR¼

CRV � Rþ R Z þ T

@Z

@T

� �r;n

( )2

Z þ r@Z

@r

� �� 1pR¼ RT

pZ � 1ð Þ


where V*i is the molar volume of pure component i. Other thermodynamic

properties for ideal mixtures are defined by:

�U idm ¼

Xi

xiU�i ; ð2:78Þ

H idm ¼

Xi

xiH�i ; ð2:79Þ

Sidm ¼

Xi

xiS�i � R

Xi

xi ln xi; ð2:80Þ

Aidm ¼

Xi

xiA�i þ RT

Xi

xi ln xi; ð2:81Þ

and

Gidm ¼

Xi

xiG�i þ RT

Xi

xi ln xi: ð2:82Þ

From eqs 2.77 to 2.82, the mixing properties are obtained as follows:

DmixVm ¼ V idm �

Xi

xiV�i ¼ 0; ð2:83Þ

DmixUm ¼ U idm �

Xi

xiU�i ¼ 0; ð2:84Þ

DmixHm ¼ H idm �

Xi

xiH�i ¼ 0; ð2:85Þ

DmixSm ¼ Sidm �

Xi

xiS�i ¼ �R

Xi

xi ln xi; ð2:86Þ

DmixAm ¼ Aidm �

Xi

xiA�i ¼ RT

Xi

xi ln xi; ð2:87Þ

andDmixGm ¼ Gid

m �Xi

xiG�i ¼ RT

Xi

xi ln xi: ð2:88Þ

Again an important aspect in this definition is the choice of the independent vari-ables. From eq 2.76 two different definitions of excess functions can be obtained:

ME T ;V ; nð Þ ¼M T ;V ; nð Þ �Mid T ;V ; nð Þ ð2:89Þand

ME T ; p; nð Þ ¼M T ; p; nð Þ �Mid T ; p; nð Þ: ð2:90Þ

The two approaches are related by:

MEm T ;V ; nð Þ ¼ME

m T ; p; nð Þ þZppr

@Midm

@p

� �T ;x

dp; ð2:91Þ

18 Chapter 2

in this case p is the pressure for which the molar volume of the ideal solution isthe same as that of the real solution at given temperature and composition.4

This pressure is obtained by solving:

Vm T ; p;xð Þ ¼Xi

xiV�i T ; prð Þ: ð2:92Þ

A required feature of the model is that eq 2.92 can be solved for the pressure.For eq 2.91, Abbott and Nass4 proposed an approximate relation suitable forpractical purposes:

MEm T ;V ; nð Þ ¼ME

m T ; p; nð Þ þ @Mid

@p

� �T ;x

VEmP

i

xikiV�i

0@

1A: ð2:93Þ

In eq 2.93, k*i is the isothermal compressibility of pure i:

k�i ¼ �1

V�i

@V�i@p

� �T

: ð2:94Þ

Based on eq 2.89, Abbott and Nass4 have summarized expressions for thethermodynamic properties.Excess functions and residual functions are related. From eqs 2.37 and 2.90 it

can be shown the following equality holds:

ME T ; p; nð Þ ¼MR T ; p; nð Þ � Mid T ; p; nð Þ �Mpg T ; p; nð Þ�

: ð2:95Þ

Based on eqs 2.52 to 2.56 and 2.78 to 2.82, it can be shown for an arbitraryextensive thermodynamic property the following relation holds:

ME T ; p; nð Þ ¼MR T ; p; nð Þ

�Xi

xiMi T ; p; nð Þ �Xi

xiMpgi T ; p; nð Þ

( ): ð2:96Þ

2.6 Partial Molar Properties

A partial molar property Mi of an arbitrary extensive thermodynamic propertyM¼M(T, p, n1, n2, � � �, ni) is defined by the equation:

Mi ¼@M

@ni

� �T ;P;nj

: ð2:97Þ

Partial molar properties give information about the change of the total prop-erty due to addition of an infinitely small amount of substance of species i to themixture. From eq 2.16 it becomes apparent that, by definition, the chemical


potential is the partial molar Gibbs function, that is, mi¼Gi. For this arbitrarythermodynamic Property M the following expressions can be derived:2

dM ¼ @M

@p

� �T ;n

dpþ @M

@T

� �dT þ

Xi

Midni; ð2:98Þ

where M ¼Pi

xiMi, or equivalently,

Mm ¼Xi

xiMi: ð2:99Þ

From eqs 2.98 and 2.99 the generalized Gibbs-Duhem equation is readilyobtained:

@M

@p

� �T ;n

dpþ @M

@T

� �dT �

Xi

nidMi ¼ 0: ð2:100Þ

In the case when M¼G, eq 2.12 is recovered.

2.7 Fugacity and Fugacity Coefficients

The fugacity p of a real fluid mixture with constant composition and at constanttemperature is defined by the equation:

dG ¼ RTd ln p: ð2:101Þ

Combination of this equation with its ideal gas equivalent leads to:

d G� Gpgð Þ ¼ dGR ¼ RTd lnp

p

� �¼ RTd lnf: ð2:102Þ

In this equation GR is the residual Gibbs energy and f is the fugacity coeffi-cient. Integration of eq 2.102 yields:

GR ¼ RT lnf: ð2:103Þ

Comparison of eq 2.101 and its equivalent in Table 2.4 leads to:

lnf ¼Zp0

Z � 1ð Þd p

p

� �: ð2:104Þ

This expression shows that an equation of state can be used to evaluate thefugacity coefficient. Similar expressions hold for a pure component i:

dGi ¼ RTd ln pi; ð2:105Þ

20 Chapter 2

and

GRi ¼ RT lnfi: ð2:106Þ

In a real solution for species i the defining equation is:

dGi ¼ RTd ln ^pi: ð2:107Þ

From the definition of the residual Gibbs energy GR¼G–Gpg, it follows that:

GRi ¼ Gi � G

pgi ; ð2:108Þ

and for an ideal gas at constant temperature it holds that:

mi ¼ Gpgi þ RT ln yi: ð2:109Þ

From eqs 2.107 through 2.109 the following expression results:

d Gi � Gpgið Þ ¼ dGR

i ¼ RTd ln~piyip

!¼ RTd ln fi: ð2:110Þ

From eq 2.110 the relationship between the fugacity pi and the fugacitycoefficient fi is as follows:

fi ¼~piyip

: ð2:111Þ

Integration of eq 2.110 leads to a similar expression as represented by eq 2.106:

GRi ¼ RT ln fi: ð2:112Þ

For M¼G, eq 2.97 can be rewritten as:

GRi ¼

@ nGR� �@ni

� �T ;p;nj

: ð2:113Þ

Substitution of eqs 2.106 and 2.110 into eq 2.113 gives:

ln fi ¼@ lnfð Þ@ni

� �T ;p;nj

: ð2:114Þ

Since lnfi is related to ln fi as a partial molar property, from eq 2.99 the fol-lowing relation can be cast:

lnf ¼Xi

xi ln fi: ð2:115Þ


At constant temperature and pressure, from eq 2.100 the Gibbs-Duhem equationcan be obtained:

Xi

xid ln fi

�¼ 0: ð2:116Þ

Fugacities and fugacity-coefficients of mixtures can be evaluated from allequation of state models. In the case the equation of state has pressure andtemperature as the independent variables the relationships that can be appliedare as follows:

RT ln fi ¼ RT ln~piyip

!¼Zp0

Vi �RT

p

� �dp: ð2:117Þ

In eq 2.117 the partial molar volume Vi is evaluated from the equation ofstate and eq 2.97. For a pure substance the partial molar volume Vi isequivalent to the molar volume V*

i and eq 2.117 simplifies to eq 2.104. Mostequation-of-state models have temperature and volume as the independentvariables. For the evaluation of the fugacity and fugacity-coefficient in mixturesat constant temperature and composition can be obtained from the following:

RT ln fi ¼ RT ln~piyip

!¼ZNV

@p

@ni

� �T ;V ;nj

�RT

V

" #dV � RT lnZ: ð2:118Þ

In eq 2.118, Z¼ pV/nRT is the compression factor of the mixture; the partialderivative in eq 2.118 can be obtained from the equation of state used. For apure substance eq 2.118 reduces to:

RT lnfi ¼ RT lnp

p¼ZNV

p

ni� RT

V

� dV � RT lnZ þ RT Z � 1ð Þ: ð2:119Þ

For an extensive treatment of the fugacity concept, the reader should refer toreferences 1,2 and 6.

2.8 Activity Coefficients

Although activity coefficients, in general, are most conveniently evaluated frommodels which are specifically designed for the condensed phase only, this sec-tion demonstrates how the concept of activity is related to a similar formalismintroduced for fugacity. Equation 2.106 in Section 2.7 relates the residual Gibbsfunction and the fugacity. The excess Gibbs function (Section 2.5) is related tothe activity-coefficient, which may be useful in describing the non-ideality of acondensed phase.

22 Chapter 2

From eqs 2.90 and 2.97 the following relationship is obtained:

GEi ¼ Gi � Gid

i : ð2:120Þ

Integration of eq 2.106 at constant temperature and pressure from the purestate of component i, where Gi¼G*

i and pi¼ pi, to a composition in the solu-tion yields:

Gi � G�i ¼ RT ln~pi~pi

!: ð2:121Þ

Application of eq 2.97 to eq 2.82 gives:

Gidi ¼ G�i þ RT ln xi: ð2:122Þ

Consequently, from eqs 2.120 and 2.121 the following is obtained:

GEm ¼ RT ln

~pixi~pi

!¼ RT ln fi; ð2:123Þ

where the activity coefficient of component i in the solution is defined by fi¼ pi/(xipi). Thus the activity coefficient is related to the excess Gibbs function, forexample, the partial molar excess Gibbs function of species i is related to theactivity coefficient by:

GEi ¼ RT ln fi: ð2:124Þ

Since GEi is the partial molar property of GE, consequently, ln fi is also a partial

molar property of G, and the following two relationships can be derived:

GE ¼ RTXi

xi ln gi; ð2:125Þ

RT ln fi ¼@ nGE� �@ni

� �T ;p;nj

: ð2:126Þ

Additionally, at constant temperature and pressure, from eq 2.100 it followsthat:

Xi

xid knfið Þ ¼ 0 ð2:127Þ

For further details the reader should refer to references 1, 2, 3, 5 and 6.


2.9 The Phase Rule

For a system consisting of C components and P phases in equilibrium thenumber of intensive variables required to specify the state of the system (that isthe number of degrees of freedom F) is given by the Gibbs phase rule:

F ¼ C � Pþ 2� R: ð2:128Þ

In this equation R represents the number of restrictions imposed on thesystem. While the value for isothermal, isobaric or isochoric changes areobvious, the restrictions imposed by chemical reactions are often more subtle.For example, liquid water will exist as a mixture of H2O, H1, H3O

1, and OH�

but, if C is taken as 4, then the requirements of electroneutrality and the ionicequilibria lead to R¼ 3 and the system still behaves, quite correctly, as a purecomponent. Two restrictions are particularly important when studying phasesin equilibrium. If P phases have the same composition, then there are (P� 1)phase boundaries across which the (C� 1) compositions must be the same andthe additional restriction is:

R ¼ P� 1ð Þ C � 1ð Þ; ð2:129Þ

and the number of degrees of freedom is:

F ¼ C 2� Pð Þ þ 1: ð2:130Þ

This makes it clear that 3 phases can have the same composition only in thespecial case of a pure component that is at the triple point. For an azeotropePaz¼ 2, then F¼ 1, and a line always results irrespective of the number ofcomponents.The second special case applies to the critical state. Here the Pc phases that

become identical at the critical state are considered separately from the Pnc

phases that behave normally. In this case, the additional restriction is R¼2Pc–1, and the number of degrees of freedom becomes:

F ¼ C � Pnc þ 2� 2Pc � 1ð Þ: ð2:131Þ

Consequently, at a critical point (F¼ 0) is unique for a pure component while,for a binary mixture, a critical line (F¼ 1) is the simplest case and criticalendpoints (Pnc¼ 1, Pnc¼ 2) are unique. Since F cannot be negative:

C � 2Pc � 3: ð2:132Þ

As a consequence, a tri-critical point cannot exist in a binary mixture. Thecombination of the restrictions for an azeotrope and a critical state shows thatit is not possible for two azeotropic phases to become identical in a critical

24 Chapter 2

state, that is, the critical and the azeotropic composition must be different. Therestrictions that follow from the phase rule are simply consequences of geo-metry and are useful because they reduce the number of variables that must beused to describe the state of a system: small values of F should result in simpleequations of state. However, the phase rule gives no guidance as to whichvariables should be chosen.

2.10 Equilibrium Conditions

For a closed system with an arbitrary number of components and phases inwhich the temperature and pressure are uniform, the following combinedstatement of the first and second laws is:2,6

dUt þ pdV t � TdSt � 0: ð2:133Þ

In eq 2.133 the superscript t refers to the total value of each property. Theinequality of eq 2.133 applies for infinitely small changes between non-equili-brium states and the equality symbol holds for infinite small changes betweenequilibrium states, that is, a reversible processes.For practical purposes within a laboratory both temperature and pressure

are the most easily controlled and measured and are the independent variables.In this case, the Gibbs function is the appropriate thermodynamic parameterand eq 2.133 can be cast as:

dGt þ StdT � V tdp � 0: ð2:134Þ

At constant temperature and pressure, eq 2.134 reduces to:

dGt� �

p;T� 0: ð2:135Þ

This equation states that at constant temperature and pressure any irreversibleprocess proceeds in such a direction that the total Gibbs energy of a closedsystem will decrease. At equilibrium the Gibbs energy has reached a minimumvalue for the given temperature and pressure.

2.10.1 Phase Equilibria

For a closed multi-component system, eq 2.135 can be used to derive theequilibrium conditions between two or more phases in a system at constanttemperature and pressure. If we indicate the various phases by a, b, g, � � �, p,and the various species by 1,2,3, � � �, C, the following equilibrium conditions interms of the chemical potential result:

mai ¼ mbi ¼ mgi ¼ � � � ¼ mpi ; i ¼ 1; 2; 3; � � � ;C: ð2:136Þ


Alternatively, it can be shown2,6 that phase equilibrium can also be defined interms of the fugacity:

~pai ¼ ~pib ¼ ~pgi ¼ � � � ¼ ~ppi ; i ¼ 1; 2; 3; � � � ;C: ð2:137Þ

For vapour-liquid equilibrium eq 2.137 becomes:

~pVi ¼ ~pli; i ¼ 1; 2; 3; � � � ;C: ð2:138Þ

Substitution if eqs 2.111 and 2.123 into this equilibrium condition gives:

yifvi p ¼ xifi~p

li; i ¼ 1; 2; 3; � � � ;C: ð2:139Þ

This formalism is known as the gamma-phi approach for calculating vapour-liquid equilibria. The fugacity coefficient fv

i of each component that accountsfor the non-ideality of the vapour phase can be evaluated from an equation ofstate model, while the activity coefficient fi to describe the non-ideal behaviourof the liquid phase can be obtained from an excess Gibbs function model.The fugacity p l

i of pure species i can be obtained from the relation:2,6,7

RT ln~pi~pi�

� �¼Zpp�i

Vidp: ð2:140Þ

For the case that the temperature is appreciably below the critical value andfor pressures that are also not too high, eq 2.140 can be approximated by:

ln~pi~pi� ¼

V li p� p�i� �RT

: ð2:141Þ

Substitution of f *i ¼f*i p

*i into eq 2.141 gives:

pli ¼ f�i p�i exp

V li p� p�i� �RT

� : ð2:142Þ

In eq 2.142 the exponential is known as the Poynting factor. The contribu-tion of this term becomes significant only at higher pressures. For an ideal gasphase (fi¼ 1), the liquid phase is an ideal solution ( fi¼ 1) and if the Poyntingfactor does not contribute because the pressure is sufficiently low then eq 2.139reduces to Raoult’s law.Equations of state, in principle, are able to describe the vapour and liquid

phase simultaneously because both fvi and fl

i can be evaluated from the model.Substitution of eq 2.111 for both the vapour and liquid phase into eq 2.138

26 Chapter 2

leads to the phase equilibrium conditions:

yifvi ¼ xif

li; i ¼ 1; 2; 3; � � � ;C: ð2:143Þ

Details of the various approaches to model vapour-liquid equilibria with\equations of state can be found in references 2 and 3 and 5 through 9.

2.10.2 Chemical Equilibria

The general criterion for chemical equilibrium eq 2.135 can be convenientlyexpressed in terms of the chemical potential of each species present in theequilibrium mixture by:

Xi

nimi ¼ 0; i ¼ 1; 2; 3; � � � ;C: ð2:144Þ

In eq 2.144 ni are the stoichiometric numbers, which for products are positiveand for negative for reactants. At constant temperature and composition, eq2.107 is:

dmi ¼ dGi ¼ RTd ln ~pi: ð2:145Þ

Integration of eq 2.145 from a standard state of pure species i to the actualstate in solution gives:

mi ¼ m�Ji þ RT ln~pi~pie

!¼ m�Ji þ RT ln ai: ð2:146Þ

In eq 2.146 ai is the activity of component i in the mixture and substitution ofeq 2.146 into the condition for chemical equilibrium gives the important rela-tion:

Yi

ai� �ni ¼ exp

�Pi

nim�Ji

RT

24

35: ð2:147Þ

The right-hand side of eq 2.147 is solely a function of temperature and can bewritten as:

DrGe ¼

Xi

nim�Ji ¼ �RT lnK Tð Þ ð2:148Þ

Equation 2.148 defines the thermodynamic equilibrium constant K, whichonly a function of temperature. Equations 2.147 and 2.148 show the


equilibrium constant is related to the activities of the species in the mixture.Since the activities of the reacting species are related to their fugacities, equa-tions of state can be used to evaluate them. Further details on chemical equi-libria can be found in references 1, 3 and 7.

2.11 Stability and the Critical State

2.11.1 Densities and Fields

Griffiths and Wheeler11 divided thermodynamic properties into two classes:fields f (for example, T, p and m) that must be uniform throughout a system atequilibrium; and densities r (for example S, V and n) which, in general, arediscontinuous across a phase boundary although they are uniform throughouteach phase. With this nomenclature, the fundamental equation 2.1 for thesecond law of thermodynamics can be written, very compactly, as:

dU ¼Xj

fjdrj ; ð2:149Þ

where the density U(S, V, n1, � � �, nC) is the thermodynamic surface and thehydrostatic field is –p rather than p. Equation 2.149 shows that conjugatedensities and fields are related by:

f ¼ @U=@rj� �

rj0; ð2:150Þ

where the subscript rj indicates that all the densities except rj are held constant.Griffiths and Wheeler used an equivalent definition:

rj ¼ �@ ~p0@~pj

� �~pj

; ð2:151Þ

where p0 is a thermodynamic potential.

2.11.2 Stability

Just as the fields, which are the first derivatives (@U/@r), characterise equili-briums, the curvature of the thermodynamics surface, which depends on thesecond derivatives (@2U/@rjqrk), determines the stability of the system. Thestability determinant for a system with C components may be written as

D S;V ; n1; : � � � ; nC�1ð Þ ¼

USS USV USn1 :::UVS UVV UVn1 :::Un1S Un1V Un1n1 :::::: ::: ::: :::

��

��; ð2:152Þ

28 Chapter 2

where the elements of the determinant are given by

Urjrk ¼@2U

@rj@rk

!rjrk

with rj and rk ¼ S; V ; n1; � � � ; nC: ð2:153Þ

In a stable system with C components, the thermodynamic surface U(S, V,n1, � � �, nC) lies above its tangent plane and has positive curvature and, conse-quently, all the (Cþ 1) determinants D(S), D(S, V), D(S, V, n1), � � �, D(S, V, n1,� � �, nC�1) are positive. Furthermore, since the variables may be chosen in anyorder, many more determinants may be formed and they are all positive.However, with C components there are only (Cþ 1) independent variables anda set of (Cþ 1) determinants is sufficient to establish the conditions for stability.For example, the system is stable provided the following hold:

DðSÞ ¼ @T

@S

� �V ;n

¼ T

CV� 0; ð2:154Þ

DðS;VÞ ¼ DðS;VÞDðSÞ DðSÞ ¼ �DðSÞ @p

@V

� �T ;n

¼ DðSÞ 1

VkT� 0; ð2:155Þ

DðS;V ; n1Þ ¼DðS;V ; n1ÞDðS;VÞ DðS;VÞ ¼ DðS;VÞ @m

@n1

� �T ;p;n1

� 0; ð2:156Þ

and so on. The ratios of the determinants are obtained from

D r1; : � � � ;rj� �

D r1; : � � � ; rj�1� � ¼ @pj

@rj

!~pioj ;rk4j

; ð2:157Þ

which was derived by Gibbs.12

2.11.3 Critical State

The critical sate is the limit of stability at which all the determinants that werepositive in Section 2.11.2 become zero. However, in the usual case where atransition between two phases is terminated, the critical state imposes only 3additional restrictions, irrespective of the number of components. Similarly,although all the discontinuities in the densities vanish because the phases becomeidentical, it is sufficient to consider the behaviour of the system with respect to asingle density and to formulate the restrictions in terms of higher-order derivatives

@2U

@r2j

!rj

¼ 0;@3U

@r3j

!rj

¼ 0;@4U

@r4j

!rj

40: ð2:158Þ


The conditions are often defined in other thermodynamic surfaces where thevariables more closely match an equation of state or the experimental condi-tions. For example, a gas-liquid critical point in a pure fluid is usually defined by

@p

@V

� �T ;n

¼ 0;@2p

@V2

� �T ;n

¼ 0;@3p

@V3

� �T ;n

o0; ð2:159Þ

which may be written in terms of the A(T, V, n) surface as

A2V ¼@2H

@V2

� �T ;n

¼ 1

VkT¼ 0; A3V ¼

@3A

@V3

� �T ;n

¼ 0; A4V

¼ @4A

@V4

� �T ;n

40; ð2:160Þ

where the notation introduced in eq 2.154 has been used. Temperature is assumedto be uniform and constant for both eqs 2.159 and 2.160. However, if theH(S, p,n) surface is used, then an entirely equivalent set of conditions is obtained

H2S ¼@2H

@S2

� �p;n

¼ T

CP¼ 0; H3S ¼

@3H

@S3

� �p;n

¼ 0; H4S

¼ @4H

@S4

� �p;n

40; ð2:161Þ

but now pressure is assumed to be uniform and constant. Equation 2.157 showsthe relation between these surfaces and U(S, V, n), since A2V¼D(S, V)/D(S) andH2S¼D(V, S)/D(V). Conditions equivalent to eqs 2.160 and 2.161 are obtainedfrom theU(S, V, n) surface and eq 2.158 with rj¼V or S. While eq 2.159 is morefamiliar through its use of (p, V, T) variables and association with gas-liquidcritical points in pure fluids, each set of conditions can be used to describe thesame critical state. For example eq 2.161 or rj¼S in eq 2.158 might be veryappropriate for a calorimetric study of a gas-liquid critical point.The experimental conditions of a critical state in a binary mixture are closely

matched by the Gibbs function G(T, p, n1, n2) and the relation with the U(S, V,n1, n2) surface is established with eq 2.157 in the form

DðS; v; n1Þ=DðS;VÞ ¼@m1@n1

� �T ;p;n2

; ð2:162Þ

which leads to the following conditions for the critical state:

@m1@n1

� �T ;p;n2

¼ 0;@2m1@n21

� �T ;p;n2

¼ 0;@3m1@n31

� �T ;p;n2

� 0: ð2:163Þ

30 Chapter 2

The Gibbs-Duhem equation 2.18 allows these conditions to be expressed1,10

in terms of the molar Gibbs function Gm and a mole fraction x

G2x ¼@2G

@x2

� �T ;p

¼ 0; G3x ¼@3G

@x3

� �T ;p

¼ 0; G4x ¼@4G

@x4

� �T ;p

� 0: ð2:164Þ

Since most equations of state have temperature, molar volume, and com-position as independent variables, while the Gibbs function is explicit in tem-perature, pressure, and composition; a formulation of the critical conditions interms of the Helmholtz function is required. The following equations allow atransformation between G(T, p, x) and A(T, V, x).1,10

G2x ¼ A2x � AVxð Þ2=A2V ; ð2:165Þ

and

G3x ¼ A3x � 3AV2x AVx=A2Vð Þ þ 3A2Vx AVx=A2Vð Þ2�A3V AVx=A2Vð Þ: ð2:166Þ

The Helmholtz function A(T, V, x) and the derivatives required for eqs 2.165and 2.166 may be obtained from any equation of state that gives the pressurethrough

A V ;T ; xð Þ ¼ 1� xð ÞA�1 Vr;Tð Þ þ xA�2 Vr;Tð Þ

þ RT 1� xð Þ ln 1� xð Þ þ x ln xf g �ZVVr

pdV : ð2:167Þ

In this equation A*1(Vr,T) and A*

2(Vr,T) are the molar Helmholtz functions ofthe pure components and Vr is a reference volume.Multicomponent systems are handled in a similar way. For example, a

ternary mixture can be described in terms of four variables and eq 2.157 gives

D S;V ; n1; n2ð Þ=D S;V ; n1ð Þ ¼ @m2@n2

� �T ;p;m1 ;n3

; ð2:168Þ

and the conditions for a critical state are therefore

@m2@n2

� �T ;p;m1;n3

¼ 0;@2m2@n22

� �T ;p;m1;n3

¼ 0;@3m2@n2

� �T ;p;m1;n3

40: ð2:169Þ

The defining equations for higher-order critical points are straightforward interms of the Gibbs function and the composition variables. For instance, for a


tri-critical point in a (pseudo) binary mixture the following 2Pc� 1¼ 5 condi-tions have to be satisfied1,10:

G2x ¼ 0; G3x ¼ 0; G4x ¼ 0; G5x ¼ 0; G6x40: ð2:170Þ

For higher-order critical points, the transformation equations from theGibbs into the Helmholtz function become extremely complex.

References

1. M. Modell and R. C. Reid, Thermodynamics and its Applications, 2nd ed.,Prentice Hall, New York, 1983.

2. H. C. Van Ness and M. M. Abbott, Classical Thermodynamics of None-lectrolyte Solutions, McGraw-Hill, New York, 1982.

3. S. M. Walas, Phase Equilibrium in Chemical Engineering, Butterworth,1985.

4. M. M. Abbott and K. K. Nass, Equations of State and Classical SolutionThermodynamics: Survey of the Connection, in: K. C. Chao and R. L.Robinson, eds., Equations of State: Theories and Applications, ACS Sym-posium Series 300, American Chemical Society, Washington DC, 1986.

5. B. Poling, J. Prausnitz and J. P. O’Connell, The Properties of Gases andLiquids, 5th ed., McGraw-Hill, New York, 2001.

6. J. M. Prausnitz, R. N. Lichtenthaler and E. Gomes de Azevedo, MolecularThermodynamics of Fluid-Phase Equilibria, 2nd edition, Prentice Hall, 1986.

7. J. M. Smith, H. C. Van Ness and M. M. Abbott, Introduction to ChemicalEngineering Thermodynamics, 5th edition, McGraw-Hill Book Company,1996.

8. S. Malanowski and A. Anderko,Modelling Phase Equilibria, John Wiley &Sons, Inc., 1992.

9. A. Anderko, Fluid Phase Equilib., 1990, 61, 145–225.10. J. S. Rowlinson and F. L. Swinton, Liquids and Liquid Mixtures, 3rd ed.,

Butterworth Publishers, 1982.

32 Chapter 2

CHAPTER 3

The Virial Equation of State

J. P. MARTIN TRUSLER

Department of Chemical Engineering, Imperial College London, SouthKensington Campus, London SW7 2AZ, U.K.

3.1 Introduction

The virial equation of state is a power series expansion for the pressure p of areal gas in terms of the amount-of-substance density rn:

p ¼ rnRT 1þ Brn þ Cr2n þ � � ��

: ð3:1Þ

Here, T is the thermodynamic temperature, R is the universal gas constant,rn¼ n/V, n is the amount of substance, V is the volume, and 1, B, C, � � � arecalled virial coefficients. The virial series is also conveniently written in terms ofthe compression factor Z:

Z ¼ p= rnRTð Þ ¼ 1þ Brn þ Cr2n þ � � � : ð3:2Þ

Since the leading term on the right of eq 3.1 is the pressure of a perfect gas, thesecond and high-order virial coefficients (B, C, � � �) describe gas imperfections.These coefficients depend upon temperature and upon the nature of the gas; soin a mixture, they depend upon the composition.The virial equation is limited to gases at low or moderate densities, while

many other equations exist that purport to cover the entire fluid region.Nevertheless, the virial equation possess a number of key strengths including:

� a rigorous basis in statistical thermodynamics;� the exactly known composition dependence of the virial coefficients;




33

� a large database of experimentally-determined and critically-assessedsecond virial coefficients for pure gases1,2 and mixtures;2,3

� experimental values of third virial coefficients for many gases and a fewmixtures;1,2,3

� well-founded predictive methods for the second and third virial coefficientsthat can be used when experimental values are lacking.

3.1.1 Temperature Dependence of the Virial Coefficients

The temperature dependence of the second virial coefficient is exemplified bythe experimental data for argon,4 which are plotted in dimensionless form inFigure 3.1. At high temperatures, B is positive and varies only slowly withtemperature while, at lower temperatures, it becomes large and negative. Thetemperature TB at which B¼ 0, known as the Boyle temperature, varies greatlywith the nature of the substance and is only experimentally accessible for a fewsmall molecules. At higher temperatures still, the second virial coefficientapproaches a nearly constant value but does eventually decline at very hightemperatures as observed experimentally for helium.The temperature dependence of the third virial coefficient is illustrated in

Figure 3.2, again taking argon as the example. Like B, C is a slowly varyingpositive quantity at high temperatures and a rapidly varying negative one atlow temperatures; in between these extremes it passes through a maximum.All gases exhibit the same general features illustrated in Figures 3.1 and 3.2.

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0 2 4 6 8 10T/T c

B (

pc/R

Tc )

-1

0

0 1 2

B (

pc/R

Tc )

T c/T

Figure 3.1 The dimensionless second virial coefficient B(pc/RT c) of argon as a func-tion of reduced temperature T/T c, where pc is the critical pressure and T c

is the critical temperature. Values computed from the equation of state ofTegeler et al.4 The insert shows the same data on an inverse temperaturescale.

34 Chapter 3

The temperature dependence of the higher virial coefficients is not wellestablished experimentally but theoretical evaluations for model systems indi-cate that these virial coefficients too are positive at high temperatures with arapid divergence towards large negative values at low temperatures.5

3.1.2 Composition Dependence of the Virial Coefficients

The dependence of the virial coefficients upon composition may be deducedexactly by noting that, in a mixture composed of n componentsw with amountsof substance n1, n2, n3 � � � nn present in volume V, the pressure must be afunction of the (nþ 1) variables T, n1/V, n2/V, n3/V � � � nn/V and an isothermalexpansion of p in powers of n1/V, n2/V, n3/V � � � nn/V must exist. When thatexpansion is carried out, eq 3.1 is again recovered with amount-of-substance

density rn ¼Pni¼1

ni=V and with virial coefficients that depend upon composition

as follows:

Bmix ¼Pni¼1

Pnj¼1

xixjBij

Cmix ¼Pni¼1

Pnj¼1

Pnk¼1

xixjxkCijk

etc:

9>>>>=>>>>;

ð3:3Þ

Here, xi is the mole fraction of component i in the mixture, and Bij, Cijk � � � areso-called interaction virial coefficients. Clearly Bii, Ciii etc. pertain to the pure

-0.2

-0.1

0.0

0 2 4 6 8 10T/T c

C (pc /R

Tc )2

Figure 3.2 The dimensionless third virial coefficient C(pc/RTc)2 of argon as a functionof reduced temperature T/Tc, where pc is the critical pressure and Tc is thecritical temperature. Values computed from the equation of state ofTegeler et al.4

wn has been used rather than C as recommended by JUPAC, to avoid confusion with the third virialcoefficient.

35The Virial Equation of State

species i, while unlike terms such as Bij (iaj) pertain to interactions betweenunlike molecules.

3.1.3 Convergence of the Virial Series

The virial series is not convergent for all experimentally realisable densities; forexample, it certainly diverges in the vicinity of the critical point. The true radiusof convergence is unknown in general and the experimental evidence is some-what ambiguous. It is certainly possible to fit experimental compression-factorisotherms that extend over a wide range of densities to a truncated form of eq3.1. At supercritical temperatures, such representations appear to be satisfac-tory at densities well in excess of the critical while, at subcritical temperatures,good results may be obtained up to the density of the saturated vapour.However, it is often argued that the coefficients obtained in this way are not thetrue virial coefficients; indeed, the value of B determined in a fit to high-densitydata may differ noticeably from that obtained by analysis of precise low-densitydata. Whether or not such differences should be attributed solely to experi-mental uncertainties, or to the limitations of eq 3.1 itself, is really not clear.Whatever the ultimate radius of convergence might be, a more practical issue

is to establish the region in which the series is rapidly convergent and, specifi-cally, the magnitude of the relative error d arising from truncation of the virialseries after a specified number of terms. This question is also difficult to answerin the general but examples may be computed for any fluid described by a wide-ranging equation of state of high accuracy which reduces to viral form at lowdensities. For example, Figure 3.3 shows results along two isochores for

0.001

0.01

0.1

1

1 1.5 2 2.5 3

T/T c

102 |

δ|

c�n = 0.1�n

c�n = 0.5�n

Figure 3.3 Relative error d arising from truncation of the virial equation after thethird term along two isochores at supercritical temperatures: ———,methane;6 — — —, argon;4 - - - -, carbon dioxide.7

36 Chapter 3

methane,6 argon,4 and carbon dioxide7 when eq 3.1, truncated after the term inC, is compared with the corresponding full equation of state. In these cases,|d|r2 � 10�4 for rn¼ 0.1rc,n, deteriorating to |d|r2 � 10�2 at rn¼ 0.5rc,n andT/TcZ 1.05, where rc,n is the critical molar density. In Figure 3.4, the trun-cation error is illustrated for the same three gases along the saturated-vapourcurve at temperatures between the triple point and the critical point. The rapiddecline in the saturated vapour density is such that the series converges more-and-more rapidly as the temperature is reduced. Figures 3.3 and 3.4 both showrapid deterioration in the approach to the critical point itself. These indicativeresults are also supported by theoretical expectations of the magnitude of Dand higher virial coefficients for the case of the Lennard-Jones (12,6) fluid.5

3.1.4 The Pressure Series

It is often convenient to use (T, p) as the independent variables in place of(T, rn) and, for this purpose, an expansion of Z in p is used which may bewritten as:

Z ¼ 1þ B0pþ C 0p2 þ � � � ð3:4Þ

The coefficients of this series are uniquely related to the virial coefficients asmay be shown by eliminating p from the right hand side of eq 3.3 using eq 3.1and collecting terms, leading to B0 ¼B/RT and C0 ¼ (C�B2)/(RT)2; Table 3.1compares the coefficients of the density (eq 3.2) and pressure (eq 3.4) explicitexpansions of Z. The composition dependence of the coefficients B0, C0, � � � in amixture may be determined by combining these relations with eq 3.3.

0.01

0.1

1

10

0.5 0.6 0.7 0.8 0.9 1

T/T c

102 |δ

|

Figure 3.4 Relative error d arising from truncation of the virial equation after thethird term along the saturated vapour curve at temperatures from thetriple point to the critical point: ———, methane;6 — ——, argon;4 - - - -,carbon dioxide.7


3.2 Theoretical Background

As already mentioned, one of the merits of the virial equation is that it has afirm foundation in statistical thermodynamics and molecular theory. The the-oretical derivation of the series has been described in numerous texts and willnot be discussed in detail here.8–13 The most complete derivation for a mixturecontaining an arbitrary number of components is made by means of anexpansion of the grand partition function.13 This leads to expressions for thevirial coefficients in terms of cluster integrals involving two molecules for B,three molecules for C etc. These expressions are completely general and involveno restrictive assumptions about the nature of molecular interactions. Never-theless, to simplify the expressions for the virial coefficients, a number ofassumptions are often made as follows:

� the molecules are rigid bodies� they obey classical Newtonian mechanics� the intermolecular potential energy of a pair of molecules depend only

upon the separation of the centres of mass� the intermolecular potential energy of a cluster of molecules is the sum of

that calculated for each unique pair in the cluster considered in isolation(pair additivity assumption)

The first of these assumption, often implicitly made, affects the virial coef-ficients of all orders, as do the second and third assumptions. The fourthassumption affects the third and high-order virial coefficients.It should be emphasised that none of the assumptions outlined above is

either necessary or even correct, but taken together they lead to simple andinformative results. In particular, under these simplifications, the second virialcoefficient is given by

B12 ¼ �2pNA

ZN0

f12r212dr12; ð3:5Þ

where

fij ¼ expð�uij=kBTÞ � 1 ð3:6Þ

Table 3.1 Relations between coefficients in the density and pressure explicitexpansions of Z.

Pressure Series Density Series

B0 ¼B/RT B¼RTB0

C0 ¼ (C�B2)/(RT)2 C¼ (RT)2(C0 þB02)D0 ¼ (D� 3BCþ 2B3)/(RT)3 D¼ (RT)3(D0 þ 3B0C0 þB03)

38 Chapter 3

is the Mayer function. In these equations, uij is the intermolecular potentialenergy of molecules i and j, which may be of the same species or different, rij isthe separation of their centres of mass, NA is Avagadro’s constant, and kB isBoltzmann’s constant. Similarly, the third virial coefficient is given by

C123 ¼ �ð82N2A=3Þ

Zf12f13f23 r12r13r23dr12dr13dr23 ð3:7Þ

and involves an integral over the positions of three molecules.

3.2.1 Virial Coefficients of Hard-Core-Square-Well Molecules

Eqs 3.5 and 3.7 are easily evaluated by numerical quadrature for any assumedintermolecular potential-energy function uij. In a few simple cases, analyticalresults may be obtained and we consider here the case of the hard-core-square-well potential defined by

u12 ¼N; r12osu12 ¼ ��; s � r12 � lsu12 ¼ 0; r124ls

9=; ð3:8Þ

where s is the diameter of the hard-cores, e is the depth of the potential-energywell, and ls is the range of the intermolecular potential. Combining eqs 3.5 and3.8 one obtains the following simple expression for the second virial coefficient:

B ¼ b0 1� l3 � 1� �

D� �

: ð3:9Þ

where b0¼ (2pNAs3/3) and D¼ {exp(e/kBT)� 1}. Although the hard-core-

square-well potential is a very crude approximation to the true intermolecularpotential-energy function, it turns out that eq 3.9 offers an excellent repre-sentation of the second virial coefficient, even for non-spherical and polarmolecules, provided that e, s and l are treated as adjustable parameters. Thethird virial coefficient of the hard-core-square-well potential is given by14

C ¼18b20f5� ðl

6 � 18l4 þ 32l3 � 15ÞD� ð2l6 � 36l4

þ 32l3 þ 18l2 � 16ÞD2 � ð6l6 � 18l4 þ 18l2 � 6ÞD3g: ð3:10Þ

Eq 3.10 suffers not only from the crude nature of the pair-potential modelbut also from the neglect of three-body forces; nevertheless, it too can provide areasonable representation of experimental third virial coefficients when theparameters are adjusted freely. Typically, the parameters of the pair-potentialthat give best agreement with experimental data for C are significantly differentfrom those that best represent B.14


3.3 Thermodynamic Properties of Gases

Practical expressions for the thermodynamic properties of real gases mayobtained by decomposition of the properties into perfect-gas and residualcontributions, followed by evaluation of the latter in terms of the virial equa-tion of state. This separation is motivated by the availability of widely-applicable theory for the prediction, estimation and correlation of perfect-gasproperties, enabling those terms to be written down and evaluated essentiallyexactly. Residual properties may then be evaluated in terms of any applicableform of the equation of state.

3.3.1 Perfect-gas and Residual Properties

For generality, we consider again a mixture of n components characterised byamounts of substance n1, n2, n3 � � � nn (denoted by the vector nn), temperature T,pressure p and volume V. If (T, V, nn) are taken as the independent variables, athermodynamic property X(T, V, nn) may be written in the form

XðT ;V ; nnÞ ¼ XpgðT ;V ; nnÞ þ X resðT ;V ; nnÞ ð3:11Þ

where Xpg (T, V, nn) denotes the property of a hypothetical perfect gas with thespecified values of (T, V, nn) and Xres (T, V, nn) is the residual term. A similardecomposition may be applied with (T, p, nn) as the independent variables,

XðT ; p; nnÞ ¼ XpgðT ; p; nnÞ þ X resðT ; p; nnÞ ð3:12Þ

but, although X (T, V, nn)¼X (T, p, nn), the perfect-gas terms and the residualterms in eqs 3.11 and 3.12 generally differ. This is a consequence of the factthat, while (T, V, p, nn) characterise the state of the real fluid, the state of thehypothetical perfect gas depends upon whether (T, V, nn) or (T, p, nn) arespecified. The difference between Xpg (T, p, nn) and Xpg (T, V, nn) may beobtained by noting that the volume of the hypothetical perfect gas is nRT/p in

the one case, where n ¼Pni¼1

ni, but V in the other so that

XpgðT ; p; nnÞ ¼ XpgðT ;V ; nnÞ �ZV

nRT=p

@Xpg=@Vð ÞT ;ndV ð3:13Þ

and

X resðT ; p; nnÞ ¼ XresðT ;V ; nnÞ þZV

nRT=p

@Xpg=@Vð ÞT ;ndV ð3:14Þ

In the case of properties for which the perfect-gas term depends only on (T, nn),one then has Xpg (T, V, nn)¼Xpg (T, p, nn)

40 Chapter 3

The usual choice is to adopt (T, p, nn) as the independent variables and thusfollow eq 3.12. Nevertheless, equations of state are usually formulated with (T,V, nn) as the independent variables, in which case one must first evaluate Xres

(T, V, nn) and then apply eq 3.14 to obtain the desired residual properties at thespecified temperature, pressure and composition. A necessary step in thisprocedure is evaluation, from the chosen equation of state, of the volumecorresponding to the specified (T, p, nn).

3.3.2 Helmholtz Energy and Gibbs Energy

All of the thermodynamic properties of a homogeneous phase may be obtainedfrom the Helmholtz energy or from the Gibbs energy. When (T, V, nn) are theindependent variables, the Helmholtz energy A (T, V, nn) is the appropriatechoice and the fundamental thermodynamic equation for the phase is

dA ¼ �SdT � pdV þXni¼1

midni ð3:15Þ

where S is entropy and mi is the chemical potential of component i. The partialderivatives of A with respect to T, V or ni then give -S, -p or mi respectively.Once these quantities are obtained, the other state functions such as enthalpyH, energy U and Gibbs energy G follow from the appropriate combinations ofA, TS and pV, while quantities such as heat capacity and compressibility maybe obtained from second derivatives of the Helmholtz energy combined, wherenecessary, with the Maxwell relations.When (T, p, nn) are the independent variables, the properties of a homo-

geneous phase are best obtained from the Gibbs energy G(T, p, nn) and thefundamental equation in the form

dG ¼ �SdT þ VdpþXni¼1

midni: ð3:16Þ

S, V and mi are then obtained from the first-order partial derivatives, other statefunctions from combinations of G, TS and pV, and the remaining propertiesfrom second-order partial derivatives of G.

3.3.3 Perfect-Gas Properties

The molar Helmholtz energy Apgm ¼ Apg=n of a pure perfect gas may be obtained

by integration of eq 3.15 subject to the equation of state, p ¼ �ð@Apgm =@VmÞ ¼

nRT=V , and an expression for the perfect-gas molar heat capacity at constantvolume, C

pgV ðTÞ ¼ Tð@Spg

m =@TÞV . Starting from a reference state defined bytemperature T1 and amount-of-substance density rn

�J, the result is13

Apgm ¼

ZTT0

CpgV ;mdT � T

ZTT0

CpgV ;md lnTð Þ þ RT ln rn

�r�Jn

� �þU�Jm � TS

�Jm ; ð3:17Þ


Table

3.2

Perfect-gasthermodynamic

properties

ofpure

gases.Here,

p�J¼RT� Jr n�J

andH� J m¼U� J mþRT.

Pure

gaswithgiven

(T,r n,xu )

Pure

gaswithgiven

(T,p,xnu)

Mixture

Apg

m¼ZT T�

Cpg

V;mdT�T

ZT T�

Cpg

V;mdlnT

þRTlnðr

n=r� J nÞþ

U� J m�TS� J m

Apg

m¼Gpg

m�RT

Apg

mix¼Xu i¼

1

xiA

pg

iþRTXu i¼

1

xilnxi

Gpg

m¼Apg

mþRT

Gpg

m¼ZT T�

Cpg

p;mdT�T

ZT T�

Cpg

p;mdlnT

þRTlnðp=p� JÞþ

H� J m�TS� J m

Gpg

mix¼Xu i¼

1

xiG

pg

iþRTXu i¼

1

xilnxi

Spg

m¼

S� mþZT T�

Cpg

V;mdlnT�Rlnðr

n=r� J nÞ

Spg

m¼

S� mþZT T�

Cpg

p;mdlnTþRlnðp=p� JÞ

Spg

mix¼Xu i¼

1

xiS

pg

i�RXu i¼

1

xilnxi

42 Chapter 3

Upg

m¼

U� J mþZT T�

Cpg

V;mdT

Upg

m¼

H� mþZT T�

Cpg

p;mdT�RT

Upg

mix¼Xu i¼

1

xiU

pg

i

Hpg

m¼

U� J mþRTþZT T�

Cpg

V;mdT

Hpg

m¼

H� J mþZT T�

Cpg

p;mdT

Hpg

mix¼Xu i¼

1

xiH

pg

i

Cpg

V,m¼Cpg

V,m

(T)

Cpg

V,m¼Cpg

p,m

(T)�R

Cpg

V;m

ix¼Xu i¼

1

xiC

pg

V;iðTÞ

Cpg

p,m¼Cpg

V,m

(T)þR

Cpg

p,m¼Cpg

p,m

(T)

Cpg

p;m

ix¼Xu i¼

1

xiC

pg

V;iðTÞ

mpg

s�m*

pg

s¼Gpg

mmp

gs�m*

pg

s¼Gpg

mm s�m*

pg

sþRTln

xs


where U�Jm and S

�Jm are defined values of molar energy and molar entropy at the

chosen reference state. For a mixture of n components, the molar Helmholtzenergy is given by9

ApgmixðT ;rn;x

nÞ ¼Xni¼1

xiApgi þ RT

Xni¼1

xi lnxi; ð3:18Þ

where Apgi is the molar Helmholtz energy of pure i and xn denotes the set of mole

fractions x1, x2, x3 � � � xn.A parallel route to the same properties starts with the molar Gibbs energy

Gpgm ¼ Gpg=n of a pure perfect gas. This is obtained by integration of eq 3.16

starting from a reference state defined by temperature T1 and pressure p�J at

which Gpgm ¼ RT þU�Jm � TS�Jm . The result is:

Gpgm ¼

ZTT0

Cpgp;mdT � T

ZTT0

Cpgp;md lnTð Þ þ RT ln p=p�J

� �þ RT þU

�Jm � TS

�Jm ; ð3:19Þ

where Cpgp;m ¼ C

pgV ;m þ R is the isobaric perfect-gas heat capacity. Mixture

properties may then be obtained by means of the equation

GpgmixðT ; p;x

nÞ ¼Xni¼1

xiGpgi þ RT

Xni¼1

xi ln xi: ð3:20Þ

In order to make the two approaches consistent, one must choose the samereference state, and this requires that p

�J¼RT�Jr�Jn.

The principal perfect-gas thermodynamic properties of pure substances andmulti-component mixtures determined from these relations are summarised inTable 3.2.13

3.3.4 Residual Properties

The residual part of the Helmholtz energy for a phase of constant compositionat given temperature and molar density may be obtained by combining theidentity

Aresm T ;rnð Þ ¼

Zrn0

@Am=@rnð ÞT� @Apgm

�@rn

� �T

� �drn; ð3:21Þ

in which ð@Am=@rnÞT ¼ p=r2n, with eq 3.1 for the pressure. All other residualproperties may then be derived by manipulation of the result, and the corre-sponding residual properties at specified temperature and pressure may beobtained by invoking eq 3.14. Table 3.2 gives expressions for Ares

m and for the

44 Chapter 3

other five common residual thermodynamic functions Sresm , Ures

m , CresV,m, H

resm and

Gresm in terms of the virial coefficients, with (T, p, xn) as the given variables.13

The residual part of the Gibbs energy may be obtained in terms of thepressure series, eq 3.4, in a manner analogous to that used to obtain Ares

m . Table3.2 gives the expansions of Gres

m , Sresm , Hres

m , Cresp,m, U

resm , Ares

m and mress in terms ofthe coefficients of that series, again with (T, p, xn) as the given variables.Also given in Table 3.2 is the residual part of the chemical potential ms of

component s in a multi-component mixture at specified (T, p, xn). The partialfugacity ps of component s in a mixture, often used in phase equilibrium cal-culations, is defined by the relation

lnð~ps=pÞ ¼ ðms � m�pgs Þ=RT ; ð3:22Þ

where m�pgs ; is the perfect-gas chemical potential of pure s at the temperatureand pressure in question. In turn, ps may be conveniently expressed in terms ofthe dimensionless partial fugacity coefficient fs¼ (ps/xs p) which, in view of eqs3.20 and 3.22, is related to the residual chemical potential at specified (T, p, xn)as follows:

RT lnfsðT ;rn;xnÞ ¼ mress ðT ; p; xnÞ ð3:23Þ

Another property of considerable interest is the speed of sound u. Sinceu2¼ (@p/@r)s, it is possible to develop u2 as a power series in amount-of-sub-stance density:

u2 ¼ ðRTgpg=MÞð1þ barn þ gar2n þ � � �Þ: ð3:24Þ

Here, gpg ¼ Cpgp;m=C

pgV ;m,M is the molar mass, and ba, ga, � � � are the second, third

� � � acoustic virial coefficients of the gas. The second and third acoustic virialcoefficients are related to the ordinary virial coefficients as follows:15

ba ¼ 2Bþ 2ðgpg � 1ÞTB0 þ ðgpg � 1Þ2T2B 0 0=gpg

ga ¼ ½Bþ ð2gpg � 1ÞTB0 þ ðgpg � 1ÞT2B0 0=gpg2ðgpg � 1Þ2=gpgþ½ð1þ 2gpgÞC þ fðgpgÞ2 � 1gTC 0 þ 1

2ðgpg � 1Þ2T2C 0 0=gpg

9=;: ð3:25Þ

The corresponding expansion in powers of pressure is usually written as

u2 ¼ A0 þ A1pþ A2p2 þ � � � ; ð3:26Þ

where A0¼ (RTgpg/M), (M/gpg)A1¼ ba, and (M/gpg)A2, (ga�Bba)/RT.13

Other thermodynamic properties of gases may be obtained through standardmanipulations of the relations given in Tables 3.2 and 3.3.


Table3.3

Residualthermodynamicproperties

ofpure

gasesandgaseousmixtureswith(T

,p,xn )astheindependentvariables.

From

equation(3.1)

From

equation(3.4)

Ares

m¼

RT

Br nþ

1 2Cr2 nþ��

��

RTlnZ

Ares

m¼�RT

1 2C0 p

2þ��

��

Gres

m¼

RT

2Br nþ

3 2Cr2 nþ��

��

RTlnZ

Gres

m¼

RT

B0 pþ

1 2C0 p

2þ��

��

Sres

m¼�R

BþB1

ðÞr

nþ

1 2CþC

1ð

Þr2 nþ��

�� þ

RlnZ

Sres

m¼�R

B0 þ

B0 1

�� pþ

1 2C0 þ

C0 1

�� p2þ��

��

Ures

m¼�RT

B1r nþ

1 2C

1r2 nþ��

��

Ures

m¼�RT

B0 þ

B0 1

�� pþ

C0 þ

1 2C0 1

�� p2þ��

��

Hres

m¼

RT

B�B1

ðÞr

nþ

C�

1 2C

1

�� r2 nþ��

��

Hres

m¼�RT

B0 1pþ

1 2C0 1p2þ��

��

Cres

V;m¼�R

2B1þB2

ðÞr

nþ

1 22C

1þC

2ð

Þr2 nþ��

��

Cres

p;m¼�R

2B0 1þB0 2

�� pþ

1 22C0 1þC0 2

�� p2þ��

��

mres s¼

RT

2X i

xiB

is

! r nþ

3 2

X i

X j

xix

jCijs

! r2 nþ��

()

�RTlnZ

mres s¼

RT

2X i

xiB

is�B

() pþ

3 2

X i

X j

xix

jCijs�2BX i

xiB

isþ

3 2B2�C

() p

2þ��

"#

*HereB1¼T(dB/dT),B2¼T2(d

2B/dT2)etcforapure

gas,andB¼Bmix,B1¼T(dBmix/dT),etcforamixture.

46 Chapter 3

3.4 Estimation of Second and Third Virial Coefficients

It has already been mentioned that a large body of experimental data exist forthe second virial coefficients of gas and mixtures.1–3 Although beyond the scopeof this chapter, experimental methods for the determination of virial coeffi-cients have been reviewed by a number of authors and a good summary cov-ering modern methods may be found in references 2 and 3.

3.4.1 Application of Intermolecular Potential-energy Functions

Second virial coefficients may be correlated in terms of eq 3.9, the hard-core-square-well formula, and the resulting model may be used for interpolation and,with caution, extrapolation of the data with respect to temperature.16–20 Other,somewhat more realistic, intermolecular potential models have also been used tocorrelate experimental second virial coefficients. Examples include the hard-core-Lennard-Jones, the Maitland-Smith, and generalised Mie models.21–24

Each of these models has at least one ‘shape’ parameter in addition to thelength- and energy-scaling parameters s and e, and adjustment of this parameteris generally necessary to achieve an accurate correlation over an extended rangeof temperature. Third virial coefficients have also been correlated in terms of thehard-core-square-well model, eq 3.10,14 or by combining one of the pairpotentials mentioned above with an additional three-body potential.23 In thelatter case, the parameters of the pair potential may be the same in the repre-sentation of both B and C while, in the former case, it is necessary to haveseparate parameter sets for B and C. The parameters in model intermolecularpotential-energy functions may be optimised against second (and third) acousticvirial coefficients and the resulting models used in turn to compute ordinarysecond (and third) virial coefficients. Indeed all of the examples cited hereinvolve the analysis of acoustic virial coefficients either alone or in combinationwith ordinary virial coefficients.16–24

The same intermolecular potential-energy functions may be applied to cor-relate interaction virial coefficients. In the case where there are no, or insuffi-cient, experimental data, but adequate data exist for the pure components, itmay be possible to estimate with useful accuracy the unlike interaction para-meters from combining rules such as:

s12 ¼ 12ðs11 þ s22Þ

�12 ¼ ð�11�22Þ1=2�: ð3:27Þ

The first of these relations is the Berthelot rule, and the second is the Lorentzrule. Combining rules are also required for other parameters in thechosen model. For example, the range parameter l12 in the unlike hard-core-square-well potential might be estimated as an arithmetic mean of l11 and l12.(Table 3.3)


3.4.2 Corresponding-states Methods

Although the pair potentials models commonly used have the advantage ofbeing able to correlate second virial coefficients with high accuracy, they havethe disadvantage of requiring three molecular parameters. In cases where virialcoefficients have to be estimated in the absence of any direct measurements, amodel based on the principle of corresponding states is more robust. In itssimplest form, the principle applies to systems whose intermolecular pairpotentials may be written in the form u(r)¼ e F(r/s), where e and s are scalingparameters which characterise a particular substance and F is a universalfunction. Systems which obey this relation are said to be conformal. In allconformal systems to which classical statistical mechanics applies, the reducedsecond virial coefficient, B� ¼ B=ð2

3pNAs3Þ, is a universal function of T*¼ kT/e.

Thus, the second virial coefficient of one conformal substance (labelled i) maybe estimated from that of another (labelled 0) if the ratios ei/e0 and si/s0 areknown. From a theoretical point of view, the most satisfactory way of relatingthe scaling parameters to measurable properties is by means of the Boyletemperature and the so-called Boyle volume VB, equal to T(dB/dT) at T¼TB.In terms of these quantities, B/VB is, according to the principle, a universalfunction of T/TB and, if pair-wise additivity of the intermolecular forces isassumed, C/(VB)2 is another universal function of T/TB. This method ofselecting the scaling parameters has the disadvantage of requiring some mea-surements of B(T) in the first place. Furthermore, TB is inconveniently high formost substances.Practical correlations of virial coefficients employ as scaling parameters the

critical temperature Tc and the characteristic molar volume RTc/pc, where pc isthe critical pressure, and seek to represent B(pc/RTc) and C(pc/RTc)

2 as uni-versal functions of the new reduced temperature Tr¼T/Tc. Although theprinciple, as stated above, applies to only a small number of simple fluids,Pitzer25 showed that many different kinds of molecular complexity may beaccounted for by the inclusion of a third parameter o which he called theacentric factor. This parameter is defined in terms of the vapour pressure psat,by the equation

o ¼ �1� log10 psat Tr ¼ 0:7ð Þ=pcf g ð3:28Þ

such that it is essentially zero for the simple fluids Ar, Kr and Xe. For otherfluids, values between 0 and about 0.4 are usually found. The second virialcoefficient is given in this extended principle of corresponding states by

B pc=RTcð Þ ¼ B0 þ oB1 ð3:29Þ

where B0 and B1 are dimensionless functions of Tr. Empirical correlation for B0

and B1 were given by Pitzer and Curl26 in their original formulation, andimproved expressions were later proposed by Tsonopoulos.27,28 Updated cor-relations have recently been published by Meng et. al.29 These are based on the

48 Chapter 3

most recent comprehensive review of the available data for pure non-polargases and are given by:

B0 ¼ 0:13356� 0:30252=Tr � 0:15668=T2r � 0:00724=T3

r � 0:00022=T8r

B1 ¼ 0:17404� 0:15581=Tr þ 0:38183=T2r � 0:44044=T3

r � 0:00541=T8r

�ð3:30Þ

As it stands, eq 3.29 is useful only for essentially non-polar gases but Tsono-poulos28 showed that the addition of a third term, B2, to the right-hand side ofthat equation permitted the results for polar gases to be correlated also.27 Thisthird term is expressed in terms of a reduced dipole moment, defined by

mR ¼ ðpc=101:325 kPaÞðm=3:33564 10�30C �mÞ2ðTc=KÞ�2; ð3:31Þ

where m is the dipole moment. Weber30 considered the form of B2 for smallpolar halogenated alkanes in the context of the original correlations for B0 andB1 given by Tsonopoulos, and concluded that B2 could be represented by asingle term in B2¼ � 910�7 m2R provided that the expression for B1 wasslightly modified for those gases. Meng et al.,29 working with the expressionsfor B1 and B2 obtained for non-polar gases, eq 3.30, found that that data forsmall polar halogenated alkanes were best fitted when B2 was given by:

B2 ¼ �1:1524 104m2R þ 7:2238 109m4R � 1:8701 1015m6R: ð3:32Þ

However, other non-associated polar gases required an alternative form givenby:

B2 ¼ �3:0309 104m2R þ 9:503 109m4R � 1:2469 1015m6R: ð3:33Þ

In order to obtain interaction second virial coefficients for mixtures, somemethod is required for determining the acentric factor oij and the pseudo-cri-tical constants Tc,ij and pc,ij pertaining to the unlike interactions. In the presentcase, extended van der Waals one-fluid mixing rules are applied in terms ofwhich

oij ¼1

2oi þ oj

� �Tc;ij ¼ 1� kij

� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiTc;iTc;j

ppc;ij ¼ 4Tc;ij pc;iTc;i=Tc;ið Þ þ pc;jTc;j

�Tc;j

� �� .Vc;ið Þ1=3þ Vc;j

� �1=3n o ð3:34Þ

where kij is a binary interaction parameter, which may be optimised againstexperimental data, and Vc,i is the critical molar volume of pure i. When anexperimental value is unavailable, Vc,I¼ (RTc,iZc,i/pc,i) may be estimated fromthe correlation of the critical compression factor proposed by Lee and Kesler:Zc,i¼ 0.2905� 0.085oi.

31 Optimal values of kij for a large number of non-polar


mixtures have been reported, together with correlations that can be used toestimate kij for other binary systems formed from inorganic gases and n-alkanes.32 For interactions involving polar molecules, the term B2 is againincluded with mR obtained from eq 3.31 with m2 replaced by the product m1m2 ofthe dipole moments of the two unlike species.The third virial coefficients of pure non-polar gases have also been correlated

using a three-parameter corresponding-states model by Orbey and Vera.33

However, in order to obtain a reliable correlation for the third virial coefficientsof both polar and non-polar substances it is again necessary to introduce anadditional parameter. Liu and Xiang present a correlation of this kind in whichthe fourth parameter is the critical compression factor Zc; their model is:34

CðpcÞ2=ðZcRTcÞ2 ¼ C0 þ oC1 þ ðZc � 0:29ÞC2

C0 ¼ 0:1623538þ 0:308744=T3r � 0:01790184=T6

r � 0:02789157=T11r

C1 ¼ �0:5390344þ 1:783526=T3r � 1:055391=T6

r þ 0:09955867=T11r

C2 ¼ 34:22804� 74:76559=T3r þ 279:9220=T6

r � 62:85431=T11r

9>>>=>>>;ð3:35Þ

This method is claimed to be comparable in accuracy to the methods of Orbeyand Vera33 for non-polar gases and to a method described by Weber30 for polarsubstances.Several methods have been proposed for the estimation of the interaction

third virial coefficients Cijk in mixtures. Orbey and Vera33 follow Chueh andPrausnitz35 in proposing the simple relation

Cijk ¼ CijCikCjk

� �1=3 ð3:36Þ

in which Cij is evaluated from eqs 3.35 with the acentric factor and pseudo-critical constants pertaining to the binary pair i and j. In a test against accurateexperimental results for several non-polar binary mixtures,36 this method wasfound to give satisfactory estimates.Although the available estimation methods for second and third virial

coefficients are not highly accurate, gas densities and partial fugacity coeffi-cients estimated by such methods for non-polar and slightly-polar pure gasesand mixtures may be accurate enough for many engineering purposes.33,36

Furthermore, as the corresponding-states method may be applied knowingonly the critical constants, acentric factor and dipole moment for each com-ponent, it can be applied to a very wide range of substances. Perhaps the mostvaluable area of application for the virial equation is in the estimation ofvapour-liquid equilibria in highly non-ideal mixtures by the so-called ‘g - f’approach.37 In this method, the chemical potential of components in the liquidphase is treated with an activity coefficient model while that in the gas phase istreated with an equation of state. In this context, the virial equation is generallysuperior to cubic equations of state and is no more complicated to implement.

50 Chapter 3

References

1. J. H. Dymond, K. N. Marsh, R. C. Wilhoit and K. C. Wong, ‘‘VirialCoefficients of Pure Gases and Mixtures’’ (Landolt-Bornstein-Group IVPhysical Chemistry, Volume 21A Virial Coefficients of Pure Gases), ed. M.Frenkel, K. N. Marsh, Springer-Verlag, Heidelberg, 2002.

2. J. H. Dymond and E. B. Smith, The Virial Coefficients of Pure Gases andMixtures, Clarendon Press, Oxford, 1980.

3. J. H. Dymond, K. N. Marsh and R. C. Wilhoit, ‘‘Virial Coefficients of PureGases and Mixtures’’ (Landolt-Bornstein-Group IV Physical Chemistry,Volume 21B Virial Coefficients of Mixtures), M. Frenkel, K. N. Marsh,Springer-Verlag, Heidelberg, 2003.

4. Ch. Tegeler, R. Span and R. W. Wagner, J. Phys. Chem. Ref. Data, 1999,28, 779–850.

5. J. A. Barker, P. J. Leonard and A. Pompe, J. Chem. Phys., 1966, 44,4206–4211.

6. U. Setzmann andW.Wagner, J. Phys. Chem. Ref. Data, 1991, 20, 1061–1151.7. R. Span and W. Wagner, J. Phys. Chem. Ref. Data, 1996, 25, 1509–1596.8. E. A. Mason and T. H. Spurling, The Virial Equation of State, Pergamon

Press, Oxford, 1969.9. T. L. Hill, An Introduction to Statistical Thermodynamics, Addison-Wesley,

Reading, 1960.10. J. O. Hirschfelder, C. F. Curtiss and R. B. Bird, Molecular Theory of Gases

and Liquids (corrected edition), John Wiley, New York, 1964.11. J. E. Kilpatrick, J. Chem. Phys., 1953, 21, 274–278.12. J. E. Mayer and M. G. Mayer, Statistical Mechanics, John Wiley, New

York, 1940.13. J. P. M. Trusler, in Equations of State for Fluids and Fluid Mixtures. Part 1,

ed. J. V. Sengers, R. F. Kayser, C. J. Peters, H. J. White, Elsevier,Amsterdam, 2000; Ch. 3.

14. J. J. Hurly, D. R. Defibaugh and M. R. Moldover, Int. J. Thermophys.,2000, 21, 739–765.

15. J. P. M. Trusler, Physical Acoustics and Metrology of Fluids, Adam Hilger,Bristol, 1991, p. 9.

16. M. B. Ewing, A. R. H. Goodwin, M. L. McGlashan and J. P. M. Trusler,J. Chem. Thermodyn., 1987, 19, 721–739; J. Chem. Thermodyn. 1988, 20,234–256.

17. M. B. Ewing, A. R. H. Goodwin and J. P. M. Trusler, J. Chem. Thermo-dyn., 1989, 21, 867–877.

18. A. R. H. Goodwin and M. R. Moldover, J. Chem. Phys., 1990, 93,2741–2753; 1991, 95, 5230-5235; 1991, 95, 5236–5242.

19. K. A. Gillis and M. R. Moldover, Int. J. Thermophys., 1996, 17, 1305–1324.20. J. J. Hurly, J. W. Schmidt and S. J. Boyes, Int. J. Thermophys., 1997, 18,

579–634.21. M. B. Ewing and J. P. M. Trusler, Physica A, 1992, 184, 415–436; Physica

A 1992, 184, 437–450.


22. J. P. M. Trusler and M. P. Zarari, J. Chem. Thermodyn., 1995, 27, 771–778.23. J. P. M. Trusler, W. A. Wakeham and M. P. Zarari, Mol. Phys., 1997, 90,

695–704.24. J. J. Hurly, D. R. Defibaugh and M. R. Moldover, Int. J. Thermophys.,

2000, 21, 739–765.25. K. S. Pitzer, J. Am. Chem. Soc., 1955, 77, 3427–3433.26. K. S. Pitzer and R. F. Curl Jr., J. Am. Chem. Soc., 1957, 79, 2369–2370.27. C. Tsonopoulos, AIChE J., 1974, 20, 263–272; 1975, 21, 827-829; 1978, 24,

1112–1115.28. C. Tsonopoulos, Adv. Chem. Ser., 1979, 182, 143–162.29. L. Meng, Y.Y. Duan and L. Li, Fluid Phase Equilib., 2004, 226, 109–120.30. L. A. Weber, Int. J. Thermophys., 1994, 15, 461–482.31. B. I. Lee and M. G. Kesler, AIChE J., 1975, 21, 510–527.32. L. Meng and Y.-Y. Duan, Fluid Phase Equilib., 2005, 238, 229–238.33. H. Orbey and J. H. Vera, AIChE J., 1983, 29, 107–113.34. D. X. Liu and H. W. Xiang, Int. J. Thermophys., 2003, 24, 1667–1680.35. P. L. Chueh and J. M. Prausnitz, AIChE J., 1967, 13, 896–902.36. H. B. Brugge, L. Yurtlas, J. C. Holste and K. R. Hall, Fluid Phase Equilib.,

1989, 51, 187–196.37. M. J. Assael, J. P. M. Trusler and T. F. Tsolakis, Thermophysical Proper-

ties of Fluids, Imperial College Press, London, 1986.

52 Chapter 3

CHAPTER 4

Cubic and Generalized van derWaals Equations of State

IOANNIS G. ECONOMOUa, b

aNational Centre for Scientific Research ‘‘Demokritos’’, Institute of PhysicalChemistry, Molecular Thermodynamics and Modeling of MaterialsLaboratory, GR-15310 Aghia Paraskevi Attikis, Greece; b The PetroleumInstitute, Department of Chemical Engineering, PO Box 2533, Abu Dhabi,United Arab Emirates

4.1 Introduction

Equations of state are used widely for the calculation of thermodynamicproperties and phase equilibria of pure components and of mixtures in industryand academia. An accurate knowledge of these properties over a wide range oftemperature, pressure and composition is critical for the design and optimi-zation of a broad range of processes in a variety of industrial segments thatinclude oil and gas, bulk and specialty chemicals including polymers, phar-maceuticals and cosmetics and for environmental control.The most popular class of equations of state are the so-called cubic that

originate from van der Waals equation of state. Van der Waals proposed hisequation of state in 18731 and it was the first thermodynamic model applicableto both the gas and liquid state of fluids. The van der Waals equation of statehas been the basis for literally hundreds of equation of state over the last 137years. Extensions and modifications of the models have been directed towardextended range of temperature and pressure to include subcritical, near-criticaland supercritical conditions, to fluids that are of variable molecular size from




53

small spherical molecules (i.e., gases) to long-chain molecules (i.e., heavyhydrocarbons and polymers) and to fluids whose molecules exhibit non-polar,polar and hydrogen-bonding interactions. In parallel, the model has beenextended to multicomponent mixtures of components that may be similar orvery dissimilar in terms of molecular size, shape and interactions.Cubic equations of state were introduced to the oil and gas industry in the

1960s and 1970s with the development of the first industrial process simulators.The cubic equations of state attracted the interest and the acceptance ofengineers because of: (a) their relative simplicity in terms of mathematicalformulation, and (b) their reasonable uncertainty for the correlation and pre-diction of thermodynamic properties and phase equilibria for both pure com-ponents and, more importantly, mixtures. Developments in applied statisticalmechanics during the 1980s and onwards and the introduction of higher-orderequations of state (some of them are reviewed in other chapters of this book)resulted in advances in cubic equations of state through the introduction ofmore accurate repulsive and attractive terms, more sophisticated mixing rules,refined approaches to calculate binary-interaction parameter(s), etc. The basicphysical concepts in the development of these equations of state are the same asthose in the van der Waals equation of state and so these models are oftencharacterized as generalized van der Waals equations of state.The tremendous increase of computing power resulted in more sophisticated

process simulators with applications to chemical industry, and, more recently,to pharmaceuticals and cosmetics industry. Here again, the underlying ther-modynamic model is often a cubic equation of state with appropriate mod-ifications in order to be applicable to the fluid systems of interest.Over the past 40 years, numerous excellent review papers and book chapters

on cubic and generalized van der Waals equations of state have appeared in theliterature.2–5 The aim of this chapter is to provide the general formulation ofthe cubic equation of state, to discuss some of the major developments of cubicand higher-order generalized van der Waals equations of state starting from thework of Redlich and Kwong and associated developments with respect tomixing rules. The latter is discussed in more detail in Chapter 5 of this book.The last part of this chapter is devoted to specific applications of cubic equa-tions of state to industrial problems with emphasis to applications that haveappeared during the last two decades. The chapter closes with an introductionto the calculation of transport properties with the aid of cubic equationsof state.

4.2 Cubic Equation of State Formulation

4.2.1 The van der Waals Equation of State (1873)

The history of equations of state starts in 1662 with Boyle who concluded thatfor a given gas at a fixed temperature; the product of pressure and volume, PV,is a constant. In 1873, during the course of his Ph.D. thesis, van der Waals

54 Chapter 4

proposed his equation of state, according to which the pressure, P, the tem-perature, T, and the molar volume, Vm, of a fluid (either gas or liquid) are inter-related according to the expression:

P ¼ RT

Vm � b� a

V2m

; ð4:1Þ

where R is the gas constant, a the attraction parameter and b the repulsionparameter. In other words, in the right-hand side (rhs) of eq 4.1, the first termaccounts for the effect of repulsions and the second term for the effect ofattractions to the pressure of the fluid, respectively. By re-writing eq 4.1 in apolynomial form, a third-order (cubic) equation with respect to volume isobtained. A simple and empirical, yet reasonably precise, formulation of thevan der Waals equation of state can be obtained from statistical mechanics byassuming a hard-sphere core potential for repulsions and a Lennard-Jones typeattractive potential beyond the hard core6 that is proportional to the separationdistance r according to r�6. In this way, the model receives a physical justifi-cation and the parameter a is associated with the intermolecular attractiveinteraction parameter e while parameter b with the hard core volume of themolecules.For engineering applications with real fluids, the parameters a and b are

calculated by imposing the critical-point conditions to the equation of state thatare as follows:

@P

@Vm

� �Tc

¼ @2P

@V2m

� �Tc

¼ 0; ð4:2Þ

resulting in the following:

a ¼ 27

64

R2 Tcð Þ2

Pc; ð4:3Þ

and

b ¼ RTc

8Pc; ð4:4Þ

where the subscript c denotes the value of the property at the critical point.Equations 4.1, 4.3 and 4.4 provide only a qualitative description of PVTproperties and vapour – liquid equilibrium (VLE) of real fluids, even of thesimplest ones such as argon or methane. This inefficiency of the model has ledto many additions and improvements of the van der Waals equation of stateduring the last century. A thorough discussion of all these models is beyond thescope of this chapter. We have chosen to present here five of the most inno-vative and most widely used cubic equations of state.

55Cubic and Generalized van der Waals Equations of State

4.2.2 The Redlich and Kwong Equation of State (1949)

Early in the history of development of equations of state, it was realized thatthe parameter a in the van der Waals equation of state had to be temperaturedependent in order to correlate accurately the PVT properties of gases. Out oftens of modifications proposed in the first half of the 20th century, the mostnotable one was the equation proposed by Redlich and Kwong in 1949.7 TheRedlich-Kwong equation of state is written as:

P ¼ RT

Vm � b� aaVm Vm þ bð Þ ; ð4:5Þ

where

a ¼ 1

T0:5: ð4:6Þ

By applying the critical-point conditions (eq 4.2), parameters a and b are cal-culated in terms of the critical properties of the fluid are:

a ¼ OaR2 Tcð Þ2:5

Pc¼ 0:42748

R2 Tcð Þ2:5

Pc; ð4:7Þ

and

b ¼ ObRTc

Pc¼ 0:08664

RTc

Pc: ð4:8Þ

An improved agreement is obtained between experimental data and modelcalculations if the coefficients Oa and Ob become substance dependent. Fur-thermore, by making Oa and Ob temperature-dependent, the accuracy of themodel is further improved. Walas8 provided a list of 16 different approachesregarding the adjustment of the a and b parameters in the Redlich-Kwongequation of state. A major limitation of the Redlich-Kwong equation of state isits inability to correlate liquid-phase thermodynamic properties and, conse-quently, predict VLE. Subsequent cubic equations of state presented belowprovide an improvement in this respect.

4.2.3 The Soave, Redlich and Kwong Equation of State (1972)

Soave9 proposed a significant improvement to the Redlich-Kwong equationof state by introducing a more complex temperature dependence to theattraction parameter that is also a function of the acentric factor o. The Soave-Redlich-Kwong equation of state is given by eq 4.5 where a and b are given by

56 Chapter 4

eqs 4.7 and 4.8, respectively, and

a ¼ 1þ 1� T0:5r

� �0:480þ 1:574o� 0:176o2� �� 2

; ð4:9Þ

where Tr¼T/Tc. The functional form of a in eq 4.9 was formulated in order toprovide a best fit to the vapour pressure of hydrocarbons. In his original paper,Soave9 showed that the new cubic equation of state was a significantimprovement over Redlich-Kwong for pure hydrocarbons and for hydro-carbon mixtures VLE. Soave’s new model received great attention by academiaand industry (primarily oil and gas). Besides the higher uncertainty, thedevelopment of the Soave-Redlich-Kwong equation of state also coincidedwith the introduction of the first generation process simulators that had a needfor robust and precise models for thermodynamic property estimation.

4.2.4 The Peng and Robinson Equation of State (1976)

The success of Soave-Redlich-Kwong equation of state in correlating pure andmixed hydrocarbon phase equilibria generated more interest to the academiccommunity and resulted in the proposal of further improvements to the cubicequation of state in the mid-1970s and onwards. Here we refer to the post-1970cubic equation of state as a ‘‘modern’’ cubic equation of state. The most celebratedand popular modification was the one attributed to Peng and Robinson in 1976.10

The Peng-Robinson cubic equation of state uses a different expression for thedenominator of the attractive term and a different functional form for the para-meters a and b. The Peng-Robinson equation of state is given by the following:

P ¼ RT

Vm � b� aaVm Vm þ bð Þ þ bðVm � bÞ ; ð4:10Þ

where

a ¼ 0:45724R2 Tcð Þ2

Pc; ð4:11Þ

a ¼ 1þ 1� T0:5r

� �0:37464þ 1:54226o� 0:26992o2� �� 2

; ð4:12Þ

and

b ¼ 0:07780RTc

Pc: ð4:13Þ

Peng-Robinson and Soave-Redlich-Kwong equations of state are by far the mostpopular cubic equations of state in academia and industry today. They are widelyused to calculate the properties of pure components and, more importantly,binary, ternary and multicomponent mixtures including both low and high pres-sure VLE, liquid-liquid equilibrium (LLE) and single phase primary and


derivative thermodynamic properties. Representative specific applications arediscussed in Section 4.3.

4.2.5 The Patel and Teja (PT) Equation of State (1982)

All of the cubic equations of state presented so far are generally termed two-parameter equations of state. In this respect, each one of them predicts aconstant compressibility factor at the critical point (Zc¼PcVc/RTc), irrespec-tive of the nature of the compound: for van der Waals Zc¼ 0.375, for Redlich-Kwong and Soave-Redlich-Kwong Zc¼ 0.333 and for Peng-RobinsonZc¼ 0.307. The actual values may vary significantly, especially for polar andassociating fluids: for methane Zc¼ 0.286, for propane Zc¼ 0.276, for pentaneand benzene Zc¼ 0.268 and for water Zc¼ 0.229.11 To correct for this defi-ciency, a number of authors have proposed three or four adjustable parametersto the cubic equation of state. The most popular three-parameter cubic equa-tion of state was proposed by Patel and Teja12 is given by:

P ¼ RT

Vm � b� aaVm Vm þ bð Þ þ c Vm � bð Þ ; ð4:14Þ

where the parameters a, a, b and c are functions of Tc, Pc and of two newadjustable parameters F and zc, for which generalized correlations with respectto acentric factor (o) have been proposed. For relatively low molar massnon-polar substances (oE0), zc approaches 0.329 and the model is comparableto the Soave-Redlich-Kwong equation of state, while for components with oclose to 0.3, zc approaches 0.307 and the model behaves like the Peng-Robinsonequation of state.

4.2.6 The a Parameter

Apart from the van der Waals equation of state, all of the cubic equations ofstate discussed so far contain a temperature-dependent parameter (most of thetimes called a) in the attractive part of the equation. Over the years, thefunctional form of the parameter a increased substantially from a simplea¼T�0.5 in the Redlich-Kwong equation of state (eq 4.5) to a complex func-tion of reduced temperature and acentric factor as in eqs 4.9 and 4.12. It hasbeen recognized that a complicated temperature dependent a improves pre-dictions of vapour pressure of substances, and especially of the polar ones.Valderrama5 lists 23 different expressions proposed for a parameter.Twu and co-workers13 abandoned the functional form for a written as a

polynomial with respect to acentric factor, introduced by Soave (eq 4.9) andslightly modified by Peng and Robinson (eq 4.12), and proposed the expression:

a ¼ TNðM�1Þr exp L 1� TNM

r

� �� ; ð4:15Þ

58 Chapter 4

where L, M and N are substance-dependent parameters. Equation 4.15,incorporated into the van der Waals, Redlich-Kwong and Peng-Robinsonequations of state, was applied to more than 1 000 pure substances, andparameters were regressed to vapour pressure and liquid heat capacity datafrom the triple point to the critical point. The accuracy obtained from all threemodels regarding vapour pressure was almost within the experimental uncer-tainty and very similar to each other. This indicates that with the appropriateselection of an a function, all cubic equations of state provide similar results, atleast for the pure-component properties examined.Later on, Twu and co-workers14 refined their expression for a by proposing

the following functional form for Peng-Robinson equation of state:

a ¼ að0Þ þ o að1Þ � að0Þ �

; ð4:16Þ

where

að0Þ ¼ T�0:171813r exp 0:125283 1� T1:77634r

� �� for Tro1;

að0Þ ¼ T�0:792615r exp 0:401219 1� T�0:992615r

� �� for Tr41;

ð4:17Þ

að1Þ ¼ T�0:607352r exp 0:511614 1� T2:20517r

� �� for Tro1; and

að1Þ ¼ T�1:98471r exp 0:024955 1� T�9:98471r

� �� for Tr41:

ð4:18Þ

Equations 4.16 to 4.18 were developed by regressing experimental dataprimarily for alkanes with the addition of cyclohexane and benzene, inorder to evaluate the generality of the form. The average deviation betweenexperimental data and model correlation of vapour pressure from the triplepoint to the critical point for these new expressions was 3.28% comparedto 12.08% with the original expression in Peng-Robinson equation ofstate (eq 4.12). Equation 4.16 was also applied to the Redlich-Kwong equationof state with different parameter values in eqs 4.17 and 4.18.15 The uncertaintyof the estimates obtained from the Redlich-Kwong equation of state is verysimilar to those of the Peng-Robinson equation of state, verifying that thefunctional form for a plays a dominant role in the correlation of vapour-pressure data.For the case of polar fluids such as water and methanol, Mathias and

Copeman16 proposed an improved expression for the parameter a compared tothe original expression of Peng and Robinson.10 The new expression is writtenin terms of the reduced temperature Tr of the polar fluid.

4.2.7 Volume Translation

Predictions of saturated liquid densities of pure fluids from the Soave-Redlich-Kwong equation of state and, to a lesser extent, the Peng-Robsinson equationof state deviate from experimental data. This should be expected given the


differences between model predictions and experimental data in Zc discussedabove. In order to improve the model accuracy, a volume translation can beintroduced.3 Peneloux et al.17 proposed a volume translated Soave-Redlich-Kwong equation of state through the introduction of a translated volumeparameter t. In this way, the equation of state is given by:

P ¼ RT

Vm þ t� b� aa

Vm þ tð Þ Vm þ tþ bð Þ ; ð4:19Þ

where t is a substance-dependent parameter. A correlation with respect to Tc, Pc

and the Rackett compressibility factor, ZRA was proposed.17 The volume trans-lation resulted in almost an order of magnitude decrease of the relative deviationbetween experimental data and Soave-Redlich-Kwong predictions of the saturatedliquid densities for 233 compounds.17 At the same time, it has no effect on thevapour-pressure predictions. The volume translation can be applied to any cubicequation of state.3 Application to the Peng-Robinson equation of state also resultedin an improvement in the uncertainty of the predicted thermodynamic properties.18

4.2.8 The Elliott, Suresh and Donohue (ESD) Equation of State

(1990)

In all of the equations of state discussed so far, the repulsive term has remainedunchanged and equal to that proposed by van der Waals and given by the firstterm on the r.h.s. of eq 4.1. Thanks to the development of molecular simulationmethods starting in the 1960s, we are able today to quantify the effects ofdifferent interactions on the thermodynamic properties of a fluid. A simplecomparison of the van der Waals repulsive term against molecular simulationdata for hard spheres reveals the inaccuracy of the former. A more accuratesimple repulsive term was proposed by Elliott et al.19 and was incorporated intoa cubic equation of state that also accounts for the shape (non-sphericity) of themolecules. The Elliott-Suresh-Donohue equation of state is given by:

P ¼ RT

Vm1þ 4cV�

Vm � 1:9V�

� �� RT

Vm

9:49qV�Y

Vm þ 1:7745V�Yð4:20Þ

where c is the shape factor, V* is the characteristic size parameter, q accounts forthe effect of shape on the attractive part of the equation of state andY is an energyparameter. Elliott et al.19 correlated c, q, V* and Y with Tc, Pc and o. The variousterms in eq 4.20 were based on statistical mechanics with appropriate simplifi-cations in order to retain the cubic nature of the equation of state. For pure fluids,the uncertainty of the estimates obtained from the Elliott-Suresh-Donohueequation of state is similar to that obtained from Soave-Redlich-Kwong equationof state. However, the Elliott-Suresh-Donohue equation of state when applied tofluid mixtures of associating components provides results with a significantimprovement in the uncertainty of the estimated values.20

60 Chapter 4

4.2.9 Higher-Order Equations of State Rooted to the Cubic

Equations of State

Thanks to their simplicity and success in the correlation and predictionof thermodynamic properties of pure fluids and mixtures, cubic equations ofstate were used as the basis for the development of more sophisticatedequations of state that account explicitly for different types of intermolecularinteractions (strong polar and hydrogen bonding interactions). Theseequations of state are higher order than cubic, but they retain the ‘‘cubicnature’’, in the sense that they have three real roots at subcritical conditionsand one real root at supercritical conditions. Most of these models were pro-posed with the aim to treat associating fluids. Elliott et al.19 used a chemical-equilibrium scheme to account for the formation of oligomers due to hydrogenbonding and they incorporated it into the ESD equation of state. Kontogeorgiset al.21 incorporated a first-order perturbation theory for associating fluids intothe Soave-Redlich-Kwong equation of state and proposed the so-called CubicPlus Association equation of state (given hereinafter the acronym CPA), that isgiven by:

P ¼ RT

Vm � b� aaVm Vm þ bð Þ � RT

XA

1

XA� 1

2

� @XA

@Vm: ð4:21Þ

The last term in the r.h.s. of eq 4.21 is based on the results of StatisticalAssociating Fluid Theory (SAFT)22 where XA is the fraction of molecules nothydrogen bonded at site A, given by the expression:

XA ¼ 1

1þ 1V

PB

XBDAB: ð4:22Þ

The summations in eqs 4.21 and 4.22 run over all hydrogen-bonding sites of themolecule. The DAB is the strength of association between sites A and B. Theadditional term in eq 4.21 introduces two new substance-specific parameters inthe equation of state, which are the energy and the volume of association. Inprinciple, these parameters can be deduced from spectroscopic measurementsor quantum mechanical calculations. In practice though, they are fitted toexperimental thermodynamic data. A thorough presentation of SAFT andother associating theories is given in Chapter 8.Calculation of XA and its density derivative is a rather time-consuming

process that requires a trial and error procedure. Michelsen and Hendriks23

proposed a mathematical formulation that results in substantial reduction incomputing time, thus making CPA (and all SAFT-based models) suitable forengineering applications. The CPA equation of state has been applied to abroad range of associating pure fluids and mixtures.24


4.2.10 Extension of Cubic Equations of State to Mixtures

The use of the cubic equation of state for fluid mixtures is much moreimportant for engineering applications. In this respect, appropriate mixingrules are needed for the calculation of a, b and other equation of state para-meters of the mixture based on its composition. An entire chapter of this bookis devoted to mixing rules. Consequently, only a brief introduction is given herefor completeness with the applications discussed later on.For an N-component mixture, the simplest set of mixing rules for parameters

a and b in van der Waals, Redlich-Kwong and more recent two-parameterequations of state are as follows:

am ¼XNi¼1

XNj¼1

xixjaij ; ð4:23Þ

andbm ¼

XNi¼1

XNj¼1

xixjbij : ð4:24Þ

These mixing rules are known as one-fluid van der Waals mixing rules. Forthree-parameter equations of state (such as Patel-Teja), a similar mixing rule isused for c. For unlike interactions (iaj), appropriate combining rules are usedfor the calculation of aij, bij and cij. More specifically, the geometric mean isused for the calculation of aij and the arithmetic mean for the calculation of bijand cij, so that the following relationships are used:

aij ¼ffiffiffiffiffiffiffiffiffiffiaiiajjp

; ð4:25Þ

bij ¼ bii þ bjj� �

=2; ð4:26Þ

andcij ¼ cii þ cjj

� �=2: ð4:27Þ

For Soave-Redlich-Kwong, Peng-Robinson and more recent equations ofstate, eq 4.25, applies to aa, often expressed as a(T). Calculations using eqs 4.23to 4.27 are based only on pure component properties without invoking anyinformation concerning the properties of mixtures. It has been recognized thatthe use of a binary interaction parameter, kij is required to ‘‘correct’’ the geo-metric rule of eq 4.25, so that:

aij ¼ffiffiffiffiffiffiffiffiffiffiaiiajjp

1� kij� �

; ð4:28Þ

results in significant improvement in VLE calculations for binary mixtures.This adjustable parameter, or as it is often called an interaction parameter, isobtained by regression to experimental VLE data for the mixture of interest. Inprinciple, a single mixture isotherm or even a single data point is sufficient toevaluate kij. However, for practical applications, data for several isotherms are

62 Chapter 4

used to calculate kij. In order to reduce the uncertainty for mixtures where oneof the components is supercritical, a temperature-dependent kij is used. Inaddition, binary parameters have been introduced for the combining rules ineqs 4.26 and 4.27. Although some improvement in the uncertainty of correla-tion is obtained, such an approach increases the number of adjustable para-meters and in general this approach should be avoided.An elegant approach to develop mixing rules for cubic equation of state is

based on the idea to match the excess Gibbs function (GE) of the mixture aspredicted by the equation of state with the one predicted by a Gibbs functionor, equivalently, an activity coefficient model at a reference pressure. Theequation is solved to obtain the attractive energy parameter, a. Huron andVidal25 chose the infinite pressure as the reference and the Non-Random TwoLiquid (known by the acronym NRTL) for activity coefficient model for GE.Mollerup26questioned the validity of using infinite pressure, and instead pro-posed that a low pressure is used to equate GE expressions from the two models.In this respect, he derived mixing rules for the Redlich-Kwong equation of statebased on Wilson, NRTL and UNIversal QUAsi Chemical (UNIQUAC)activity coefficient models. Combination of the cubic equation of state with agroup contribution activity coefficient model results in a fully predictive model(no further parameter needs to be adjusted), assuming that all group – groupinteraction parameters are known. This approach was applied by Holderbaumand Gmehling to Soave-Redlich-Kwong equation of state using UNIversalFunctional Activity Coefficient (UNIFAC) to develop the so-called PredictiveSoave-Redlich-Kwong equation of state.27 In addition, Gmehling and co-workers proposed the combination of the volume-translated Peng-Robinsonequation of state with UNIFAC and applied it to correlate the phase behaviourof polymer solutions.28,29 Furthermore, Tassios and co-workers30 proposed anew mixing rule for the PR equation of state based on the linear combination ofVidal and of Michelsen mixing rules (acronym LCVM). LCVM was shown tobe more accurate than Predictive Soave-Redlich-Kwong in predicting the VLEof mixtures of asymmetric components, such as (CO2þ alkane) and of polarmixtures. In addition, the same research group proposed the so-called Uni-versal Mixing Rule (UMR) by combining the translated-modified Peng-Robinson equation of state with UNIFAC.31 This approach was shown to bevery accurate for a broad range of fluid mixtures, including very asymmetricsystems such as polymer solutions.32

Wong and Sandler33 followed a different approach by matching the Helm-holtz function at infinite pressure from the cubic equation of state and from anactivity coefficient model. This approach ensures consistency with statisticalmechanics requirements that the second virial coefficient of a mixture has aquadratic dependence on composition. For the case of the Peng-Robinsonequation of state, the Wong-Sandler mixing rules are:

am ¼ bmXNi¼1

xiai

biþ AE

NðxÞ

0:62323

" #; ð4:29Þ


bm ¼

PNi¼1

PNj¼1

xixj b� a

RT

�ij

1þ AENðxÞ

RT�PNi¼1

xiai

biRT

� � ; ð4:30Þ

and

b� a

RT

�ij¼

bi �ai

RT

�þ bj �

aj

RT

�2

1� kij� �

; ð4:31Þ

where AEN(x) is given by the activity coefficient model used. Wong and Sandler

used the Non Random Two Liquid model for their calculations, so that localcomposition effects were incorporated in the model.33–34

4.3 Applications

4.3.1 Pure Components

The ‘‘modern’’ cubic equations of state provide reliable predictions for pure-component thermodynamic properties at conditions where the substance is agas, liquid or supercritical. Walas8 and Valderrama5 provided a thoroughevaluation and recommendations on the use of cubic equation of state forprimary and derivative properties. Vapour pressures for non-polar and slightlypolar fluids can be calculated precisely from any of the ‘‘modern’’ cubicequations of state presented above (Soave-Redlich-Kwong, Peng-Robinson orPatel-Teja). The use of a complex function for a (such as those proposed byTwu and co-workers13–15) results in a significant improvement in uncertainty ofthe predicted values. For associating fluids (such as water and alcohols), ahigher-order equation of state with explicit account for association, such aseither the Elliott-Suresh-Donohue or CPA equations of state, are preferred.For saturated liquid volumes, a three-parameter cubic equation of state (suchas Patel-Teja) should be used, whereas for saturated vapour volumes any‘‘modern’’ cubic equation of state can be used.Enthalpy and entropy of gases at low pressure can be calculated accurately

from the Soave-Redlich-Kwong, Peng-Robinson or Patel-Teja equations ofstate; at moderate and high pressure the Peng-Robinson or Patel-Teja equa-tions of state are recommended. On the other hand, for liquid phase enthalpyand entropy none of the cubic equations of state can provide precise results.5

Empirical correlations, such as that proposed by Lee-Kesler,8 are much moreprecise.In the critical region, cubic equations of state predictions deviate from

experimental data. Although cubic equation of state parameters are calculatedfrom eq 4.2 so that Tc and Pc are reproduced ‘‘exactly’’, all these models aremean-field theories and do not account for critical phenomena. Consequently,significant deviations between experimental data and model predictions shouldbe expected for pure-component thermodynamic properties. In Chapter 10, amore thorough analysis of thermodynamic properties in the critical region is

64 Chapter 4

provided. For supercritical conditions, a complex a function generally improvesthe agreement of predictions with experiments.

4.3.2 Oil and Gas Industry – Hydrocarbons and Petroleum

Fractions

The description of hydrocarbon mixture VLE at low and high pressure is ofmajor importance to the oil industry. For such mixtures, any of the ‘‘modern’’cubic equations of state (such as Redlich-Kwong, Peng-Robinson or Patel-Teja) provide precise predictions when used with a temperature-independentbinary interaction parameter of relatively small value (in most cases in between� 0.1 and 0.1). For the case of non-polar hydrocarbon mixtures of similar size,even kij¼ 0.0 results in excellent prediction of VLE.The optimum design of coal liquefaction relies on the precise knowledge of

high pressure VLE of relatively high molar mass hydrocarbons, often referredto as heavy hydrocarbons, which are usually also polyaromatic, with hydrogenand with methane that are relatively of low molar mass. Modelling of suchmixtures is challenging, primarily because of the large molecular asymmetrybetween hydrogen or methane molecules and heavy hydrocarbon molecules.Tsonopoulos and co-workers35,36 have performed an extensive comparison ofthe uncertainty of the predicted VLE for these mixtures obtained from theRedlich-Kwong, Peng-Robinson and Joffe and Zudkevitch’s modification ofthe Redlich-Kwong equations of state.37 In Redlich-Kwong-Joffe-Zudkevitch,the constants a and b in eq 4.5 are functions of temperature (the term T0.5 in thedenominator of the attractive part of the equation of state does not exist). Theactual values of the constants are determined by simultaneously matchingliquid density and forcing the vapour and liquid fugacities to be equal at thepure component’s vapour pressure. Furthermore, above the critical tempera-ture, a and b are set equal to their respective values at the critical temperature.As Tsonopoulos et al.35,36 argue, this approach makes a and b independent ofthe critical constants for subcritical substances, which is an advantage for high-boiling substances.35

Tsonopoulos and co-workers36 correlated the VLE of 30 (hydrogenþhydrocarbon) and (hydrogenþ diluent) mixtures at low and high pressures withRedlich-Kwong-Joffe-Zudkevitch, Soave-Redlich-Kwong and Peng-Robinsonand concluded that although all three equations of state perform well using atemperature-independent binary interaction parameter, the Redlich-Kwong-Joffe-Zudkevitch equation of state provides overall more precise estimates; inthis case diluent refers to nitrogen, carbon dioxide etc. A representativeexample is shown in Figure 4.1 for (H2þ 1,2,3,4-tetrahydronaphthalene) wherethe K-values (Ki¼ yi/xi) of the two components are shown as a function ofpressure at four temperatures within the range (462.75 to 662.25) K.35 Theaverage deviation of the K, that varies by two orders of magnitude over thistemperature range, are 3.6% for hydrogen and 6.6% for 1,2,3,4-tetra-hydronaphthalene. In Figure 4.1, the optimum value for the binary interaction


parameter Cij¼ 0.24, is an indication of the very different intermolecularinteractions between unlike molecules compared to like molecules. The binaryinteraction parameter value was in the same range for all the other hydro-genþ heavy hydrocarbon mixtures examined by Tsonopoulos et al.35

A more difficult and industrially important test for an equation of state is theability to predict the VLE of ternary and multicomponent mixtures usinginteractions parameters fitted to corresponding binary data. Tsonopoulos andco-workers35,36 analyzed experimental VLE data for 6 ternary mixtures andshowed that Redlich-Kwong-Joffe-Zudkevitch is again slightly more precisethan the Soave-Redlich-Kwong and Peng-Robinson equations of state.Significant more challenging mixtures to model are those involving petro-

leum fractions. The challenges arise from the following:

(a) petroleum fractions often can not be characterized in full detail. In oilindustry, the normal boiling point (Tb) and the density (specificgravity) at T¼ 288.71K are used to define a fraction. The criticalconstants and the acentric factor need to be determined for a fractionso that it can be treated as a pseudo-component in the equation ofstate. In this way, a typical petroleum cut is treated as a multi-com-ponent mixture. Tsonopoulos et al.35 have provided a comprehensiveanalysis of the methods used by the oil industry to characterize pet-roleum fractions;

(b) petroleum fractions consist of non-polar and polar compounds (ben-zene and poly-aromatic derivatives) so that weak van der Waals and

T = 462.75KT = 541.85K

T = 621.6K

T = 662.25K

T= 662.25K

T = 621.6K

T= 541.85K

T= 462.75K

10 1

lg(P/P°) lg(P/P°)

10 1

K(H

2)

K(C

10H

12)

102

10-2

10-3

10-1

10

1

Figure 4.1 K(H2) (LEFT) and K(C10H12) (RIGHT) as a function of lg(P/P1) whereP1¼ 1MPa for (hydrogenþ 1,2,3,4-Tetrahydronaphthalene) at temperaturesbetween (462.75 and 662.25) K. J, n and &, measurements; and ––––––,estimates obtained from the Redlich-Kwong-Joffe-Zudkevitch equation ofstate with Cij¼ 0.24.35 With permission from John Wiley & Sons, Inc.

66 Chapter 4

strong dipolar and quadrupolar interactions are substantial and affectthe thermodynamics of the system.

Tsonopoulos et al.35 performed a thorough analysis of the accuracy of Redlich-Kwong-Joffe-Zudkevitch equation of state with particular emphasis on theuncertainty of the predicted VLE for coal liquids with hydrogen and methane.Five different coal liquids with a broad range of normal boiling temperaturewere examined. Each coal liquid was represented by about 10 fractions. Binary-interaction parameter values were taken from correlation of mixture data withso called model (that in this case refers to compositionally well defined) sub-stances. In Figure 4.2, a representative example is shown. The volatility of oneof the coal liquids examined (namely Illinois Coal Liquid I) is presented atdifferent temperature and pressures. Experimental data and predictionsobtained from the Redlich-Kwong-Joffe-Zudkevitch equation of state for theoverall weight fraction vapourized are compared. Two different correlations

P/MPa10 15 18

T = 728K

T = 644 K

w

0.4

0.3

0.2

0.1

0

Figure 4.2 Mass fraction w of Illinois Coal Liquid I vapourized as a function ofpressure P at temperatures of 644K and 728K. J and K, measuredvalues; ––––––, predictions obtained from the Redlich-Kwong-Joffe-Zudkevitch equation of state with the modified Maxwell–Bonnell; and– – – – – , Redlich-Kwong-Joffe-Zudkevitch equation of state with themodified Riedel for the coal-liquid vapour pressure.35 With permissionfrom John Wiley & Sons, Inc.


widely used in petroleum industry were employed here for the calculation ofcoal-liquid vapour pressure, the modified Maxwell–Bonnell and the modifiedRiedel. In all cases, the agreement between experimental data and Redlich-Kwong-Joffe-Zudkevitch predictions is considered to be very good.In a different industrial application, a cubic equation of state was used to

predict retrograde condensation which is very important, among others, innatural-gas storage, transportation and processing.38 Natural gas consists pri-marily of methane (often with a mole fraction of 40.8), other light and inter-mediate alkanes, and diluents gases (typically nitrogen and, to a lesser extent,carbon dioxide, helium, etc.). Retrograde condensation consists of the forma-tion of a liquid phase containing the higher molar mass compounds that occursupon pressure reduction and this condensation should be avoided particularly inthe reservoir and production tubulars. The typical temperature range of interestis (250 to 310)K at pressures between (1 and 7)MPa. Voulgaris et al.38 appliedthe Peng-Robinson equation of state with the van der Waals mixing rules topredict the VLE of (methaneþ alkane) (or aromatic hydrocarbon) and (carbondioxideþ alkane) (or aromatic hydrocarbon) mixtures. Cubic equation of statecalculations were compared to calculations from a simplified equation of statebased on perturbation theory, namely the Simplified Perturbed Hard ChainTheory (SPHCT). The Peng-Robinson equation of state provided a precisedescription of the phase behaviour of the mixtures. Specifically, with the use of abinary-interaction parameter Peng-Robinson equation of state resulted inliquid-phase composition that differed from measurement by less than 2.5% formost binary mixtures with a lower uncertainty in the gas-phase. In Figure 4.3,uncertainty of the two models has been compared. Overall, Peng-Robinsonequation of state is found to provide values that are differ less from the mea-sured values than estimates obtained from SPHCT and, consequently, the Peng-Robinson equation of state is the model recommended by Voulgaris et al.38 forcondensates. Furthermore, a comparison between van der Waals and Wong-Sandler mixing rules in the context of Peng-Robinson equation of state for thecorrelation of hydrocarbon mixture VLE revealed that there is no advantage inusing the more complicated latter mixing rules over the simpler van der Waals.39

Recently, Nasrifar and Bolland40 proposed a modified Soave-Redlich-Kwong cubic equation of state that was shown to be more accurate than boththe original Soave-Redlich-Kwong and Peng-Robinson equations of state atpredicting the compressibility factor and the speed of sound of natural gasmixtures. Furthermore, the proposed equation40 was shown to be able topredict (binary interaction parameters set to zero) values of K-values for thesemixtures that were in excellent agreement with experiment. A representativeexample is shown in Figure 4.4.

4.3.3 Chemical Industry – Polar and Hydrogen Bonding Fluids

Predictions and correlation of VLE and LLE of mixtures containing polarsubstances with a cubic equation of state requires the use of advanced mixing

68 Chapter 4

rules (presented previously and in Chapter 5) as well as inclusion of binaryinteraction parameters. This topic has been reviewed by Voutsas et al.41 Inthis chapter, we will focus our attention to a very important class of mixturesthat contain hydrogen bonding and, in some cases, polar fluids, namely(waterþ hydrocarbon).The mixture (waterþ hydrocarbon) exhibits non-ideal thermodynamic

behaviour owing to the very different intermolecular interactions that occur

Figure 4.3 Comparison of the measured composition of (methaneþ alkane) com-pared with composition estimates obtained from both the Peng-Robinsonequation of state and the Simplified Perturbed Hard Chain Theory. (a),liquid-phase composition, and (b) vapour-phase composition.38 Withpermission from the American Chemical Society.


between these components. Water molecules form hydrogen bonds whereashydrocarbon molecules interact through weak van der Waals forces and, forthe case of aromatics and polyaromatics, quadrupolar forces. A precisedescription of the phase equilibrium of such mixtures is thus challenging andalso of high technological importance for refining and petrochemical processes,such as the design of distillation towers.The mutual solubility of water and a hydrocarbon, and in particular the

hydrocarbon solubility, is very small over a broad temperature range.

Figure 4.4 K as a function of pressure P for N2, CH4, CO2, C2H6, C3H8, C5H12,C7H16 and C10H22 in natural gas mixture at T¼ 366.44K.40 J, mea-surements; ––––––, predictions obtained from the modified Soave-Red-lich-Kwong equation of state. With permission from Elsevier.

70 Chapter 4

Economou and Tsonopoulos42 examined the precision of the Redlich-Kwong-Joffe-Zudkevitch and two higher-order equations of state, with roots in per-turbation theory, namely the Associated Perturbed Anisotropic Chain Theory(APACT) and SAFT, for the correlation of the (liquidþ liquid) equilibrium(LLE) of water with 10 different hydrocarbons (alkanes, alk-1-enes, and aro-matics) using a single temperature-independent interaction parameter. Inter-estingly enough, Redlich-Kwong-Joffe-Zudkevitch was found the most preciseof these models when compared to experimental data for the correlation ofsolubility of water in hydrocarbon. The average absolute deviation betweenexperiment and model was 6.9% for Redlich-Kwong-Joffe-Zudkevitch, 9.3%and 11.9% for APACT with a two- and a three-hydrogen bonding site modelfor water, and between (9.3 and 16.0)% for the four different SAFT modelexamined.42 However, prediction obtained from the Redlich-Kwong-Joffe-Zudkevitch equation of state for the hydrocarbon solubility in water wereseveral orders of magnitude below the experimental data.In an attempt to improve the behaviour of the cubic equation of state, the

more elaborate Huron–Vidal mixing rules25 were used with the Peng-Robinsonequation of state.42 As shown in Figure 4.5 for the (vapourþ liquidþ liquid)(VLLE) equilibrium of (waterþ hex-1-ene) the Huron–Vidal mixing rulesimproved significantly the predicted solubility of hex-1-ene over the standardvan der Waals mixing rules; however, there was also a significant decrease in theability of the model to correlate of solubility of water.A different approach was used by Luedecke and Prausnitz43 who proposed a

density-dependent correction to the van der Waals mixing rule for the attractiveparameter of the equation of state. The Luedecke and Prausnitz43 mixing rule isgiven by:

am ¼XNi¼1

XNj¼1

xixj aiiajj� �1=2

1� kij� �

þ rRT

XNi¼1

XNj¼1j 6¼1

xixj xiciðjÞ þ xjcjðiÞ�

: ð4:32Þ

This correction introduces two additional temperature-independent interactionparameters for each binary mixture, that are ci(j) and cj(i). This approach resultsin a precise representation of the binary (waterþ hydrocarbon) LLE and VLE.However, when the model was extended to ternary mixtures, it was foundunable to adequately represent the measured LLE.An explicit account of hydrogen bonding in water by the equation of state

results in substantial improvement of the correlation of (waterþ hydrocarbon)LLE.44 In Figure 4.6, LLE for (waterþ hexane) is shown. The CPA equation ofstate correlates the water solubility with an Absolute Average Deviation of4.5% and reasonable agreement is obtained between experiment and calcula-tions for hexane solubility. Unfortunately, the minimum of the solubility ofhydrocarbon cannot be captured with a single temperature-independent binaryinteraction parameter.The CPA equation of state has been used successfully to model VLE and

LLE of (polarþ non-polar), (polarþ polar), (associatingþ non-polar) and


(associatingþ polar). A review of the model and its applications has been givenby Kontogeorgis and co-workers.24 The model is more complex than a cubicequation of state. For mixtures containing one associating component, thehydrogen-bonding term can be calculated explicitly and the computing time isvery similar to the that required by a cubic equation of state for a similarcalculation.45

A representative example of the phase equilibrium of a binary mix-ture containing one associating component is shown in Figure 4.7 for

T/K

x

10-1

10-2

10-3

100

10-4

10-5

10-6

10-7

10-8

10-9

250 300 350 400 450 500

water in vapour

water in hex 1-ene

hex 1-ene in water

Figure 4.5 Mole fraction x as a function of temperature T for (waterþ hex-1-ene)VLLE at the three-phase equilibrium pressure.42 J and m, measuredvalues; – – – – , predictions obtained from the Peng-Robinson equation ofstate with the van der Waals mixing rules; and ––––––, predictionsobtained from the Peng-Robinson equation of state with the Huron –Vidal mixing rules. With permission from Elsevier.

72 Chapter 4

(ethaneþmethanol).46 The CPA equation of state provides predictions (with-out adjustment of the binary parameters) of the VLE at lower pressures, LLEat higher pressures and the VLLE at P¼ 4.091MPa. The precision of theSoave-Redlich-Kwong equation of state is similar for VLE to that found for theCPA but significantly worse for VLLE and LLE. Thus, for engineeringapplications involving polar and associating components, the CPA is therecommended method to be used for phase equilibrium calculations.A relatively new class of compounds that has received attention over the last

decade as providing environmentally benign reaction and separation media inchemical industry is ionic liquids. Ionic liquids are molten salts with meltingpoints close to room temperature. Their most remarkable property is that theirvapour pressure is negligibly small, so that ionic liquids are non-volatile, non-flammable and odorless. The precise description of the phase equilibrium ofionic liquids with pure and mixed solvents is important for process design.

250 300 350 400 450 500 550 600

T/K

x10-1

10-2

10-3

100

10-4

10-5

10-6

10-7

10-8

10-9

Figure 4.6 Mole fraction x as a function of temperature T for (waterþ hexane) LLEat three phase equilibrium pressure.44E, measured values for hexane inthe water phase; ’, measured values for water in the hexane rich phase;––––––, predictions obtained from the Cubic Plus Association equation ofstate with kij ¼ 0.05 fit to the water solubility in hexane; and – – – – – – – ,predictions obtained from the Statistical Associating Fluid Theory equa-tion of state with the kij¼ 0.407 fit to the water solubility in hexane. Withpermission from the American Chemical Society.


Chapter 11 provides an analysis of ionic liquids thermodynamic properties.Here, only a representative example is provided. A modified Redlich-Kwongcubic equation of state was used to correlate VLE and predict the VLLE of(fluorocarbonþ ionic liquid).47 Because the ionic liquids have no measurablevapour pressure, the equation of state pure-component parameters were fit tothe liquid density data and critical constants.47 To correlate the experimentalVLE data at temperature over the range (283 to 348)K, Shiflett and Yokozeki47

used three binary interaction parameters. These parameters were used, withoutfurther adjustment, to predict the VLLE of these mixtures. In Figure 4.8,experimental data and correlation are shown for (1,1,1,2-tetrafluoroethaneþ 1-butyl-3-methylimidazolium hexafluorophosphate [bmim1][PF6

�]).

4.3.4 Polymers

Although originally developed to model thermodynamic properties of volatilefluids, cubic equations of state have also been applied to polymer mixtures. Sako,Wu and Prausnitz48 proposed a cubic equation of state for polymers based on theSoave-Redlich-Kwong equation of state with the Prigogine’s parameter c toaccount for the non-sphericity of chain molecules. The Sako-Wu-Prausnitz

Figure 4.7 Pressure P as a function of mole fraction of ethane x(C2H6) of(C2H6þCH3OH) at T¼ 298.15K.46 VLE below P¼ 4.091MPa; LLEabove P¼ 4.091MPa and VLLE at P¼ 4.091MPa.B, n and J, mea-sured values; – – – – – – – , estimates obtained from the Soave-Redlich-Kwong equation of state; and ––––––, estimates obtained from the CubicPlus Association equation of state. With permission from the InstitutFrancais du Petrole.

74 Chapter 4

equation is given by:

P ¼ RT Vm � bþ bcð ÞVm Vm � bð Þ � aa

Vm Vm þ bð Þ : ð4:33Þ

When c¼ 1, eq 4.33 reduces to the Soave-Redlich-Kwong equation of state(that is eq 4.5). In Prigogine’s theory, 3c is the total number of external degreesof freedom available per molecule. This parameter c is not the same as that usedin the Patel-Teja equation of state (given by eq 4.14). This parameter has beenused by Prausnitz, Donohue and co-workers in higher-order equations of state,such as the Perturbed-Hard-Chain Theory (PHCT)49 and its extension topolar fluids.50 For a given fluid, parameters c, a and b are adjusted to fit themeasured vapour pressure and saturated liquid density. Sako et al.48 showedthat their equation of state can be used to correlate, significantly more preciselythan the Soave-Redlich-Kwong equation of state, the saturated liquid densityof alkanes and the vapour pressure of alkanes with greater than 8 carbonatoms.For polymers, the parameter c is adjusted to fit the melting-line density over

an extended temperature and pressure range, a is obtained from London’stheory for dispersion interactions and b from van der Waals volume using

Figure 4.8 (P, x) section for (1,1,1,2-tetrafluoroethaneþ [bmim1][PF�6 ] mixture) attemperatures between (283.15 and 355)K.47 J, measured VLE;’ and m,measured VLLE; ––––––, estimated VLE obtained from the modifiedRedlich-Kwong equation of state; – – – – – – – , estimated VLLE obtainedfrom the modified Redlich-Kwong equation of state. With permissionfrom the American Chemical Society.


Bondi’s method. With this approach, the melting density data for polyethene,polypropene and poly-isobutene were fit with an average deviation fromexperimentally determined values of about 1% while for polymers containingpolar monomer units, including butan-2-yl acetate, poly(ethenebenzene) andpoly(1-ethene-2-methylbenzene), the deviation between experiment and Sako-Wu-Prausnitz correlation was between (3.0 and 3.5)%.48

The Sako-Wu-Prausnitz equation of state was applied to correlate (etha-neþ polyethene) data at the high-pressure of interest to polyalkene technology.In low-density polyethene (LDPE), polymerization takes place at relativelyhigh temperatures of between (453 and 573)K and very high pressures (100 to300)MPa.48,51 Models that precisely represent the phase behaviour at theseconditions are important for the optimum design of the process, including thesubsequent separation at lower pressure.51 Low pressure (solventþ polymer)phase equilibrium data are often correlated with activity coefficient models thatdo not account for the pressure effect and, thus, cannot be applied at highpressure.52 Using a single binary interaction parameter, kij, the Sako-Wu-Prausnitz equation of state provided a precise correlation of (ethaneþpolyethene) data at pressures up to 180MPa. As Figure 4.9 shows, the model

0.20 0.4 0.6

w

200

150

100

50

P/M

Pa

Figure 4.9 Pressure P as a function of polyethene mass fraction w in (ethe-neþ polyethene) at T¼ 403.2K.48 J, measurements. Estimates wereobtained from the SWP correlation with k12¼ 0.07 and the polymermodelled as monodisperse: ––––––, poyethene molar mass M¼ 9 kgmol�1; – – – – – – – , poyethene molar mass M¼ 9.5 kgmol�1; and --------,poyethene molar mass M¼ 10 kgmol�1. With permission from JohnWiley & Sons, Inc.

76 Chapter 4

was able to capture the effect of polymer molecular mass on the phase equili-bria. Consequently, the model can be used to design a relevant fractionationprocess for a polydisperse polymer sample.48

Tassios and co-workers53 proposed a simple approach to use the van derWaals equation of state to polymers: The two parameters a and b were fitted tomelt density at two different temperatures and zero pressure. Using the van derWaals mixing rules with a single binary interaction parameter, the model wasable to correlate with sufficient precision the VLE and LLE of the polymersolutions and also the LLE of the polymer blends.54 Two representativeexamples for the latter are shown in Figure 4.10. The phase diagram for apoly(ethenebenzene)þ poly(buta-1,3-diene) blend exhibits an Upper CriticalSolution Temperature (UCST) behaviour that depends strongly on themolar mass of poly(ethenebenzene) as shown in Figure 4.10(a).54 A correlationof the experimental data is obtained with a kij that varies with the molarmass. In Figure 4.10(b), experimental data and the van der Waals correlationare shown for the poly(ethenebenzene)þ poly(methoxyethene) blend thatexhibits a Lower Critical Solution Temperature (LCST). The LCST beha-viour is primarily attributed to specific interactions between unlike molecules,here between the p electrons of the aromatic ring and the oxygen atom inpoly(methoxyethene). These interactions are not explicitly accounted for witha cubic equation of state and so a temperature-dependent kij is needed tocorrelate precisely the measured data.This relatively simple approach in modelling polymers results in poor pre-

diction of polymer melt volumetric properties at high pressures and in unrea-listically high vapour pressures for the pure polymers. Alternatively, one may fitthe a and b parameters of cubic equations of state to polymer melting densitydata over a broad pressure range, typically up to a pressure of 200MPa. Louliand Tassios55 applied the Peng-Robinson equation of state to (poly-merþ solvent) VLE with this approach to obtain polymer parameters. Calcu-lations were performed with the van der Waals mixing rules, the modifiedHuron-Vidal mixing rules and the Zhong and Masuoka mixing rules. TheZhong-Masuoka mixing rules are very similar to Wong-Sandler mixing rules;the only difference is that the excess Helmholtz function at infinite pressureterm in the former is set equal to zero. Louli and Tassios55 concluded thatZhong-Masuoka mixing rules provided the best representation of the experi-mental data with a single temperature-independent kij.

4.3.5 Transport Properties

All of the applications of cubic and non-cubic equations of state presented sofar refer to equilibrium thermodynamics. Cubic equations of state have beenalso used for the calculation of transport properties of pure components andmixtures, including viscosity, diffusion coefficient and thermal conductivity.Some recent viscosity calculations will be presented here.


The viscosity of a liquid mixture, Z, can be calculated from the Eyring’sabsolute rate theory:

Z ¼ ZVð ÞIDV

expG 6¼;E

RT

� �; ð4:34Þ

where (ZV)ID is the kinematic viscosity of ideal solution calculated from asimple mixing rule based on individual component kinematic viscosities andGa,E is the excess Gibbs function of the activated state. The latter quantity

Figure 4.10 Temperature T as a function of volume fraction of poly(ethenebenzene) v(TOP), or mass fraction w of poly(ethenebenzene) (BOTTOM), withestimates obtained from the van der Waals equation of state.54 TOP:{poly(ethenebenzene)þ poly(buta-1,3-diene)}. ’, K, m measurementsand estimates ––––––, – – – – – – –, and - � - � - for the same compositionobtained form the van der Waals equation of state. BOTTOM: {poly(ethenebenzene)þ poly(methoxyethene)}. ’, and K measurements andestimates ––––––, – – – – – – – for the same composition obtained form thevan der Waals equation of state. Reproduced with permission of theAmerican Institute of Chemical Engineers.54

78 Chapter 4

depends on the excess Gibbs function, GE, through the expression:

G 6¼;E ¼ kGE; ð4:35Þ

where k is a system-dependent constant. In this respect, a cubic equation ofstate or an activity coefficient model can be used to calculate GE in eq 4.35 andthus, viscosity through eq 4.34. Tochigi et al.56 used a cubic equation of statewith modified Huron-Vidal mixing rules for the calculation of volume and theNRTL model to calculate GE for binary hydrocarbon mixtures, alcohols andwater. A representative example for the viscosity of (methanolþwater) atP¼ 0.1MPa is shown in Figure 4.11. The model was used to predict viscosity athigher pressures where experimental data were not available.56 A similarapproach was used by Weirong and Lempe57 to calculate the viscosity forbinary and ternary mixtures of aqueous and other mixtures of associatingfluids.Quinones-Cisneros et al.58 proposed the friction theory (the so called f-the-

ory) to predict viscosity using an equation of state. According to f-theory, theviscosity of dense fluids is a mechanical property rather than a transportproperty. Consequently, the total viscosity of a dense fluid can be written as thesum of a dilute-gas term Zo and a friction term Zf through:

Z ¼ Zo þ Zf : ð4:36Þ

Figure 4.11 Comparison of viscosity Z estimated from the cubic equation of sta-teþNRTL correlation and measured values for (methanolþwater)at P¼ 0.1MPa.56 K, T¼ 285K; m, T¼ 300K; ’, T¼ 315K; .,T¼ 330K; ––––––, estimated values. Reproduced with permission ofElsevier.56


The dilute gas term can be calculated from empirical models based on thekinetic theory of gases, while for the friction term a theory that accounts forrepulsive and attractive intermolecular interactions in dense fluids is invokedso that:58

Zf ¼ krPr þ krrP2r þ kaPa; ð4:37Þ

where Pr and Pa are the repulsive and attractive contribution to total pressureand kr, krr and ka are temperature-dependent friction coefficients. Quinones-Cisneros and co-workers used various cubic equations of state (includingthe Soave-Redlich-Kwong, Peng-Robinson and Predictive Soave-Redlich-Kwong) to calculate Pr and Pa and applied f-theory to predict the viscosity ofpure hydrocarbons, polar gases and their mixtures over an extended range oftemperature and pressure.58,59 A typical example of the predictions are shownin Figure 4.12 for viscosity of a quaternary hydrocarbon mixture with the Peng-Robinson equation of state.59 The model has been found precise over the entire

3.5

3.0

2.5

2.0

1.5

1.0

20 40 80 100 60 P/MPa

T = 333.15 K

T = 353.15 K

T = 313.15 K

�/m

Pa⋅s

Figure 4.12 Comparison of the viscosity Z estimated from f-theory using the Peng-Robinson equation of state and measured values for {0.3144C10H22þ 0.2626 C12H26þ 0.2255 C14H30þ 0.1975 C16H34} as a functionof pressure at temperatures of (313.15, 333.15 and 353.15)K.59 K,measured values; ––––––, estimated values. Reproduced with permissionof Elsevier.59

80 Chapter 4

range of conditions tested. More recently, Quinones-Cisneros and co-workersextended the f-theory to the prediction of diffusion coefficients60 and surfacetension for pure fluids and mixtures.61 In all cases, cubic equations of state wereused to account for the dense fluid contribution to the property of interest.

4.4 Conclusions

After more than 135 years since the work of van der Waals on the cubicequation of state, the generalized van der Waals equation remains an activetopic for applied research and development in Chemical Engineering. Despitetheir inherent empiricism, cubic equations of state have been extended to cal-culate precisely thermodynamic properties of fluids that exhibit very non-idealintermolecular interactions, ranging from non-polar to dipolar and quad-rupolar through associating and even electrostatic interactions. At the sametime, the models have been extended to treat mixtures of components that varysignificantly in molecular dimensions, such as (solventþ polymer). Finally,when compared with experimental data, the models are precise over a widerange of both temperature and pressure so that they can be used reliably tomodel a wide spectrum of industrial processes while also retaining the com-putational efficiency; an important parameter in process optimization andhydrocarbon reservoir modeling that often require 4106 flash calculations.In recent years, the development of cubic equations of state has benefited by

parallel developments in applied statistical mechanics, molecular theory andsimulation, primarily with respect to intermolecular interactions. Furthermore,the accumulated experience with cubic equations of state, the large databaseswith optimum pure component and binary interaction parameters and thefamiliarity of applied scientists, chemical and process engineers in industryguarantee that these models will retain their leading position in applied researchand development in the years to come.

Acknowledgements

I am thankful to Manolis Vasiliadis and Eirini Siougrou for a preliminaryliterature search on cubic equations of state and to Professor EpaminondasVoutsas and Dr. Vicky Louli for critical comments to the manuscript.

References

1. J. H. van der Waals, Over de Continuıteit van den Gas- en Vloeistoftoestand(On the Continuity of the Gaseous and Liquid State). Ph.D. Dissertation,Leiden University, 1873.

2. C. Tsonopoulos and J. M. Prausnitz, Cryogenics, 1969, 9, 315–327.3. J. J. Martin, Ind. Eng. Chem. Fundam., 1979, 18, 81–97.4. C.-C. Chen and P. M. Mathias, AIChE J., 2002, 48, 194–200.5. J. O. Valderrama, Ind. Eng. Chem. Res., 2003, 42, 1603–1618.


6. T. L. Hill, An Introduction to Statistical Thermodynamics, Addison-Wesley,Reading, Mass., 1960.

7. O. Redlich and J. N. S. Kwong, Chem. Rev., 1949, 44, 233–244.8. S. M. Walas, Phase Equilibria in Chemical Engineering, Butterworth Pub-

lishers, Stoneham, Mass., 1985.9. G. Soave, Chem. Eng. Sci., 1972, 27, 1197–1203.

10. D. Peng and D. B. Robinson, Ind. Eng. Chem. Fundam., 1976, 15, 59–64.11. B. E. Poling, J. M. Prausnitz and J. P. O’ Connell, The Properties of Gases

and Liquids, 5th Edition, McGraw-Hill, New York, 2000.12. N. C. Patel and A. S. Teja, Chem. Eng. Sci., 1982, 37, 463–473.13. C. H. Twu, D. Bluck, J. R. Cunningham and J. E. Coon, Fluid Phase

Equilib., 1991, 69, 33–50.14. C. H. Twu, J. E. Coon and J. R. Cunningham, Fluid Phase Equilib., 1995,

105, 49–59.15. C. H. Twu, J. E. Coon and J. R. Cunningham, Fluid Phase Equilib., 1995,

105, 61–69.16. P. M. Mathias and T. W. Copeman, Fluid Phase Equilib., 1983, 13, 91–108.17. A. Peneloux, E. Rauzy and R. Freze, Fluid Phase Equilib., 1982, 8, 7–23.18. J.-M. Yu and B. C.-Y. Lu, Fluid Phase Equilib., 1987, 34, 1–19.19. J. R. Elliott Jr, S. J. Suresh and M. D. Donohue, Ind. Eng. Chem. Res.,

1990, 29, 1476–1485.20. S. J. Suresh and J. R. Elliott Jr., Ind. Eng. Chem. Res., 1991, 30, 524–532.21. G. M. Kontogeorgis, E. C. Voutsas, I. V. Yakoumis and D. P. Tassios, Ind.

Eng. Chem. Res., 1996, 35, 4310–4318.22. W. G. Chapman, K. E. Gubbins, G. Jackson and M. Radosz, Ind. Eng.

Chem. Res., 1990, 29, 1709–1721.23. M. L. Michelsen and E. M. Hendriks, Fluid Phase Equilib., 2001, 180,

165–174.24. G. M. Kontogeorgis, M. L. Michelsen, G. K. Folas, S. Derawi, N. von

Solms and E. H. Stenby, Ind. Eng. Chem. Res., 2006, 45, 4855–4868; ibid.2006, 45, 4869–4878.

25. M.-J. Huron and J. Vidal, Fluid Phase Equilib., 1979, 3, 255–271.26. J. Mollerup, Fluid Phase Equilib., 1986, 25, 323–327.27. T. Holderbaum and J. Gmehling, Fluid Phase Equilib., 1986, 70, 251–265.28. J. Ahlers and J. Gmehling, Ind. Eng. Chem. Res., 2002, 41, 3489–3498.29. L.-S. Wang, J. Ahlers and J. Gmehling, Ind. Eng. Chem. Res., 2003, 42,

6205–6211.30. C. Boukouvalas, N. Spiliotis, P. Coutsikos, N. Tzouvaras and D. Tassios,

Fluid Phase Equilib., 1986, 92, 75–106.31. E. Voutsas, K. Magoulas and D. Tassios, Ind. Eng. Chem. Res., 2004, 43,

6238–6246.32. E. Voutsas, V. Louli, C. Boukouvalas, K. Magoulas and D. Tassios, Fluid

Phase Equilib., 2006, 241, 216–228.33. D. S. H. Wong and S. I. Sandler, AIChE J., 1992, 38, 671–680.34. D. S. H. Wong and S. I. Sandler, Ind. Eng. Chem. Res., 1992, 31,

2033–2039.

82 Chapter 4

35. C. Tsonopoulos, J. L. Heidman and S.-C. Hwang, Thermodynamic andTransport Properties of Coal Liquids, John Wiley & Sons, New York, 1986.

36. R. D. Gray Jr, J. L. Heidman, S.-C. Hwang and C. Tsonopoulos, FluidPhase Equilib., 1983, 13, 59–76.

37. D. Zudkevitch and J. Joffe, AIChE J., 1970, 16, 112–119.38. M. E. Voulgaris, C. J. Peters and J. de Swaan Arons, Ind. Eng. Chem. Res.,

1998, 37, 1696–1706.39. M. E. Voulgaris, Prediction and Verification of Hydrocarbon Liquid Drop

Out of Lean Natural Gas, PhD Dissertation, Delft University of Technol-ogy, 1995.

40. K. Nasrifar and O. Bolland, J. Petrol. Sci. Eng., 2006, 51, 253–266.41. E. C. Voutsas, P. Coutsikos and G. M. Kontogeorgis, In Computer Aided

Property Estimation for Process and Product Design, Chapter 5, Elsevier,Amsterdam, 2004.

42. I. G. Economou and C. Tsonopoulos, Chem. Eng. Sci., 1997, 52, 511–525.43. D. Luedecke and J. M. Prausnitz, Fluid Phase Equilib., 1985, 22, 1–19.44. E. C. Voutsas, G. C. Boulougouris, I. G. Economou and D. P. Tassios, Ind.

Eng. Chem. Res., 2000, 39, 797–804.45. T. Kraska, Ind. Eng. Chem. Res., 1998, 37, 4889–4892.46. J. C. de Hemptinne, P. Mougin, A. Barreau, L. Ruffine, S. Tamouza and R.

Inchekel, Oil & Gas Sci. Tech., Rev. IFP, 2006, 61, 363–386.47. M. B. Shiflett and A. Yokozeki, J. Chem. Eng. Data, 2006, 51, 1931–1939.48. T. Sako, A. H. Wu and J. M. Prausnitz, J. App. Pol. Sci., 1989, 38,

1839–1858.49. M. D. Donohue and J. M. Prausnitz, AIChE J., 1978, 24, 849–860.50. P. Vimalchand and M. D. Donohue, Ind. Eng. Chem. Fundam., 1985, 24,

246–257.51. B. Folie and M. Radosz, Ind. Eng. Chem. Res., 1995, 34, 1501–1516.52. G. M. Kontogeorgis, In Computer Aided Property Estimation for Process

and Product Design, Chapter 7, Elsevier, Amsterdam, 2004.53. G. M. Kontogeorgis, V. I. Harismiadis, A. Fredenslund and D. P. Tassios,

Fluid Phase Equilib., 1994, 96, 65–92.54. V. I. Harismiadis, A. R. D. van Bergen, A. Saraiva, G. M. Kontogeorgis,

A. Fredenslund and D. P. Tassios, AIChE J., 1996, 42, 3170–3180.55. V. Louli and D. Tassios, Fluid Phase Equilib., 2000, 168, 165–182.56. K. Tochigi, T. Okamura and V. K. Rattan, Fluid Phase Equilib., 2007, 257,

228–232.57. J. I. Weirong and D. A. Lempe, Chinese J. Chem. Eng., 2006, 14, 770–779.58. S. E. Quinones-Cisneros, C. K. Zeberg-Mikkelsen and E. H. Stenby, Fluid

Phase Equilib., 2000, 169, 249–276.59. S. E. Quinones-Cisneros, C. K. Zeberg-Mikkelsen and E. H. Stenby, Fluid

Phase Equilib., 2001, 178, 1–16.60. T. Kraska, S. E. Quinones-Cisneros and U. K. Deiters, J. Supercrit. Fluids,

2007, 42, 212–218.61. S. E. Quinones-Cisneros, U. K. Deiters, R. E. Rozas and T. Kraska,

J. Phys. Chem. B, 2009, 113, 3504–3511.


CHAPTER 5

Mixing and Combining Rules

ANTHONY R. H. GOODWINa AND STANLEY I.SANDLERb

a Schlumberger Technology Corporation, Sugar Land, TX 77478, U.S.A.;b Centre for Molecular and Engineering Thermodynamics, Department ofChemical Engineering, University of Delaware, Newark, DE 19716, U.S.A.

5.1 Introduction

Equations of state are used in engineering to predict the thermodynamicproperties in particular the phase behaviour of pure substances and mixtures.However, since there is neither an exact statistical-mechanical solution relatingthe properties of dense fluids to their intermolecular potentials, nor detailedinformation available on intermolecular potential functions, all equations ofstate are, at least partially, empirical in nature. The equations of state in com-mon use within both industry and academia are described elsewhere in this bookand can be arbitrarily classified as follows: (1), cubic equations derived from theobservation of van der Waals that are described in Chapter 4; (2), those basedon the virial equation discussed in Chapter 3; (3), equations based on generalresults obtained from statistical mechanics and computer simulations mentionedin Chapter 8; and (4), those obtained by selecting, based on statistical means,terms that best represent the available measurements obtained from a broadrange of experiments as outlined in Chapter 12. The methods used for mixturesare also alluded to in these chapters and in Chapter 6.The development of an equation of state typically commences with the

representation of the thermodynamic properties of pure fluids and the func-tions are then extended to provide estimates of the properties of mixtures. Thepurpose of this chapter is to provide the methods used to permit this calculation




84

to proceed for the items numbered 1 through 4 listed in the previousparagraph. This procedure involves the introduction of mixing and combiningrules.Mixing rules are used to obtain numerical estimates for the parameters

in an equation of state for a specified mixture from the same parameterswhen the same equation of state is used to represents the properties of the puresubstances. However, in the description of a mixture parameters appear thatresult from the interactions between unlike species. These are obtainedusing combining rules. By using mixing and combining rules measurementsare only required for the pure substances and not the infinite number ofmixtures. When these mixing and combining rules are used with p(Vm, T)equations of state they provide the link between the microscopic and themacroscopic. The certainty with which the predictions resulting from theuse of an equation of state with its mixing and combining rules can be evaluatedwith experimental data and additional adjustable parameters added when thereare sufficient experimental data. Therefore, development of an equation ofstate for mixtures is largely reduced to the elucidation of the mixing andcombining rules to describe the thermodynamic properties, especially the phaseboundaries.The plethora of both equations of state and of mixing and combining rules

means there are a multitude of options available and that some adopted arepurely empirical. Consequently, the task of providing a comprehensive list ofall mixing and combining rules is rather daunting. Thus one basis for theinclusion of the rules in this chapter were their frequent appearance in thearchival literature, which does not necessarily imply that the rules are optimalor even correct. In addition, some of the rules included may also be sufficientlyrecent to give the appearance of an important technical advance and so havebeen mentioned herein. This chapter repeats some and attempts to build on thework of Orbey and Sandler.1 The reader requiring a rather more extensivereview of mixing and combining rules should consult the recent work ofKontogeorgis and Folas.2

The methods most frequently used to predict the properties of mixtures forover 100 years have inevitably undergone only minor additions and correctionsto, it is claimed, improve the representation of experimental data for specificcategories of substances. It is, however, possible that completely differentalternatives to these traditional approaches are required, particularly for amethod to be both predictive and applicable over a wide range of fluids andconditions.3 Such methods might arise from future research and methods basedon statistical mechanics and quantum-mechanical calculations4,5 are ultimatelysought rather than empiricism.

5.2 The Virial Equation of State

One of the exact results from statistical mechanics, discussed in Chapter 3,6 isthe virial equation of state expressed in terms of the definition of the

85Mixing and Combining Rules

compression factor Z by

Z ¼ pVm

RT¼ 1þ B Tð Þ

Vmþ C Tð Þ

V2m

þ � � � ; ð5:1Þ

where p is the pressure, Vm the molar volume, R the gas constant, T the tem-perature and B(T), C(T), � � �, the second, third, � � � � , virial coefficients. Thesecond virial coefficient is only a function of temperature and related to thepair-interaction energy u(r) that depends only on the separation r of the centresof mass of two spherical molecules through

B Tð Þ ¼ 2pLZN0

1� exp �u rð Þ=kTf g½ � r2dr: ð5:2Þ

In eq 5.2, k is Boltzmann’s constant, L Avogadros’s number and r theseparation. There are analogous expressions to eq 5.2 for higher virial coeffi-cients that contain multiple integrals over distances between molecular cen-tres.7,8 In general, the interaction of two molecules depends on the distancesbetween the centres of mass and also the angles that describe the orientation ofone molecule with respect to another; the formulae required are more complexthan eq 5.2. The extent to which the interaction energy of a triplet differs fromthe sum of three pair-interaction energies, that is the extent of non-pairwiseadditivity, is still an area of active research.For mixtures of mole fraction x, eq 3.3 of Chapter 3 provides the composition

dependence of the virial coefficients B and C that are given, respectively, by

B T ; xð Þ ¼Xi

Xj

xixjBij Tð Þ; ð5:3Þ

andC T ;xð Þ ¼

Xi

Xj

Xk

xixjxkCijk Tð Þ: ð5:4Þ

Equations 5.3 and 5.4 are exact low-density boundary conditions that must besatisfied by other mixture equations when expanded into the virial form. For amixture {(1� x)Aþ xB}, eqs 5.3 and 5.4 become

B T ;xð Þ ¼ 1� xð Þ2BA þ 2 1� xð ÞxBAB þ x2BB ð5:5Þ

and

C T ; xð Þ ¼ 1� xð Þ3CA þ 3 1� xð Þ2xCAAB þ 3 1� xð Þx2CABB þ x3CB: ð5:6Þ

In eq 5.5, BA and BB are the second virial coefficients of pure A and pure B, andBAB the virial coefficient that results from pair-wise molecular interactions

86 Chapter 5

between A and B. The C’s of eq 5.6 are similarly related to the simultaneousinteractions among three molecules.

5.3 Cubic Equations of State

Cubic equations of state in various forms are ubiquitous in the chemical andpetroleum industries. It is argued this is because of the computational timerequired to estimate phase equilibrium that can become significant when thereare more than 106 calculations to be performed as is the case for either chemicalprocesses or hydrocarbon reservoir engineering.The van der Waals equation, which is the basis for numerous cubic equations

described in Chapter 4, is for a pure fluid given by:

p ¼ RT

Vm � b� a

V2m

; ð5:7Þ

where parameter b represents the volume excluded by the molecules and iscalled the repulsive term while the parameter a and the second term representthe attractive interactions between the molecules. Equation 5.7 can be cast interms of powers of (b/Vm) as

pVm

RT¼XNi¼0

b

Vm

� �i

� a

RTVm: ð5:8Þ

The general form of a cubic equation of state for a pure substance is:

p ¼ RT

Vm � b� a

V2m þ Vmcb� c� 1ð Þb2 ; ð5:9Þ

and is, apart from the temperature dependence of a, similar to the proposal ofClausius.9 When c of eq 5.9 is unity, the van der Waals expression given by eq5.7 (eq 4.1 of Chapter 4) is obtained. In the case of c¼ 1 and a is replaced with atemperature-dependent function a(T), the Redlich-Kwong10 equation (eq 4.5originally intended for application to the gas phase) is obtained as is the Soave-Redlich-Kwong11 equation given by eq 4.9 when a is replaced by aa(T). Theequation proposed by Gibbons and Laughton12 is also recovered when c¼ 1and the appropriate functions for a(T) are used. When c of eq 5.9 is equal 2, thePeng-Robinson13 equation (primarily intended for vapour-liquid equilibria) eq4.10 is obtained from eq 5.9. Equations of state have also been proposed withc¼ 3 by Harmens14 and when c is another adjustable parameter9,15 by Patel andTeja;16 these are three-parameter equations of state. Modification of therepulsive term gives rise to non-cubic equations. For the sake of simplicity, inthe remainder of this section, the discussion will revolve around the van derWaals equation. Cubic equations of state have been reviewed elsewhere by


Valderrama.17 In general, there is neither a direct relationship between theattractive part of the intermolecular potential and the parameter a in a cubicequation of state nor is a real molecule a hard sphere; such analogies comfortthe user rather than state scientific fact.18

The van der Waals equation of state often gives no more than qualitativeagreement with experiment but never leads to unphysical behaviour exceptperhaps at the critical point where, in common with many analytic equations ofstate, it lacks the appropriate singularity; a problem that is addressed inChapter 10. Nevertheless it is not surprising that the van der Waals equationand the other numerous related equations have been used extensively forengineering calculations.For a pure substance, the parameters a and b from eq 5.7, as described in

Chapter 4, are determined by imposing the conditions

@p

@Vm

� �T

¼ 0 and@2p

@V2m

� �T

¼ 0 ð5:10Þ

at T¼Tc and p¼ pc where Tc is the critical temperature and pc the criticalpressure. The procedure results in eqs 4.3 and 4.4 and provides a means ofdetermining a and b from pc and Tc which is particularly convenient for engi-neering purposes. Improvements in the representation of experimental data byan equation of state are typically provided by the introduction of additionalparameters, such as, the acentric factor o or a temperature or density depen-dence for the parameters.It is to the extension of this approach for pure fluids and the estimation of the

thermodynamic properties of mixtures that we now turn. This requires theintroduction of mixing rules to provide a(x) and b(x), that are now functions ofmole fraction x, from the values of a and b for pure substances. The expressionsfor a(x) and b(x) include the interactions between unlike molecules, andmethods are then required to determine the parameters aij and bij for moleculesi and j from the values of a and b for the pure fluids. This step is achieved usingcombining rules.

5.3.1 Mixing Rules

The van der Waals one–fluid theory for mixtures assumes the properties of amixture can be represented by a hypothetical pure fluid. Thus the thermo-dynamic behaviour of a mixture of constant composition is assumed to beisomorphic to that of a one-component fluid; this assumption is not true nearthe critical point where the thermodynamic behaviour of a mixture at constantthermodynamic potential is isomorphic with that of a one-component fluid andthis is discussed further in Chapter 10.The molecular basis of the van der Waals one-fluid approximation was

developed by Reid and Leland19 and is discussed in Chapter 6. In this case, theintermolecular potential was assumed to be composed of a hard-sphere

88 Chapter 5

combined with long-range attraction20–22 and by use of the mean-densityapproximation led to an expression for the compression factor that is an infiniteseries in T�n. The van der Waals one-fluid theory is obtained by using only thefirst two terms of this series to give:

�xs3x ¼XCi¼0

XCj¼0

xixj�3ij ð5:11Þ

and

s3x ¼XCi¼0

XCj¼0

xixjs3ij : ð5:12Þ

In eqs 5.11 and 5.12 the subscript x denotes the hypothetical pure fluid.Assuming s3 of eq 5.12 is proportional to b of eq 5.7 for a mixture and e of eq5.11 is proportional to a/b, eqs 5.11, eq. 5.12 can be transformed to the mixingrules for van der Waals equation that are quadratic in mole fraction of:

aðxÞ ¼XCi¼0

XCj¼0

xixjaij ð5:13Þ

and

bðxÞ ¼XCi¼0

XCj¼0

xixjbij : ð5:14Þ

Equation 5.14 is often approximated by

bðxÞ ¼XCi¼0

xibi: ð5:15Þ

The summations in eqs 5.13, 5.14 and 5.15 are over all components C. Forthree-parameter cubic equations of state, the parameter c(x) is usually assumedto be given by

cðxÞ ¼XCi¼0

XCj¼0

xixjcij : ð5:16Þ

Equations 5.13 and 5.14 (and in some cases also eq 5.16) provide the means ofestimating the parameters required for a mixture by recourse solely to thecritical properties of the pure substances (see Chapter 4).


The evolution of equation of state mixing rules has been reported byCopeman and Mathias23 and the mixing rules used in cubic equations of statehave been discussed by Knapp et al.24 However, except for the van der Waalsequation, there is no direct relationship between the two parameters in theequation of state and those of the intermolecular potential. Also there is only atenuous basis for similar mixing rules for other cubic equations, and littleevidence that any are superior to the van der Waals one-fluid rules.The composition dependence of a two-parameter cubic equation of state

should conform to the boundary condition established by eq 5.3 of

B Tð Þ ¼Xi

Xj

xixjBij Tð Þ ¼ bðxÞ � aðxÞRT¼Xi

Xj

xixj b� a

RT

� �ijð5:17Þ

and also satisfy eq 5.4 with

C Tð Þ ¼Xi

Xj

Xk

xixjxkCijk Tð Þ ¼ bðxÞ2: ð5:18Þ

In practice, it has proven impossible to set a composition dependence of b thatsimultaneously satisfies both eqs 5.3 and 5.4.25

For a binary mixture (1� x)Aþ xB eqs 5.13 and 5.14 become:

b xð Þ ¼ 1� xð Þ2bA þ 2 1� xð ÞxbAB þ x2bB ð5:19Þ

and

a xð Þ ¼ 1� xð Þ2aA þ 2 1� xð ÞxaAB þ x2aB: ð5:20Þ

To evaluate the quantities bAB and aAB from the values of a and b for each ofthe pure components requires combining rules that will be addressed in the nextsection.There are numerous alternative forms for a and b provided in the literature.

For example, Luedecke and Prausnitz26 provided a semi-empirical density-dependent mixing rule for a given by:

aðxÞ ¼XCi¼1

XCj¼1

xixj aiiajj� �1=2

1� kij� �

þ rRT

XCi¼1

XCj¼1j 6¼i

xixj xiciðjÞ þ xjcjðiÞ�

:

ð5:21Þ

where r is the density and the ci(j) and cj(i) are two temperature-independentinteraction parameters for each binary mixture. Equation 5.21 has the theo-retically correct quadratic composition dependence required to comply with eq5.17 for B(T). Alternative expressions for a(x) have been proposed by numer-ous authors including Mollerup,27 Whiting and Prausnitz28 and Mathias and

90 Chapter 5

Copeman.29 Most provide the frame work for handling multi-componentmixtures. Vooka et al.30 compared the (vapourþ liquid) equilibria estimatedfrom cubic equations of state with measured values and preferred mixing rulesof the form of eq 5.21.In general the van der Waals one-fluid mixing rules are applicable to mixtures

that exhibit relatively moderate departure from ideal systems. For example, thePeng-Robinson equation of state, as modified by Stryjek and Vera,31 representthe vapour-liquid equilibrium behaviour of (CO2þ butane)32 with a singlebinary-interaction parameter but, the (gasþ liquid) equilibria of (acetoneþwater) at a temperature of 473K33 cannot be represented adequately with onebinary interaction parameter with these mixing and combining rules; there aremany other examples in the literature to both support and refute this claim.An understanding of the inability of the van der Waals one-fluid mixing rules

to describe nonideal mixtures can be obtained from the relationship betweenthe molar excess Gibbs energy of mixing, GE

m, and fugacity coefficients, f,obtained from an equation of state given by:

GEm ¼ RT lnf�

Xi

xi lnf�t

!: ð5:22Þ

In eq 5.22 f and f*i are the fugacity coefficients of the solution and of the pure

component i, respectively, that are calculated from

lnf ¼ ln~p T ; pÞð Þ

p

�¼ 1

RT

ZVm

N

RT

Vm� p

� �dVm � lnZ þ Z � 1ð Þ: ð5:23Þ

When equation 5.22 is cast in terms of eq 5.7 for the van der Waals equation ofstate with eq 5.23 for a binary mixture of mole fractions x1 and x2 we obtain:

GEm

RT¼ ln

p

RT

Y2i¼1

RTp� bij

� �xiRTp � b

� �24

35

8<:

9=;� x1x2

RT x1b1 þ x2b2ð Þ

� a1b2

b1

� �1=2

� a2b1

b2

� �1=2( )2

þ 2x1x2 a1a2ð Þ 1� k12ð Þx1b1 þ x2b2

:

ð5:24Þ

Equation 5.24 has three contributions of which the second is analogous withthe excess Gibbs-function of regular-solution theory and the third is similar tothat of augmented regular-solution theory.34 A conclusion of this observationis that a cubic equation of state with the van der Waals mixing and combiningrules can only represent mixtures with moderate departure from solution ide-ality as described by augmented regular-solution theory. Some mixtures ofindustrial interest exhibit greater solution non-idealities and thus demand quitedifferent mixing rules for cubic equations of state and it is to these that we will


turn in the next section. From eq 5.24, it is clear when k12¼ 0 that GEma0 and

thus k12¼ 0 does not reproduce an ideal solution. Indeed, use of k12¼ 0 for(acetoneþwater) can lead to phase-behaviour predictions that are unrealisticand in the absence of measurements for the mixture against which to adjust k12choosing k12¼ 0 as the default may give rise to erroneous results.

5.3.2 Combining Rules

It is to the discussion of combining rules that we now turn by way of adigression regarding intermolecular potentials. In the absence of experimentalmeasurements for the mixtures of the components of interest estimates of theparameters must be made by other means.Using the Lennard-Jones intermolecular potential,35 which accounts for the

repulsive and attractive forces, for the interaction of spherical substances A andB in (AþB), uAB(r) is given by:

uAB rð Þ ¼ 4�ABsAB

rAB

� �12

� sAB

rAB

� �6( )

; ð5:25Þ

and is frequently used in computer simulation. For a ternary mixture ofspherical molecules, it is assumed that u(rAB, rBC, rCA) is given by the sum ofthree pair-interaction energies {u(rAB) þ u(rBC)þ u(rCA)} of which the first termin the summation is given by eq 5.25. The parameter eAB of eq 5.25 defines thedepth of the potential well and sAB is the separation distance at the potentialminimum. Combining rules at the molecular level are required to determine eAB

and sAB from the pure-component values, and it is the discussion of these thatwe now turn to because they provide back-ground information for this andother sections of this Chapter.The parameter sAB for un-like interactions between molecules A and B is

most often determined from the rule proposed by Lorentz,36 which is based onthe collision of hard spheres, resulting sAB being given by the arithmetic meanof the pure-component values with:

sAB ¼sA þ sB

2: ð5:26Þ

The parameter eAB is obtained from the expression of Berthelot42 for thegeometric mean of the pure-component parameters of:

�AB ¼ �A�Bð Þ1=2: ð5:27Þ

Equation 5.27 arises from consideration of the London37 theory of disper-sion.38–41 Equations 5.26 and 5.27 are collectively known as the Lorentz-Berthelot combining rules; they are known to fail particularly in the case ofhighly non-ideal mixtures.40,43–47

92 Chapter 5

Because the core volume b of eq 5.7 is proportional to s3 of eq 5.26 and a isproportional to the depth of the potential well given by eq 5.27, eqs 5.26 and5.27 can be recast as

bAB ¼b1=3A þ b

1=3A

� �38

ð5:28Þ

and

aAB ¼ aAaBð Þ1=2; ð5:29Þ

respectively. Equations 5.28 and 5.29 provide the means to estimate both aAB

and bAB. Molecules are not hard spheres so that eq 5.28 is corrected, particu-larly to estimate phase boundaries, by the addition of a parameter bAB.Equation 5.29 is also modified by a parameter kAB for the same reason. Thesemodifications lead to the actual forms of eqs 5.28 and 5.29 that are routinelyused in engineering calculations:

bAB ¼ 1� bABð Þb1=3A þ b

1=3A

� �38

ð5:30Þ

and

aAB ¼ 1� kABð Þ aAaBð Þ1=2: ð5:31Þ

The parameters bAB of eq 5.30 and kAB of eq 5.31 are frequently called inter-action parameters. Equation 5.30 is often cast as

bAB ¼ 0:5 1� bABð Þ bA þ bBð Þ; ð5:32Þ

because in this form the combined equation of state, mixing and combiningrules provide estimates of the properties of the mixture that differ less from theexperimental measurements than when eq 5.30 is used.For a three-parameter cubic equation of state, the cAB is typically given by

cAB ¼ 0:5 1� dABð Þ cA þ cBð Þ: ð5:33Þ

In the absence of a theoretical basis or means to estimate the interactionparameters kAB and bAB, these are determined by regression to measurementsfor each binary mixture, for example, kAB can be determined from measure-ments of the second virial coefficient for gaseous mixtures.The combining rules obtained from consideration of intermolecular poten-

tials provide, at least by analogies similar to those invoked here, both infor-mation regarding the functional forms of the interaction parameters andestimates of their values. Some of the alternative forms proposed for eqs 5.26


and 5.27 will be mentioned in the following, and these have also beenreviewed by Schnabel et al.48 For almost ideal mixtures, for example, alkanemixtures, kABE0, while for other mixtures, kABa0 and also depends ontemperature.We now turn to introduce some alternative sources of combining rules.

Hudson and McCoubrey49 derived a combining rule for eAB by relating theattractive part of the Lennard-Jones potential to London-dispersion37 andobtained the expression:

�AB ¼ �A�Bð Þ1=2 2 sAsBð Þ1=2

sA þ sB

( )62 EAEBð Þ1=2

EA þ EB; ð5:34Þ

where EA and EB are the ionization energy, which is the minimum energyrequired to eject an electron from a neutral atom or molecule in its groundstate; an expression similar to eq 5.34 was reported by Prausnitz.50 Equation5.34 reduces to eq 5.27 when EA¼EB and sA¼ sB. For molecules with verydifferent effective diameters eq 5.34 may be cast as

�AB ¼ �A�Bð Þ1=2 26s3As3B

sA þ sBð Þ6; ð5:35Þ

or when sAEsB it can be assumed |sA� sB|/sB¼ r to give by expansion:

�AB ¼ �A�Bð Þ1=2 1� 3

4r2 þ 3

4r3 þ � � �

� �: ð5:36Þ

Comparison of eqs 5.34 and 5.31 suggests that kAB can be obtained from:

kAB ¼ 1þ 2 sAAsBBð Þ1=2

sAA þ sBB

( )62 EAEBð Þ1=2

EA þ EB; ð5:37Þ

in the absence of other suitable data, for example, (vapourþ liquid) equilibriameasurements. Equation 5.37, through EA and EB, provides insight to theplausible temperature dependence of kAB. Equation 5.34 can be used to predictthe interaction virial coefficient BAB as shown in ref 49.If it is assumed that s3 is proportional to the critical volume Vc and e/k is

proportional to the critical temperature Tc then eq 5.34 can be cast as:49

Tc;AB ¼ Tc;ATc;Bð Þ1=2 2 Vc;m;AVc;m;Bð Þ1=2

Vc;m;A þ Vc;m;B

( )62 EAEBð Þ1=2

EA þ EB; ð5:38Þ

where the effective critical properties are discussed further in Section 5.4.2.Prausnitz and co-workers51–53 used eq 5.38 for the calculation of

94 Chapter 5

(vapourþ liquid) equilibria at elevated pressure for asymmetric mixtures that isof (carbon dioxideþ hydrocarbons). Equation 5.38 can be simplified to

Tc;AB ¼ Tc;ATc;Bð Þ1=2 2 Vc;m;AVc;m;Bð Þ1=2

Vc;m;A þ Vc;m;B

( )6

; ð5:39Þ

and assuming a of eq 5.7 is proportional to es3 that is also proportional toTcVc,m then eq 5.39 reduces to eq 5.29.The combining rule reported by Kohler54 for eAB is

�AB ¼ 27�A�BsAsB

sA þ sB

� �aAaB

a2Bs6A�A þ a2As

6B�B

� �; ð5:40Þ

where a is the quantity relating the induced electric dipole moment, PA, to theapplied electric field strength, E by PA¼ aEA. However, the requirement toknow the polarizability has limited the practical application of eq 5.40.Fender and Halsey55 proposed for eAB:

�AB ¼2�A�B�B þ �B

: ð5:41Þ

Equation 5.26 was also used for sAB in refs 49, 54 and 55.Hiza and co-workers56–58 used regression analysis with experimental data to

propose corrections to eqs 5.28 and 5.29 that also included the ionizationenergy from the ground state of the neutral atom or molecule and suggested thefollowing:

sAB ¼ 1þ 0:025kABð Þ sA þ sB2

� �; ð5:42Þ

�AB ¼ 1� 0:18kABð Þ �A�Bð Þ1=2; and ð5:43Þ

kAB ¼ EA � EBð Þ1=2ln EA=EBð Þ: ð5:44Þ

In eq 5.44, the ionization energies are arranged so that EA4EB.The combining rules proposed by Sikora59 are:

�AB ¼ 215EAEB

EA þ EBð Þ2�Aa12A �Bs

12B

� �1=2�Aa12A� �1=13þ �Bs12B

� �1=13n o1=13�A�Bð Þ1=2 ð5:45Þ

and

sAB ¼ 2�13=12 s12=13A þ s12=13B

� �13=12: ð5:46Þ


Kong60 adapted an approach first advocated by Smith61 to use the combiningrules:

�AB ¼ 213�As6A�Bs

6B

�As12A� �1=13þ �Bs12B

� �1=13n o13ð5:47Þ

and

sAB ¼�As12A� �1=13þ �Bs12B

� �1=13n o13

213 �As6A�Bs6B

� �1=2264

3751=6

: ð5:48Þ

Halgren62 proposed the combining rules:

�AB ¼4�A�B

�Að Þ1=2þ �Bð Þ1=2n o ð5:49Þ

and

sAB ¼s3A þ s3Bs2A þ s2B

: ð5:50Þ

Waldman and Hagler63 suggested

�AB ¼2s2As

3B

s6A þ s6B�A�Bð Þ1=2 ð5:51Þ

and

sAB ¼s6A þ s6B

2

� 1=6: ð5:52Þ

From the Mie potential,64 given by

u rABð Þ ¼ m

m� n

m

n

� � nm�n�AB

sAB

rAB

� �m

� sAB

rAB

� �n �; ð5:53Þ

Coutinho et al.65 proposed for eAB the form

�AB ¼ �A�Bð Þ1=2 sAsBð Þ1=2

sAB

( )n2 EAEBð Þ1=2

EA þ EB: ð5:54Þ

96 Chapter 5

Equation 5.54 reduces to the form of the Lennard-Jones potential given by eq5.25 when m¼ 12 and n¼ 6. Coutinho et al.66 assumed s3pb, e/kpTc andepa/b and obtained aAB of the cubic equation of state from eq 5.54 to give:

aAB ¼ aAaBð Þ1=2 bAbBð Þ1=2

bAB

( )n�332 EAEBð Þ1=2

EA þ EB; ð5:55Þ

and when EAEEB, the kAB is given by

kAB ¼ 1� bAbBð Þ1=2

bAB

( ) n=3ð Þ�2

: ð5:56Þ

Other combining rules can be obtained from eq 5.55 and some of these are asfollows:66 (a), with n¼ 3 the rules reported by Chueh and Prausnitz;52 (b), withn¼ 3.75 those reported by Plocker et al.;67 (c), with n¼ 5.25 the work of Radoszet al.;68 and (d), with n ¼ 7.2 the methods reported by Trebble and Sigmund.69

Coutinho et al.66 have also discussed the variation of kAB with respect to themolar mass, dipole moment and molecular dimensions, and have also per-mitted n to be an adjustable parameter determined by regression to measuredthermodynamic properties.Clearly, from the partial list of alternative combining rules provided by eqs

5.34 to 5.54 there are a multitude of methods that can be applied to determinethe properties of mixtures from those of pure substances with combining rulesand quadratic mixing rules. Nevertheless, the Lorentz-Berthelot combiningrules given by eqs 5.26 and 5.27 are still frequently used and form the basis formost engineering calculations; they have the benefit of relying on data for puresubstance even in the absence of experimental data for the mixture that is withkAB of eq 5.31 equal 0.

5.3.3 Non-Quadratic Mixing and Combining Rules

The desire to apply cubic equations of state to non-ideal systems, which hadonly been adequately correlated by activity-coefficient models, has resulted inadditional composition dependence for the a parameter through the kij; themixing and combining rules for b have generally remained unchanged. Pana-giotopoulos and Reid70 suggested the kij of eq 5.31 be given by

kij ¼ Kij � Kij � Kji

� �xi; ð5:57Þ

while Adachi and Sugie71 report

kij ¼ Kij þ lij xi � xj� �

: ð5:58Þ


Sanfoval et al.72 give

kij ¼ Kijxi þ Kijxi þ 0:5 Kij � Kji

� �1� xi � xj� �

: ð5:59Þ

Schwartzentruber and Renon73,74 have suggested kij be given by

kij ¼ Kij þ lijmijxi �mjixj

mijxi þmjixjxi þ xj� �

; ð5:60Þ

where Kij¼Kji, lji¼ lij, mji¼ 1�mij and Kii¼ lii¼ 0 and a(x) is no longerquadratic in mole fraction. By an appropriate choice of parameters, thesecombining rules reduce to one another, and to eq 5.31, and they have beenshown able to correlate vapour-liquid equilibrium data for non-ideal binarymixtures.Despite the apparent success of eqs 5.57 through 5.60 at representing the

properties of fluids, there are some inherent problems with the approach ofintroducing additional, what are essentially, adjustable parameters. Thus, it isnot surprising that eqs 5.57 through 5.60 fail to satisfy the boundary conditionsgiven by eqs 5.3 to 5.4. In addition, as Michelsen and Kistenmacher75 clearlydescribe, eqs 5.57 through 5.60 provides the incorrect treatment of multi-component mixtures containing two or more very similar components; mixingrules of the form of eq 5.31 with constants for kij do comply with ref 75. From apractical perspective, this last failure is highly relevant because there are manyindustrially important multi-component mixtures for which two of the com-ponents are very similar and also have similar equation of state parameters. Asthe number of components in the mixture increases, the mole fractions of eachdecreases and eqs 5.57 to 5.60 lead to reduced contributions from the termswith the higher-order composition dependencies; this effectively reducesequations of the form of 5.57 to 5.60 to quadratic mixing rules as the number ofcomponents in the mixture increases.For multi-component mixtures Mathias et al.76 proposed a(x) should have

the form:

aðxÞ ¼Xni¼1

xiXnj¼1

xj 1� kij� �

aiaj� �1=2

þXCi¼1

xiXCj¼1

xj 1� kij� �

aiaj� �1=2n o1=3

l1=3ij

" #3ð5:61Þ

to comply with the requirements of ref. 75.Zabaloy77 presented a cubic mixing rule for a(x) given by

aðxÞ ¼XCi¼1

XCj¼1

XCk¼1

xixjxkaijk; ð5:62Þ

98 Chapter 5

where aijk is given by

aijk ¼ 1� kijk� �

aiajak� �1=3

: ð5:63Þ

Methods were introduced in ref. 77, based initially on the work of Mathiaset al.,76 to estimate the interaction parameter kijk from existing values of kij.When the approach has been adopted, it apparently satisfies the invariance testof ref. 75 but, because of the cubic dependence on mole fraction, the conditionsgiven by eq 5.17 are not met. Valderrama et al.78 have proposed a general non-quadratic mixing rule that also fails to comply with the requirements ofMichelsen and Kisternmacher.75

The relationship between an activity coefficient, f, and the fugacity coeffi-cients, f, obtained from an equation of state is

fi ¼ fi T ; p;xð Þ�f�i T ; pð Þ

n o; ð5:64Þ

where fi T ; p;xð Þ is the fugacity coefficient of species in a mixture and f�i T ; pð Þis the pure-component fugacity coefficient in the same state of aggregation asthe mixture. Both fi T ; p;xð Þ and f�i T ; pð Þ are obtained from an equation ofstate. Consequently, the two interaction parameters per binary mixture (in theclass of combining rules being considered here) can be obtained from theactivity coefficients over the whole composition range or the values at specifiedcompositions, such as the two infinite-dilution activity coefficients of amixture74,79,80 eliminating the requirement for (vapourþ liquid) equilibriameasurements over the whole composition range. Equation 5.64 at the twoinfinite-dilution limits gives two equations, for the two parameters in eqs 5.57to 5.60, in terms of the pure-component equation of state parameters and theinfinite-dilution activity coefficients. In this case, it is possible to predict thephase behaviour of some highly non-ideal systems successfully using onlyinfinite-dilution activity-coefficient information.In the development of mixing and combining rules with activity coefficient

data the requirement to correctly predict the composition dependence of thesecond virial coefficient is usually ignored because the requirement is con-sidered to affect only the relatively low-density regime. However, this isincorrect because it contributes to the error in the calculated fugacities at alldensities. The reason for this statement arises from the expression for thefugacity coefficient of a substance in a mixture given by

lnfi ¼1

RT

ZVm

N

RT

Vm� @p

@ni

� �T ;Vm ;nj 6¼i

" #dVm � lnZ; ð5:65Þ

because the integration is from zero density to the density of interest, andincludes a partial derivative with respect to composition. In eq 5.65, ni is thenumber of moles of substance i in the mixture and Vm is the total molar volume.


Any error in the virial coefficient of the pure component will also introduce anerror into the calculated fugacity independent of the choice of mixing andcombining rules.There is a need to predict the properties of mixtures that depart significantly

from ideal while also complying with the requirements of eqs 5.13 and 5.14.Thus alternative mixing and combining rules are required and these are thetopic of Section 5.3.4.

5.3.4 Mixing Rules that Combine an Equation of State with an

Activity-Coefficient Model

Many mixtures of interest in the chemical industry exhibit strong non-ideality, greater than results from regular-solution theory, and have tradi-tionally been described by activity coefficient (or excess molar Gibbsfunction) models for the liquid phase, and an equation of state model for thevapour phase. However, the activity-coefficient description has numerousdrawbacks that include: (1) the inability to define standard states for super-critical components; (2) critical phenomena cannot be predicted because adifferent model is used for the vapour and liquid phases; (3) the model para-meters are highly temperature dependent; and (4) the approach cannot predictvalues for the density, enthalpy and entropy from the same model. In view ofthese drawbacks, there is a need for equation of state models that candescribe greater degrees of non-ideality than is possible with the van der Waalsone-fluid approach. The methods adopted rely on the suggestion that theexcess molar Gibbs function of mixing obtained from the proposed equation ofstate DGE

m(eos) be equal to that obtained from an activity coefficient modelDGE

m(acm):

DGEm eosð Þ ¼ DGE

m acmð Þ: ð5:66Þ

It is to the consideration of methods that generally adopt the concept providedby eq 5.66 that we now turn.

5.3.4.1 The Huron-Vidal Model

The first model that combined an equation of state and activity-coefficientmodel by use of eq 5.66 was described by Vidal81 and later Huron and Vidal82

and has the form

GEm f ;T ; p!N;xð Þ ¼ GE

m eos;T ; p!N;xð Þ; ð5:67Þ

where GEm f ;T ; p!N;xð Þ and GE

m eos;T ; p!N;xð Þ are the excess molarGibbs functions as the pressure tends to infinity (that is at liquid-like densities)calculated from an activity-coefficient model and an equation of state,

100 Chapter 5

respectively. The excess molar Gibbs function can be expressed by

GEm ¼ AE

m þ pVEm; ð5:68Þ

where AEm is the excess molar Helmholtz function that for a liquid is almost

independent of pressure and VEm the excess molar volume of mixing. As the

pressure p-N for a cubic equation of state of the form of eq 5.7 Vi¼ bi, and inorder to use eq 5.67 it is necessary to assume eq 5.32 for a multi-componentmixture of

bðxÞ ¼XCi¼1

xibi; ð5:69Þ

so that

VEm ¼ V �

XCi¼1

xiVi ¼ b�XCi¼1

xibi ¼ 0: ð5:70Þ

Equations 5.68 and 5.70 provide the two expressions required to determine thetwo parameters resulting in the mixing rule for a of:

aðxÞ ¼ bðxÞXCi¼1

xiai

bi

� ��DGE

m

" #: ð5:71Þ

In eq 5.71, D is a constant that depends on the particular equation of state usedand GE

m is an excess Gibbs function of mixing obtained from an activity coef-ficient model. Activity coefficients are usually obtained from measurements of(vapourþ liquid) equilibria at a pressure relatively low compared with therequirement of eq 5.67 for which p-N; the activity coefficients are tabulated,for example, those in the DECHEMA Chemistry Data Series.32 This distinc-tion in pressure is particularly important because the excess molar Gibbsfunction of mixing, obtained from experiment and estimated from an equationof state, depends on pressure; dðGE

m=RTÞ=dpo0:002MPa�1 for (metha-nolþ benzene) at a temperature of 373K. Equation 5.71 does not satisfy thequadratic composition dependence required by the boundary condition ofeq 5.3. However, equations 5.70 and 5.71 form the mixing rules that have beenused to describe the (vapourþ liquid) equilibria of non-ideal systems, such as(propanoneþwater),33 successfully; in this particular case the three-parameterNon-Random Two Liquid (known by the acronym NRTL) activity-coefficientmodel was used for GE

m and the value depends significantly on temperature tothe extent that the model, while useful for correlation of data, cannot be used toextrapolate reliably to other temperatures.


5.3.4.2 Combination of GEm and Equations of State at pE0

The excess molar Gibbs function is obtained from the equation of statefrom

GEm eos;T ; p;xð Þ

RT¼ lnf T ; p;xð Þ �

Xi

xi lnfi T ; pð Þ

¼ Z T ; p; xð Þ �Xi

xiZi T ; pð Þ" #

� lnZ T ; p; xð Þ �Xi

xi lnZi T ; pð Þ" #

�ZVm T ;p;xð Þ

N

ZðxÞ � 1

VmðxÞdVmðxÞ �

Xi

xi

ZVm;i T ;pð Þ

N

Zi � 1

VidVi

0B@

1CA:ð5:72Þ

The equivalent expression for the excess molar Helmholtz function of mixing is

AEm eos;T ; p;xð Þ

RT¼�

Xi

xiZ T ; p;xð ÞZi T ; pð Þ

�ZVm T ;p;xð Þ

N

ZðxÞ � 1

VmðxÞdVmðxÞ �

Xi

xi

ZVm;i T ;pð Þ

N

Zi � 1

VidVi

0B@

1CA:ð5:73Þ

From the definition of an excess-property of mixing, it is necessary that thepure components and the mixture be in the same state. It is obvious from eqs5.72 and 5.73 that the excess Gibbs or Helmholtz functions of mixing obtainedfrom an equation of state are a function of pressure while the GE

m obtained froman activity coefficient model is independent of pressure. Therefore, the equalityof eq 5.66 between GE

m (or AEm) from an equation of state and an activity-

coefficient model must be at the same pressure.For the van der Waals cubic equations eqs 5.72 and 5.73 take the form

GEm eos;T ; p;xð Þ

RT¼Z T ; p; xð Þ �

Xi

xiZi T ; pð Þ �Xi

xi lnZ T ; p;xð ÞZi T ; pð Þ

�

�Xi

xi ln 1� bðxÞ=VmðxÞf g= 1� bi=Vm;ið Þ½ �

þ aðxÞbðxÞRT

�C VmðxÞf g �

Xi

xiai

biRT

� �C Vm;ið Þ;

ð5:74Þ

102 Chapter 5

and

AEm eos;T ; p;xð Þ

RT¼�

Xi

xi lnZ T ; p;xð ÞZi T ; pð Þ

�

�Xi

xi ln 1� bðxÞ=VmðxÞf g= 1� bi=Vm;ið Þ½ �

þ aðxÞbðxÞRT

�C VmðxÞf g �

Xi

xiai

biRT

� �C Vm;ið Þ:

ð5:75Þ

Here CV{m(x)} is the molar volume for the specific equation of state; forthe Peng-Robinson equation C Vm xð Þf g ¼ 1

�2ffiffiffi2p� �

ln Vm þ 1�ffiffiffi2p� �

b� ��

Vm þ 1þffiffiffi2p� �

b� �

g. In the limit of infinite pressure, Vi-bi and V(x)-b(x) sothat b(x)¼C(Vm,i¼ bi)¼C*, and for the Peng-Robinson equation C� ¼ln

ffiffiffi2p� 1

� �� ffiffiffi2p¼ �0:62323.

To overcome the effect of using different pressures for the equation of stateand activity-coefficient models, Mollerup83 suggested using the axiom

GEm f ;T ; p ¼ 0;xð Þ ¼ GE

m eos;T ; p ¼ 0;xð Þ: ð5:76Þ

Equation 5.74 at p¼ 0 is

GEm eos; p ¼ 0ð Þ

RT¼�

Xi

xi lnVmðxÞ � bðxÞVm;i � bm;i

��Xi

xi ln bðxÞ=bif g

þ aðxÞbðxÞRT

�C VmðxÞf g �

Xi

xiai

biRT

� �C Vm;ið Þ:

ð5:77Þ

The second equation for the mixing rules is given by eq 5.69; in the absence ofthe divergence of the excess molar Gibbs function as the pressure tends toinfinity it is unnecessary to impose the use of eq 5.69 and other choices, such as,b� a/RT could be used.The concept of eq 5.76 was adopted by Heidemann and Kokal84 and

revealed, in the absence of an estimated liquid molar volume at zero-pressure,an apparent zero-pressure liquid volume was obtained.84 Michelsen85,86 over-came the deficiency of ref 84 by writing eq 5.77 as

q aðxÞf g ¼XCi¼1

xiqai þGE

m T ; p ¼ 0;xð ÞRT

þXCi¼1

xi lnbðxÞbi

�; ð5:78Þ

where

qfaðxÞg ¼ � lnVmðT ; p ¼ 0Þ

bðxÞ

�þ aðxÞbðxÞRT CfVmðp ¼ 0Þg ð5:79Þ


and selecting a value of a(¼ a/bRT) for which a liquid root was found. At atemperature for which there is no liquid root to the equation of state for one ormore of the components in the mixture a linear extrapolation of a was used asfollows:

qðaÞ ¼ q0 þ q1a: ð5:80Þ

Equation 5.80 provided a and eq 5.79 was replaced by

aðxÞbðxÞRT ¼ a ¼

XCi¼1

xiai þ1

q1

GEm T ; p ¼ 0;xð Þ

RTþXCi¼1

xi lnbðxÞbi

�" #; ð5:81Þ

and when combined with eq 5.69 forms the mixing rules known by the acronymMHV1.85 When a quadratic function is used to extrapolate a of the form

qðaÞ ¼ q0 þ q1aþ q2a2; ð5:82Þ

with the constants q0, q1 and q2 chosen to insure continuity of q(a) and deri-vatives one obtains

q1aðxÞ

bðxÞRT �XCi¼1

xiai

biRT

( )þ q2

aðxÞbðxÞRT

�2

�XCi¼1

xiai

biRT

�2" #

¼ GEm T ; p ¼ 0;xð Þ

RTþXCi¼1

xi lnbðxÞbi

� �;

ð5:83Þ

which together with eqs 5.69 are the mixing rules known by the acronymMHV2.86,87 Neither MHV1 nor MHV2 satisfy eq 5.17.Tochigi et al.,88 proposed a modified mixing rule consistent with eq 5.17 by

combining eqs 5.77 or 5.78 and 5.82 eliminating the binary interaction para-meter of eq 5.17 to give

aðxÞRT¼ bðxÞ

Xi

xiai

biRTþ 1

q1

GEm g;T ; p ¼ 0;xð Þ

RTþXi

xi lnbðxÞbi

�" # !

ð5:84Þ

and

bðxÞ ¼

Pi

xi ln bi � aiRT

� �

1�Pi

xiai

biRTþ 1

q1

GEm g;T ; p ¼ 0; xð Þ

RTþXi

xi lnbðxÞbi

�" # : ð5:85Þ

These mixing rules are equivalent to MHV1.

104 Chapter 5

Hoderbaum and Gmehling89 proposed the Soave-Redlich-Kwong equationof state be combined with UNIFAC and produced the Predictive Soave-Red-lich-Kwong (and given the acronym PSRK) equation for which a(x) is given by

aðxÞ ¼ RTbðxÞ 1

0:64663

GEm

RTþXCi¼1

xi lnbðxÞbi

� �( )þXCi¼1

xiai

biRT

" #: ð5:86Þ

5.3.4.3 The Wong-Sandler Model

The Wong and Sandler90 mixing rules used the modified Peng-Robinson91 orany other cubic equation of state and permit the use of tabulated92 GE

m such asthe DECHEMA Data Series.32 The mixing rules are comprised of the right-hand side of eq 5.17 with the combining rule

bðxÞ � aðxÞRT

�ij

¼ 1

2bi �

ai

RT

� �þ bj �

aj

RT

� �h i1� kij� �

; ð5:87Þ

which introduces a second-virial-coefficient binary-interaction parameter kijand ensures the correct composition dependence of the second virial coefficient.The second equation was obtained from the observation that the excess molarGibbs function of mixing defined by

GEm ¼ DmixGm � RT 1� xð Þ ln 1� xð Þ þ x ln xf g ð5:88Þ

at a vapour pressure psat low enough so that virial coefficients higher than Bmay be neglected and where psatEp so that the partial molar volume in theliquid is independent of pressure, so that

GEm ¼ð1� xlÞRT ln 1� xgð Þpsat=ð1� xlÞpsat;A

� þ xlRT xgpsat=x

lpsat;B�

þ ð1� xlÞðBAA � V�lA Þðpsat � psat;AÞ þ xlðBBB � V�lB Þðpsat � psat;AÞ

þ 1� xl� �

ðxgÞ2 þ xlð1� xgÞ2n o

2dABpsat

þ VEmðT ; xlÞðp� psatÞ;

ð5:89Þ

and the approximation:

GEm T ; p ¼ 0:1MPa; xð ÞEAE

m T ; p ¼ 0:1MPa; xð Þ ð5:90Þ

can be used. At liquid densities AEm is essentially independent of pressure and

the approximation

AEm T ; p ¼ 0:1MPa; xð ÞEAE

m T ; p440:1MPa; xð Þ; ð5:91Þ


also holds. The second equation required to determine a and b is obtained fromeq 5.66 in the form

AEm eos;T ; p!N;xð Þ ¼ AE

m g;T ; p!N;xð Þ¼ AE

m f ;T ; p � 0:1MPa;xð Þ ¼ GEm f ;T ; p � 0:1MPa;xð Þ;

ð5:92Þ

where f refers to the property obtained from an excess-energy model. Com-bining eqs 5.17 and 5.92 gives the rules

bðxÞ � aðxÞRT

�¼Xi

Xj

xixj b� a

RT

� �ij

ð5:93Þ

andGE

m fð ÞDRT

¼ aðxÞbðxÞRT �

XCi¼1

xiai

biRT

� �; ð5:94Þ

where D is a constant and the cross term in eq 5.93 is obtained from eq 5.87.Any excess molar Gibbs function may be used in eq 5.94. This approachprovides a mixing rule that is independent of density. When compared to priorad-hoc density-dependent rules23,93,94 for the cubic equations of state themethod does not violate the one-fluid model with different density dependencesfor pure fluids and mixtures.The Wong-Sandler mixing rules extend the use of cubic equations of state to

mixtures that were previously only correlated with activity-coefficient models. Formany mixtures, the Gibbs-function model parameters in the equation of state couldbe taken to be independent of temperature, thereby allowing extrapolation of phasebehaviour over wide ranges of temperature and pressure. For example, for (etha-nolþwater)95 the activity-coefficient model reported in DECHEMA32 is at apressure of 0.4MPa and this model provides reasonable predictions of the phaseboundaries at pressures up to 20MPa. This means the method can be used withUNIversal Functional Activity Coefficient (known by the acronym UNIFAC) andother group-contribution methods96 to predict properties at elevated pressure.It is desirable for the excess molar Gibbs-function mixing rules to converge

smoothly with the mixing rules of the conventional van der Waals one-fluidmodel because within multi-component mixtures some binary pairs will formhighly non-ideal mixtures that require mixing rules, such as those of Wong andSandler, while other binary pairs in the same mixture are almost ideal and canbe adequately described by use of eqs 5.13 and 5.14. Orbey and Sandler97

showed the Wong-Sandler mixing rule can converge to the van der Waals one-fluid theory by retaining the mixing rules given by eqs 5.93 and 5.94 andreplacing the combining rule of eq 5.87 with

b� a

RT

� �ij¼ 1

2bi � bj� �

�ffiffiffiffiffiffiffiffiaiajp

RT1� kij� �

: ð5:95Þ

106 Chapter 5

Equation 5.95 introduces the binary interaction parameter in a manner similarto that in eq 5.31. In this case, a modified form of the NRTL was used for theexcess-Gibbs-function so that:

GEm

RT¼Xi

xi

Pj

xjGjitjiPk

xkGki

0B@

1CA; ð5:96Þ

with

Gij ¼ bj exp �atji� �

; ð5:97Þ

where bj is the volume parameter in the equation of state for species j. The useof the modified NRTL was suggested by Huron and Vidal81 previously. Forbinary mixtures, eqs 5.95, 5.96 and 5.97 have four dimensionless parameters a,t12, t21, and k12 that can all be used to correlate the phase behaviour of complexmixtures. Fewer adjustable parameters may also be used, for example, values ofa and k12 permit solution of eq 5.96 in the limit of infinite dilution for t21 giving

t21 ¼ ln gN12 � t12b1

b2exp �at12ð Þ; ð5:98Þ

where gN12 is the infinite-dilution activity coefficient of species 1 in 2. Predictingthe properties of mixtures in the absence of experimentally determined values,eq 5.98 can be used with infinite-dilution activity coefficients obtained fromUNIFAC. For binary pairs in multi-component mixtures that exhibit slightlynon-ideal behaviour setting a¼ 0.1 and solving gives

t21 ¼C

RT

2ffiffiffiffiffiffiffiffiffia1a2p

b1 þ b21� kij� �

� a1

b1

� ; ð5:99Þ

for t12 and returns the van der Waals one-fluid mixing rule with a single binary-interaction parameter kij. Equation 5.99 is not unique, and other expressionsthat lead to the van der Waals mixing rules can be obtained.97

Orbey and Sandler98 also showed the Wong-Sandler mixing rule with theFlory-Huggins model for the excess molar Gibbs function can be used topredict the (vapourþ liquid) equilibria of ethane with low-molar mass poly-mers. With three parameters the mixture (etheneþ tetracontane) has beencorrelated by the same approach.99

The Wong-Sandler mixing rule has also been applied to (hydro-genþ hydrocarbon),100 however, the results obtained are dependent on thefunction used to represent the temperature variation of the parameter a in theequation of state; there are several methods proposed and described in Chapter4. Usually, the form of a(T) is obtained from the vapour pressure that is poorlydefined at high reduced temperatures. When one component of the mixture is


hydrogen, the temperature dependence, including those proposed for the Peng-Robinson and Soave-Redlich-Kwong equations of state, lead to erroneousresults from the Wong-Sandler mixing rule owing to the low critical tempera-ture of hydrogen. Similar issues arise with other so-called asymmetric mixturescontaining substances with very different molar mass particularly those of verylow molar mass components.The b(x) parameter in the Wong-Sandler model is, from eqs 5.93 and 5.94,

given by

bðxÞ ¼

Xi

Xj

xixj bij �aij

RT

� �

1�Xi

xiai

biRTþ GE

m

CRT

!:

ð5:100Þ

The denominator of eq 5.100 contains three terms. The GEm can be negative or

positive and vanishes at high temperatures and for the purpose of discussioncan be neglected. Equation 5.100 requires (ai/biRT) c1 for all components ofthe mixture to prevent the denominator becoming zero or change sign for acomposition of the (vapourþ liquid). The condition (ai/biRT) c1 is obtainedby requiring for all temperatures and components in the mixture that

bi �ai

RTo0: ð5:101Þ

For the Peng-Robinson equation of state eq 5.101 provides

a ¼ 0:45725R2T2

c

pca Tð Þ; ð5:102Þ

b ¼ 0:0778RTc

pcð5:103Þ

anda

bRT¼ 0:45724

0:0778

� �a Tð ÞTr¼ 5:87712

a Tð ÞTr

; ð5:104Þ

where subscript c denotes the critical property, Tr¼T/Tc, and a(T) is thetemperature dependent function. Equation 5.101 requires a(T)ZTr/5.87712for the Peng-Robinson equation of state and for the majority of the cubicequations of state this requirement is met at Tro2. However, the restrictionmay not be met at higher Tr because of the temperature dependence of a(T).For mixtures with one or more supercritical components, the singularity can beremoved, without introduction of additional uncertainties in the calculatedphase behaviour or thermodynamic properties, by introduction of the

108 Chapter 5

expression a(T)¼Tr/M, where M is a parameter; for the Peng-Robinsonequation of state M¼ 5.87712.Orbey and Sandler101 assumed for all fluids a universal parameter u

relates the liquid molar volumes to the hard-core volumes, b through Vm¼ ub.The parameter u is positive and greater than unity so that eq 5.74 can bewritten as:

AEm eos;T ; p; xð Þ

RT¼�

Xi

xi lnub T ; p; xð Þubi T ; pð Þ

� �Xi

xi ln 1� uð Þ= 1� uð Þ½ �

þ aðxÞbðxÞRT

�C ub T ; p;xð Þf g �

Xi

xiai

biRT

� �C ubið Þ

¼ �Xi

xi lnb T ; p; xð Þbi T ; pð Þ

� þ aðxÞ

bðxÞRT

�C ubðxÞf g

�Xi

xiai

biRT

� �C ubið Þ

ð5:105Þ

At both infinite pressure and at very low temperatures u¼ l {so that C(b)¼C*}and results in:

AEm eos;T ; p; xð Þ

RT¼ �

Xi

xi lnb T ; p;xð Þbi T ; pð Þ

� þ C�

RT

aðxÞbðxÞ �

Xi

xiai

bi

� �" #: ð5:106Þ

Equation 5.106 uses one degree of freedom to determine both a and bfor the equation of state and can be coupled with either eqs 5.14, 5.15 or 5.17 toobtain a new mixing rule. The first alternative, eq 5.14, when combinedwith eq 5.106, leads to an algebraically simple mixing rule that is very similar tothe MHV1 model, but with the ql replaced with C*, and, like that model,does not satisfy the second-virial-coefficient composition dependence. For thePeng-Robinson equation of state, ql¼ � 0.53, while C*¼ � 0.62323. However,this small difference is significant when extrapolating or predictingvapour-liquid equilibria. In general, for phase-equilibria predictions, theWong-Sandler model provides a better representation of measurements thanthe MHV2 particularly for multi-component mixtures when the completemodel is required to extrapolate in temperature and data for binary mixturesare used.102

5.3.4.4 Linear Combination of Huron-Vidal-Michelsen

Boukouvalas et al.103 proposed a mixing rule by forming the following linearcombination of the Huron-Vidal and Michelsen models known by the acronym


LCVM and given by

aðxÞbðxÞRT ¼

lC�þ 1� l

q1

� �GE

m fð ÞRT

þ 1� lq1

Xi

xi lnbðxÞbi

�þXi

xiai

biRT: ð5:107Þ

In deriving eq 5.107, the pressure dependence of the excess Gibbs function ofmixing (which is why activity-coefficient parameters had to be re-correlatedwhen using the Huron-Vidal model) has been ignored. The assumption is thatGEm(f) of the Huron-Vidal model, which is evaluated at infinite pressure, and

GEm (f) of the Michelsen model, which is evaluated at zero pressure, are iden-

tical. They have shown that use of the UNIFAC in the mixing rule, with anappropriate choice of the additional parameter l (0.36 for the original UNI-FAC model, and in the range from 0.65 to 0.75 for the modified UNIFACmodel), led to reasonable predictions of phase boundaries for many mixtures.

5.3.4.5 Universal Mixing Rule for GEm

Voutas et al.104 have proposed a mixing rule that is universal for all cubicequations of state with

aðxÞ ¼ 1

E

GEmð f ;T ; p ¼ 0Þ

RTþXCi¼1

xiai; ð5:108Þ

and eq 5.14 for b(x) where bij is given by

bij ¼b1=2i þ b

1=2i

2

!s

: ð5:109Þ

In eq 5.108 E is a coefficient that depends on the cubic equation of state usedand in eq 5.109 s¼ 2. The GE

m(f,T,p¼ 0) is given by

GEmðf ;T ; p ¼ 0Þ ¼ 5

XCi¼1

xiqi ln

qi

, PCj¼1

qj

!( )

rPi

, PCj¼1

xjri

!( )266664

377775þ GE

m;r ð5:110Þ

where r and q are the van der Waals volume and area of molecule i, respectively.The first term on the right hand side of eq 5.110 is the Staverman-Guggenheimcontribution to GE

m and GEm,r is the residual part of the UNIFAC molar Gibbs

function. When eqs 5.108 through 5.110 were combined with the cubic equationof state reported by Magoulas and Tassios,105 which was based on the

110 Chapter 5

Peng-Robinson, prediction of the properties of mixtures such as (hex-aneþ hexadecane) is obtained with reasonable precision when compared withexperiment.For any of the methods to be useful requires values of Tc, pc and o (see

Chapter 3 and Section 5.3.2) to estimate a and b for the pure compounds.Values of Tc, pc and o can be obtained from the American Institute of Che-mical Engineers Design Institute for Physical Properties DIPPR106 or otherhandbooks107 or estimated from sources such as refs 108, 109 and 110. Thetemperature dependence of a is given by Mathias and Copeman29 for thePSRK, MHV2 and LCVM while for Wong-Sandler, the Stryjek and Vera31

method is used for the Peng-Robinson equation of state. Estimates of thevapour pressure can be obtained, for example, from ref 108, 110, 111 and 112.An alternative review of excess Gibbs mixing function and combining rules is

provided by Voutsas et al.113 and it is clear that each cubic equation of state haslimitations.

5.4 Multi-Parameter Equations of State

5.4.1 Benedict, Webb, and Rubin Equation of State

The virial equation of state given by eq 5.1 applies to gases and has beendiscussed in Chapter 3. The composition dependence for the second andthird virial coefficients are obtained from statistical mechanics and givenby eqs 5.3 and 5.4. Consequently, the virial equation has formed the basisfor the development of other semi-empirical equations of state capable ofcorrelating both (p, V, T) and phase behaviour; some approaches arediscussed in Chapter 12. One example of this form of equation is the Benedict,Webb and Rubin114 (known by the acronym BWR) equation of stategiven by:

Z ¼1þB0 � A0= RTð Þ � C0= RTð Þ3h i

Vmþ b� a= RTð Þ½ �

V2m

þ aaRTþ c

RT3V2m

1þ gV2

m

� �exp � g

V2m

� �;

ð5:111Þ

that was originally developed to represent the (p, V, T) data of methane, ethane,propane, and butane and provide estimates of density, enthalpy, fugacity andvapour pressure; equation 5.111 was extended to include eight hydrocarbons upto heptane.115 In eq 5.111, the a, b, c, A0, B0, C0, a and g are parametersadjusted to represent the available experimental measurements; in this sectionthe g is not the activity coefficient introduced in section 5.3.4.1. The BWRequation is considered a closed form of the virial equation because of theexponential term that can be expanded as an infinite series in reciprocal molarvolume. The exponential term makes a large contribution to the equation of


state at high density and in the critical region. Expanding the BWRequation gives

Z ¼ 1þB0 � A0= RTð Þ � C0= RTð Þ3h i

Vmþ

b� a= RTð Þ þ c�

RT3� ��

V2m

� gRT3V4

m

þ � � � : ð5:112Þ

The BWR equation has been used to calculate the thermodynamic properties ofmixtures based on the idea that both the mixture and the pure-fluid equationsshould satisfy the same equation of state and provide the correct compositiondependence of as many virial coefficients as possible. This was achieved withmixing rules similar to those obtained from the van der Waals one-fluid theorythat are as follows:116

A0 ¼XCi¼0

XCj¼0

xixjA0;ij ; ð5:113Þ

B0 ¼XCi¼0

XCj¼0

xixjB0;ij ; ð5:114Þ

C0 ¼XCi¼0

XCj¼0

xixjC0;ij ; ð5:115Þ

a ¼XCi¼1

XCj¼1

XCk¼1


b ¼XCi¼1

XCj¼1

XCk¼1

xixjxkbijk; ð5:117Þ

c ¼XCi¼1

XCj¼1

XCk¼1

xixjxkcijk; ð5:118Þ

a ¼XCi¼1

XCj¼1

XCk¼1


and

g ¼XCi¼0

XCj¼0

xixjgij : ð5:120Þ

112 Chapter 5

Equation 5.114 has been, depending on the requirements, replaced by the linearform

B0 ¼XCi¼0

xiB0;i: ð5:121Þ

Several different expressions were used for the combining rules, but all arerelated to the eqs 5.26 and 5.27 and are for a, b, c and a of the form:

aijk ¼ aiajak� �1=3

; ð5:122Þ

while the combining rules for A0 C0 and g have the form

A0;ij ¼ A0;iA0;j

� �1=2; ð5:123Þ

and for B0 the function

B0;ij ¼ B0;ið Þ1=3þ B0;j

� �1=3n o3�

8 ð5:124Þ

is used. Equations 5.114 through 5.124 contain no adjustable parametersand have been found to provide satisfactory representation of the propertiesof mixtures formed from components with molar mass lower than about0.1 kg �mol�1, but the BWR equation with the mixing and combining rules isunable to adequately correlate data for mixtures containing non-hydrocarbonsand of higher molar mass hydrocarbons. In an attempt to overcome thisdeficiency, Stotler and Benedict117 proposed the introduction of a singleinteraction parameter mij (which is similar to the interaction parameter usedwith cubic equations of state) in eq 5.113 to give

A0 ¼XCi¼1

XCj¼1

xixj A0;iA0;j

� �1=2mij : ð5:125Þ

Equation 5.125 was used by Orye118 for hydrocarbon mixtures and by Nohkaet al.119 for mixtures of refrigerants with nitrogen, argon and methane.Many modifications of the BWR equation have been proposed, most by

increasing the number of terms and thus increasing the number of adjustableparameters. One of the first such modifications was by Strobridge120

p ¼ RTrþ C1RT þ C2 þC3

Tþ C4

T2þ C5

T4

� �r2 þ C6RT þ C7ð Þr3

þ C8Tr4 þ C15r6

þ C9

T2þ C10

T3þ C11

T4

� �þ C12

T2þ C13

T3þ C14

T4

� �r2

� r3 exp �gr2

� �;

ð5:126Þ


where r is density and the Ci are parameters of the equation of state. Equation5.126 has 16 adjustable parameters (often unfortunately also known as constants).Equation 5.126 is also the basis for more recent modifications of the BWRincluding those reported by Bender121 that has 20 parameters adjusted to best fitavailable data; Morsy122 that used 10 adjustable parameters; Starling123 with 11adjustable parameters; Jacobsen and Stewart124 with 32 adjustable parameters; Leeand Kesler125 with 12 adjustable parameters; Nishiumi and Saito126 with 15adjustable parameters; Schmidt andWagner127 that used 32 adjustable parameters;and the AGA Natural Gas Equation Number 8 provided by Starling128 with 53adjustable parameters. Each of these equations used mixing and combining rulessimilar to eqs 5.113 to 5.124. Not surprisingly, for pure fluids, increasing thenumber of adjustable parameters decreases the differences between calculated andexperimentally determined thermodynamic properties. The reader is referred toChapter 12 and ref 129 for additional details for this class of equations of state.The source and uncertainty of the measurements used to determine the

coefficients of eq 5.111 for each substance results in many different values ofthe parameters reported by different authors for the same substance; varyingthe weighting of each measurement in the regression analysis has the sameresult. Equations 5.113 through 5.124 show how the coefficients of the puresubstances are used to determine the parameters for a mixture, and this isparticularly difficult to do when the parameters obtained differ considerably inmagnitude between components and from alternate sources for the samecomponent. This matter will be discussed shortly.

5.4.2 Generalization with the Acentric Factor

The non-uniqueness of the parameters of extended virial equations can be over-come by application of the concepts introduced by the principle of correspondingstates discussed in Chapter 6 and replacing temperature and density with reducedproperties that are either universal constants or generalized in terms of some fluidproperty, such as, the acentric factor o (see Chapter 3). Joffe130 generalized theeight parameters of the BWR equation by following the suggestion of Kamer-lingh Onnes131 and expressing the reduced pressure (pr¼ p/pc) as a function ofreduced temperature (Tr¼T/Tc) and ideal reduced volume {pcVc/(RTc)} witheight constants. Opfell et al.132 used the BWR equation in terms of reducedpressure and reduced temperature, and represented each parameter except g as alinear function of o. Edmister et al.133 extended this concept and used quadraticfunctions ofo for seven of the eight parameters and required the product of a anda be constant. These relatively minor changes improved the ability of the equationto correlate measured thermodynamic properties. Starling and Han134 provide amore extensive generalization of Starling’s modified BWR equation given by

p ¼ RTrþ B0RT � A0 �C0

T2þD0

T3� E0

T4

� �r2 þ bRT � a� d

T

� �r3

þ a aþ d

T

� �r6 þ cr3

T21þ gr2� �

exp �gr2� �

:

ð5:127Þ

114 Chapter 5

In eq 5.127, the A0, B0, C0, D0, E0, a, b, c, d, a and g are, with the exception of E0,each a linear function of o and the adjustable parameters; to give a total of 22parameters. For a mixture of C components the relationships are as follows:134

A0 ¼ RXCi¼1

Tc;i A1 þ B1oið Þ�rc;i; ð5:128Þ

B0 ¼XCi¼1

A2 þ B2oið Þ�rc;i; ð5:129Þ

C0 ¼ RXCi¼1

T3c;i A3 þ B3oið Þ

�rc;i; ð5:130Þ

D0 ¼ RXCi¼1


.r2c;i; ð5:131Þ

E0 ¼ RXCi¼1

T5c;i A5 þ B5oi exp �3:8oið Þf g

�rc;i; ð5:132Þ

a ¼ RXCi¼1

Tc;i A6 þ B6oið Þ.r2c;i; ð5:133Þ

b ¼XCi¼1

A7 þ B7oið Þ.r2c;i; ð5:134Þ

c ¼ RXCi¼1


.r2c;i; ð5:135Þ

d ¼ RXCi¼1


.r2c;i; ð5:136Þ

a ¼XCi¼1

A10 þ B10oið Þ.r3c;i ð5:137Þ

and

g ¼XCi¼1

A11 þ B11oið Þ.r2c;i: ð5:138Þ

The equation of state of Starling and Han134 used the mixing and combining rulesof eqs 5.113 to 5.124 except for the following

A0 ¼XCi¼1

XCj¼1

xixj A0;iA0;j

� �1=21� kij� �

; ð5:139Þ


C0 ¼XCi¼1

XCj¼1

xixj C0;iC0;j

� �1=21� kij� �3

; ð5:140Þ

D0 ¼XCi¼1

XCj¼1

xixj D0;iD0;j

� �1=21� kij� �4

; ð5:141Þ

E0 ¼XCi¼1

XCj¼1

xixj E0;iE0;j

� �1=21� kij� �5

; ð5:142Þ

and

d ¼XCi¼1

xid1=3i

!3

: ð5:143Þ

Unfortunately, equations of state with this generalized form provide neither themolar volume (or the compression factor) nor other properties that are linearfunctions of the acentric factor as observed experimentally.Pitzer and co-workers135–137 and Curl and Pitzer138 developed a correlation

for the compression factor for hydrocarbons and inorganic gases in terms of thereduced pressure and reduced temperature that is a linear function of acentricfactor and is of the form

Z Tr; pr;oð Þ ¼ Z0 Tr; prð Þ þ oZ1 Tr; prð Þ; ð5:144Þ

where Z0 is the compressibility factor of a fluid for which o¼ 0 (for example, anoble gas) and Z1 a departure function. The Z0 and Z1 are functions of onlyreduced pressure and reduced temperature. Lee and Kesler125 expanded thisidea and recommended

Z Tr; pr;oð Þ ¼ Z0 Tr; prð Þ þ oZr Tr; prð Þ � Z0 Tr; prð Þ

or; ð5:145Þ

where Z0 and Zr are the compression factors for two reference fluids with o¼ 0and or¼ 0.3978 (essentially octane) at the same reduced conditions. Lee andKesler125 used the reduced form of the modified BWR equation of state torepresent both Z0 and Zr with:

Z ¼ prVr

Tr

� �¼ 1þ B

Vrþ C

V2r

þ C

V5r

þ c4

T3r V

2r

bþ gV2

r

� �exp � g

V2r

� �; ð5:146Þ

where

Vr ¼Vmpc

RTc; ð5:147Þ

116 Chapter 5

B ¼ b1 �b2

Tr� b3

T2r

� b4

T3r

; ð5:148Þ

C ¼ c1 �c2

Trþ c3

T3r

ð5:149Þ

and

D ¼ d1 þd2

Tr: ð5:150Þ

Lee and Kesler125 used mixing rules for the effective critical parameters of

Tc ¼1

Vc

XCi¼1

XCj¼1

xixjVc;i;jTc;i;j ; ð5:151Þ

Vc ¼XCi¼1

XCj¼1

xixjVc;i;j ð5:152Þ

and

o ¼XCi¼1

xioi; ð5:153Þ

with combining rules of

Vc;i;j ¼V

1=3c;i þ V

1=3c;j

2

!3

; ð5:154Þ

Vc;i ¼ZcRTc;i

pc;i; ð5:155Þ

Tc;i;j ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiTc;iTc;j

p; ð5:156Þ

Zc;i ¼ 0:2905� 0:085oi ð5:157Þ

and

pc;i ¼0:2905� 0:085oið ÞRTc

Vc;i: ð5:158Þ

The mixing and combining rules of eqs 5.151 through 5.158 provide theparameters for the mixture from those of the pure substance in terms of the


critical temperature, critical pressure, and acentric factor. This is in contrast toall other equations presented so far for which the mixing rules operate on theequation of state parameters for the pure substances to obtain the parametersrequired for the mixture. To apply the Lee-Kesler equation to vapour-liquidequilibria, Joffe139 proposed the addition of an interaction parameter toeqs 5.151 through 5.153, which Ploecker et al.140 and Oellrich et al.141 did aswell by modifying the mixing rule for Tc to

Tc ¼1

V1=4c

Xni¼1

Xnj¼1

xixjV1=4c;i;j Tc;i;j ; ð5:159Þ

where

Tc;i;j ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiTc;iTc;j

pki;j : ð5:160Þ

Mixing rules for the equation of state parameters and the effective criticalproperties are the only two used for the extended virial equations of state.Unfortunately, these mixing and combining rules are only suitable for mixturesof hydrocarbons and for hydrocarbons with inorganic gases. Extension of theextended virial equations of state to non-ideal mixtures can presumably beaccommodated with different mixing and combining rules; a matter to beaddressed by further research.

5.4.3 Helmholtz-Function Equations of State

Chapter 12 describes Helmholtz-function based multi-parameter equations ofstate and provides the mixing and combining rules used to extend the equationsfor pure fluids to mixtures. In this approach, the Helmholtz function equationsof state for the components in a mixture are combined with an excess functionto effectively account for the interactions between unlike molecules.142–146 Ofparticular importance to this field is the work of Kunz et al.147,148 that provideda Helmholtz-function equation of state to predict the thermodynamic proper-ties of mixtures with chemical composition found for natural gas. This equationof state is known by the acronym GERG-2004; this research was sponsored bymembers of Groupe Europeen de Recherches Gazieres.The molar Helmholtz function for mixtures Am is assumed to be the sum of

contributions from the ideal mixture Aidm and an excess AE

m given by eq 12.59:

Am r;T ; xð Þ ¼ Aidm r;T ;xð Þ þ AE

m r;T ;xð Þ: ð5:161Þ

The Helmholtz function for the ideal mixture is represented by

Aidm ¼

Xni¼1

xi Apgm;i rn;Tð Þ þ Ar

m;i d; tð Þ þ RT ln xi

h i; ð5:162Þ

118 Chapter 5

where, C is the number of components in the mixture, Apgm,i is the ideal-gas

Helmholtz function for component i, and Ari is the pure-fluid residual Helm-

holtz energy of component i evaluated at a reduced density and temperature.The term AE

m of eq 5.161 is given by

AEm

RT¼ aE ¼

XC�1i¼1

XCj¼iþ1

xixjFij

�XKpol

k¼1Nkd

dkttk þXKpolþKexp

k¼Kpolþ1Nkd

dkttk exp �Zk d� �kð Þ2�bk d� gkð Þ� �2

435;

ð5:163Þ

where the coefficients and exponents are determined by nonlinear regression tomeasured values of the properties of mixtures. For binary mixtures for whichthere are adequate experimental measurements, for example, (metha-neþ ethane), the parameters of eq 5.163 are adjusted to provide an equationthat is specific to the mixture. For other mixtures, particularly those binarymixtures for which there are insufficient experimental data, the parameter Fij ofeq 5.163 is used to relate AE

m of one binary mixture to that of another, and thisformalism permits use of the same coefficients.All single-phase thermodynamic properties can be calculated from the

Helmholtz energy as described in Chapter 12 Section 12.5 with eqs 12.62 and12.63 that are:

a0 ¼XCi¼1

xiA0

i rn;Tð ÞRT

þ ln xi

� ð5:164Þ

and

ar ¼XCi¼1

xiari d; tð Þ þ aE d; t;xð Þ; ð5:165Þ

where the derivatives are taken at constant composition. The reduced values ofdensity and temperature for the mixture are given by

d ¼ r=rr xð Þ ð5:166Þ

and

t ¼ Tr xð Þ=T ; ð5:167Þ

where r and T are the density and temperature of the mixture, and rr(x) andTr(x) are given by:

1

rr xð Þ¼XCi¼1

XCj¼1

xixjbVm;ijgVm ;ij

xi þ xj

b2Vm ;ijxi þ xj

1

8

1

r1=3c;i

þ 1

r1=3c;j

!3

ð5:168Þ


and

Tr xð Þ ¼XCi¼1

XCj¼1

xixjbT ;ijgT ;ijxi þ xj

b2T ;ijxi þ xj

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiTc;iTc;j

p; ð5:169Þ

that are quadratic in mole fraction x. The four adjustable parameters bVm;ij ,bT,ij, gVm;ij and gT,ij of eqs 5.168 and 5.169, which are determined simultaneouslywith the other parameters in the nonlinear regression to the available experi-mental measurements, permit arbitrary symmetry for the reducing functions;these parameters are not the critical properties of the mixture. The asymmetriccomposition dependence is based on the excess Gibbs function of

GEm ¼ x1x2

AB

Ax1 þ Bx2: ð5:170Þ

To ensure mole-fraction symmetry the following conditions are imposed:

gVm;ij ¼ gVm ;ji; gT ;ij ¼ gT ;ji;bVm;ij ¼ 1�bVm;ji and bT ;ij ¼ 1

�bT ;ji: ð5:171Þ

The numerator xiþ xj of eqs 5.168 and 5.169 is important solely for multi-component mixtures (as in binary mixtures these sum to one). Setting para-meters b and g equal to unity reduces eqs 5.168 and 5.169 to the Lorentz-Berthelot combining rules of eqs 5.26 and 5.27 for the effective critical para-meters and provide, in the absence of measurements other than the criticalproperties of the pure components, a means of estimating the properties of themixture. At least for binary mixtures (1� x)A þ xB formed from componentsof natural gas it was concluded in ref. 148 that the linear rule

Tc;AB ¼1

2Tc;A þ Tc;Bð Þ; ð5:172Þ

was preferred. Combining eq 5.172 with eq 5.169 for the case of b¼ g¼ 1 resultsin

Tc;AB ¼ xATc;A þ xBTc;B; ð5:173Þ

The parameters gVm;ij and gT,ij can be estimated with bVm;ij ¼ bT ;;ij ¼ 1 from

gVm;ij ¼ 4r�1c;i þ r�1c;j

� �r�1=3c;i þ r�1=3c;j

� �3 ð5:174Þ

and

gT ;ij ¼1

2

Tc;i þ Tc;j

� �Tc;iTc;j

� �0:5 : ð5:175Þ

120 Chapter 5

The suitability of a mixing function based on the analysis of binary mixtures foruse with multi-component mixture was determined by the methods reported byMichelsen and Kistenmacher,75 Mathias et al.149 and Avlonitis et al.150 Basedon the work Mathias et al.149 an alternative set of mixing rules

Tr xð Þ ¼XCi¼1

XCj¼1

xixjfT ;ijTc;ij þXCi¼1

xiXCj¼1

xjl1=3T ;ijT

1=3c;ij

!3

ð5:176Þ

and

Vm;r xð Þ ¼XCi¼1

XCj¼1

xixjfVm ;ijVm;ij þXCi¼1

xiXCj¼1

xjl1=3Vm;ij

V1=3m;ij

!3

ð5:177Þ

were proposed in ref. 148.

5.5 Mixing Rules for Hard Spheres and Association

Theory and computer simulation provides information in addition to the virialequation of state that can be used to develop mixing and combining rules.The equation of state for the pure hard–sphere fluid can be represented by theequation of state of Carnahan-Starling:151

p ¼ RT

Vm

1þ Zþ Z2 � Z3

1� Zð Þ3

" #; ð5:178Þ

where Z¼ prs3/6, s is the hard-sphere diameter and r the density. For mixturesof hard spheres of different diameters the expression given by Mansooriet al.152,153 is commonly used

p ¼ RT

Vm

6

pZ0

1� Z3þ 3Z1Z2

1� Z3ð Þ2þ 3Z32

1� Z3ð Þ3þ 3Z3Z

32

1� Z3ð Þ3

" #; ð5:179Þ

where Zi ¼ p=6ð ÞPi

risji . The combining rules for each Zi are different owing to

the power to which s is raised, and there are no binary interaction parameters(such as eq 5.30) for the b in the van der Waals equation of eq 5.7. Equation5.179 has the same density dependence as eq 5.178 and when all componentsare identical eq 5.179 reduces to eq 5.178.To match the results of molecular-dynamics computer simulations for

square-well molecules, Alder et al.154 used a double-power-series expansion inreduced inverse temperature and volume for the attractive part of an equationof state. Modifications of their attractive term have been used in a number ofequations of state, such as the augmented and perturbed-hard-sphere equa-tions, the perturbed-hard-chain equation, and the BACK equations of state.155


The general form of this attractive term is

p attrð Þ ¼ �Xn

Xm

Anmu

kT

� �n V0

V

� �m

; ð5:180Þ

where the Amn are adjustable parameters, V0 is the close-packed volume and u isan effective well depth, both of which depend upon temperature when theseequations are used to describe real fluids. The mixing rules commonly used inthis case are

V0 ¼Xi

Xj

xixjV0;i;j ð5:181Þ

and

u ¼Xi

Xj

xixjui;j ð5:182Þ

with the combining rules

V0;i;j ¼ 1� li;j� � V

1=3ii þ V

1=3jj

2

!3

ð5:183Þ

and

u ¼ ffiffiffiffiffiffiffiffiffiffiuiiujjp

1� kij� �

: ð5:184Þ

In all the equations that include the double power-series expansion (or varia-tions thereof), such as the family of perturbed-hard-chain equations, theparameters are related to molecular rather than critical properties, and themixing and combining rules are quadratic in composition for the attractiveterm and based on hard-sphere theory for the repulsive term.Molecules are not hard spheres and other equations of the same general form

as eq 5.179 have been proposed for molecules of various geometrical shapes.For example, hard convex bodies Boublik156 gives

pV

RT¼ Z ¼

1þ 3a� 2ð Þxþ 3a2 � 3aþ 1� �

x2 � a2x3

1� xð Þ3; ð5:185Þ

where a is the surface integral of the radius of curvature divided by three timesthe molecular volume, and

x ¼ pffiffiffi2p

V0

6V; ð5:186Þ

122 Chapter 5

where V0 is the hard-core volume. However, there is no obvious or theoreticallybased mixing rule to extend this equation to mixtures of non-sphericalmolecules.

5.5.1 Mixing and Combining Rules for SAFT

The last type of equation of state that we will consider is the Statistical Asso-ciating-Fluid Theory (SAFT) first proposed by Chapman et al.157 and Huangand Radosz,158 and for which there are now many variants as discussed inChapter 8.In the SAFT model, a molecule is considered to consist of a collection of

segments, and the Helmholtz function is written as:

A ¼ Aid þ Aseg þ Achain þ Aassoc; ð5:187Þ

which is a sum of contributions from the ideal gas, Aid, the intermolecularinteractions between segments, Aseg, the formation of chains from segments,Achain, and association due to hydrogen bonding and donor-acceptor interac-tions, Aassoc. The sum of the Carnahan-Starling and Alder et al.154 expressionsfor interactions among spheres is used for the segment term Aseg. The first-order perturbation expression of Wertheim159–163 is used to account for asso-ciation Aassoc and, based on the work of Chapman,164 also for chain formationAchain. The segment Helmholtz energy per mole of molecules, Aseg, is

Aseg ¼ rAseg0 ¼ r Ahs

0 þ Adisp0

� �; ð5:188Þ

where r is the number of segments per chain, and the subscript 0 refers to aproperty of a single segment. The segment properties are further assumed to bea combination of a hard-sphere term represented by the Helmholtz functionderived from the Carnahan-Starling expression and a dispersion term givenby the expression of Alder et al.154 with parameters modified by Chen andKreglewski.155 The contribution to the molar Helmholtz function for chainformation comes from an expression due to Chapman164 and Chapmanet al.157 that are based on Wertheim’s first-order perturbation theory159–163

Achain

RT¼ 1� rð Þ ln 2� rZ

2 1� rZð Þ3

" #: ð5:189Þ

The association term from Wertheim’s theory is given by

Aassoc

RT¼XA

lnXA � XA

2

� þM

2; ð5:190Þ


where the sum is over all association sites, M is the number of association siteson each molecule and XA is the mole fraction of molecules which are notbonded at site A.The application of this equation of state to mixtures requires the replacement

of the Carnahan-Starling term with the expression reported by Mansoori etal.,165 and more complicated chain and association terms. In the SAFT equa-tion, and other equations of this type, each of the terms has its own theoreti-cally-based mixing rule that is different from the mixing rules for other terms inthe same equation. For example, the mean attractive energy associated with thefirst–order perturbation is treated by Galindo et al.166 SAFT and the relatedmethods can be considered molecular-based equations of state for associatingfluids. SAFT is reviewed in Chapter 8 and by Muller and Gubbins.167

For mixtures,168,169 only the dispersion part of the segment Helmholtz energyrequires the use of combining rules; the composition dependence is built intothe chain and association terms by the statistical thermodynamics. Gross andSadowski170 presented a new parameterization of SAFT with constants for awide range of substances that does not suffer from the numerical incon-sistencies171 of the Huang-Radosz model.168,169

Many reports of using the SAFT equation of state for mixtures are based onthe use of eqs 5.26 and 5.27, the Lorentz-Berthelot combining rules, that areknown to fail for non-ideal mixtures.44,45,172 An adjustable parameter kij hasbeen included, as it was with eq 5.27, to improve the representation of the phaseboundaries; the value of kij is specific to each mixture and is typically deter-mined by a fit to experimental data. Alternative combining rules have beeninvestigated by Schnabel et al.48 who concluded that the combining rulereported by Hudson and McCoubrey49 (and given by eq 5.34) was preferred.Haslam et al.43 have generalized eq 5.34 so that it applies to potentials otherthan the Lennard-Jones potential and in particular to the square-well potentialas used in SAFT-VR that is discussed further in Chapter 8. Haslam et al.43

provided correlations for the ionization potential of pure compound required ineq 5.34 and proposed a correlation between the molecular ionization potentialsand the model chain length for non-spherical molecules.A form analogous to eq 5.37 was used to predict the kij by Haslem et al.43 and

these values were used with SAFT-VR equation of state to predict the phaseboundaries of (CH4þC8H18) with surprisingly good results when comparedwith experimental data using kij¼ 0. For (CF4þC4H10) the SAFT-VR withpredicted kij was able to correctly predict the phase behaviour.43 Huynh etal.173,174 have used the form proposed in ref 43 to predict binary parametersbased on the pseudo-ionization energy of functional groups in a group-con-tribution method (see Chapter 8).The molecular basis of the SAFT equation of state, makes it possible to use

the interaction parameters between the same molecular pairs within a homo-logous series. This idea has been used by others to predict the phase behaviourof (alkanesþwater),175–177 (alkanesþ perfluoroalkanes),178 (alkanesþ carbondioxide),179,180 (alkanesþ hydrogen chloride),181 (alkanesþ nitrogen)182 andaqueous solutions of surfactants.183–185

124 Chapter 5

5.5.2 Cubic Plus Association Equation of State

The cubic plus association186 equation of state, discussed in Chapter 4, is acombination of the Soave-Redlich-Kwong equation with first-order perturba-tion theory for associating fluids that is essentially the same as in the SAFTequation, and is given by Michelsen and Hendricks.187 The cubic plus asso-ciation equation of state has been reviewed in refs 188, 189 and 190. Because ofthe association term, similar to that used for SAFT, the equation is no longercubic. The cubic plus association equation of state is as follows:

Z ¼ Vm

Vm � b� aðTÞRT Vm � bð Þ �

1

21þ r

d ln g

dr

� �Xi¼1

xiXAi

1� XAið Þ; ð5:191Þ

where g is the radial distribution function at contact, xi the mole fraction ofcomponent i and XAi

the mole fraction of molecule i non bonded at site A, thatis the monomer mole fraction, given by

XAi¼ 1þ r

Xj

xjXBj

XBjDAiBj

0@

1A: ð5:192Þ

The radial distribution function g for the reference fluid has been approximatedby188,189

g rð Þ ¼ 1� 1:9br=4f g�1: ð5:193Þ

The term DAiBj in eq 5.192 is the association strength between site A onmolecule i and site B on molecule j given by:

DAiBj ¼ g rð Þ exp�AiBj

RT

� �� 1

�bijb

AiBj ; ð5:194Þ

where �AiBj and bAiBj are the cross-association energy and volume, respectively,between site A on molecule i and site B on molecule j.The cubic association equation requires mixing rules for the a and b of the

SRK and those typically used are eq 5.13 and eq 5.15 with combining rulesgiven by eqs 5.31 and 5.32 with bAB¼ 1. Combining rules for the associationterms are only required for the cross-association energy �AiBj and volume bAiBj

Suresh and Beckman191 used

�AiBj ¼ �Ai þ �Bj� �1=2

and bAiBj ¼ aij bAi þ bBj� �1=2 ð5:195Þ

where aij is a parameters to adjust for the imperfect agreement betweenexperimental and predicted phase boundaries. Fu and Sandler192 recommendedfor a simplified form of SAFT


�AiBj ¼ �Ai þ �Bj� �1=2

and bAiBj ¼ bAi þ bBj� ��

2 ð5:196Þ

with no adjustable parameters. Voustsas et al.193 considered the rules

�AiBj ¼ �Ai þ �Bj� ��

2 and bAiBj ¼ bAi þ bBj� ��

2; ð5:197Þ

�AiBj ¼ �Ai þ �Bj� ��

2 and bAiBj ¼ bAi þ bBj� �1=2 ð5:198Þ

and

�AiBj ¼ �Ai þ �Bj� �1=2

and bAiBj ¼ bAi þ bBj� �1=2

: ð5:199Þ

Equations 5.196 through 5.199 are obtained from eqs 5.27 and 5.26.For the Elliott-Suresh-Donohue equation,194 which will not be discussed

further here, the combining rules for DAiBj of eq 5.192 were proposed by Sureshand Elliott195 as

DAiBj ¼ DBiAj ¼ DAiDBj� �1=2 ð5:200Þ

and an alternative modified rule196 of

DAiBj ¼ DBiAj ¼ 1� kij� �

DAiDBj� �1=2

: ð5:201Þ

Combining rules given by eqs 5.200 and 5.201 have been used with the cubicplus association equation. Wolbach and Sandler197–199 provided ab initio jus-tification for eq 5.200. For (methanolþwater) Koh et al.200 also validated theuse of either eq 5.198 or 5.200. For (vapourþ liquid), (liquidþ liquid) and(solidþ liquid) equilibria ref 189 lists the combining rules preferred for 18cross-associating mixtures.Derawi et al.201,202 found eqs 5.197 and 5.199 were preferred for the optimal

representation of the phase boundaries for (methanolþwater), (ethane-1,2-diolþwaer), {(2-hydroxyethoxy)ethan-2-ol þwater} and (2-[2-(2-hydroxyethoxy)ethoxy]ethanol þwater); ref. 201 defines the acronyms for the combining rulesused in the literature. The phase boundaries of (waterþ ethanolþCO2) wereconsidered in the context of the cubic plus association equation by Perakiset al.203 with combining rules given by eqs 5.198 and eqs 5.199 where theexpression for �AiBj included an additional interaction parameter lij as a mul-tiplying factor (1� lij); the parameters kij and lij are adjusted to reproduce thephase equilibrium of binary mixtures.The cubic plus association equation has been applied to oil reservoir

hydrocarbons with the ubiquitous water using the combining rule given by eq5.200.204 The phase equilibria of (waterþCO2) is important for carbonsequestration in geological formations and requires methods to predict thephase boundaries and solubility. Pappa et al.205 estimated the phase boundaries

126 Chapter 5

for (waterþCO2) using the following methods: (1) cubic plus associationequation formed from the Peng-Robinson using combining rules given by eq5.199 where �AiBj included the factor (1� lij); (2) the Peng-Robinson withmixing rules given by eqs 5.13 and 5.14 and combining rules eqs 5.31 and 5.32;and (3) the Peng-Robinson equation with universal mixing rule for GE

m of eq5.108 and eq 5.14 for b(x) where bij is given by eq 5.109. Option 1 was found tobest represent the measured phase boundaries.For mixtures of hydrocarbons and alcohols, often involved in hydrate

inhibition, the cubic plus association equation of state has been used to estimatethe (vapourþ liquid)206 and (liquidþ liquid)207 equilibria. In both cases nocombining rules are needed for the association parameters �AiBj and bAiBj for(alcoholþ hydrocarbon).

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2794.196. S. J. Suresh and J. R. Elliott Jr., Ind. Eng. Chem. Res., 1991, 30, 524–532.197. J. P. Wolbach and S. I. Sandler, AIChE J., 1997, 43, 1589–1596.198. J. P. Wolbach and S. I. Sandler, AIChE J., 1997, 43, 1597–1604.199. J. P. Wolbach and S. I. Sandler, Ind. Eng. Chem. Res., 1997, 36,

4041–4051.200. C. A. Koh, H. Tanaka, J. M. Walsh, K. E. Gubbins and J. Zollweg, Fluid

Phase Equilib., 1993, 83, 51.201. S. O. Derawi, G. M. Kontogeorgis, M. L. Michelsen and E. H. Stenby,

Ind. Eng. Chem. Res., 2003, 42, 1470–1477.202. S. O. Derawi, M. L. Michelsen, G. M. Kontogeorgis and E. H. Stenby,

Fluid Phase Equilib., 2003, 209, 163–184.203. C. Perakis, E. Voutsas, K. Magoulas and D. Tassios, Fluid Phase Equilib,

2006, 243, 142–150.204. W. Yan, G. M. Kontogeorgis and E. H. Stenby, Fluid Phase Equilib.,

2009, 276, 75–85.205. G. D. Pappa, C. Perakis, I. N. Tsimpanogiannis and E. C. Voutsas, Fluid

Phase Equilib., 2009, 284, 56–63.206. I. V. Yakoumis, G. M. Kontogeorgis, E. C. Voutsas and D. P. Tassios,

Fluid Phase Equilib., 1997, 130, 31–47.207. I. V. Yakoumis, G. M. Kontogeorgis, E. C. Voutsas and D. P. Tassios,

Fluid Phase Equilib., 1997, 132, 61–75.

134 Chapter 5

CHAPTER 6

The Corresponding-StatesPrinciple

JAMES F. ELY

Chemical Engineering Department, Colorado School of Mines, Golden, CO80401-1887, USA

6.1 Introduction

During the past years, there have been great advances in the theory of densefluids and in the application of these theories to complex molecular systems.These advances, which include both integral-equation and statistical-mechan-ical perturbation theory, have been brought about primarily by the advent offaster and cheaper computers. As pointed out by Kreglewski,1 however, theseresults seem hopelessly complex for a chemical engineer who is generallyinterested in simple and practical solutions. With this complexity in mind, themost powerful tool available today (just as 35 years ago) for making highlyaccurate, yet mathematically simple, predictions of the thermophysical prop-erties of fluids and fluid mixtures is the corresponding-states principle. Thepower of the corresponding-states principle is that it allows the prediction offluid properties with a minimum amount of information for the system ofinterest, given a detailed knowledge of few reference systems. The principle iswell founded in molecular theory but certainly is not new. Its fundamentals andapplications to pure-fluids and mixtures have been reviewed in almost all of therecently published thermodynamic and statistical mechanics books.2–5 As dis-cussed by Leland and Chappelear6 in their review of the corresponding-statesprinciple more than 30 years ago, the basic concept of corresponding states is toapply dimensional analysis to the configurational portion of the statistical




135

mechanical partition function. The end result of this analysis is the expressionof residual thermodynamic properties in terms of dimensionless groups. On anempirical basis, the corresponding-states principle was originally proposed byvan der Waals who observed that the reduced form of his equation of statecould be written for all fluids. All modern generalized engineering equations ofstate are examples of applications of this principle.The original, two-parameter corresponding-states principle leads to an

equation of state which expresses the residual compressibility factor (or com-pression factor) Zr in terms of a universal function of the dimensionless tem-perature and molar volume (or density):

Zr � pVm

RT� 1 ¼ FðV�;T�Þ; ð6:1Þ

where p is the pressure, R the gas constant and V* and T* are the dimensionlessvolume and temperature, respectively. Starting from a molecular basis V*

would be identical to V/Ns3 where s is the intermolecular-potential distanceparameter and T* would be given by kBT/e where kB is Boltzmann’s constantand e is the intermolecular-potential well depth. If one invokes the stabilitycriteria for a pure-fluid critical point, the dimensionless volume and tempera-ture would be given by V/Vc and T/Tc, respectively, where the superscript cdenotes a value at the critical point. We note that this two-parameter corre-sponding-states principle can be applied to any polynomial equation of statewhich has a liquid-vapour critical point.6

Equation 6.1 implies all substances obey the same reduced equation of stateand we can make a slight transformation of this result to relate directly theproperties of one fluid to another. For two fluids j and 0 which obey the simplecorresponding-states principle we can write from eq 6.1

Zrj ðV�;T�Þ ¼ Zr

0ðV�;T�Þ ¼ FðV�;T�Þ ð6:2Þ

or

Zrj ðVj ;TjÞ ¼ Zr

0ðV0;T0Þ ð6:3Þ

where V0 and T0 are related to their corresponding values in the j-fluid by

V0 ¼ Vj=hj and T0 ¼ Tj=fj ð6:4Þ

where hj¼Vc,j/Vc,0¼ s3j /s30 and fi¼Tc,j/Tc,0¼ ej/e0. The quantities fj and hj are

known as equivalent-substance reducing ratios.Experimental evidence has shown that the two-parameter corresponding-

states equation is obeyed only by the higher molar mass noble gases (Ar, Krand Xe) and nearly spherical molecules such as methane, nitrogen and oxygen.In order to extend the corresponding-states theory to a larger spectrum offluids, additional characterization parameters have been introduced into the

136 Chapter 6

basic two-parameter corresponding-states given by eq 6.1. Two main approa-ches have been followed in this parameterization. The first is to introduce theadditional characterization parameters and then perform a multi-parameterfirst-order Taylor series expansion of the compressibility factor about theparameters. Mathematically this gives,

ZðV�;T�; l1; l2; � � � ; lnÞ ¼ ZðV�;T�Þj li¼0f gþXni¼1

@Z

@li

� �li¼0

li ð6:5Þ

where the {li} are the characterization parameters. The derivatives appearing inthis equation are typically evaluated by making finite-difference approxima-tions using reference fluids that differ in the parameter of interest. A goodexample of the use of a single additional characterization parameter is thePitzer acentric factor o.7–9 In this case

@Z

@o

� �o¼0¼ ZðT�;V�;o ¼ o1Þ � ZðT�;V�;o ¼ 0Þ

o1; ð6:6Þ

and the corresponding-states model becomes

ZðT�;V�;oÞ ¼ZðT�;V�;o ¼ 0Þ

þ ZðT�;V�;o ¼ o1Þ � ZðT�;V�;o ¼ 0Þo1

o: ð6:7Þ

Examples of this approach and its generalizations include those of Pitzer,8–11

Lee and Kesler,12–14 Teja and co-workers15–20 and, more recently, Johnson andRowley,15–22 Pai-Panandiker et al.,23 Sun and Ely,24 and Malanowski andAnderko.25 References 21 and 25 present concise overviews of the differenttechniques used in corresponding states, giving special attention to the devel-opment of four-parameter models which are capable of describing fluids whichexhibit large deviations from the simple two-parameter corresponding-statesprinciple due to effects arising from size/shape and polarity/association. Inaddition, Poling, Prausnitz and O’Connell26 also summarize a large number ofcorresponding-states-like correlations that use this approach.The second approach is to extend the simple two-parameter corresponding-

states principle at its molecular origin. This is accomplished by making theintermolecular potential parameters functions of the additional characteriza-tion parameters {li} and the thermodynamic state, for example, the density rand temperature T. This can be justified theoretically on the basis of resultsobtained by performing angle averaging on a non-spherical model potentialand by ‘‘apparent’’ three-body effects in the intermolecular pair potential. Thenet result of this substitution is a corresponding-states model that has the samemathematical form as the simple two-parameter model, but the definitions ofthe dimensionless volume and temperature are more complex. In particular the

137The Corresponding-States Principle

dimensionless volume becomes

V� ¼ V

Ns3ðr;T ; lif gÞ¼ V

Vcjðr;T ; lif gÞ; ð6:8Þ

and the dimensionless temperature takes the form

T� ¼ kT

eðr;T ; lif gÞ¼ T

TcWðr;T ; lif gÞ: ð6:9Þ

The quantities y and j which appear in these equations are referred to as shapefactors and we refer to this method as the extended corresponding-states theory(ECST). Several review papers have been published that focus on thisapproach, the most extensive of which are those of Leland and Chappelear,6

Rowlinson and Watson27 and Mentzer et al.28

Thus far we have only introduced the pure-fluid corresponding-statesprinciple which, as mentioned above, has a rigorous basis in molecular theory.The extension of this theory to mixtures cannot, however, be made withoutfurther approximation and the problem of rigorous, yet tractable, prediction ofmixture properties remains unsolved. These approximations take the form ofmixing rules which are the topic of Chapter 5 in this volume. We willonly discuss mixing rules from an illustrative basis to show problems that canarise in the implementation of a corresponding-states model. In that regard, wewill focus our discussions on the one-fluid theories and primarily the van derWaals one-fluid theory proposed by Leland et al.29,30 The essence of this modelis that the properties of a mixture are first equated to those of a hypotheticalpure-fluid whose properties are then evaluated via the pure-fluid correspond-ing-states principle. The one-fluid theory is capable of providing highly preciseresults, especially when the extended pure-fluid corresponding-states formalismis used.In the remainder of this review, we have chosen to focus on the corre-

sponding-states nature of generalized engineering equations of state and themore general shape-factor-based extended corresponding-states principle. Westart by reviewing the basic molecular theory of pure-fluid corresponding statesand then describe in some detail the extended corresponding-states principle.Some time will be spent discussing the methods of calculating and/or predictingshape factors. Part of this discussion will be spent examining common engi-neering equations to extract the dependence of their shape factors on tem-perature, volume and other characterization parameters. We will also try toillustrate the dependence of the ‘experimental’ shape factors on temperature,volume and for example, dipole moment by studying highly accurate equationsof state for both non-polar (for example, hydrocarbons) and polar materials(for example, refrigerants and water). Finally we discuss the implementation ofthe extended corresponding-states principle for mixture calculations anddemonstrate some of the successes and difficulties encountered in the applica-tion of the model.

138 Chapter 6

6.2 Theoretical Considerations

The direct calculation of thermophysical properties from statistical-mechanicsinvolves not only extraordinary mathematical complexity, but also detailedknowledge of the interactions between the molecules. Although consider-able progress has been made in developing molecularly based predictivemethods, most of the results have been obtained using more or less drasticapproximations and are computationally complex. Also, most of these corre-lations are developed for specific types of fluids in certain regions of thephase diagram. Examples include the statistical-mechanical perturbation the-ories and integral-equation theories. The corresponding-states theoryprovides an alternative route to calculate thermophysical properties since ituses experimentally measured properties of one or more reference fluids torepresent the solution of the configurational part of the partition function.In this section we briefly review the molecular basis of the corresponding-statestheory.As a detailed derivation can be found in the literature,3,4,8 only the basic

assumptions of simple corresponding states are stated here:

1. The partition function can be factorized into a density-independentintra-molecular contribution and a density-dependent configurationalcontribution.

2. The configurational contribution can be treated using classical statis-tical mechanics.

3. The intermolecular pair potential may be written as a productof an energy parameter and a universal function f, which dependsonly on the distance between the molecules, r, for example, u(r)¼ ef(r/s) where e is the potential well depth and s is the collisiondiameter.

The first assumption is generally valid at low densities and even at highdensities for simple fluids but it does not necessarily apply to polyatomicfluids or associating molecules. The second assumption excludes fluids thatexhibit quantum behaviour. The third assumption is the most restrictivesince it excludes all non-spherically symmetric molecules and multibodyeffects.Given these assumptions the corresponding-states principle may be easily

derived from scaling arguments applied to the residual canonical-ensemblepartition function, Qr¼Q/Qid. Using standard notation31

Qr N;V ;Tð Þ ¼ 1

VN

Z� � �Z

e�UN ðrN Þ=kT drN

¼ s3N

VN

Z� � �Z

e�U�Nðr�N Þ=kT dr�

N

;

¼FðV�;T�Þ

ð6:10Þ


where UN is the configurational energy. Since the Helmholtz function A is givenby -kT ln Q(N, V, T), this equation reduces to

Ar

eT�¼ �FðV�;T�Þ: ð6:11Þ

Applying eq. 6.11 to two conformal fluids (fluids which obey assumptions 1 to 3given above) results in our basic working relation in corresponding-statestheory, namely that if two fluids are conformal, their dimensionless residualHelmholtz functions are identical when evaluated at equivalent conditions

Arj ðVj ;TjÞ ¼ fjA

r0ðV0;T0Þ ¼ fjA

r0ðVj=hj ;Tj=fjÞ; ð6:12Þ

where we have used the definition of the equivalent-substance reducing ratios fjand hj presented in section 6.1. Relations between other thermodynamicproperties of the two conformal fluids can be obtained by straightforwarddifferentiation. For example, for the pressure one finds

pjðVj ;TjÞ ¼fj

hjp0ðV0;T0Þ ¼

fj

hjp0ðVj=hj ;Tj=fjÞ: ð6:13Þ

In order to extend the simple molecular corresponding-states principle tonon-spherical fluids, two approaches are possible. The first simply amounts tointroducing models for the non-spherical interactions into the intermolecularpotential. For example, the intermolecular potential between two axiallysymmetric molecules whose electrostatic interactions can be represented aspoint dipoles and quadrupoles can be modeled as3

u r; y1; y2;f12ð Þ ¼ ef0ðr=sÞ þm2

r3fm y1; y2;f12ð Þ þY2

r5fY y1; y2;f12ð Þ; ð6:14Þ

where y1, y2 and f12 are angles describing the relative orientation of the twomolecules and m and Y are the dipole and quadrupole moments, respectively.When this potential is used in the evaluation of the configurational energy andthe residual partition function is made dimensionless one finds

Ar

eT�¼ �FðV�;T�; m�2 ;Y�2Þ; ð6:15Þ

and the corresponding-states working equation becomes

Arj ðVj ;Tj; m2j ;Y

2j Þ ¼fjAr

0 V0;T0; m20;Y20

� �¼fjAr

0

Vj

hj;Tj

fj;m2jfj hj

;Y2

j

fj h5=3j

!:

ð6:16Þ

140 Chapter 6

From a theoretical view point, this approach is perfectly viable. However froma practical point of view, neither theory nor experiment has provided quantitativedetails about how the equation of state depends on the multipole moments. Thuseq 6.16 cannot be used directly in any corresponding-states predictions.The second approach to extending the molecular corresponding-states

principle to non-spherical molecules was suggested by the work on angle-averaged potentials by Rushbrooke,32 Pople33,34 and Cook and Rowlinson.35

For example, if the spherical portion of the potential of an axial dipolarmolecule can be represented by the Lennard-Jones (12-6) model with

uLJ0 ¼ 4e0s0r

� �12� s0

r

� �6� ; ð6:17Þ

and the dipole can be modelled as a point dipole as in eq 6.14, Boltzmannaveraging of the potential yields an effective spherical Lennard-Jones potentialfor which the parameters are temperature dependent and are given by

eðTÞ ¼ e0 1þ m4

12kTe0s60

� �and sðTÞ ¼ s0 1þ m4

12kTe0s60

� ��1=6: ð6:18Þ

Allowing the potential parameters to be temperature dependent retains theform of the simple, two-parameter corresponding-states relationship exceptthat the equivalent-substance reducing ratios become functions of temperature.In particular,

Arj ðVj ;TjÞ ¼ fjðTjÞAr

0ðV0;T0Þ ¼ fjðTjÞAr0ðVj=hj ;Tj=fjÞ: ð6:19Þ

The temperature dependence of the equivalent-substance reducing ratiossomewhat complicates the thermodynamics of the thermal properties calcu-lated from this model. For example, the internal energy of the fluid j is relatedto that of the reference fluid 0 by the equation

Urj

RTj¼ 1� Tj

fj

@fj@T

� �� Ur

0

RT0� Tj

hj

@hj@T

� �Zr

0: ð6:20Þ

In applying this formalism, one must have knowledge of the reference fluidproperties (denoted by subscript 0) and the equivalent-substance reducingratios. These reducing ratios are typically expressed in terms of the molecularshape factors y and j that are defined as

fj ¼Tc;j

Tc;0

WðTj ; m2j ;Y2j ; . . .Þ and hj ¼

Vc;j

Vc;0

jðTj ;m2j ;Y2j ; . . .Þ: ð6:21Þ


6.3 Determination of Shape Factors

The theoretical basis for the molecular shape factors was derived in section 6.2.That analysis, which led to temperature-dependent shape factors, represents anidealized case where the non-spherical potential parameters may be incorpo-rated with the spherical parameters through angle averaging. Although thatapproach is correct in certain circumstances, it is of limited practical use sincethe intermolecular potential function for real fluids is not known precisely.Hence, one is forced to use macroscopic thermodynamic measurements todetermine the shape factors and then try to develop a generalized correlationfor them which depend on known molecular parameters. We shall refer to theshape factors determined from experimental data as the apparent or exact shapefactors and their generalized correlation as the correlated shape factors.The shape factors are weak functions of temperature and, in principle,

density and can be visualized as distorting scales that force the two fluids toconformality. Although there is no direct theoretical evidence for the densitydependence of the shape factors, mathematical solutions for exact shape factorsfound by equating the dimensionless residual compressibility factor andHelmholtz energy of two pure-fluids exhibit weak density dependence.The first attempt to find exact shape factors is due to Leach,36,37 who equated

the residual compressibility factor and fugacity coefficient of two fluids, with

zrj ðVj ;TjÞ ¼zr0ðVj=hj ;Tj=fjÞ and

frj ðVj ;TjÞ ¼fr

0ðVj=hj ;Tj=fjÞ;ð6:22Þ

that are solved for the equivalent-substance reducing ratios, fj and hj . Ineq 6.22, the superscript r denotes a residual (real minus ideal) property evaluatedat the temperature and molar volume of the system. For example, fr¼ f/rRTwhere f is the fugacity of the fluid. At low density these equations becomeidentical and the apparent shape factors were found by simultaneously solvingcorresponding-states relationships for the second and third virial coefficients

BjðTjÞ ¼hjBðT=fjÞCjðTjÞ ¼h2j CðT=fjÞ:

ð6:23Þ

As emphasized by Leland and Chappelear,6 the shape factors determined fromthe solutions to eq 6.22 depend on both density and temperature and, as such,cannot be related to any sort of intermolecular pair potential. An explanationfor the apparent density dependence can be partially attributed to the role ofthree-body intermolecular forces which are not considered in the basic corre-sponding-states model. In particular, it has been shown that if one wish tosimultaneously represent gas phase and condensed phase properties three bodyforces must be included in the calculations.38–43 One method of achieving asimultaneous representation of properties is, however, the use of an effective

142 Chapter 6

pair potential which is a function of density.44 Massih and Mansoori havediscussed the statistical-mechanical basis of the shape factors.45

Using methane as reference and a large number of pure normal hydrocarbonsfrom CH4 to C15H32, Leach

36 obtained solutions to these systems of equationsand empirically fitted the results in terms of the acentric factor and the criticalparameters. The set of correlated shape factors that were obtained is given by:

y ¼1þ ðoj � o0Þ a1 � a2 lnT�j þ a3 �a4T�j

!V�j � 0:5� �" #

j ¼Zc;0

Zc;j

1þ ðoj � o0Þ b1 V�j � b2� �

lnT�j þ b3 V�j � b4� �h in o

;

ð6:24Þ

where y and j are the shape factors, o the acentric factor, T* the temperature,V* the volume and Zc the critical compressibility factor. The subscripts j and 0indicate the fluid of interest and the reference fluid (methane), respectively. Thevalues of the parameters reported by Leach, et al.37 for the y shape factor werea1¼ 0.0892, a2¼ 0.8493, a3¼ 0.3063 and a4¼ 0.4506 with the j parametersbeing b1¼ � 0.9462, b2¼ 0.7663, b3¼ 0.3903 and b4 ¼ 1.0177. When V*

j42.0its value is set equal to 2.0; when V*

jo0.5 its value is set equal to 0.5. These limitscorrespond to the virial region and dense-liquid regions, respectively, where theapparent shape factors are independent of density. For other values of V*

j

between these limits, the shape factors were found to be density dependent. Ifthe apparent shape factors are defined according to eq 6.24, two parameters inaddition to Tc and Vc are introduced, oj and Zc,j, giving rise to a four parametercorresponding-states model.In 1981, Ely and Hanley46 developed an extended corresponding-states

theory for the viscosity of hydrocarbon mixtures. In conjunction with thatwork they developed a wide-range reference-fluid equation of state for methaneand a new set of correlation parameters for the apparent shape factors. Thefunctional form of that correlation was the same as determined by Leach, et al.,eq 6.24, but the parameters were somewhat different owing to the expandedrange of temperature. The values of the parameters reported by Ely and Hanleyfor the y shape factor were a1¼ 0.090569, a2¼ 0.862762, a3¼ 0.316636 anda4¼ 0.465684 with the j parameters being b1¼ � 0.93281, b2¼ 0.754639,b3¼ 0.394901 and b4¼ 1.023545.More recently Estela-Uribe and Trusler47 have developed a shape-factor

model specifically designed for application to natural gas components andsystems. In that model methane is the reference fluid48 and

y ¼1þ ðoj � o0Þ A1ðtÞ þ A2ðtÞe�d2 þCyðt; dÞ

h iand

j ¼Zc;0

Zc;j

!1þ ðoj � o0Þ A3ðtÞ þ A4ðtÞe�d

2 þCjðt; dÞh in o

;ð6:25Þ


where t¼Tc/T and d¼ r/rc and the temperature-dependent coefficients aregiven by an expression

AiðtÞ ¼ ai;1 � ai;2 ln t; ð6:26Þ

and the C terms are near critical correction terms given by

Cy ¼b1de�b2 ðd�1Þ2þðt�1�1Þ2½ � and

Cj ¼c1de�c2 ðd�1Þ2þðt�1�1Þ2½ �:

ð6:27Þ

Reference 47 reports optimized parameters for 13 natural gas components aswell as a set of generalized parameters.

6.3.1 Other Reference Fluids

A disadvantage of the original Leach shape-factor approach is that the referencefluid is fixed as methane. This introduces errors in the method when using it tocalculate properties of fluids which have very different properties from methaneor when the reduced temperature of the fluid of interest (target fluid) is less thanthe triple-point temperature of methane. Ely and Hanley attempted to overcomethis latter problem by developing an equation of state for methane which had afluid region extrapolated to a temperature of 40K (T*¼ 0.21). Another solutionto this problem was adopted by Leach36 and later by Rowlinson27 who showedthat it is possible to convert shape factors relative to one reference fluid intoshape factors relative to another reference fluid, thereby allowing two differentreference fluids to be used in the corresponding-states calculations. For example,Leach et al.37 used methane and pentane in their original extended corre-sponding-states model. In developing the transformation equations, we assumethat we know the shape factors or the equivalent-substance reducing ratios offluids i and j relative to some reference fluid which we shall denote with subscript0. The two fluids are at an equivalent corresponding state when Ti/fi0¼Tj/fj0¼T0 and Vi/hi0¼Vj/hj0¼V0. A second subscript has been added to clarify thereference fluid. Rewriting these equations we obtain relationships between thestate parameters of the i fluid with respect to fluid j, e.g., Ti¼ fi0 �Tj/fj0 andVi¼ hi0 �Vj/hj0. Finally, if we define the equivalent-substance reducing ratios for irelative to j as fij¼ fi0/fj0 and hij¼ hi0/hj0 we obtain the same functional mathe-matical relationships between the state points of the i and j fluids as we startedwith for the i and j fluids relative to the reference 0. The difference, however, isthat fij and hij involve the state points of both fluids, rather than just the statepoint of fluid i. The shape factors which are associated with fij and hij were calledrelative shape factors by Leach et al. and are given mathematically by

yijðT�i ;V�i Þ ¼yi0ðT�i ;V�i Þyj0ðT�j ;V�j Þ

¼ yj0ðT�i ;V�i Þ

yj0yj0T�iyi0

;jj0V

�i

ji0

� �0BB@

1CCA; ð6:28Þ

144 Chapter 6

and

jijðT�i ;V�i Þ ¼ji0ðT�i ;V�i Þjj0ðT�j ;V�j Þ

¼jj0ðT�i ;V�i Þ

jj0

Wj0T�iWi0

;jj0V

�i

ji0

� �0BB@

1CCA: ð6:29Þ

Equations 6.28 and 6.29 are non-linear and must be solved numerically.In 1987 Younglove and Ely49 reported a wide-range equation of state for

propane based on the functional form of the 32-term modified Benedict-Webb-Rubin equation (MBWR-32) which was proposed by Jacobsen and Stewart50

for nitrogen. The advantage of that equation of state was that its range of fluidstates included the triple point of propane which is 85K (the reduced triplepoint is 0.22). This development eliminated the need to use two reference fluidsor to use an artificially extrapolated reference fluid as in the work of Ely andHanley.46 To avoid having to use the shape-factor transformation formulasgiven above, Ely re-determined the apparent shape factors relative to thepropane reference. In determining the apparent shape factors, a slightly dif-ferent method than that used by Leach et al. was incorporated. In particular, atsubcritical conditions the procedure suggested by Cullick and Ely51 was used.In this procedure the vapour pressures and saturated-liquid densities of thetarget and reference fluids are equated

psat;jðTjÞ ¼psat;jðTj=fjÞhj=fjrsat;jðTjÞ ¼rsat;jðTj=fjÞ=hj ;

ð6:30Þ

and solved simultaneously for fj and hj. At supercritical conditions the virialmethod given in eq 6.23 was used. This procedure, unlike that used in previousstudies, generates apparent shape factors which only depend on temperature.Their correlation gave the following results:

y ¼1þ ðoj � o0Þ 0:05202� 0:74981 � lnT�jh i

j ¼Zc;0

Zc;j

1þ ðoj � o0Þ �0:14359þ 0:28215 � lnT�jh in o

:ð6:31Þ

Marrucho and Ely52 developed a saturation boundary based method toevaluate the shape factors that is easily transferable between reference fluids.The method is based on the Frost-Kalkwarf vapour-pressure equation53 andthe Rackett equation54 for saturated-liquid densities. Using these relations andeqs 6.30 one finds for the y shape factor

y ¼1� C�0 þ 2ð1� T�j Þ

2=7 ln Zc;j

.Zc;0

� �� DB� þ DC� lnT�j þ B�j

.T�j

1� C�0 þ B�0

.T�j

ð6:32Þ


and for j

j ¼Zc

j

� �ð1�T�j Þ2=7Zc

0

� �ð1�T�j=yÞ2=7

: ð6:33Þ

In deriving these equations, we have assumed that y is close to one andtherefore ln yDy–1 and have defined DB*¼B*

j –B*0 and DC*¼C*

j –C*0 and

neglected the D* term in the Frost-Kalkwarf equation

ln p�sat ¼ B�1

T�� 1

� �þ C� lnT� þD�

p�satT�2� 1

� �: ð6:34Þ

Since the reference fluid parameters appear explicitly in the shape factorexpressions, this formulation is easily transferable between reference fluids. Inthe supercritical region, Marrucho and Ely52 proposed a method of calculatingthe shape factors assuming that jj¼Zc,0/Zc,j and that isochores were nearlylinear, resulting in an expression for the y shape factor

f ¼Tc;j

Tc;0

y ¼h0ðpc;j � Tc;jgc;jÞ þ ðh0gc;j � gs0ÞTj

ps0 � gs0Ts0

: ð6:35Þ

The superscript s indicates the isochore which intersects the reference fluidsaturation boundary at rs0¼ rh0, h0¼Zc,0rc,0/Zc,jrc,j and g� (@p/@T)r. The gcfor the target fluid may be obtained from the Frost-Kalkwarf equation as

gc;j ¼ a pc;jðC�j � B�j �D�=2Þ=ð1�D�ÞTc;j : ð6:36Þ

As discussed in section 6.3.2, Mollerup55 analyzed common cubic equationsof state, such as the Soave-Redlich-Kwong56 and Peng-Robinson57 equationsin terms of separating them into a shape-factor correlation and an equation forthe pure reference fluid. The shape factors obtained from the cubic equations ofstate have the advantage of being similar to the ones developed by the morecomplex correlation schemes outlined here, but are simpler to use, since theyare density independent and can be used with any reference fluid.

6.3.2 Exact Shape Factors

From the late 1970’s to the early 1980’s, an increasing number of high-accu-racy, analytic, wide-range equations of state started to appear in the literature.The availability of these highly accurate equations of state allows one to

146 Chapter 6

‘exactly’ (although numerically) solve the two-parameter corresponding-statesrelationship for the apparent molecular shape factors. The possibility of densitydependence in the shape factors does, however, complicate the thermodynamicdescription of the target fluid in terms of the reference fluid and the resultingsolution for the shape factors themselves. The basic equation from the scalingof the partition function remains the same,

arj ðVj ;TjÞ ¼ ar0ðV0;T0Þ ¼ arj ðVj=hj ;Tj=fjÞ; ð6:37Þ

where we have introduced a more compact notation which denotes a dimen-sionless residual property by a lower case letter with a superscript r to emphasizethat the property represents the difference between the real and ideal valuesevaluated at the same volume and temperature. For example, in the equationabove ar� {A(V, T)–Apg(V, T)}/RT. Using the thermodynamic relationship

dar ¼ � zr

VdV � ur

TdT ; ð6:38Þ

and the chain rule we find the following thermodynamic relations between thetarget and reference fluids:

zrj ¼ zr0 1�Hvð Þ � ur0 Fv;

urj ¼ ur0 1� FTð Þ � zr0 HT and

srj ¼ sr0 � zr0 HT �ur0 FT :

ð6:39Þ

The enthalpy and Gibbs function can be constructed from their thermo-dynamic definitions and the relations given above. In the corresponding-statesrelations summarized in eq 6.39, dimensionless derivatives of the equivalent-substance reducing ratios must be known. These derivatives are defined as

FT ¼T

fj

@fj@T

� �V

and FV ¼V

fj

@fj@V

� �T

; ð6:40Þ

with similar expressions for the derivatives of hj.Given the thermodynamic relations summarized above, it is not possible to

solve for the equivalent-substance reducing ratios without making some otherassumption, i.e., the set of equations given in eq 6.39 are under-determinedsince a knowledge of both the values and derivatives of fj and hj is required. Thesimplest assumption is to choose the solution for which zrj ¼ zr0, which requiresthe relationship

zr0 HV ¼ �ur0 FV : ð6:41Þ

Figures 6.1 to 6.6 illustrate the results of the apparent shape factor calcu-lations obtained using this technique with propane as a reference. Figure 6.1


Figure

6.1

Theshapefactors

jandyasafunctionofdim

ensionless

temperaturesT*formethanerelativeto

aMBWR-32reference

equation

forpropanewiththeMBWR-32equationofstate

formethane.49

148 Chapter 6

Figure

6.2

Theshapefactors


ensionless

temperaturesT*formethanerelativeto

aMBWR-32reference

equation

forpropanewiththeSchmidt-Wagner

equationofstate

formethane.58


Figure

6.3

Theshapefactors


ensionless

temperaturesT*forcarbondioxiderelativeto

aMBWR-32reference

equationforpropanewiththeSchmidt-Wagner

equationofstate

forcarbondioxide.59

150 Chapter 6

Figure

6.4

Theshapefactors


ensionless

temperaturesT*for1,1,1,2-tetrafluoroethanerelativeto

aMBWR-32

reference

equationforpropanewiththeMBWR-32equationofstate

for1,1,1,2-tetrafluoroethane.141


Figure

6.5

Theshapefactors


ensionless

temperaturesT*shapefactors

forwaterrelativeto

aMBWR-32

reference

equationforpropanewiththeSaul-Wagner

equationofstate

forwater.142

152 Chapter 6

shows the shape factors for methane obtained using the MBWR-32 equation ofstate of Younglove and Ely.49 Methane differs from propane in both size andmolecular shape but both are non-polar. The shape factors reflect this in theirweak temperature and volume dependencies. Figure 6.2 shows the same resultsobtained using the newer methane equation developed by Friend et al.58 Theresults are similar with the differences in the shape factors being only a fractionof 1 %. We conclude that the volume dependence of the shape factors, althoughweak, is real and not due to artefacts of the equations of state used in thecalculation.Figures 6.3 to 6.5 show the apparent-shape-factors of carbon dioxide,

1,1,1,2-tetra-fluoroethane (also known by the refrigeration acronym R134a)and water relative to propane. Carbon dioxide59 was chosen for illustrationbecause of its large quadrupole moment which would lead to an intermolecularpotential which is substantially different from that of propane. In this case,the shape factors are nearly independent of volume and weak functions of

Figure 6.6 Comparison of the slope of the temperature dependent term in the y shapefactor (see eq 6.31) as a function of acentric factor difference for variousfluids.


temperature. R134a has a large dipole moment and exhibits a stronger volumedependence in its shape factors relative to propane than is observed with CO2.Finally water is highly polar and associating and exhibits stronger temperatureand volume dependence than that observed in the other fluids. These figuresillustrate that the extended corresponding-states approach is extremely pow-erful in that it can be used to make any fluid, regardless of intermolecularpotential, conformal to a selected reference fluid. One would hope that bystudying the shape factors of various families of fluids (for example, refriger-ants and alcohols.), relative to a fixed reference fluid, behavioural trends couldbe identified and correlated with known molecular parameters. Figure 6.6illustrates this type of relationship for the shape factors of various polarcompounds as a function of acentric factor.

6.3.3 Shape Factors from Generalized Equations of State

Thus far we have discussed the determination of shape factors from equa-tions of state which have a high degree of accuracy in representing the prop-erties of the pure-fluids. Mollerup55 observed that, although this procedureoffers a very high accuracy in the determination of the shape factors, it can bevery time consuming in terms of evaluating equations of state. This led himto examine ‘simple’ generalized equations of state which are commonlyused in engineering calculations. Examples include the Redlich-KwongSoave equation,56 the Peng-Robinson equation57 and others. In our analysis,we have also included the Carnahan-Starling de Santis equation60 because ithas a temperature-dependent volume parameter and a more complex volumedependence.Although we do not normally think of these modified van der Waals types of

equations of state in terms of shape factors, they all contain a prescription fortheir determination. To illustrate this, consider the Soave-Redlich-Kwongequation discussed in Chapter 4 and given as

p ¼ RT

Vm � b� aðT�;oÞVmðVm þ bÞ ; ð6:42Þ

where Vm is the molar volume, b the volume parameter given by ObRTc/pcwhich is independent of temperature in this model and a is a parameter which isgiven by

aðT�;oÞ ¼ OaðRTcÞ2

pcaðT�;oÞ; ð6:43Þ

where

aðT�;oÞ ¼ 1�mðoÞ 1�ffiffiffiffiffiffiT�p� �h i2

: ð6:44Þ

154 Chapter 6

The parameters Oa and Ob are universal for this equation of state and m(o) isa simple quadratic function of the acentric factor. The residual compressibilityfactor and dimensionless residual Helmholtz energy are given by

zrj ðVj ;TjÞ ¼bj

Vj � bj�

aðT�j ;ojÞRTjðVj þ bjÞ

; ð6:45Þ

and

Arj

RTj¼ ln

Vj � bj

Vj

� ��aðT�j ;ojÞbjRTj

lnVj þ bj

Vj

� �: ð6:46Þ

An examination of the corresponding-states relations, eqs 6.29 and 6.31 showthat, if the equivalent-substance reducing ratios are given by

hj ¼bj

b0and fj ¼

ajb0

a0b0; ð6:47Þ

the mathematical corresponding-states principle is obeyed. For the shape fac-tors, these relations imply that jj¼Zc,0/Zc,j and

yj ¼aðT�j ;ojÞaðT�0 ;o0Þ

¼1þmðojÞ þ mðo0Þ �mðojÞ

� � ffiffiffiffiffiffiT�j

p1þmðo0Þ

( )2

; ð6:48Þ

where we have used the fact that T*0¼T*

j /yj in deriving the second relationship.As an example of a more complex modified van der Waals-type model, we

have also made shape factor calculations with the generalized Carnahan-Star-ling-De Santis (GCSD) equation which in dimensionless residual form is given by

zrj ¼4yj þ 2y2j

ð1� yjÞ3�

4aðT�j ÞyjRTjbðT�j Þð1� 4yjÞ

; ð6:49Þ

where y is the packing fraction b(T*)/4v and the parameters a(T*) and b(T*) aregiven by

aðT�Þ ¼OaðRTcÞ2

pce a1ð1�T�Þþa2ð1�T�Þ2½ �

bðT�Þ ¼ObRTc

pc1þ b1ð1� T�Þ þ b2ð1� T�Þ2h i

:

ð6:50Þ

Again the parameters Oa and Ob are universal for this equation of state withvalues of 0.461883 and 0.104999, respectively. At temperatures above the cri-tical temperature, b1 and b2 are set equal to zero. In this case (or any case in


which the volume parameter has temperature dependence), we find a set ofcoupled equations for the shape factors which cannot be solved in closed form.For the generalized Carnahan-Starling-De Santis equation

yj ¼e a1ð1�T�j Þþa2ð1�T

�j Þ

2� �e a1ð1�T�0 Þþa2ð1�T

�0Þ2½ �

1þ b1ð1� T�0 Þ þ b2ð1� T�0 Þ2

1þ b1ð1� T�j Þ þ b2ð1� T�j Þ2

!

jj ¼Zc

0

Zcj

1þ b1ð1� T�j Þ þ b2ð1� T�j Þ2

1þ b1ð1� T�0 Þ þ b2ð1� T�0 Þ2

!;

ð6:51Þ

where T*0¼T*

j /yj. In Figure 6.7 we have compared the Peng-Robinson shape-factor relations, eq 6.48 and the Carnahan-Starling de Santis relations, eq 6.51for methane, relative to a propane reference, with the shape factors calculatedusing the MBWR-32 equations described above. This figure also shows resultsobtained using the generalized shape-factor correlation, eq 6.24. These com-parisons show a relatively good agreement at sub-critical conditions but fairlysubstantial differences in the supercritical region. The Soave-Redlich-Kwongmodel predicts a j shape factor which has a constant value of 0.965, as comparedto ‘exact’ values which range between 0.92 and 1.02, as shown in Figure 6.7.

6.4 Mixtures

In order to apply the simple corresponding-states theory to mixtures of con-formal molecules, approximations must be made concerning the microscopicinteractions and resulting structure of a mixture. A basic problem arises in thisextension because the configurational energy in a mixture is a function of theposition of the molecules and the species of the molecule located at thesepositions. This should be contrasted to the pure-fluid case where the moleculesare indistinguishable and the energy of the system is only a function of mole-cular positions. Thus, for a mixture, the scaling arguments that led to eq 6.11for pure fluids, do not apply, even if the intermolecular potentials for all themixture components are conformal.The earliest theoretically based attempt to deal with this problem was to

average the configurational energy of a mixture over all possible randomassignments of species to a given position.61 This random-mixing concept leadsto an effective, hypothetical, pure fluid potential (equivalent-substance poten-tial) which can be used in the formalism developed for pure-fluids. Con-sideration of the explicit form of the conformal potential (e.g., Lennard–Jonespotential) leads to mixing rules for the equivalent-substance reducing ratios ofthe hypothetical pure-fluid, that is:

fx ¼ f xkf g; fij �

; hij ��

and hx ¼ h xkf g; fij �

; hij ��

; ð6:52Þ

where the subscript x denotes the hypothetical pure-fluid and for examplefij¼ eij/e0 where ij denotes the interaction of an ij pair in the mixture. The term

156 Chapter 6

Figure

6.7

Methaneshapefactors

jandyusingapropaneMBWR-32reference

calculatedusingseveralequationsofstate

andcorrelations.

—,‘‘exact’’values

calculatedwithMBWR-32equationsofstate;J

,CullickandEly

saturationboundary

method;

m,generalized

correlation,eq

6.28;*Carnahan-Starling-D

eSantisequation;andK

,Peng-R

obinsonequationofstate.


{xk} indicates the explicit dependence upon composition. For example, for theLennard-Jones 12-6 potential

fxh2x ¼

XCi¼1

XCj¼1

xixjfijh2ij and fxh

4x ¼

XCi¼1

XCj¼1

xixjfijh4ij : ð6:53Þ

Even though the random-mixing theory played an important role in thedevelopment of mixture theories, its predictions are not reliable due to itsunrealistic physical basis (random assignment of molecules). However, the one–fluid concept, which states that the properties of a mixture can somehow beequated to those of a hypothetical pure-fluid whose properties can be evaluatedfrom corresponding-states, has persisted and forms the basis for what arecurrently the most accurate corresponding-states models for mixtures.

6.4.1 van der Waals One-Fluid Theory

The most successful corresponding-states theory for mixtures is called the van derWaals one–fluid theory. This theory was developed on a molecular basis byLeland and co–workers29–30,62 and follows from an expansion of the properties ofa system about those of a hard sphere system. A hard-sphere system is one whosemolecules only have repulsive intermolecular potentials with no attractive con-tributions. The starting equation for the development of the van der Waals one–fluid (known by the acronym VDW–l) theory is a rigorous statistical-mechanicalresult for the equation of state of a mixture of pair wise-additive, sphericallysymmetric molecules:

Z ¼ 1� 2pr3kT

XCi¼1

XCj¼1

xixj

ZN0

u0ijðrÞgijðr; r;T ; xkf g; ekf g; skf gÞr2dr: ð6:54Þ

In eq 6.54 equation Z is the compressibility factor pV/RT, gij is the radial dis-tribution function which gives the probability of finding a molecule of type i at adistance r from a central molecule of type j, uij is the intermolecular potentialwhose parameters are eij and sij, the prime denotes differentiation with respect todistance r, k is Boltzmann’s constant and r the number density. In the devel-opment of the VDW–1 theory the intermolecular potential is assumed to becomposed of a hard-sphere term plus a long-range attraction, that is given by,

uijðrÞ ¼ uHSij þ eijF�ðr=sijÞ; ð6:55Þ

where

uHSij ¼

N r � s0 r4s:

�ð6:56Þ

F(r) of eq 6.55 is a long-range attraction contribution to the potential, such asC6/r

6. Before one can proceed, assumptions concerning the radial distribution

158 Chapter 6

functions of mixture pairs must be made. In the development of this model,Leland proposed the mean-density approximation. This approximationamounts to saying that the radial distribution function of the ij pair is identicalto that of a pure fluid evaluated at the reduced conditions of the pair and a meannumber density. Mathematically

gij r; r;T ; xkf g; ekf g; skf gð Þ ¼ g0 r=sij; r0s3x; kT=eij� �

; ð6:57Þ

where the subscript 0 denotes a pure-fluid distribution function. The next step isaccomplished using the expansion techniques developed by Kirkwood, et al.63

for the distribution function of a real fluid in terms of that of a hard-sphere fluid,namely

g0 r=sij; r0s3x; kT=eij� �

¼ gHS0 r=sij; r0s3x� �

þXNn¼1

eijkT

� �nCn r0s3x� �

; ð6:58Þ

where the Cn are complicated integrals over the hard-sphere radial distributionfunction. Substituting these results into 6.54, we find a temperature expansionfor the mixture compressibility factor

Zrmix ¼

2pr3kT

XNn¼0

XCi¼1

XCj¼1

xixjeijkT

� �ns3ijCn r0s3x

� �: ð6:59Þ

The analogous result for a hypothetical pure fluid at the same reduced numberdensity is

Zrx ¼

2pr3kT

XNn¼0

eijkT

� �ns3ijCn r0s3x

� �: ð6:60Þ

Subtracting, eq (6.59) from eq (6.60) w obtain

Zrx � Zr

mix ¼2pr3kT

XNn¼0

1

kT

� �n

enxs3x �

XCi¼1

XCj¼1

xixjenijs3ij

" #Cnðr0s3xÞ: ð6:61Þ

Thus, we have at our disposal an infinite set of terms (coefficients of T�n)from which we can choose two for the determination of the potential para-meters of the hypothetical pure fluid. In the van der Waals one fluid model, thefirst two members of the series are chosen, giving

exs3x ¼XCi¼0

XCj¼0

xixjeijs3ij and s3x ¼XCi¼0

XCj¼0

xixjs3ij : ð6:62Þ


Dividing through by the reference fluid parameters we obtain

fxhx ¼XCi¼0

XCj¼0

xixjfijhij and hx ¼XCi¼0

XCj¼0

xixjhij ð6:63Þ

These are exactly the relations that van der Waals assumed in the generalizationof his equation of state to mixtures. Within the mean-densityapproximation, we see that these mixing rules are correct to order T�2

in the compressibility factor terms that have higher-order temperature depen-dence in the reference fluid are not correctly mapped by the van der Waalsmixing rules.

6.4.2 Mixture Corresponding-States Relations

The working equations for the mixture extended corresponding-states theoryare exactly the same as in the case of a pure fluid, eq. 6.39. The expressions forthe derivatives of the equivalent-substance reducing ratios in terms of thecomponent ratios are, however, somewhat complex. In particular, applicationof the formulas given above to mixtures requires derivatives of fx and hx withrespect to temperature, density and composition. An inspection of the mixingrules and the definitions of the equivalent-substance reducing ratios show thatthe arguments of the shape factors are the effective temperatures and densitiesof components in the mixture. These, in fact, do not correspond to the tem-perature and density of the mixture unless the shape factors are identicallyunity. Thus, in a mixture, the arguments of the shape factors are themselvesfunctions of fx and hx. The dependence of these is nominally given by Tj ¼ Txfj/fx and Vj ¼Vxhj/hx. Differentiating these relations with respect to Tx oneobtains two equations:

FjðTxÞ ¼FjðTjÞ 1þ FjðTxÞ � FxðTxÞ� �

þ FjðVjÞ HjðTxÞ �HxðTxÞ� �

and

HjðTxÞ ¼HjðTjÞ 1þ FjðTxÞ � FxðTxÞ� �

þHjðVjÞ HjðTxÞ �HxðTxÞ� �

:

ð6:64Þ

Differentiation of the mixing rules with respect to temperature, Tx, yields

FxðTxÞ þHxðTxÞ ¼1

fxhx

XCi¼1

XCj¼1

xixjfijhij FiðTxÞ þqi

qijHiðTxÞ

� ; ð6:65Þ

and

HxðTxÞ ¼1

hx

XCi¼1

xixjhijqi

qijHiðTxÞ; ð6:66Þ

160 Chapter 6

where qi¼ hi1/3 and qi¼ (qiþ qj)/2. Simultaneous solution of eq 6.64 for Fj(Tx)

and Hj(Tx) and substitution into eqs 6.65 and 6.66 and subsequent solution forFx(Tx) and Hx(Tx) yields Fx(Tx)¼ 1–S7/R and Hx(Tx)¼S6/R. Similar proce-dures yield for the volume derivatives

FxðVxÞ ¼ðS2 þ S4 � S7Þ

Dxand HxðVxÞ ¼

ðS6 þ S1 � S3 þDxÞDx

: ð6:67Þ

In order to perform phase-equilibrium calculations, fugacity coefficients canbe calculated from the thermodynamic relationship

lnfi ¼ gr0 þ ur0FxðniÞ þ zr0HxðniÞ � lnZ; ð6:68Þ

and the necessary composition derivatives of the equivalent-substance reducingratios are given by

FxðnkÞ ¼SðkÞ5 S7 � S

ðkÞ5 S2 þ S4ð ÞD

and

HxðnkÞ ¼SðkÞ8 S1 þ S3ð Þ � S

ðkÞ5 S6

D;

ð6:69Þ

where k denotes component k in the mixture andDx¼ (S1þS3)S7� (S2þS4)S6.The definitions of the sums Sm that appear in these results are given in Table 6.1.

Table 6.1 Sums Required for Evaluation of Mixture Corresponding-StatesProperties.

S1 ¼1

fxhx

XCi¼1

XCj¼1

xixjfijhij 1�HiðViÞ½ �Di SðkÞ5 ¼

2xk

fxhx

XCi¼1

xifikhik

S2 ¼1

fxhx

XCi¼1

XCj¼1

xixjfijhijFiðViÞDi S6 ¼1

hx

XCi¼1

XCj¼1

xixjhijqi

qijHiðTiÞDi

S3 ¼1

fxhx

XCi¼1

XCj¼1

xixjfijhijqi

qijHiðTiÞDi S4 ¼

1

hx

XCi¼1

XCj¼1

xixjhijqi

qij1� FiðTiÞ½ �Di

S4 ¼1

fxhx

XCi¼1

XCj¼1

xixjfijhijqi

qij1� FiðTiÞ½ �Di S

ðkÞ8 ¼

2xk

xhx

XCi¼1

xihik

whereD�1i ¼ 1�HiðViÞ½ � 1� FiðTiÞ½ � � FiðViÞHiðTiÞ


6.5 Applications of Corresponding-States Theory

Historically, applications of the extended corresponding-states theory havebeen limited by the scarcity of high-accuracy thermodynamic data and wide-range equations of state. As this data situation has improved over the pasttwenty years, refinements have been made in how the model is applied. In itsoriginal form, the extended corresponding-states model proposed by Leland,et al.6 incorporated a combined reference fluid of methane in the majority ofPVT space and n–pentane in the low-temperature, high-density region. Theshape factors used in this model were described in Section 6.3. Leland andco-workers originally applied this model to predicting vapour-liquid equili-brium in non-polar mixtures with a reported uncertainty of about � 10% inthe equilibrium K-values. Later, Fisher and Leland applied the model to pre-dicting enthalpies, compressibility factors and fugacities in systems that did notdeviate greatly from ideality.64

Starting in 1969, Rowlinson and coworkers published a series of papersapplying the extended corresponding-states model to a series of fluids andproperties. In their application they used the Leach shape factors, but anextended methane equation of state that incorporated the low-temperature dataof Vennix65,66 was used as the reference fluid. In the first paper, Watson andRowlinson67 predicted bubble-point temperatures and vapour compositionsof (argonþ nitrogenþ oxygen) with satisfactory uncertainty. Gunning andRowlinson68 calculated compression factors, enthalpies, Joule-Thomsoncoefficients and VLE for various systems and concluded that the extendedcorresponding-states principle had the advantage of requiring relatively littlestarting information and could be successfully applied to a wide variety ofproperties and fluids. Its primary disadvantage was pointed out to be a highdegree of numerical complexity. In 1972, Teja and Rowlinson69 applied themethod to the prediction of critical and azeotropic states finding quantitativeagreement for mixtures with one liquid phase and at least qualitative agreementin systems that had multiple liquid phases. Teja and Kropholler70 and Teja69

extended this study in 1975 by investigating azeotropic behaviour in the mix-ture critical region. In the second study, azeotrope formation and saturated-liquid densities in (CO2þC2H6) were predicted with excellent results. In 1976,Teja and Rice71 measured densities of various (benzeneþ alkane) mixtures andcompared their measurements to extended corresponding-states predictionsusing the Leach shape factors and Bender’s methane equation of state72 as thereference fluid. The average absolute differences found between the measure-ments and predictions were less than 2% for alkanes from hexane to hexa-decane. Teja15 also applied the extended corresponding-states method tomixtures containing polar components such as ammonia and hydrogen sulfide.In 1974, Goodwin published a very high-precision, wide-range equation of

state for methane73 which was capable of providing complete thermodynamicdata from the triple point p¼ 70MPa and T¼ 500K. This reference-fluidequation of state, along with the Leach shape factors, was used in a series ofstudies of liquefied-natural-gas (LNG) properties by Mollerup and co-workers.

162 Chapter 6

In the first study,74 Mollerup and Rowlinson found that it was possible toreproduce liquefied natural gas densities to within � 0.2%, even down toreduced temperatures of 0.3. In 1975, Mollerup75 continued his study ofliquefied natural gas properties and reported results for phase equilibria, den-sities and enthalpies in both the critical- and normal-fluid regions. The methodwas also applied to natural gas, liquefied petroleum gas and related mixtures ina 1978 investigation.76,77 Mixtures studied included methane through pentaneand common inorganics such as N2, CO, CO2 and H2S. The paper reporteddensity predictions to within � 0.2%, dew- and bubble-point errors ‘‘notexceeding those of good experimental data’’ and errors in liquid-phaseenthalpies which were less than � 2 kJ � kg�1.During the mid-1970s, researchers at the National Institute of Standards and

Technology (NIST) at Boulder, Colorado undertook a series of projects withthe objectives of measuring and predicting the properties of liquefied naturalgas and related mixtures. One result from this project was an extended corre-sponding-states model for liquefied natural gas densities developed byMcCarty.78 That implementation used a 32-term, modified Benedict-Webb-Rubin equation of state for methane as the reference fluid and shape factorswhich had the same functional form as those proposed by Leach, but which hadbeen re-fit to liquefied natural gas density data. The model reproduced theavailable liquefied natural gas density data to within � 0.1 %. Eaton et al.79

used McCarty’s methane equation to predict critical lines and VLE in(methaneþ ethane).Another part of the NIST study focused on the development of predictive

extended corresponding-states models for transport properties.46,80–85 Thatwork has been reviewed recently86 and will not be included here. As mentionedin Section 4, however, the transport property work produced another reference-fluid equation of state for methane that was extrapolated to T¼ 40K so as toavoid problems with the relatively high triple point of methane. That equationwas later used by Romig and Hanley87 as the reference-fluid equation to predictthe 1PAlnQ phase equilibria of (nitrogenþ ethane).In addition to the studies mentioned here, Mentzer, et al.28,88 summarized

extended corresponding-states results for phase equilibrium—especially forsystems containing hydrogen and common inorganics. They found accuratepure-fluid predictions for non-polar compounds up to about C7H16. Formixtures they found a strong dependence on the binary interaction parametersbut once those parameters were optimized for phase equilibrium, they could beused to represent a variety of properties accurately, without furtheroptimization.An overriding conclusion from all of these studies is that, while extended

corresponding-states calculations are very accurate for systems of similarmolecules, the predictions tend to decrease in accuracy as the system of interestdeviates in size and shape from methane. More generally, as the components ina mixture become more dissimilar both in size and polarity, there is a markeddecrease in the accuracy of the predictions. With this situation in mind, aresearch program was initiated by Ely and co-workers in the mid-1980’s to


investigate the possibility of improving the extended corresponding-statesmodel in three areas: (1), more realistic reference fluids; (2), better shape-factorgeneralizations; and (3), improved mixing rules. In a sense, the most importantpart of this work has been the development of a base of precise equations ofstate that can be used as reference fluids or which can be used to generate‘exact’ shape factors as described in section 6.3. The equations have beenreported by Younglove and Ely49 and Ely and co-workers.58–59,89–93 In addi-tion, during the same time frame, equations have been reported by Jacobsen,Pennoncello and Beyerlein and coworkers at the University of Idaho,50,94–100

Wagner and co-workers,101–108 Lemmon and co-workers at NIST109–114 anddeReuck and co-workers at Imperial College.115–117 Other early work includesthe studies of Haar and Gallagher on ammonia,118 Haar, Gallagher and Kellon water119 and equations summarized by Younglove.120 Finally we note thatmany of these equations of state have focused on the development of equationsof state for alternate refrigerants.121–126 Examples of applications that haveincorporated these new equations are diverse and most are not summarizedhere. However, access to these equations (and mixture models) is readilyavailable through the NIST Standard Reference Data program, especiallythrough the NIST 23 database REFPROP (v8) program which contains aprecise corresponding-states equation of state model. We would, however, liketo briefly discuss three relatively recent applications of corresponding-statestheory, all of which have some degree of novelty.

6.5.1 Extended Corresponding-States for Natural Gas Systems

As mentioned in our discussion of shape factors, Estla-Uribe and Trusler andco-workers47,127–133 have performed a detailed study of the extended corre-sponding states approach for natural-gas systems. Initially, corresponding-states shape factors for ethane, propane, methylpropane, butane and nitrogenwere determined and correlated using methane as the reference fluid. Thecorrelations involved six parameters and were unique for each substance. A setof generalized shape factors was also generated. Using both the specific andgeneralized shape factors, the extended corresponding states predictions ofthermodynamic properties of natural gas components were in excellent agree-ment with measurements. The extension to multicomponent systems wascarried out using the VDW one-fluid model with temperature and density-dependent binary interaction parameters. Compression factors and speeds ofsound of natural gases were predicted with average deviations within� 0.036%.In 2004, a new ‘‘hybrid’’ corresponding states model for the calculation of

mixture properties was proposed. In that model, the residual Helmholtzfunction of the mixture was given by the sum of two terms: one being theresidual Helmholtz function calculated by an extended corresponding statesmodel while the other is a correction term. The extended corresponding statesmodel uses methane as the reference fluid and VDW one-fluid mixing rules. The

164 Chapter 6

correction term is temperature and density dependent and is given by a localcomposition2 mixing rule. Local compositions were calculated from a coordi-nation number model based on lattice gas theory. Using this model for binary-mixture properties, densities were calculated with an average absolute deviation(AAD) of 0.12 %; speeds of sound were calculated with an AAD of 0.16 % andbubble pressures were calculated with an AAD of 1.77 %. Also, natural gasdensities were calculated with AAD of 0.03 % and natural gas speeds of soundwere calculated with AAD of 0.049%.

6.5.2 Extended Lee-Kesler

Sun and Ely134 used a simultaneous optimization algorithm to develop anaccurate but compact engineering equation of state for wide range of fluids withone single functional form. The algorithm was based on a simulated annealingmethod, and operates on different fluids at the same time to achieve the bestaverage results. A 14-term equation of state was developed based using thealgorithm that demonstrated good precision for selected non-polar and polarfluids. The equation of state also gives good predictions for some associatingfluids such as alcohols and water. Using that equation, a four-parameter cor-responding-states principle was proposed.24 This model is in the form of theHelmholtz function and takes the reduced density, reduced temperature,acentric factor and a polarity factor as variables. Compared to other general-ized equations such as the one by Span and Wagner135,136 for non-polar fluids,and by Platzer and Maurer137 and by Wilding and Rowley138 for polar fluids,the corresponding states model developed in this work was able to preciselyrepresent 22 non-polar, polar, and associating fluids considered in this study.The approach offers the flexibility to be extended to other fluids of industrialinterest. A unique feature of this work was that the polarity factor used in themodel was correlated from quantitative structure activity relationship mole-cular descriptors.

6.5.3 Generalized Crossover Cubic Equation of State

Kiselev and Ely139 have developed a generalized cubic equation of state forpure fluids, which incorporates non-analytic scaling laws in the critical regionand in the limit r-0 is transformed into the ideal gas equation. The general-ized cubic equation of state contains 10 adjustable parameters and reproducesthe thermodynamic properties of pure fluids with low uncertainty, including theasymptotic scaling behaviour of the isochoric heat capacity in the one- and two-phase regions. The generalized cubic equation of state is based on the crossoversine model and can be extended into the metastable region for representinganalytically connected van der Waals loops. In addition, using the generalizedcubic equation of state and the decoupled-mode theory (DMT) we combinedthe generalized cubic with the decoupled mode theory, which reproduces thesingular behaviour of the thermal conductivity of pure fluids in and beyond the


critical region. In later work,140 the generalized cubic equation of state wasmodified to give better performance at high temperatures. This type of modelcould lead to new shape factor correlations which not only map one classicalfluid onto another but also map a classical fluid onto a non-classical surface.

6.6 Conclusions

The extended corresponding-states theory incorporating shape factors can pre-dict thermodynamic properties of mixtures precisely, especially if the exact orsaturation- boundary methods are used. Due to the higher level of complexity ofthis method, recent applications have been limited to studies where very lowuncertainty is of importance. Advances in the development of precise equationsof state for a wide variety of substances (for example, heavier hydrocarbons,refrigerants and polar compounds such as alcohols, water and ammonia) haveenabled researchers to extend the extended corresponding states methodology toa wide variety of systems and this trend will continue in the future.Previous studies have shown that the approach is not very well suited for the

prediction of excess properties especially if there are substantial differences inmolecular size. Since the corresponding-states principle for pure fluids is exact,this failure can be attributed to a failure of the van der Waals one-fluid mixingrules to correctly map the composition dependence of higher-order temperatureterms in the reference-fluid equation of state. Efforts like those of Estala-Uribeet al.47,133 are underway to develop more sophisticated mixing rules for cal-culating the equivalent-substance reducing ratios in mixtures, but the ultimatesolution may involve abandonment of the one-fluid approach.

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CHAPTER 7

Thermodynamics of Fluids atMeso and Nano Scales

MIKHAIL A. ANISIMOV AND CHRISTOPHERE. BERTRAND

Department of Chemical & Biomolecular Engineering and Institute forPhysical Science & Technology, University of Maryland, College Park, MD20742, U.S.A. and The Petroleum Institute, Abu Dhabi, U.A.E.

7.1 Introduction

In contemporary process and product design, engineers often deal with systemsor phenomena for which traditional thermodynamics may be insufficient, as instrongly fluctuating and nano-size systems, system with nano-scale hetero-geneities, or mesoscopic dissipative structures. However, many such cases canbe successfully treated with mesoscopic thermodynamics. Mesoscopic thermo-dynamics can be defined as a semi-phenomenological approach to describingsystems where a length-intermediate to the atomistic and macroscopic scales-emerges and explicitly affects the thermodynamic properties and phasebehaviour.1

Traditionally, the thermodynamics of fluids used in engineering is essentiallymacroscopic. Fluids are treated as homogeneous; molecular structure andfluctuations are ignored. Size and surface effects disappear in the thermo-dynamic limit in which the volume V and the number of particles N tend toinfinity while the molecular density of the substance, r¼N/V, remains finite.Macroscopic thermodynamics often eliminates the size of the system by redu-cing the extensive thermodynamic properties by the number of particles, mass,or volume. The actual scale is restored only in the stage of engineering design.




172

These approximations are usually sufficient, even, for instance, when con-sidering a 1mm3 of water, which contains 3 � 1019 molecules.However, there are at least three categories of popular systems in which these

assumptions may breakdown:

1. Soft condensed-matter materials, such as complex fluids (polymersolutions and melts, microemulsions, gels, and liquid crystals),2–7 whichare characterized by the existence of one or more mesoscopic lengthscales.

2. Systems that are essentially finite (practically speaking, sub-micron ornano-size) in one or more dimensions.8–10 Examples include nano-particles (if the particle is considered as a separate system), pores, filmsand layers, fibers and threads.

3. Systems in a near-critical state (i.e., in the vicinity of a critical orsecond-order phase-transition point).11–13 Such systems are mesosco-pically inhomogeneous as a result of thermal fluctuations. The meso-scopic length scale associated with these fluctuations is known as thecorrelation length.

Each of these categories has been the subject of extensive experimental andtheoretical investigations over the last decades. It would be an unrealistic taskto cover in one article all applications of meso-thermodynamics. In particular,we do not consider such important topics as the thermodynamics of adsorp-tion,14 wetting transitions,15 microphase separation in polymers,16 gels,4,17 orphase equilibria in confined fluids.18–21 Nor do we discuss the increasinglyinformative simulations of meso-scale systems (see, for example, refs 8, 22 and23). Instead, in this Chapter we demonstrate only a few characteristic appli-cations of meso-thermodynamics to each category, while emphasizing uni-versality rather than specific details of the phenomena.Representation of a micro-heterogeneous system as an ‘‘ensemble’’ of small

open systems was introduced and elaborated by Hill.24,25 As an example, Hillrelates the Helmholtz energy A of a system of N nanoparticles to the Helmholtzenergy of a single nanoparticle a as:

A ¼ Naþ F1ðp;TÞN2=3 þ F2ðp;TÞ lnN þ F3ðp;TÞ; ð7:1Þ

where F1, F2, and F3 are functions of pressure p and temperature T only. Thisexpression reduces to the explicitly extensive result A¼Na in the thermo-dynamic limit. The part F1(p,T )N2/3 represents surface effects. Some phe-nomena considered in this Chapter, such as micellization or nucleation, areindeed driven by a competition between the bulk and surface contributions tothe energy. In other examples, like polymer solutions or near-critical states, thebulk energy itself depends on a mesoscopic length scale, which makes the bulkproperties position dependent. Treatment of mesoscopically heterogeneousfluids can be unified with the concept of a local Helmholtz (or Gibbs) energydensity. In this Chapter we demonstrate the concept of the local Helmholtz

173Thermodynamics of Fluids at Meso and Nano Scales

energy density on a variety of phenomena from smooth interfaces and modu-lated phases to critical phenomena and spinodal decomposition.

7.2 Thermodynamic Approach to Meso-Heterogeneous

Systems

Two major features must be introduced into the standard thermodynamicframework in order to apply thermodynamics at meso-scales: fluctuations andlocal (coordinate-dependent) properties. Thermodynamics of equilibriumfluctuations is a well developed science and we will briefly address this topic inthis section. Incorporating the local inhomogeneities is another task of meso-scopic thermodynamics. In this section we introduce a phenomenologicalapproach, which is restricted to fluids with smoothly varying properties, knownas ‘‘local’’ or ‘‘quasi’’ thermodynamics,26 and which dates back to van derWaals.27,28

7.2.1 Equilibrium Fluctuations

Fluctuations are spontaneous and random deviations of thermodynamicproperties from their average equilibrium values. These deviations are causedby thermal molecular motion. Macroscopic thermodynamics ignores fluctua-tions because they do not affect thermodynamic properties in the thermo-dynamic limit and they are usually insignificant in finite ‘‘macroscopic’’systems. However, the situation changes when the system becomes very smallor when it is near the limit of thermodynamic stability. In these two cases,fluctuations may become very large and may play a significant role in deter-mining thermodynamic properties.A general approach for introducing fluctuations into thermodynamics is

given by statistical mechanics.29,30 Let us consider an arbitrary, small portion ofan isolated fluid. This small portion, referred to as ‘‘the system’’, has a fixedvolume V and is in equilibrium with the surrounding fluid at temperature T andchemical potential m. The thermal molecular motion of the fluid particles causesfluctuations of the thermodynamic properties of the system. These fluctuationsexist in ‘‘violation’’ of the Second Law since they decrease the total entropy St

of the fluid. Hence, the probability density of a fluctuation is

W / expDSt

kB

� �; ð7:2Þ

where kB is Boltzmann’s constant. The change in the total entropy of the fluidDSt can be divided into two contributions: DSt¼VdsþDS; one from the sys-tem (Vds), where s is the entropy density of the system, and another from thesurroundings (DS). The entropy change in the surroundings is related to theheat flow Q from the surroundings into the system by DS¼ �Q/T. The First

174 Chapter 7

Law can be used to express the heat flow Q in terms of r and the energy densityu as Q¼V(du–mdr). The probably density of a fluctuation then becomes

W / exp �V du� Tds� mdrð ÞkBT

� �: ð7:3Þ

Assuming that the fluctuations are small, du can be expanded to second order inthe quantities ds and dr such that

du� Tds� mdr ¼ 1

2

@2u

@s2dsð Þ2þ2 @

2u

@r@sdrdsð Þ þ @

2u

@r2drð Þ2

� �

¼ 1

2d@u

@s

� �rdsþ d

@u

@r

� �s

dr

" #¼ 1

2dTdsþ dmdrð Þ:

ð7:4Þ

Furthermore, the fluctuations ds and dm can be expanded as

ds ¼ rCV

TdT � @m

@T

� �rdr

dm ¼ @m@T

� �rdT þ 1

r2kTdr;

ð7:5Þ

where CV is the isochoric heat capacity per molecule and kT¼ � (1/V)(@V/@p)Tis the isothermal compressibility. Substituting these expressions into theprobability density, we arrive at an expression for the probability distributionof the fluctuations, which is Gaussian with zero mean:

W / exp � 1

kBT

NCV

2TdT2 þ V

2r2kTdr2

� �� : ð7:6Þ

This Gaussian distribution is consistent with the assumptions that fluctuationsare small, random, and non-interacting. The mean square fluctuations are givenby the standard deviation of the Gaussian

drð Þ2D E

¼ kBTr2kTV

; dTð Þ2D E

¼ kBT2

NCV; drdTh i ¼ 0: ð7:7Þ

Alternatively, we could have expressed w in terms of ds instead of dT. This leadsto the relationships

dsð Þ2D E

¼ kBr2Cm

V; drdsh i ¼ kBT

V

@r@T

� �m; ð7:8Þ


where Cm is the heat capacity per molecule at constant chemical potential.Thermodynamic derivatives at constant chemical potential are not uniquelydefined as they depend on a reference value of the absolute entropy. In parti-cular, with an appropriate choice of the entropy value, (@r/@T)m may becomezero and dr and ds may not be correlated.How important can fluctuations be for a nanoparticle? Consider a nano

droplet of water (in ambient conditions) suspended in oil. A droplet with theradius of 1 nm contains about 130 water molecules. Since the compressibility ofwater is lower than the compressibility of oil, the fluctuations (expansion/contraction) of the water droplet will be controlled by the lower compressibilityof water. If one neglects the mutual solubility of oil and water, the number ofmolecules in the droplet is constant, while the volume can fluctuate. Therelative mean-square fluctuation of volume is given as

dVð Þ2D E

V2¼ kBTkT

V: ð7:9Þ

For a droplet of radius 1 nm and kTD5.10�10 Pa�1, the volume will fluctuatearound its average value with a standard deviation of about 2 %. This estimateis merely qualitative at the nanoscale where the compressibility depends on thedroplet size and the volume fluctuations are affected by surface tension.The preceding relationships have the common feature that they relate the

mean square fluctuations of thermodynamic variables to the correspondingthermodynamic susceptibilities, i.e., the second derivatives of appropriatethermodynamic potentials. A general thermodynamic expression for fluctua-tions of a generalized density j in a homogeneous fluid reads

ðdjÞ2D E

¼ kBT

Vw; ð7:10Þ

where w is a generalized susceptibility. One can notice that the mean-squarefluctuations become large not only when the volume is small, but also when thesusceptibility is large. In particular, the fluctuations diverge at the limit ofthermodynamic stability where w�1¼ 0.

7.2.2 Local Helmholtz Energy

Furthermore, the concept of susceptibility can be extended to inhomogeneoussystems where the thermodynamic properties are functions of the coordinatevector x.29,30 Let ji(x) be a generalized density, and let hi(x) be the field con-jugate to ji(x) such that

hiðxÞ ¼df

d jiðxÞh i ; ð7:11Þ

176 Chapter 7

where f¼ fv0/kBT is the dimensionless Helmholtz energy per unit volume, v0 isthe molecular volume, and the d stands for a functional derivative. Forexample, if we consider the number density of the fluid, the conjugate field is thechemical potential. The generalized dimensionless susceptibility is defined bythe functional derivative

wij x;x0ð Þ ¼ d jiðxÞh i

dhjðx0Þ

!hk

¼ d2fd jiðxÞh id jjðx0Þ

� � : ð7:12Þ

For an isotropic fluid, the susceptibility must be invariant under rotations andtranslations. These conditions imply that the susceptibility should only be afunction of the distance |x–x0|. As the distance |x–x0| between two fluctuationsincreases, the random thermal motion increasingly degrades the correlations.This implies lim|x–x0|-Nhdji(x)djj(x

0)i¼ 0. For a distance much larger than thedistance between the neighboring molecules, the spatial correlation function isgiven by the Ornstein-Zernike exponential decay:29,30

djiðxÞdjjðx0Þ� �

/ e� x�x0j j=x

x� x0j j ; ð7:13Þ

where the length scale x over which the correlation function decays to zero iscalled the correlation length.Common scattering experiments (electromagnetic and neutron) can measure

the susceptibility and the correlation length in fluids. These experiments do not,however, measure the susceptibility as a function of distance. Instead theymeasure the wave-number dependent susceptibility corresponding to a singleFourier component of the susceptibility. The wave number in these experimentsis q¼ [(4pn)/l]sin(y/2), where l is the wave length of radiation, n is therefractive index, and y is the angle of scattering. The Ornstein-Zernike wavenumber-dependent susceptibility reads.29,30

wijðqÞ ¼1

V

Zeiq� x�x

0j jwijð x� x0j jÞdV ¼wq¼0

1þ x2q2; ð7:14Þ

where wq¼0 is the dimensionless susceptibility in the limit q-0 (identical tothe macroscopic thermodynamic susceptibility @hjii/@hj when the orderingfield hj is taken to be uniform). Since the correlation length is much larger thanthe average distance between molecules, the susceptibility is an essentiallymesoscopic property, which depends on the ‘‘instrumental’’ scale q�1.In a typical light-scattering experiment this scale can be as small as 30 nm.To study the susceptibility at smaller scales, one needs to use neutron or X-rayscattering.The simplest way to connect the mesoscopic susceptibility with other ther-

modynamic properties and introduce smooth heterogeneities into mesoscopicthermodynamics is to consider a local (coordinate-dependent) thermodynamic


potential. If the density of a fluid varies in space, r¼ r(x), then the localHelmholtz energy density will also depend on the density gradient, in additionto the temperature and the density. In first approximation, the total Helmholtzenergy will then be given by

A ¼Z

f ðr;TÞ þ 1

2m rrj j2

� �dV ; ð7:15Þ

wherem is a phenomenological constant (the ‘‘influence parameter’’) associatedwith the range of intermolecular interactions, f is the portion of the Helmholtzenergy density that is independent of the gradients and which can be obtainedby integration of the susceptibility in the limit q-0.In particular, near the vapour-liquid critical point the reduced function f¼

f/rckBTc is represented by a Landau expansion.29

f ¼ 1

2a0DTðDrÞ2 þ

1

4!u0ðDrÞ4 þ � � � ; ð7:16Þ

where a0 and u0 are constants, DT¼ (T–Tc)/Tc and Dr¼ (r – rc)/rc with thecritical temperature and density denoted by Tc and rc. Expanding the van derWaals equation of state near the critical point one can easily find a0¼ 9/4 andu0¼ 27/4. Along the critical isochore above the critical temperature, the sus-ceptibility diverges as w¼ (a0DT)

�1 and the correlation length as x¼ x0(DT)�1/2,

where x0 ¼ffiffiffiffiffiffiffiffiffiffiffic0=a0

pwith c0¼m/rckBTc. Below the critical temperature (in the

two-phase region) the densities of the coexisting phases behave as Dr¼ � (6a0/u0)|DT|

�1/2, where � refers to the liquid and vapour branches of thecoexistence, respectively, and the correlation length now becomesx ¼ x0=

ffiffiffi2p

DT�� 1=2. With the expression for f given by eq 7.16, the inte-

grand in eq 7.15 is known as the Landau-Ginzburg functional. Any analyticequation of state, such as the van der Waals equation and all its modific-ations, can be represented near the critical point by the Landau expansion.The Landau-Ginzburg functional is an essentially mean-field concept.While it contains the correlation length diverging at the critical point, theasymptotic critical behaviour of the thermodynamic properties remains thesame as in the van der Waals theory. The existence of the gradient term inthe Landau-Ginzburg functional causes only corrections to the van der Waalsbehaviour; the corrections are assumed to be small with respect to the leadingmean-field behaviour. The condition, which determines the validity ofmean-field theory, requires the mean square fluctuations of the density in thevolume Bx3 to be much smaller than the thermodynamic density change.Specifically,13

Drð Þ2D E

ffi v0w

4p=3ð Þx3ffi

DT�� 1=24p=3ð Þa0x30

oo Drð Þ2ffi 6a0

u0DT�� : ð7:17Þ

178 Chapter 7

Inequality (7.17) means that the fluctuation can be neglected when

DT�� 44 u20

64p2a40

v0

x30

!2

� NG: ð7:18Þ

The inequality (7.18) is known as the Ginzburg criterion and the combinationof the parameters on the right side of the inequality as the Ginzburg numberNG. Apart from the numerically small prefactor 1/64p2, which depends on theproperty (such as the density, susceptibility, or heat capacity) considered, thereare two physical parameters which control the Ginzburg number, the range ofinteraction x0 and the coupling constant u0. The Ginzburg number becomeszero, and the fluctuations become unimportant, when the range of inter-molecular interactions is infinite or in the special case u0¼ 0 (a tricritical point).For molecular fluids with short-range interactions, one should not expect themean-field approximation to be valid in the critical region, |DT|{1, where thecritical fluctuations fully determine the thermodynamic properties.

7.3 Applications of Meso-Thermodynamics

7.3.1 Van der Waals Theory of a Smooth Interface

Van der Waals was first to realize that the density varies continuously across afluid-fluid interface.27 The fact that interfaces vary smoothly suggests thatinterfacial properties can be calculated with the Landau-Ginzburg functional.The following approach is originally due to van der Waals, but was subse-quently reformulated by Landau and Lifshitz,28,31 and later was rediscoveredand extended by Cahn and Hilliard.32

Consider vapour-liquid coexistence in a simple fluid. To find the equilibriuminterface between two coexisting phases, one needs to minimize the Helmholtzenergy subject to the condition that the particle number is fixed N ¼

RrdV .

This is equivalent to maximizing the excess pressure pex¼ � [fþ (m/2)(rr)2]þ mcxcr, where mcxc is the chemical potential along the vapour-liquidcoexistence curve. Consider a planar density profile which only varies along thez-direction (against gravity). The expression for the excess grand thermo-dynamic potential O¼ –pV is

DO ¼ A

Zf ðr;TÞ � mcxcrþ

1

2m

@r@z

� �2" #

dz; ð7:19Þ

where A is the surface area. The extremal condition reduces to the standardEuler-Lagrange equation,

m@2r@z2¼ @ f � mcxcrð Þ

@r: ð7:20Þ


For a specific equation of state the differential equation (7.20) can be solved toyield an expression for the interfacial profile. The relationship between theinterfacial profile and the surface tension takes the form s ¼

Rmð@r=@zÞ2dz.

Let f be the truncated Landau expansion given by eq 7.16. The equilibriumvalues of the liquid and vapour densities set the boundary conditions corre-sponding to the bulk phases. Taking the bulk liquid phase to be located atpositive infinity, and solving the differential equation subject to this boundarycondition, we find

DrðzÞ ¼ Dr �Nð Þ tanh z

2x

� �: ð7:21Þ

This profile is shown in Figure 7.1. Note the interfacial thickness appears to beproportional to the correlation length. This implies that the thickness of theinterface becomes infinite at the critical point, while the difference in the gasand liquid densities disappears. Using the above expression for the interfacialprofile, we arrive at the classical van der Waals result26–28

s ¼ 4ffiffiffi2p

rckBTcc0a

30

u20

� �1=2DT�� 3=2: ð7:22Þ

However, as we shall discuss in Section 7.4.3, the actual temperature depen-dence of the fluid interfacial tension is modified by the critical fluctuations.

Figure 7.1 Interfacial density profile Dr/Dr(�N) as a function of z/x. — , the mean-field (van der Waals) approximation given by eq 7.21; .... , calculated inrenormalization-group theory and given by eq 7.70.

180 Chapter 7

Heretofore we have only addressed the properties of planar interfaces. For acurved surface, the radius of curvature affects the interfacial properties, inparticular the interfacial profile and the surface tension. Consider a sphericalbubble of vapour surrounded by liquid. In this case, the Laplace equationrelates the pressure difference between inside the droplet and outside the dro-plet to the surface tension and the radius of curvature R as Dp¼ 2s/R. Thecurvature-dependent surface tension can be expanded in powers of the curva-ture as

sðRÞ ¼ sN 1� 2dRþ � � �

� �; ð7:23Þ

where sN is the surface tension of the planar interface and d is the coefficient ofthe first curvature correction to the surface tension known as Tolman’slength.33

Fisher and Wortis have shown that Tolman’s length is zero for ‘‘symmetric’’fluid coexistence and non-zero for ‘‘asymmetric’’ fluid coexistence.34 Symmetricfluids are represented by the lattice-gas (Ising) model in which the shape of thecoexistence curve is perfectly symmetric with respect to the critical isochore.26

Real fluids always possess some degree of asymmetry.35 Asymmetry in thevapour-liquid coexistence in helium, especially in 3He, is very small, but notzero.36 In the mean-field approximation, the asymmetry in the vapour-liquidcoexistence is represented by the rectilinear diameter:

rd ¼r00 þ r0

2rc¼ 1þD DT

�� : ð7:24Þ

For the van der Waals equation of state D¼ 2/5.It has been suggested recently that a ratio of the ‘‘excess density’’

(Drd¼ r00 þ r0– 2rc) and the difference between the densities of the coexistingphases (r00– r0) can be related to Tolman’s length (d) and the thickness ofinterface (2x) as

d2x

E� cdDrd

r00 � r0; ð7:25Þ

where cd is a universal constant.37 Equation (7.25) unambiguously relates Tol-man’s length to the asymmetry of fluid phase coexistence and defines the sign ofTolman’s length as negative for liquid droplets and positive for bubbles, pro-vided that the slope of the ‘‘diameter’’ of the coexistence is negative. Since in themean-field theory the ratio (r00–r0)/Drd and the thickness of the interface dependon temperature near the critical point in the same manner,26 the mean-fieldTolman length in simple fluids always remains finite and microscopic.34,38,39

Fisher and Wortis also calculated the shape for the interfacial profile for anasymmetric mean-field equation of state, which in addition to the symmetricbackground profile contains an asymmetric correction, shown in Figure 7.2,


DrðzÞ ¼ Dr�N tanh zð Þ þD DTj j tanh2 zð Þ þ log cosh zð Þ½ �cosh2 zð Þ

( ); ð7:26Þ

where z¼ z/2x.34 Using this profile, they showed that Tolman’s lengthapproaches a finite value near the critical point,

dx0¼ � 5D

6ffiffiffi2p

B0

; ð7:27Þ

where B0¼ 6a0/u0. Comparison of eqs 7.25 and 7.27 gives the mean-field valueof the constant cd ¼ 5=6

ffiffiffi2p

in eq 7.27. Since for the van der Waals fluid D¼ 2/5and B0¼ 2,35 d ¼ 1=6

ffiffiffi2p

x0, being just a fraction of a molecular size. Thus,one might conclude that Tolman’s length is too small to be important inengineering practice. However, as we show in the following sections, Tolman’slength may become significant or even diverge in meso-heterogeneous fluids.

7.3.2 Polymer Chain in a Dilute Solution

A long flexible polymer chain is a typical object addressed by meso thermo-dynamics. In dilute solutions a polymer chain can exhibit either a random walkor a self-avoiding walk.4 These two regimes are separated by the theta point, thepoint of polymer-solvent phase separation in the limit of infinite degree ofpolymerization.40 The random walk occurs when the polymer chain and thesolvent form either ideal or quasi-ideal solutions. The ideal polymer chainexhibits Gaussian fluctuations of the distance R between the two ends of the

Figure 7.2 Drd(z)/D|DT| correction to the density profile caused by asymmetry influid phase coexistence given by eq 7.26 as a function of z/2x.

182 Chapter 7

chain (made up of np links of a length r0) hR2i¼ r20np, where the degree ofpolymerization np serves as the polymer-chain ‘‘susceptibility’’. For the randomwalk, the radius of gyration of a chain is defined as

Rg ¼ffiffiffiffiffiffiffiffiffiffiR2h i

q¼ r0ðnpÞ1=2; ð7:28Þ

The Gaussian probability distribution function, as discussed in Section 7.2.1,for a 3-dimensional random walk is

WðR; npÞ / expDSkB

� �¼ exp � 3R2

2 R2h i

� �: ð7:29Þ

As follows from eq 7.29, the configurational entropy change per polymer chain is

DS ¼ � 3

2kB

R2

R2g

; ð7:30Þ

Stretching the ideal chain, therefore, increases the Gibbs energy by

DG ¼ 3

2kBT

R2

r20np: ð7:31Þ

Real polymer solutions are not ideal. In the mean-field Flory-Huggins model40,41

of a polymer solution the virial expansion of the osmotic pressure reads:

P ¼ Pv0

kBT¼ f

npþ Bf2 þ Cf3 � � � ; ð7:32Þ

where the osmotic pressure is defined as P¼ (m01–m1)/v0 with m01 being the che-mical potential of the pure solvent and m1 the chemical potential of the solvent insolution, f the volume fraction of polymer, v0 the molecular volume of thesolvent, and B and C the second and third virial coefficients, respectively. Whilethe third virial coefficient is a constant, the second virial coefficient depends onthe interaction between the monomers and solvent molecules:

B ¼ 1

2� w

kBT: ð7:33Þ

The interaction parameter w determines the theta temperature, defined by B¼ 0,

Y ¼ 2w

kB: ð7:34Þ

When B40, the chain monomers repel each other and the chain swells insolution (good solvent). When Bo0, the chain monomers attract each other and


the chain collapses to form a compact globule. When B¼ 0, attractions andrepulsions cancel each other and the chain exhibits a random walk. The thetapoint is defined by the simultaneous vanishing of the first (1/np) and second (B)virial coefficients.The repulsion between the monomer units which leads to this behaviour is

characterized by the excluded volume v0Bnp. Thus for a good solvent, instead offinding eq 7.28, one should expect

Rg ¼ r0ðnpÞn ð7:35Þ

with v41/2. The nonideal part of the osmotic pressure is

DP ¼ Pv0

kBT� fnp¼ Bf2: ð7:36Þ

Considering the volume occupied by a single chain and evaluating thepolymer volume fraction as f¼ 3v0np/4pR

3 with v0¼ (4p/3)r30, one obtains theexcess (non-ideal) part GE of the Gibbs energy as

GE

kBT¼ 4p

3R3DP ¼ r30B

ðnpÞ2

R3: ð7:37Þ

The n-dependence of Rg is obtained by minimizing the total Gibbs energy

DGkBT

¼ 3R2

2npr20

þ r30BðnpÞ2

R3; ð7:38Þ

with respect to R:

Rg ¼ r0B1=5ðnpÞ3=5: ð7:39Þ

In three dimensions v¼ 3/5 which is in close agreement with scatteringexperiments42 and with an estimate from renormalization-group theory.4 In anarbitrary number d of dimensions v¼ 3/(dþ 2). If the excluded volume becomesvery small, as when the system approaches the theta condition B¼ 0, eq 7.35will not be valid any more and the Rg dependence on np will exhibit crossover toeq 7.28.

7.3.3 Building a Nanoparticle Through Self Assembly

One way to manufacture nano-size particles is to create conditions whichpromote the self assembly of amphiphilic molecules in aqueous solutions.43,44

Amphiphilic molecules contain hydrophilic heads and hydrophobic tails. Two

184 Chapter 7

simple arrangements allow the tails to be isolated from the water: a sphericalmicelle and a bilayer. A closed spherical surface formed by a bilayer with thehydrophobic tails screened by the hydrophilic heads is known as a vesicle.Consider a population of spherical nano-size micelles with a stable aggre-

gation number N in a solution which also contains water and individualamphiphilic molecules (monomers). The Gibbs energy (per molecule) of thesolution is

G ¼ m1y1 þ m2y2 þ m3y3; ð7:40Þ

where mi and yi are the chemical potentials and molecular fractions with thesubscripts (1, 2, 3) referring to the solvent (water), solute (unassembledamphiphilic monomers), and micelles, respectively. Let f¼ Ny3 be the fractionof the monomers assembled in micelles and y¼ y2þf be the total fraction ofthe amphiphilic molecules in the solution. The equilibrium value of f is foundfrom the condition

@G

@f¼ m3

�N� m2 ¼ 0 ð7:41Þ

which is equivalent to equality of chemical potentials of the free amphiphilicmolecules in an ideal solution and the amphiphilic molecules in micelles:

m02 þ kBT lnðy� fÞ ¼ 1�Nm03 þ

kBT�N

lnf�N: ð7:42Þ

From eq 7.42, the equilibrium fraction of amphiphilic molecules in micelles is

f ¼ y� f�N

� �1= �N

ycmc; ð7:43Þ

where the parameter

ycmc ¼ expm03= �N � m02

kBT

� �ð7:44Þ

is defined as the critical micelle concentration (CMC). The difference of the twostandard chemical potentials, (m03/N)� m02, represents the equilibrium workrequired to transfer an amphiphilic molecule from the micelle to the free state.For large N, fDy� ycmc. It is also seen that the aggregation process is roundedat the CMC with a characteristic parameter 1� (f/N)1/N{1 at Nc1. At N-N micellization becomes ‘‘sharp’’ and identical to equilibrium polymeriza-tion;45 in this limit micellization can be regarded as a special second-orderphase transition.46


The actual aggregation number N fluctuates around N. In first approxima-tion, both the average aggregation number and the fluctuations are controlledby the same physical parameters, the effective interfacial tension s and thegeometry of the amphiphilic molecules. Consider a simple model of a sphericalmicelle with radius R, aggregation number N, and surface area per hydrophilichead ah¼ 4pR2/N. Both R and N may fluctuate. The chemical potential permonomer in the micelle m03/N contains two competing parts. The first one,representing the surface energy, grows linearly with a as the hydrophobic tailsbecome increasingly exposed to water. The second one, inversely proportionalto a, represents the repulsion between the hard cores of the hydrophilic headswith the area ah,min upon tight packing. When combined, these parts yield

m03N¼ sah þ

ah;minkBT

ah: ð7:45Þ

Minimization of the chemical potential with respect to N gives the average areaper head

�ah ¼4pR2

N¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiah;minkBT

s

r: ð7:46Þ

For a typical value of the oil-water interfacial tension sD(20 to 30)mnm�1, thelength of the hydrophobic tail lDRD2 nm, and the hard-core areaah,minD3.10�19m2, eq 7.46 gives ND40.The fluctuations of R and N around their optimal values RD40 The devia-

tion of the chemical (minimal) value due to a fluctuation is

m3ðNÞN� m3ð �NÞ

�N¼ 1

2

@2ðm03=NÞ@a2h

� �ah¼�ah

ðah � �ahÞ2 ð7:47Þ

where (@2m3/@a2h)ah¼ah plays the role of the inverse susceptibility. For large N, eqs

7.45 and 7.47 lead to mean-square Gaussian fluctuations of the area per molecule,

ðah � �ahÞ2D E

¼ kBT�N

@2ðm03=NÞ@a2h

� ��1ah¼�ah

¼ kBT �ah�N2s

; ð7:48Þ

of the aggregation number at constant R¼ R,

ðN � �NÞ2D E

¼ kBT�N

@2ðm03=NÞ@N2

� ��1N¼ �N

¼ N2kBT

8p �R2s; ð7:49Þ

and of the curvature at constant N¼ N

ðR� �RÞ2D E

¼ kBT�N

@2ðm03=NÞ@R2

� ��1R¼ �R

¼ kBT

32ps: ð7:50Þ

186 Chapter 7

For a typical micelle with N¼ 10–100, the mean-square-root fluctuation of theaggregation number is expected to vary from about (10 to 1) %. The fluctuationsof the surface area of a micelle, which are controlled by the interfacial tension, canbe compared to the bulk fluctuations in a liquid nano droplet (Section 7.2.1),which are controlled by the bulk compressibility of the droplet. Since in micelles allmolecules are exposed to the surface, the bulk effects can be neglected if kT{R/s.The example considered in this section suggests that the size and stability of

nanoparticles built by micellization is controlled by two major factors, geo-metry of the amphiphilic molecules and the interfacial tension between thehydrophobic tails and water. If the shape of the molecules is not suitable for theformation of a spherical micelle, the micelle will not be stable with respect tofluctuations of curvature at constant aggregation number and cylindrical orworm-like micelles with high polydispersity can be formed.2,43,44

7.3.4 Modulated Fluid Phases

Aqueous solutions of didodecyldimethylammonium bromide (DDAB) exhibit aninteresting self-assembly phenomenon.47 Each amphiphilic molecule of DDAB hastwo hydrophobic tails, so that DDAB molecules spontaneously form invertedmicelles in hydrocarbons and nano-size spherical vesicles in dilute aqueous solu-tions. However, in aqueous solutions, at the DDAB concentration of about103moldm3, the spherical vesicles self-assemble into a multilayer ‘‘onion-like’’structure shown in Figure 7.3. It is also notable that at the same DDAB con-centration, the interfacial tension between water and a hydrocarbon (octane)almost vanishes, becoming less than 0.1mNm�1 and suggesting that the formationof the multilayer structure may be accompanied by water-oil interface instability.A variety of modulated phases are observed in multicomponent fluids con-

taining amphiphilic molecules.43,44 Such phases may be anisotropic (smecticliquid crystals) or isotropic (bicontinuous ‘‘sponge’’ phases in microemulsions).However, one common way to explain modulated phases is to apply theLandau-Ginzburg functional and consider a Lifshitz-point mechanism.48 Let anoil-water-surfactant mixture, with water/oil composition j, exhibit liquid-liquidcritical separation at the critical temperature Tc(c) and composition jc(c), bothof which depend on the surfactant concentration c. If the surfactant is equallysoluble in water and oil, the critical composition does not depend on c, which inthis case will represent the surfactant chemical potential (BkBTlnc). In thepresence of amphiphilic molecules the coefficient m in eq 7.15 (m¼ rckBTcc0)that characterizes the Helmholtz energy response to the emergence of inho-mogeneities may vanish or become negative; thus higher-order gradient termsmust be added to the functional to ensure thermodynamic stability.For simplicity, we consider a one-dimensional modulation.49 In lowest

approximation, the Gibbs energy of the solution is

G ¼ kBT

Z1

v0

1

2a0DTDj2 þ 1

4!u0Dj4 þ 1

2c0

djdz

� �2

þ d0

2

d2jdz2

� �2" #

dV ; ð7:51Þ


where DT¼ [T–Tc(c)]/Tc(c) and Dj¼j�jc(c). In the one-phase region, wherec040, DT40, j¼ 0. The critical temperature of demixing decreases in linearapproximation with addition of surfactant as Tc(c)¼Tc(0)þ (dTc/dc)c. In thetwo-phase region, where DTo0, Dj¼ � (� 6a0DT/u0)

1/2. Surfactant tends tosmooth heterogeneities, changing the coefficient c0 as c0(c)¼ c0(c¼ 0)�Lc,again in linear approximation, where L is a positive constant.50 The Lifshitzpoint is defined by two conditions, DT¼ 0 and c0¼ 0. In the range of negativec0, a modulated (lamellar) phase becomes stable. If one approximates themodulation of the order parameter by a sinusoid,

j zð Þ ¼ j0cos~qz; ð7:52Þ

where q is the wave number of modulation, the Gibbs energy of the modulatedphase, obtained by performing the integration in eq 7.51, is

G ¼ Gv0

kBT¼ 1

2X qð ÞDj2 þ 1

64u0Dj4; ð7:53Þ

Figure 7.3 EM micrograph of a replica from 8 � 10–4molDdm�3 DDAB aqueoussolution.47

188 Chapter 7

where X(q)¼ (1/2)(a0DTþ c0q2þ d0q

4). When integrating the local Helmholtzenergy density we have used hcos2qzi¼ 1/2,hcos4qzi¼ 3/8. For negative c0, X(q)has minimum at q¼ q0�(� c0/2d0)

1/2, the equilibrium wave number of mod-ulation. At the Lifshitz point q0¼ 0; the distance between lamellae becomesinfinite. Minimization of eq 7.53 yields the amplitude of modulationj0¼ � (� 16X(q0)/u0)

1/2. The condition

X ~q0ð Þ ¼ a0DT �c202d0

� �¼ 0 ð7:54Þ

defines the line of second-order transitions between the isotropic and modu-lated phases. For negative c0, upon cooling, the modulated phase will entercoexistence with two isotropic liquid phases, oil-rich and water-rich. Theequation for the line of the three-phase coexistence (first-order transition line inT–c space) is found by equating the Gibbs energies of the water/oil two-phaseregion, G¼ � (3/2)(a0DT)

2/u0, and the Gibbs energy of the modulated phase,G¼ � 4[X(q0)]

2/u0:

DT ¼ 3

3�ffiffiffi6p c20

4a0d0: ð7:55Þ

A phase diagram of an oil/water/surfactant mixture with a Lifshitz point isshown in Figure 7.4.Three-dimensional modulation in fluids corresponds to a so-called isotropic

Lifshitz-point,50–52 which can describe three-phase equilibrium with a middlemicroemulsion (sponge) phase and which can thus be treated as a special tri-critical point. Both the interfacial tensions between the water-rich and micro-emulsion phase and between oil-rich and microemulsion phases will vanish atthe tricritical Lifshitz point much faster than at a tricritical point in ordinarymulticomponent fluid mixtures,53 since the gradient-term coefficient c0 alsovanishes.54

7.4 Meso-Thermodynamics of Criticality

7.4.1 Critical Fluctuations

Thermodynamic behaviour of fluids near critical points is discussed in detail inChapter 10. Here we demonstrate how the near-critical anomalies emerge andare interrelated through a characteristic mesoscopic length scale, the correla-tion length of the critical fluctuations. Near the critical point, the correlationlength becomes very large, and the fluctuations of density or concentration playa dominant role in determining the physical properties. The procedure forincorporating the effect of critical fluctuations is provided by renormalizationgroup (RG) theory.55,56


In statistical mechanics, the Helmholtz energy, is calculated through thecanonical partition function Z :29,30

A

kBT¼ � lnZ ¼ � ln

XrðxÞ

exp �H½rðxÞ�kBT

� ; ð7:56Þ

where H[r(x)] is the effective Hamiltonian, which is assumed to have the sameform at meso-scales as the Landau-Ginzburg local Helmholtz energy given byeqs 7.15 and 7.16:

H

kBT¼Z

1

v0

1

2a0DTðDrÞ2 þ

1

4!u0ðDrÞ4 þ

1

2c0 rrj j2

� �dV ; ð7:57Þ

If the density fluctuations are ignored and the density is given by its averagevalue r(x)¼hr(x)i, the mean-field Helmholtz energy is recovered. The RGprocedure involves two steps. First the sum in the partition function is per-formed only over density configurations corresponding to short wavelengths.Next, the length scales in the effective Hamiltonian are rescaled to compensatefor the missing wavelengths. The combined effect is the generation of a neweffective Hamiltonian that looks like the previous Hamiltonian but witheffective coefficients. Near the critical point the correlation length is the onlyrelevant length scale in the system. Therefore, at the critical point the system

Figure 7.4 T/Tc(0) as a function of c showing schematically phase diagram of oil/water/surfactant solution. The Lifshitz point (LP) separates the liquid-liquid critical curve (on the left) and the phase coexistence (thick lowercurve); dashed curve, continuation of the critical line into the metastableregion; thin solid curve, second-order transition between the homo-geneous and modulated phases shown to the right of the LP.

190 Chapter 7

should be unaffected by a scale transformation, i.e., the critical point is a fixedpoint of the RG transformation. This is the physical origin of the critical pointuniversality. In simple words, the macroscopic properties of fluids can beuniversally explained and calculated through a single fluctuation-inducedmesoscopic length-scale.Experimentally, it is well established that asymptotically close to the critical

point all physical properties obey simple power laws.13,57 The universal powersin these laws are called critical exponents, the values of which can be calculatedfrom RG recursion relations. The phenomenological approach that interrelatesthe critical power laws is called scaling theory.12 In particular, the isochoricheat capacity diverges at the vapour-liquid critical point of one-componentfluids along the critical isochore as

CV

kB¼ A�0 DT

�� aþ background; ð7:58Þ

where DT¼ (T–Tc)/Tc is the temperature distance from the critical temperatureTc, aD0.109 the universal critical exponent for the heat capacity, A�0 thesystem-dependent critical amplitude, and B the analytic background. Thesuperscript � refers to either the one-phase region (þ ) or the two-phase region(� ). Remarkably, the spectacular divergence of the isochoric heat capacity,shown in Figure 7.5, is solely attributed to the critical fluctuations. In the van

Figure 7.5 Isochoric specific heat capacity cv as a function of (T�Tc) for SF6 alongthe critical isochore performed in microgravity during the GermanSpacelab Mission.58


der Waals (mean field) theory the isochoric heat capacity does not diverge atthe critical point and a¼ 0.Accurate light-scattering experiments59,60 have shown that the correlation

length of fluctuations diverges at the critical point of a fluid along the criticalisochore as

x ¼ x�0 DT�� n ; ð7:59Þ

where x�0 is the critical amplitude of the correlation length (the range ofintermolecular interactions, also known as the ‘‘direct correlation length’’) andvD0.630 is the critical exponent for the correlation length. The scaling theoryrelates all the physical properties of fluids and fluid mixtures to the divergingcorrelation length.

7.4.2 Scaling Relations

The main ideas and results of scaling theory can be demonstrated through asimplified mesoscopic model.13 Let us assume that the near-critical state (in theone-phase region) is a d-dimensional ideal ‘‘lattice gas’’ of fluctuation clusterswith a mesoscopic lattice spacing 2x and size L. Then the excess pressure, reads

Dp ffi LdkBTc

gxd; ð7:60Þ

where we assume g¼ 2d. With the lattice spacing diverging along the criticalisochore in accordance with eq 7.59, the excess entropy per molecule and theisochoric heat capacity asymptotically become

DS ¼ 1

r@Dp@T

� �mffi kBnd

2d xþ0 drc DT

�� dn�1 ð7:61Þ

CV

kB¼ T

kB

@DS@T

� �rffi nd nd � 1ð Þ

2d xþ0 drc DT

�� dn�2: ð7:62Þ

Comparing eqs 7.62 and 7.58, one obtains the universal scaling relationsbetween the universal critical exponents a and v and between the system-dependent critical amplitudes Aþ0 and xþ0 :

a ¼ 2� nd; ð7:63Þ

rcAþ0 xþ0 d¼ Rx: ð7:64Þ

In three dimensions a¼ 2� 3vD0.110 and Rx¼ rcAþ0 (xþ0 )3D0.21. The uni-

versal ratio (7.64) expresses two-scale factor universality.61 While the scaling

192 Chapter 7

relations (7.63) and (7.64) obtained from the simplified cluster model are in fullagreement with the RG theory, the value of the amplitude ratio RxD0.21predicted by the cluster model is about 20 % higher than the theoretical valueRxD0.17;57 the latter is confirmed by experiments.13,62

7.4.3 Near-Critical Interface

Another example of a property that can be estimated through the mesoscopicsize of fluctuations is the surface tension. As the surface tension vanishes at thecritical point, the vapour-liquid interface becomes diffuse and eventually dis-appears. Such an interface is mesoscopic, with the width extending, in practice,from nanometers to microns. As accurate experiments show (as illustrated inFigure 7.6), the surface tension asymptotically vanishes as

s ¼ s0 DT�� W; ð7:65Þ

where WD1.26 is the universal critical exponent for the surface tension and s0 isthe system-dependent amplitude. In the mean-field van der Waals theory,which was discussed in Section 7.3.2, WD3/2.

TΔ

10010-110-3 10-210-3

10-4

10-2

10-1

101

10-0

102

�/mN

⋅m-1

Figure 7.6 Surface tension s as a function of |DTˆ| for xenon.63 —, the slope WD1.26.


Scaling theory predicts that the width of the interface diverges as the cor-relation length.26 In the spirit of the simplified scaling model, given by eq 7.60,the surface tension for bulk fluids (d¼ 3), or the line tension for two-dimen-sional fluids (d¼ 2), can be estimated as a product of the density of the grandthermodynamic potential in the two-phase region, which is (for d¼ 3) abouttwice larger than in the one-phase region since A�0 D1.91Aþ0 ,64 and the widthof the interface, proportional to the correlation length. Assuming the width ofthe interface is approximately equal to 2x and using the actual RG universalvalue RxD0.17 to calculate the universal amplitude gD9.8 in eq 7.60, oneobtains

s ffi �2xDp ffi RskBTc

xþ0 d�1 DT

�� ðd�1Þn ; ð7:66Þ

where Rs¼ 2RxA�0 /Aþ0 vd(vd� 1) is a universal constant. By comparing eqs

7.65 and 7.66, one obtains the universal scaling relations between the universalcritical exponents v and W and between the system-dependent critical ampli-tudes s0 and xþ0 :

W ¼ ðd � 1Þn; ð7:67Þ

s0 xþ0 d�1kBTc

¼ Rs: ð7:68Þ

In three dimensions, W¼ 2vD1.26 and Rs¼ s0(xþ0 )2/kBTcD0.39. Not only are

the scaling relations (7.67) and (7.68) obtained from the simplified model inagreement with RG theory, the estimated value of the amplitude ratio RsD0.39is close to the universal theoretical value R0D0.37.65 This value is confirmed bythe most reliable experiments on fluids.66 Relationships between other ther-modynamic properties and the diverging correlation length can be obtained in asimilar fashion.Near-critical fluctuations modify not only the temperature dependence of the

surface tension but also the shape of the density/concentration profile. RGtheory shows that the universal expression for the order-parameter profile nearthe critical point can be written in terms of a universal scaling function,26

jj0

¼ Cðz=2xÞ; ð7:69Þ

like in the van der Waals theory (Section 7.3.2), but with the function C(z/2x)modified by the critical fluctuations. Ohta and Kawasaki67 and Jasnow68 cal-culated the scaling function to first order in the epsilon (e¼ 4� d) expansion:

Cðz=2xÞ ¼ tanhð�z=2xÞ 1þ 2a

3þ asec h2ð�z=2xÞ

� ��1=2; ð7:70Þ

194 Chapter 7

where a ¼ffiffiffi3p=6

pe. As seen in Figure 7.1, the fluctuations make the actual

profile smoother than the mean-field one.

7.4.4 Divergence of Tolman’s Length

A specific effect resulting from the critical fluctuations of the interfacial prop-erties is the divergence of Tolman’s length, the first curvature correction to thesurface tension. It has been recently shown37 that a proper description ofasymmetry in fluid phase behaviour by complete scaling69 (see also ref. 35 andthose listed in Chapter 10), yields a much stronger divergence of Tolman’slength at the critical point than previously believed.34,70 This divergence, withamplitude depending on the degree of asymmetry in phase behaviour, is purelya fluctuation-induced effect which does not exist in any mean-field model.Far away from the critical temperature, the mean of the vapour and liquid

densities is represented in first approximation by a rectilinear diameter. How-ever, close to the critical point, the critical fluctuations modify not only theshape of the coexistence boundary, which becomes

r00 � r0

2rcffi �B0 DT

�� b ð7:71Þ

with bD0.326, but also the mean of the densities given by eq 7.24, making it‘‘singular,’’ with its temperature derivative diverging at the critical point. Beingmodified by fluctuations, the excess density Drd¼ (r0 0 þ r0)/2rc� 1 splits intotwo diverging terms:35,69

Drd ffi aeffB20 DT�� 2b� beff

A�01� a

DT�� 1�a; ð7:72Þ

where aeff and beff are ‘‘effective’’ system-dependent asymmetry coefficients tobe evaluated from a mean-field equation of state.35 As typical of scalingapproaches, Tolman’s length can be estimated as the ratio of the excessadsorption DGBDrd at the surface of tension and the total adsorptionGB2x(r0 0 � r0)D4B0|DT|

b at the near-critical interface,37

d2xffi �cd aeffB0 DT

�� b� beffA�0

B0 1� að Þ DT�� 1�a�b� �

ð7:73Þ

(with a universal amplitude cdD2/3). Since the width of the interface diverges as|DT|�v, the second term in eq 7.73 causes d to diverge weakly with an exponent1� a� b� vD� 0.065, a result well known from earlier studies.34,70 The firstterm diverges more strongly (yet more weakly than the interfacial thickness),with an exponent b� vD� 0.304; it can be shown to be the only term ofpractical significance for highly asymmetric fluids, such as polymer solutions.71

An even more significant divergence of Tolman’s length is predicted for two-dimensional phase separation. Since in two dimensions the ‘‘Ising’’ critical


exponents are b¼ 1/8 and v¼ 1,29 Tolman’s length is expected to diverge verystrongly as dp� |DT|b�vD|DT|�7/8 with amplitude determined by the level ofasymmetry in the two-dimensional phase coexistence.

7.5 Competition of Meso-Scales

If a fluid possesses two or more mesoscopic length scales, a competitionbetween these scales may cause crossover between different behaviours, eachassociated with a particular meso-scale. In this section, we discuss only twoexamples of such competition: near-critical polymer solutions and near-criticalfinite-size fluids.

7.5.1 Crossover to Tricriticality in Polymer Solutions

An example of competition between mesoscales occurs in a polymer solutionnear the critical point of phase separation. In this system, two mesoscales canbe tuned independently: the correlation length of the critical fluctuations by theproximity to the critical point of demixing and the radius of gyration Rg by thedegree of polymerization np.The following approach combines classical concepts of the Flory-Huggins

theory,40,41 scaling theory,12 and de Gennes’ ideas on the tricritical nature ofthe theta point.4 De Gennes has shown that the theta point in a polymer-solventsystem is a tricritical point. According to a phenomenological definition, atricritical point separates a line of second-order transitions (lambda line) and aline of first-order transitions.29,72 This definition can be applied to the thetapoint as follows: the states above the theta temperature at the zero volumefraction of polymer, shown by the thick vertical line in Figure 7.8, correspondto special ‘‘critical states’’, self-avoiding-walk singularities, associated with thebehaviour of long polymer chains at infinite dilution. This line is analogous tothe lambda line in 4He (second-order transitions to superfluidity). Below thetheta temperature the system is phase separated and the polymer chain entirelycollapses into one phase (the dashed curve in Figure 7.7); the effect can beregarded as a first-order transition. The polymer chain is characterized by anorder parameter w associated with the concentration of the chain ends (theprobability for the ends to meet each other). This is a ‘‘vector-like’’ orderparameter, while the square of the probability, the polymer volume fraction f,is the observable scalar property. Theta-point tricriticality emerges as a result ofa coupling between the two order parameters, f and w. For the infinite chain,the probability tends to zero while the correlation length (radius of gyration)diverges. A ‘‘vector-like’’ ordering scaling field h1, conjugate to the orderparameter w, is zero along the lambda line but becomes nonzero below thetheta temperature and everywhere else at finite degrees of polymerization. Thesecond scaling field h2, a scalar conjugate to f, also vanishes in the limit ofinfinite np.

196 Chapter 7

Figure 7.7 Schematic phase diagram of polymer solution. The dotted lines indicatepaths at different critical compositions.

Figure 7.8 Fractional temperature difference (T�Tc)/T where Tc is the criticaltemperature as a function of the polymer volume fraction f, showingthe asymmetric coexistence curve for a polymer solution with a degreeof polymerization np¼ 104 near the critical point as follows frominterpolation between the critical regime and theta-point regimegiven by eq 7.85. -----, the Flory-Huggins phase behaviour at np-N.� � �� , the crossover ‘‘diameter’’ of the coexistence. In the ‘‘critical’’regime (enlarged in inset) the coexistence curve is asymptotically describedby eq 7.83, while the ‘‘diameter’’ recovers the singular behaviourDfdBx2b.


In the mean-field approximation the field-dependent potential (osmoticpressure) can be presented in the form of a truncated Landau expansion inpowers of the order parameter w:73,74

�P ¼ h2c2 þ lc4 þ uc6 � h1 � c ð7:74Þ

The dimensionless osmotic compressibility is related to a dimensionless Gibbsenergy of mixing (per monomer), DG, by a Legendre transformation:P ¼ f @DG

@f

� �� DG. This yields the expression

DG ¼ fnp

lnfþ lf2 þ uf3; ð7:75Þ

Equation (7.74) is equivalent to the virial expansion of the osmotic pressurenear the theta point given by eq 7.32 with

f ¼ c2; ð7:76Þ

h1 ¼ w 2h2 þ 4lc2 þ 6uc4

: ð7:77Þ

h2 ¼1

np� 2lc2 � 3uc4; ð7:78Þ

and

l ¼ B ¼ l0T �Y

T: ð7:79Þ

In the Flory-Huggins model l0¼ 1/2 and u¼C¼ 1/6.The critical demixing point can be found from the stability conditions

@2G

@f2

!T ;np

¼ @3G

@f3

!T ;np

¼ 0; ð7:80Þ

which yield the critical parameters as a function of np (at large np):

Y� Tc

Tc¼ 6u

l0fc; ð7:81Þ

fc ¼1ffiffiffiffiffiffiffiffiffi6unp

p : ð7:82Þ

As was elucidated byWidom,75 there are two regimes in polymer solutions, the‘‘critical’’ regime and the ‘‘polymer’’ regime, with different behaviours in each.These two regimes are distinguished by the Widom variable x¼ (1/2)(np)

1/2|DT|.

198 Chapter 7

In the ‘‘critical’’ regime, where x{1 (with |DT| small enough for any givennpc1), the liquid-liquid coexistence is affected by the critical fluctuations andasymptotically described by73,74

f00 � f0

2fc

¼ Bcxb; ð7:83Þ

where Bc is a constant. In the tricritical ‘‘polymer’’ regime, where xc1 (verylarge np at any given |DT|{1), the phase coexistence becomes angle-like:75

f00 � f0

2fc

¼ f00

2fc

ffi Bpx: ð7:84Þ

In the Flory-Huggins model Bp¼ 3/2. Crossover between the ‘‘critical’’ and‘‘polymer’’ regimes can be achieved by a simple interpolation71

f00 � f0

2fc

E �Bxb 1þ xð Þ1�b; ð7:85Þ

where, for the sake of simplicity, we adopt Bp¼Bcr¼ B.RG theory predicts that the tricritical theta-point behaviour is mean-

field with logarithmic corrections caused by the fluctuations of the infinitelylong polymer chain. This prediction is supported by experiments.74

It means that the Widom variable x plays the same role for polymer solutionsas |DT|/NG in the Ginzburg criterion (compare with inequality. (7.18)):at x{1 the behaviour is critical and dominated by the fluctuations ofconcentration, while at xc1 the behaviour is tricritical and mean-field (Flory-like).The physical meaning of the Widom variable can be explained in terms of

competition between the two correlation lengths belonging to the two com-peting order parameters, namely, the radius of gyration and the mesoscopiccorrelation length of the critical fluctuations. Indeed, in the mean-fieldapproximation xDr0(np)

1/2|DT|�1/2 and x¼ (1/2)(np)1/2|DT|¼ (1/2)R2

g/x2. In the

critical regime, x is the largest mesoscopic length scale, while in the polymerregime Rg becomes larger than x. The critical correlation length x exhibitscrossover between the mean-field theta-point behaviour (xc1) and the criticalbehaviour x{1. A simple interpolation yields71

x ¼ Rgx�n 1þ xð Þn�1=2: ð7:86Þ

In the polymer regime the correlation length of concentration fluctua-tions becomes mean-field, xDr0(np)

1/4|DT|�1/2, tending to infinity at any givenDT when np-N. Below the theta point (in two-phase region) in the limitof np-N, the correlation length of the polymer order-parameter fluctua-tions, z, is not equal to the radius of gyration, since Rg-N, but becomes


independent of np:4,71

z ffi Rg

xffi 2r0T

Y� T/ f: ð7:87Þ

A more rigorous approach to crossover between the polymer and criticalregimes shows that the crossover variable x itself changes from xp(np)

1/2

|DT| in the polymer regime to xp(np)1/2|DT|2v (with 2vD1.2) in the critical

regime.74 This feature makes crossover expressions implicit and morecomplex, without, however, significantly changing the shape of the crossoverfunctions.

7.5.2 Tolman’s Length in Polymer Solutions

A polymer solution is a remarkable example of highly asymmetric fluid coex-istence. Equation (7.25) can be adapted for polymer solutions in the form

d

2~xffi �cd

Dfd

f00 � f0; ð7:88Þ

where 2x is the width of interface in a polymer solution. The ratio (7.88)behaves very differently in the ‘‘critical’’ and ‘‘polymer’’ regimes. Consider adroplet of polymer-rich phase, with concentration f00, in coexistence withsolution with concentration f0. In this particular case the sign in Eq. (7.88) willbe negative. Scaling arguments71 suggest that in the ‘‘critical’’ regime (whenDT-0) Tolman’s length should diverge at the critical point, in the samemanner as in simple fluids (but with an np-dependent amplitude); whereas in the‘‘polymer’’ regime (when np-N) Tolman’s length and the thickness of theinterface should not depend on np.

71,76 As shown in ref.,71 the interfacialthickness in a polymer solution, x can be approximated by a simple crossoverinterpolation between the two correlation lengths, x¼Rgx

�v (which is np andDT dependent) and zDRg/x (which is np independent), as

~x ffi Rgx�n

1þ xð Þ1�n; ð7:89Þ

while the asymmetric part of the volume fraction is

DfdBx2b 1þ xð Þ1�2b: ð7:90Þ

From eqs 7.85 and 7.88–7.90, and assuming cdBD1, we obtain

d~xffi � xb

1þ xð Þb: ð7:91Þ

200 Chapter 7

Hence, the crossover Tolman’s length in polymer solutions reads

d ffi �Rgxb�n

1þ xð Þb�nþ1: ð7:92Þ

Since bD0.326 and vD0.630, in the critical regime Tolman’s length diverges atthe critical point with an amplitude increasing with an increase of np asdDRgx

b�vDr0n0.348 |DT |�0.304. In the polymer regime dD�Rg/xDr0T/(Y–T)

and d/xD� 1, i.e., Tolman’s length follows the divergence of the width of theinterface. If the ‘‘critical’’ regime were mean-field with b¼ v¼ 1/2, Tolman’slength, as follows from Eq. 7.92, would not depend on temperature: dD�Rg,while the result in the tricritical polymer regime is, in first approximation,unaffected by fluctuations.For typical r0D0.2 nm and npD104, the radius of gyration is about 20 nm

and Tolman’s length at |DT|D10�4 reaches the absolute value of about 100 nm,while the thickness of the interface is about a micron. More remarkably, in the‘‘polymer’’ regime, even not very close to the tricritical phase separation (e.g.,about 3K away from the theta temperature, |DT|D10�2), Tolman’s length forthe polystyrene-cyclohexane solution is already mesoscopic, being about 50 nm,following the width of the interface of about 100 nm. In these conditions, asfollows from eq 7.91, for a polymer-rich droplet with the radius RD500 nm, thefirst correction to the interfacial tension is expected to be positive and about20%. Figure 7.9 illustrates the crossover behaviour of Tolman’s length between

Figure 7.9 Dimensionless Tolman’s length � d/r0 as function of (Tc�T)/Tc exhi-biting crossover between the ‘‘critical’’ and ‘‘polymer’’ regimes, calculatedfrom eq 7.92, for a polymer-rich droplet with the degree of polymerizationnp¼ 104.


the ‘‘critical’’ and ‘‘polymer’’ regimes for npD104. The two asymptotic beha-viours cross each other at xD1 which reflects the assumption that the char-acteristic microscopic length scale r0 represents both the size of the monomerand the range of interactions.

7.5.3 Finite-size Scaling

The competition between the correlation length and a finite system size can beformulated in the language of finite-size scaling.8,77–79 The basic idea of finite-size scaling is simple. The correlation length x, which diverges at the criticalpoint in accordance with eq 7.59, competes with a characteristic size L of thesystem. If a fluid system is spatially restricted, for instance by confining it to aporous medium with pore size L, the near-critical fluid properties are affected inseveral ways. The diverging physical properties are suppressed. Instead, theseproperties exhibit maxima where the maximum values depend on the con-finement geometry.Voronov and co-workers investigated the isobaric heat capacity of fluid

mixtures in various porous media near the liquid-liquid critical point80 and theisochoric heat capacity near the gas-liquid critical point.81 The experimentaldata80 on the isobaric heat capacity of a 2,6-dimethylpyridine-water mixtureat the critical composition near the liquid-liquid critical point are shown inFigure 7.10. While in the bulk sample the heat capacity obeys the power lawgiven by eq 7.58, in porous media the heat capacity remains finite and itsmaximum is shifted with respect to the critical temperature in the bulk sample.The magnitude of the anomaly depends on the pore size: in 10 nm porous glassthe anomaly virtually vanishes.According to finite-size scaling, the temperature, at which the maxima occur,

Tmax(L) is shifted relative to the bulk critical temperature. The temperatureshift scales with the system size as

tm � TmaxðLÞ � Tcj j=Tc / L�1=n :

The experimental results80 shown in Figure 7.11 confirm this prediction.The finite-size scaling expression for the anomalous part C(L,x) of the finite-

size heat capacity near the critical point reads79

CðL; xÞ ¼ CNðxÞCðx=LÞ; ð7:93Þ

where CN(x) is the bulk heat capacity anomaly and C(x/L) is a universal finite-size scaling function for the heat capacity, At (x/L)o1, C(x/L)-1, and theheat capacity obeys the power law (7.58). At (x/L)c1, C(x/L)-(x/L)�a/v andthe heat capacity only depends on the size: C(L)¼ (A� /a)(L/x�0 )a/v. This pre-diction is also confirmed by experiment.80

If the system is restricted in only one dimension, by creating a thin film withthe width L, for example, one should expect the finite-size scaling to competewith the crossover between three-dimensional and two-dimensional critical

202 Chapter 7

behaviour. The critical part of the density of the grand thermodynamicpotential will read

Dp x;Lð Þ / kBTc

x3F x=Lð Þ: ð7:94Þ

At (x/L){1, F(x/L)-1 and Dp(x,L)px�3p|DT|2�a. This behaviour corre-sponds to the 3d limit. At (x/L)c1, the fluid behaves as a two-dimensionalsystem: d-2, a-0, v-1, the scaling function becomes F(x/L)-(x/L)ln(x/L)and Dp(x,L)p1/Lx2ln(x,L)p� |DT|2ln|DT|, providing the famous logarithmicdivergence of the heat capacity at the two dimensional critical point.In a similar fashion one can describe a finite-size susceptibility, predicted as

wðL; xÞ ¼ wNðxÞUðx=LÞ; ð7:95Þ

Figure 7.10 Finite-size heat capacity at constant pressure Cp,m as a function oftemperature t near the lower critical point of 2,6-dimethylpyridine aqu-eous solution at the critical composition.80 þ , in the bulk; , 250 nmporous nickel; & 100 nm porous glass; m, 10 nm porous glass.


where the asymptotic behaviour of the scaling function U(x/L) is given by

ðx=LÞoo1 Uðx=LÞ ! 1 w ¼ wNðxÞ / xg=n

ðx=LÞ441 Uðx=LÞ ! ðx=LÞ�g=n w ¼ wL / Lg=n : ð7:96Þ

Meso-scale heterogeneities can be probed by the intensity of electromagneticor neutron scattering at a selected wave number q, the instrumental scale. Agood example of the scale-dependent meso-thermodynamic property is theisothermal compressibility of fluids or osmotic susceptibility of binary liquidsnear the critical point of phase separation.74 In the limit of zero wave numberand/or when the correlation length x is small (xq{1) the intensity becomes thethermodynamic susceptibility, which diverges at the critical point as

w ¼ wNðxÞ / xg=n : ð7:97Þ

However, the susceptibility becomes spatially dependent and thus finiteat xqc1. This means that at small scales the fluctuations are suppressedand the thermodynamic properties are controlled by the instrumental scaleL¼ q�1. When the correlation length x considerably exceeds L, x is replacedby L. Figure 7.12 shows the osmotic susceptibility of polystyrene-cyclo-hexane solutions obtained for various temperatures, and two length scalesE1/q.74

Figure 7.11 Shift in the heat-capacity lntm maxima near the lower critical point of2,6-dimethylpyridine aqueous solution at the critical composition as afunction of pore size L.80 Slope of the solid line is � 1/vD� 1.59.þ , experimental; K, simulation.80

204 Chapter 7

7.6 Non-Equilibrium Meso-Thermodynamics of Fluid

Phase Separation

The concepts of meso-thermodynamics can be extended to some non-equilibriumphenomena. In particular, like the thermodynamic properties, transport coeffi-cients, such as the diffusion coefficient, become spatially dependent at meso-scales.82 Moreover, away from equilibrium, generic long-range correlationsemerge even in simple molecular fluids, making the famous concept of local equili-brium, at least, questionable.83,84 In this section we focus only on one applicationof mesoscopic nonequilibrium thermodynamics in fluids: fluid phase separation.Mesoscopic nonequilibrium thermodynamics of fluids near phase separation

deals with three characteristic examples of how mesoscopic length scales mayaffect dynamics. One example is the relaxation of the near-critical fluctuations,which are long-lived because they become very large. Another example ishomogeneous nucleation, where a characteristic length scale emerges as a resultof competition between the bulk energy and the surface energy. The thirdexample is spinodal decomposition, in which a mesoscopic heterogeneousstructure results from the competition between negative diffusion and a positivegradient term in the local Helmholtz energy.

Figure 7.12 Mesoscale osmotic compressibility measured by intensity of light scat-tering I/Ir where Ir is a reference intensity versus (T�Tc)/T at two dif-ferent wave numbers in {polystyrene (npD2.103)þ cyclohexane}solution.74 The scattering angles 1501 and 301 corresponds to theinstrumental length scales q�1¼LD36 nm and about 500 nm, respec-tively. —, the scaling predictions in accordance with eq 7.95; ------, theosmotic compressibility in the thermodynamic limit q¼ 0.


7.6.1 Relaxation of Fluctuations

When a selected portion of a fluid exhibits a fluctuation, it is no longer inequilibrium with its surroundings. Consequently, the net thermodynamic forceacting on a fluctuation is non-zero, and tends to drive the system back toequilibrium via the transport of mass and energy. The ‘‘relaxation’’ of fluc-tuations, occurs according to the macroscopic laws of hydrodynamics, with themacroscopic transport coefficients. This equality of macroscopic and meso-scopic relaxation is known as Onsager’s principle.85 For example, in theabsence of chemical reactions, the hydrodynamic equation expressing theconservation of concentration c in a binary fluid is

@ðdcÞ@tþr � J ¼ 0; ð7:98Þ

where J is the concentration current. The concentration current serves torestore homogeneity in the absence of other driving forces, and in linear ordercan therefore be related to the gradient of the solute/solvent chemical-potentialdifference m21.

86

J ¼ �Drm21 ¼ �a_ @m21

@c

� �p;T

rðdcÞ; ð7:99Þ

where a_ is an Onsager kinetic coefficient, also known as the ‘‘mobility’’, The

product a_ @m21@c

� �p;T

defines the diffusion coefficient D. Similarly, the relaxation

of energy fluctuations is controlled by the thermal diffusivity, k/rCp, where k isthe thermal conductivity, and the relaxation of velocity fluctuations is con-trolled by the viscosity, etc.Dynamic scattering techniques probe the relaxation of fluctuations at a scale

determined by the instrumental wave number q. In particular, the relaxationtime of concentration fluctuations is found to be

tD ¼1

Dq2: ð7:100Þ

The same expression describes the relaxation time of energy fluctuations withthe thermal diffusivity replacing the diffusion coefficient. For a sphericalBrownian particle, the diffusion coefficient is given by the Stokes-Einsteinformula86

D ¼ kBT

6pZR; ð7:101Þ

where Z is the shear viscosity, and R is the radius of the Brownian particle.

206 Chapter 7

7.6.2 Critical Slowing Down

At the critical point, the size of fluctuations of density or concentration, x,diverges and the time of their relaxation becomes infinite. The slow relaxationnear the critical point is known as ‘‘critical slowing down’’ and the theory thatdescribes this phenomenon is known as ‘‘dynamic scaling’’.30,87 The basic ideaof dynamic scaling is simple: a fluctuation the size of the correlation length hasa lifetime proportional to the ‘‘volume’’ of the fluctuation:

tcBx2

Dffi 6Zx3

kBT: ð7:102Þ

In eq 7.102, the diffusion coefficient, which controls the life time of the fluc-tuations, is assumed to follow eq 7.101 for a Brownian particle, where theradius R is replaced by the correlation length x. The shear viscosity alsodiverges at the critical point as Zpxz, but very weakly, with zD0.07.88–90

Equation (7.102) ignores a spatial dependence of the diffusion coefficient.Specifically, near the liquid-liquid critical point of a binary solution, the dif-fusion coefficient is given by

D ¼ kBT

6pZxK qxð Þ; ð7:103Þ

where K(qx) is a universal scaling function, which limits to a finite number veryclose to unity for qx{1, while being pqx when qxc1.82,91,92 Therefore, themacroscopic diffusion coefficient vanishes at the critical point, while themesoscopic diffusion coefficient reaches a finite value depending of the instru-mental scale q�1. The macroscopic (q-0) Onsager kinetic coefficient, known asthe ‘‘mobility’’, diverges at the critical point as

a_ ffi kT

6pZx@c

@m

� �p;T

/ x g�n�znð Þ=n ; ð7:104Þ

while in finite-size limit x is replaced by q�1. Similarly, the macroscopic thermaldiffusivity near the vapour-liquid critical point of a one-component fluid is93

DT ¼k

rCpffi kBT

6pZxK qxð Þ ð7:105Þ

and the macroscopic thermal conductivity is

k ¼ kBT

6pZxCp / x g�n�znð Þ=n : ð7:106Þ

Mesoscopic dynamics can be also affected by competition between differentmesoscales which can generate a coupling between different dynamic modes.


An interesting example of such coupling is observed in polymer solutionsbetween the relaxation of the critical concentration fluctuations and viscoelasticrelaxation.82 This coupling leads to a crossover between the diffusive criticaldynamics in the critical regime and relaxation dynamics of an ‘‘infinite’’polymer chain in the theta-point regime.

7.6.3 Homogeneous Nucleation

Below the critical temperature, in the two-phase region, fluids may exhibit twodistinctly different types of dynamic behaviour. If a quench (rapid cooling)drives the system into a state between the coexistence curve and the spinodal(the absolute limit of thermodynamic stability), then the system will be in ametastable state and will phase separate through homogeneous nucleation.94

However, if the quench drives the system inside the spinodal, then the systemwill exhibit ‘‘spinodal decomposition’’.95–97

When the system is quenched into the metastable region between the coex-istence curve and the spinodal, the resulting metastable phase may not spon-taneously decay into two-phase equilibrium. The transformation must beactivated by some perturbation, such as thermal fluctuations. Consider aspherical liquid droplet, of radius R, immersed in the uniform metastablevapour. The difference of the chemical potential in the metastable state and thechemical potential in equilibrium is Dm¼ m–mcxc. The total Gibbs energy of thedroplet is given by the sum of the bulk and surface energy contributions,

DG ¼ � 4

3pR3rDmþ 4pR2s: ð7:107Þ

The energy of the droplet has a maximum value for a mesoscopic length scale,the critical radius Rc¼ 2s/rDm. For RoRc, instead of growing, a droplet willlower its Gibbs energy by decaying back into to the homogeneous phase.However, for R4Rc the droplet continually lowers its energy by growingindefinitely until the fluid reaches equilibrium phase-separation. The curvaturedependence of the surface tension, as described in Section 7.3.1, can affect thedescription of nucleation phenomena. Moreover, in reality, nucleation isaffected by heterogeneous impurities and confining surfaces which can lowerthe activation barrier. This type of nucleation is known as heterogeneousnucleation.94

7.6.4 Spinodal Decomposition

A very different dynamic picture is observed in the unstable region, inside thespinodal. There, the system is unstable against all perturbations. To be specific,we consider a binary liquid mixture after a rapid quench into the unstableregion. In the unstable region the diffusion coefficient D ¼ a_ @m21=@cð Þp;T isnegative since (@m21/@c)p,To0, while a_ is positive. Therefore, the concentrationfluctuations will grow, creating heterogeneities. Eventually, two-phase

208 Chapter 7

equilibrium will be established. However, near the critical point and, especially,in viscous fluids, this process can be very slow.The chemical potential difference in the inhomogeneous liquid mixture is the

derivative of the Landau-Ginzburg functional, which in this case describes thelocal Gibbs energy,

G ¼ 1

V

Zf ðc;TÞ þ 1

2c0 rcj j2

� �dV ; ð7:108Þ

m21 ¼m21kBT

¼ dGdc¼ @f@c� c0r2c: ð7:109Þ

Substituting this expression into eq 7.98 we arrive at the Cahn equation forspinodal decomposition:95

@c

@t¼ a_

@m21@c

� �p;T

r2c� kBTa_c0r4c: ð7:110Þ

Taking a Fourier transform of the Cahn equation, one finds that the fluctua-tions grow exponentially as Bexp(2Gt) with a wave number-dependent rategiven by

G ¼ a_@m21@c

� �p;T

q2 � c0q4

: ð7:111Þ

The rate is positive between q¼ 0 and q¼ qc¼ c� 1/20 , and negative for all q4qc.

All fluctuations with wave number less than qc would grow exponentially. Sincein molecular solutions c1/20 is about of molecular size, it means that, essentially,all the concentration fluctuations will grow. The maximum growth rate isobserved for fluctuations with a wave number qmax¼ [(@m21/@c)p,T/c0]

1/2¼ x�1,corresponding to the correlation length x. Thus, spinodal decomposition gen-erates a nonequilibrium mesoscopic structure with a length scale determined bythe correlation length.

7.7 Conclusion

A variety of phenomena in fluids at sub-micron and nano scales, despite theirdiversity, demonstrate some generic features originating from the existence of alength scale intermediate to the size of atoms and the size one can see without amicroscope. Mesoscopic thermodynamics is a semi-phenomenological science,which explicitly introduces mesoscopic length scales into the thermodynamicproperties of fluids. Another generic feature of meso-thermodynamics isfluctuations. Mesoscopic fluctuations not only alter and enhance thermo-dynamic properties like the susceptibility, they can also lead to entirely newphenomena like the divergence of Tolman’s length. The critical Casimir


force,98,99 which is solely attributed to fluctuations, and the existence of whichwas recently proven experimentally,100 also belongs to this latter category.Furthermore, the dimensionality of a system can also have a dramatic impacton the meso-thermodynamic properties.In this chapter, we have discussed only a few examples of mesoscale phe-

nomena in fluids, for which the methods of meso-thermodynamics appeared tobe applicable. However, since coarse-grained models of meso-thermodynamics,such as the Landau-Ginzburg local Helmholtz energy, demonstrate a highdegree of universality, associated with the meso-scale structure, an extension ofthis approach to a broader variety of phenomena is very promising.

Acknowledgement is made to the ACS Petroleum Fund for financial supportof this work.

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214 Chapter 7

CHAPTER 8

SAFT Associating Fluids andFluid Mixtures

CLARE MCCABEa AND AMPARO GALINDOb

aDepartment of Chemical and Biomolecular Engineering and Department ofChemistry, Vanderbilt University, Nashville, TN 37235, U.S.A.; bDepartmentof Chemical Engineering and Centre for Process Systems Engineering,Imperial College London, London, SW7 2AZ, United Kingdom

8.1 Introduction

While we could argue that the goals of modelling fluid phase equilibria havenot changed greatly since the time of van der Waals, there is little doubt thatthe systems of interest have gradually increased in complexity; surfactants,polymers, hydrogen-bonding molecules such as water, and polyfunctionalmolecules such as amino acids and peptides are now routinely considered.The demand for effective, accurate theoretical tools grows constantly as morecomplex systems are considered and modelling is expected to play a moreprominent role in the design of new products and processes. Thus, the need todevelop a thermodynamic modelling framework that considers aniso-tropic association interactions (such as occur in hydrogen bonding fluids),molecular shape, and electrostatic interactions (Coulombic, dipolar, quadru-polar etc.) becomes increasingly acute. Wertheim’s work in the 1980’s onassociating and polymeric fluids1–4 and its implementation as an equation ofstate in the statistical associating fluid theory (SAFT) have constituted a majoradvancement towards a theoretical framework for modelling these complexfluids.




215

Associating systems present unique challenges to the determination of theirthermodynamic properties. Short-ranged attractive directional interactionslead to aggregate and network formation and as a result the behaviour ofassociating fluids deviates from that seen in so-called simple fluids. To give butone example, if water molecules did not form hydrogen bonds, water wouldmost likely be a gas at room temperature and pressure, much like methane,which has a very similar molecular mass (16 gmol�1 compared to 18 gmol�1 forwater) but does not have a permanent dipole. If water did not hydrogen bondstrongly, it would not exhibit the unusual density maximum seen in the liquidphase.5 The first attempts to model associating systems appeared as early as the1900’s and were carried out in the framework of chemical bonding by Dole-zalek.6 Association between molecules was described as a chemical reaction andso the main drawback of the approach was that the aggregate species need to bespecified a priori; while this is feasible in dimerizng fluids, and in models leadingto infinite, linear-chain aggregates, branched and three-dimensional networkssuch as those formed in aqueous systems cannot be easily described. In con-trast, in the theory of Wertheim an anisotropic intermolecular potential thatincorporates short-ranged attractive interactions to form associated aggregatesand networks is proposed; the different aggregate species are obtained as aresult of this potential (i.e. they are a product of, rather than an input to, thetheory), and in the limit of complete association the theory provides an accu-rate description of the thermodynamic properties of chain fluids.7,8

In this chapter we provide some background on the development of Wer-theim’s theory of association and then review the main versions of the SAFTequation of state used in the current literature. Rather than focus on the sys-tems that have been studied using the different versions, which was recentlyreviewed by Tan and co-authors,9 we have striven to provide a survey of thelatest theoretical developments directed at enhancing the capability of theSAFT framework for the study of increasingly complex fluids.

8.2 Statistical Mechanical Theories of Association and

Wertheim’s Theory

Approaches to developing a statistical mechanical theory of associating fluids(specifically, hydrogen bonding) and fluids undergoing reaction date back atleast to the seminal text of Hill.10 The modern theory of associating andreacting fluids begins with the work of Andersen,11,12 who developed a statis-tical mechanics approach for associating fluids using a graph-theoreticalapproach. Andersen’s approach was quite general: he assumed that thepotential between two molecules, u(1,2), which can depend on both the positionand orientation of molecules 1 and 2, has the form

uð1; 2Þ ¼ u0ð1; 2Þ þ uHBð1; 2Þ; ð8:1Þ

216 Chapter 8

where u0 is the reference potential and uHB is the hydrogen-bonding part of thepotential. The hydrogen bonding part is non-zero only for a small range ofrelative positions (and orientations, if it is not spherically symmetric), and has adeep negative potential well consistent with hydrogen bonding. The keyassumed property of the potential in Andersen’s theory is saturation at thedimer level. That is, the reference potential u0 contains an infinitely repulsiveshort-ranged part (i.e. a hard-sphere core) that prevents two molecules fromboth bonding simultaneously to a third molecule. If

uHBð1; 3Þ 6¼ 0 and uHBð2; 3Þ 6¼ 0; ð8:2Þ

then the distance between molecules 1 and 2, r12, is less than the hard-coreradius for the molecules and u0(1,2)¼N. The Mayer f-function, given by

f ð1; 2Þ ¼ exp �uð1; 2Þ=kBT½ � � 1; ð8:3Þ

where T is temperature and kB is Boltzmann’s constant, can then be written as

f ð1; 2Þ ¼ f0ð1; 2Þ þ fHBð1; 2Þ; ð8:4Þ

where

f0ð1; 2Þ ¼ exp �u0ð1; 2Þ=kBT½ � � 1;fHBð1; 2Þ ¼ exp �uð1; 2Þ=kBT½ � � exp �u0ð1; 2Þ=kBT½ �: ð8:5Þ

When eq 8.3 is substituted into the cluster expansion for the distributionfunctions in the fluid, and simplified through cancellations in graphs by takinginto account eq 8.2, the result is a formal graphical expansion for the pairdistribution function in terms of renormalized hydrogen-bond f-functions. Thisconcept of saturation at the dimer level is a key element in Wertheim’s theory,discussed below.Chandler and Pratt13 developed a similar approach based on graph theory to

study systems undergoing chemical reaction. The formal theory is quite com-plex, but the application to a simple bimolecular reaction, e.g. the chemicalequilibrium between nitrogen dioxide and di-nitrogen tetroxide (N2O4"2NO2),illustrates the results obtained. For this reaction, Chandler and Pratt illustratedtheir results by calculating the solvent effect on the chemical equilibrium con-stant,

K ¼ r2NO2=rN2O4

; ð8:6Þ

where rNO2and rN2O4

are the number densities of the product andreactant, respectively. Under simplifying assumptions, the value of K in a

217SAFT Associating Fluids and Fluid Mixtures

liquid solvent versus its value in the ideal gas state at the same temperature K0 isgiven by

K0=K ¼Z

yN2O4ðrÞbN2O4

ðrÞdr�Z

bN2O4ðrÞdr; ð8:7Þ

where yN2O4ðrÞ is the cavity distribution function for N2O4 and bN2O4

ðrÞ is abonding Boltzmann factor. For the case that the bond is highly localized, so thatbN2O4

ðrÞ can be modelled as a Dirac delta function that is non-zero only at thebond length L, eq 8.7 simplifies to

K0=K ¼ yN2O4ðr ¼ LÞ: ð8:8Þ

We note that the characterization of the degree of reaction in terms of the cavitydistribution function is also one of the results of Wertheim’s theory ofassociation.Returning to the concept of steric saturation, Høye and Olaussen14 imple-

mented Andersen’s idea11,12 for a specific model. They considered a fluidconsisting of a binary mixture of hard spheres, all of the same diameter s, sothat the like intermolecular potentials are given by

u11ðrÞ ¼ u22ðrÞ ¼N ros0 r4s

:

�ð8:9Þ

while the unlike or cross intermolecular interaction has a deep attractive regionlocated inside s/2. That is,

u12ðrÞ ¼{0 ros=2

0 r4s=2:

�ð8:10Þ

The functional form inside s/2 was not specified; however, it is assumedto be many kBT, as would be expected for a chemical bond or strong asso-ciation. The Høye-Olaussen model is a realization of the conditions givenby eqs 8.1 and 8.2, so that Andersen’s analysis can be applied. Because ofthe choice of spherically symmetric potentials, the analysis is considerablysimplified, and Høye and Olaussen14 explicitly derived formulae for theequilibrium constant.In the spirit of the Høye and Olaussen model, Cummings and Stell15 later

considered an equimolar mixture of hard spheres of type A (species 1) and B(species 2), with potentials

u11ðrÞ ¼ u22ðrÞ ¼N ros0 r4s

;

�ð8:11Þ

218 Chapter 8

and a cross-interaction that has a deep attractive region located inside s/2 ofthe form,

u12ðrÞ ¼

�1 roL� w=2��2 L� w=2oroLþ w=2�1 Lþ w=2oros0 ros

8>><>>: ð8:12Þ

centred at the distance L and of width w. Such a model system is capable offorming dimers (i.e. the reaction AþB"AB is possible, and similar to theHøye and Olaussen model, the reaction/association saturates at the level ofdimers due to excluded volume of A and B). By considering the limits e1-N,e2-N and w-0, while holding the integral of the corresponding Mayerf-function constant, Cummings and Stell defined a model in which the ABinteraction becomes a sticky shell inside the hard core located at position r¼L.In addition to steric saturation at the level of dimers, the model can be solvedanalytically in the Percus-Yevick16 (PY) approximation, making it possible tocalculate the impact of temperature and density on the degree of association/reaction. Cummings and Stell went on to extend the model to homogeneousassociation (2A"A2),

17 to reactions within a solvent,18 as well as a near-cri-tical solvent.19 A number of interesting results come from this analysis – inparticular, the solution of the PY approximation for homogeneous association,in the limit of complete association, recovers the analytic solution of thereference interaction site model (RISM)20,21 for homonuclear diatomics.Following these works, in a series of seminal papers, Wertheim1–4 performed

a general analysis of the statistical mechanics of fluids that could associate intodimers, as well as (in the general case) higher-order multimers. By using agraphical expansion, Wertheim’s approach is similar in spirit to that ofAndersen11,12 and Chandler and Pratt,13 in that cluster expansions aremanipulated in view of simplifications arising from steric considerations.Wertheim’s theory is developed within the context of a specific model forassociation, specifically

uð1; 2Þ ¼ u0ð1; 2Þ þXa

Xb

fab r2 þ db O2ð Þ � r1 � da O1ð Þ��

; ð8:13Þ

where r1, r2 are the centres of mass and O1, O2 are the orientations of particles 1and 2, da, db are vectors from the molecular centres of particles 1,2 to the centresof the association sites a, b respectively, u0 is a reference potential (typically, ahard-sphere potential with hard-sphere diameter s) and fab is the associationpotential between site a on molecule 1 and site b on molecule 2. The simplestmodel for association, assuming sites a and b to be identical, is of the form

fðxÞ ¼ {0 xoa¼ 0 x4a

; x ¼ r2 þ d O2ð Þ � r1 � d O1ð Þj j;�

ð8:14Þ


where, to ensure steric saturation, (s� a)/2od¼ |d|os/2, which is illustrated inFigure 8.1. The case of a short-ranged attractive associating interaction locatedat the edge of the hard core follows when a¼ s. In the first paper in the series,1

for the case of a single association site, Wertheim derived the Mayer clusterexpansions for thermodynamic properties in terms of a singlet density r andmonomer density r0. This includes the Helmholtz energy A, and specifically theassociation contribution, AAssoc.. In the second paper in the series,2 Wertheimderived both perturbation theory and integral equation approximations to thefull expressions obtained using the formalism developed in the first paper. Inparticular, a simple first-order perturbation theory approach yields

AAssoc:

kBT¼ A� A0

kBT¼ N lnX � X

2þ 1

2

� �; X ¼ r0

r; ð8:15Þ

where N is the total number of monomeric units, whether bonded or not, and Xis the fraction of segments that are non-bonded, determined from

rð1Þ ¼ r0ð1Þ þ r0ð1ÞZ

g0ð12ÞfAð12Þr0ð2Þd2; ð8:16Þ

2

1 3

Figure 8.1 Illustration of steric inhibition of bonding beyond the dimer level. Theblack lines represent the boundary of the hard-core potential u0(i,j) whilethe small grey spheres represent the association sites. If molecules 1 and 2are in a bonded state (overlap of the association sites), the site of molecule3 cannot overlap that of either molecule 1 or molecule 2 without experi-encing a hard sphere overlap with molecule 1 or molecule 2. Thus, three-molecule bonding is forbidden.

220 Chapter 8

where g0(12) is the pair distribution of the reference fluid, fA(12) is the Mayerf-function of the association interaction f(x), the integral is over all possiblepositions and orientations of molecule 2 and the dependence of r and r0 onposition and orientation corresponds to the general case of a spatially inho-mogeneous fluid. For the spatially uniform case, eq 8.16 becomes

r ¼ r0 þ r20

Zg0ð12ÞfAð12Þ d2 ffi r0 þ r20g0ðsÞ exp �HB=kBTð ÞK : ð8:17Þ

The second approximate expression follows when the association potential f(x)is a square well of depth eHB and when, assuming that the bonding region islimited to a small range of distances near the hard-sphere diameter (i.e. a¼ s),we invoke

Zg0ð12Þ d2 ffi g0ðsÞ

ZfA 6¼0

d2 ¼ g0ðsÞK ;K ¼ZfA 6¼0

d2; ð8:18Þ

so that K is the volume available for bonding. From the definition of X in eq8.15, eq 8.17 becomes

r0X¼ r0 þ r20g0ðsÞ exp �HB=kBTð ÞK ) X

¼ 1

1þ r0g0ðsÞ exp �HB=kBTð ÞK : ð8:19Þ

In the subsequent papers in the series, Wertheim extended his analysis to mul-tiple association sites3 and to systems undergoing polymerization.4 His keycontribution was to show that it is possible to obtain the properties of anassociating or chain fluid based on knowledge of the thermodynamic properties(the Helmholtz energy and structure) of the monomer fluid. This is the basis ofthe now well-known Wertheim thermodynamic perturbation theory, and inturn, the basis of all SAFT equations of state. Interestingly, in this series of fourpapers, Wertheim did not present a single calculated result or any numericaltests of his proposed theories.In the following years Gubbins and co-workers extended Wertheim’s theory

to mixtures of spheres and chain molecules of given length, by considering thelimit of complete association and replacing the association bonds with covalent,chain-forming bonds. These studies culminated in a paper presented in FluidPhase Equilibria titled ‘‘SAFT: equation of state solution model for associatingfluids’’.22 In this and subsequent works, Gubbins and his co-workers presentedWertheim’s key result in an equation of state form that could be used to modelfluid-phase behaviour.22–24 The SAFT approach has proven to be extremelysuccessful for modelling associating and chain-like fluids, including polymers.Today, it is arguably considered the state-of-the-art method for modelling thethermodynamic properties and phase behaviour of complex fluids and hasfound application from small molecules such as xenon, carbon dioxide and


water, through to complex copolymers, and more recently, room temperatureionic liquids and amino acids.

8.3 SAFT Equations of State

In the SAFT approach, molecules are modelled as associating chains formed ofbonded spherical segments (referred to also as monomers), with short rangedattractive sites, of the form described in the previous section, used as appro-priate to mediate association interactions (cf. Figure 8.2). The Helmholtzenergy is written as the sum of four separate contributions:22,23

A

NkBT¼ AIdeal

NkBTþ AMono:

NkBTþ AChain

NkBTþ AAssoc:

NkBT; ð8:20Þ

where AIdeal is the ideal free energy, AMono. the contribution to the free energydue to the monomer-monomer repulsion and dispersion interactions, AChain the

Figure 8.2 Schematic illustration of the perturbation scheme for a fluid within theSAFT framework. The reference fluid consists of spherical hard segmentsto which dispersion forces are added and chains can be formed throughcovalent bonds. Finally association sites allow for hydrogen bonding-likeinteractions. Adapted from Fu and Sandler and reprinted with permissionof Ind. Eng. Chem. Res.25

222 Chapter 8

contribution due to the formation of bonds between monomeric segments, andAAssoc. the contribution due to association.We consider each of the terms in eq 8.20 in turn beginning with the free

energy of an ideal gas, which is given by

AIdeal

NkBT¼ ln rL3

� �� 1; ð8:21Þ

where r¼N/V is the number density of chain molecules, V the volume of thesystem, and L the thermal de Broglie wavelength incorporating the kinetic(translational, rotational, and vibrational) contributions to the partitionfunction of the molecule. Since the ideal term is separated out, the remainingterms are noted as residual free energies. The contribution to the Helmholtzenergy due to the monomer-monomer interactions can be written as,25

AMono:

NkBT¼ m

AMono:

NskBT¼ maMono:; ð8:22Þ

where Ns is the total number of monomer segments, m is the number of seg-ments per molecule and aMono. the Helmholtz energy per monomer segment.The chain contribution is given by

AChain

NkBT¼ �ðm� 1Þ ln yMono:ðsÞ; ð8:23Þ

where yMono. (s) is the cavity distribution function of the monomer fluid and isrelated to the radial distribution function gMono. (s) by yMono:ðsÞ ¼ gMono:ðsÞexp uMono:ðrÞ

kBT

�, where uMono.(r) is the pair potential between tangentially

bonded monomers with s the monomer segment diameter.If appropriate, association interactions are included in the model and

described via short-range (square-well) bonding sites. The contribution due toassociation for s sites on a molecule is then given by:26

AAssoc:

NkBT¼Xsa¼1

lnXa �Xa

2

� �þ s

2; ð8:24Þ

where the sum is over all s sites of type a on a molecule, and Xa is the fraction ofmolecules not bonded at site a, which can be obtained from the mass-actionequation:

Xa ¼1

1þPsb¼1

rXbDa;b

: ð8:25Þ


The function Da,b, which characterizes the association between site a and site bon different molecules, can be written as

Da;b ¼ Ka:bfa:bgMono:ðsÞ; ð8:26Þ

where fa;b ¼ exp �HBa;b

.kBT

� � 1 is the Mayer f-function of the a� b site-site

bonding interaction of potential depth eHBa,b and bonding volume Ka,b.

27

In the original SAFT approach22,23 chains of Lennard-Jones (LJ) monomersegments were modelled using the equation of state for argon developedby Twu and co-workers,28 and later the expression proposed by Cottermanet al.29 In these approaches, the radial distribution function in the chain andassociation terms is evaluated at the hard-sphere contact instead of at contactfor the true monomer LJ fluid. An interesting comment on the impact of thisapproximation can be found in reference 30.Since the SAFT equation of state has a firm basis in statistical mechanical

perturbation theory for well-defined molecular models, systematic improve-ment (e.g. by improving expressions for monomer free energy and structure)and extension of the theory (e.g. by considering new monomer fluids,bonding schemes and polar interactions) is possible by direct comparisonof the theoretical predictions with computer simulation results for the samemolecular model. At each stage in the development of the SAFT theory byGubbins and co-workers, the model was carefully verified against mole-cular simulation data for associating spheres, mixtures of associating spheresand non-associating chains.24,27,31,32 Due to its role in improving and validatingthe theory, the importance of having an underlying molecular model (in con-trast to traditional engineering equations of state) cannot be overemphasized.In those cases where SAFT does not compare well with simulation, the theorycan be improved through the use of better reference systems or higher-orderperturbation theory. In comparing with experimental data, it is importantto be aware of the errors inherent in the theory, as revealed by comparisonwith computer simulation results, before attempting to estimate inter-molecular model parameters. This invaluable advantage over empiricalequations of state is one of the keys to the success of the SAFT equation andhas led to many extensions and variations of the original SAFT expressions.These essentially correspond to different choices for the monomer fluid, anddifferent theoretical approaches to the calculation of its the free energy andstructure. In what follows we describe the main SAFT based-approaches foundin the literature.

8.3.1 SAFT-HS and SAFT-HR

The simplest SAFT approach, usually referred to as SAFT-HS, describesassociating chains of hard-sphere segments with the long-range attractiveinteractions described at the van der Waals mean-field level.26,27 The Helmholtz

224 Chapter 8

energy per monomer in this case is described by,

aMono: ¼ aHS � avdwrskBT

; ð8:27Þ

where aHS is the Helmholtz energy of the hard-sphere fluid, rs is the numberdensity of monomer segments and avdw the attractive van der Waals constantcharacterising the dispersion interactions. For pure fluids aHS is determinedfrom the expression due to Carnahan and Starling,33 with the correspondinghard-sphere radial distribution function evaluated at contact used in the chainand association terms. This simple approach is particularly suited to the studyof strongly associating fluids, such as water34,35 and hydrogen fluoride,36 wherehydrogen-bonding interactions mask the simplified description of the weakerdispersion forces.34–38

The most extensively applied version of SAFT, due to Huang andRadosz39,40 and commonly denoted SAFT-HR, corresponds to a similar levelof theory. In SAFT-HR the dispersion interactions are described through theexpression of Chen and Kreglewski,41 which was fitted to argon physical-property data, with the hard-sphere radial distribution function used in thechain and association terms. SAFT-HR has been applied to study the phasebehaviour in a wide range of fluid systems and polymers.42–60 Comparisonswith SAFT-HR are often provided when a new version of SAFT is developedto demonstrate the improved ability of a new equation; however, it should benoted that when comparisons are made, it is probably more relevant to refer toone of the more recent ‘‘second-generation’’ SAFT equations describedbelow.

8.3.2 Soft-SAFT

Johnson et al’s.61 proposed equation of state to treat LJ chains in which the freeenergy of the chain fluid was obtained using the free energy and radial dis-tribution function of a monomer LJ fluid, the expressions for which were fittedto simulation data.61,62 This approach was extended to mixtures by Ghonasgi etal.63 and Blas and Vega,64 who referred to the approach as the soft-SAFTequation of state.The equation of Johnson et al.61 is one of most accurate available for LJ

chains, since it is heavily based on computer simulation data. Equally the soft-SAFT equation is very accurate for modelling mixtures of associating LJ fluids.In their extension to mixtures Blas and Vega performed extensive simulationson model homo- and heteronuclear fluids and their mixtures to test the soft-SAFT approach, before application of the theory to real fluids in a subsequentpaper was undertaken by the authors.65 Soft-SAFT has been applied to thestudy of alkanes and their binary and ternary mixtures,65–75 perfluoro-alkanes,76–82 alcohols,71,83 carbon dioxide,70,71,83–87 polymers,88,89 and morerecently room temperature ionic liquids.90,91


8.3.3 SAFT-VR

SAFT was extended to describe associating chain molecules formed from hard-core monomers with attractive potentials of variable range in the SAFT-VRequation.92,93 Although, in the original publication, several potentials of vari-able range were studied (square-well, LJ, Mie and Yukawa), typically a square-well potential is implemented in the modelling of real fluids. In the SAFT-VRapproach the monomer fluid is therefore usually a square-well (SW) fluid, withthe monomer Helmholtz energy given by,

aM ¼ aHS þ a1

kBTþ a2

kBTð Þ2; ð8:28Þ

where a1 and a2 are the first two perturbative terms associated with theattractive energy. The radial distribution function of the square-well monomerfluid, which is obtained analytically using a self-consistent method for thepressure from the Clausius virial theorem and the density derivative of theHelmholtz energy, is used in the chain and association terms. The SAFT-VRequation has been extensively tested against simulation data93–98 and success-fully used to describe the phase equilibria of a wide range of industriallyimportant systems; for example, alkanes of low molar mass through to simplepolymers,92,99–103 and their binary mixtures,103–116 perfluoroalkanes,117–121

alcohols,83,122,123 water,35,124,125 refrigerant systems,126–129 and carbon diox-ide,111,114,130–135 have all been studied.While the application of the SAFT-VR equation has primarily focused on a

square-well potential, recent work has looked at the family of m� n Miepotentials.136 The SAFT-VR Mie approach of Lafitte et al.136 when comparedto the original SW-based SAFT-VR equation, PC-SAFT (discussed below) anda LJ-based SAFT-VR, was found to provide a more accurate description ofboth the phase behaviour and derivative properties, such as condensed phaseisothermal compressibility and speed-of-sound, of alkanes, alcohols and theirmixtures, suggesting that the variable ranged repulsive term is needed todescribe derivative properties.136–138

In related work, a modified SAFT-VR equation for square well potentialstermed SAFT1 has been proposed in which a truncation term is added to themonomer free energy expansion (eq 8.28) to account for higher-order terms.139

SAFT1 has been successfully used to study both simple fluids such asalkanes,140,141 alcohols,142 polymers,143–145 and more recently room tempertureas well as ionic liquids.146

8.3.4 PC-SAFT

In contrast to the other popular versions of the SAFT equation, the PC-SAFT147,148 approach considers a hard-chain, rather than a hard-sphere,as the reference system for the application of standard high-temperature per-turbation theory to obtain the dispersion contribution. Thus the Helmholtz

226 Chapter 8

energy is written,

A

NkBT¼ AIdeal

NkBTþ AHC

NkBTþ APC

NkBTþ AAssoc:

NkBT; ð8:29Þ

where AHC corresponds to the free energy of a reference hard-sphere-chain fluid,given by equation 8.23, i.e. in the usual form of Wertheim’s theory where themonomer fluid at this point is a fluid of hard-spheres, and APC describes the chainperturbation contribution, which is usually taken to second-order. APC is basedon an earlier theory for square-well chain molecules149 and is determined from aTaylor series expansion fitted to the vapour-liquid phase envelopes of purealkanes.147 This fitting greatly enhances the accuracy of the approach in com-parison with experimental systems, but unfortunately means that it is no longerstraight forward to define the underlying intermolecular potential of the modeland so comparison against computer simulations cannot be used in the assess-ment of further theoretical developments. The association contribution used inPC-SAFT follows directly from Wertheim’s theory as given by eqs 8.24 to 8.26.In the original presentation of PC-SAFT parameters were correlated against

vapour pressure and saturated liquid density data for 78 non-associating purefluids and shown to work well in the description of mixture systems. Subse-quently the equation has been successfully applied to the study of a wide rangeof industrially important fluids from simple binary mixtures involving hydro-carbons150–173 to associating fluids,149,162,171,174–181 pharmaceuticals,182 andasphaltenes,183–187 and, in particular, polymer systems.88,188–215

In related work a simplified version of the PC-SAFT equation of state, denotedsPC-SAFT has been developed by von Solms and co-workers216 in an effort toreduce the complexity of the original equation when studying mixture systems.While the ideal and dispersion contributions to the free energy remain unchangedfrom those of the original PC-SAFT equation, the hard chain is simplified throughthe use of an average segment diameter and the use of the Carnahan-Starlingexpressions for the free energy and radial distribution function of a pure com-ponent hard-sphere system. The assumption that all of the segments in the mixturehave the same diameter in turn simplifies the calculation of the association con-tribution. The sPC-SAFT equation has been successfully used to study bothsimple fluids217–221 and a wide range of polymer systems.191,192,196,199,207,222–225

8.3.5 Summary

Comparisons between the different versions of SAFT are difficult in that manyfactors contribute to the observed agreement or disagreement with experi-mental data. For example, the type of thermodynamic data and the rangeof thermodynamic conditions used in the parameter optimization proce-dure must be consistent for a direct comparison and unfortunately suchinformation is not always clearly described in the literature. A true compa-rison between the different versions of SAFT is therefore only meaningful if a


direct comparison between the different versions of SAFT is carried out withina given study. While there have been such comparisons with SAFT-HR, alimited number of studies have compared the second-generation versions ofSAFT,83,88,122,175,181,207,226 from which it is clear that any one version is notsuperior over the others in general terms. The choice of one version of SAFTover another is often more ‘‘philosophical’’ (versions like SAFT-VR keep amore formal link with the molecular model, while PC-SAFT provides higheraccuracy in systems where dispersion dominates), than objective in terms ofoverall performance and capability. A more interesting point to note here, is theversatility of the SAFT method as highlighted by the examples in the previoussections, both in terms of the accuracy of prediction of the phase behaviour offluids and fluid mixtures ranging from very simple systems to highly complexones, and in terms of its firm molecular basis, which allows for a systematicimprovement of the theoretical approximations (the subject of the followingsections). In this respect, the method constitutes a major advance over tradi-tional cubic engineering equations.

8.4 Extensions of the SAFT Approach

8.4.1 Modelling the Critical Region

As with all equations of state that are analytical in the free energy, the SAFTequation does not accurately capture the behaviour of fluids in the criticalregion. In the vicinity of the critical point the behaviour of a fluid is stronglyinfluenced by long-range fluctuations in the density that act to lower the criticalpoint and flatten the vapour-liquid coexistence curve. Near the critical point,the thermodynamic behaviour is described by non-analytic scaling laws, withuniversal scaling functions and universal critical exponents,227 while the SAFTequation as a mean-field theory exhibits classical critical exponents, leading to aparabolic coexistence curve in the critical region. For a review of the thermo-dynamic behaviour of fluids in the critical region the reader is referred toChapter 10.In order to study critical lines and global fluid phase diagrams, one method

to overcome the theoretical over prediction of the critical point in mean-fieldapporaches is to re-scale the model parameters to the experimental criticaltemperature and pressure; however, agreement at the critical point comes at thecost of poorer agreement at low temperatures and pressures, as this approachhas the effect of shifting the coexistence curves to lower temperatures andpressures, without changing the shape of the vapour pressure curve, and resultsin poor descriptions of the coexisting densities. An accurate thermodynamicdescription of the whole fluid phase diagram is possible through the use ofso-called crossover treatments that correct the classical equation of state tosatisfy the asymptotic power laws near the critical point and incorporate acrossover to regular thermodynamic behaviour far from the critical point,where the effects of critical fluctuations become negligible.

228 Chapter 8

The modern theory of critical phenomena is based on renormalization-grouptechniques that yield asymptotic scaling laws with critical exponents and cri-tical-amplitude ratios, which for systems with short-range interactions dependon the dimensionality of the system and the number of components of the orderparameter.229 Fluids are believed to belong to the universality class of three-dimensional Ising-like systems, i.e. systems with a scalar order parameter;however, the validity of the asymptotic scaling laws for fluids near the criticalpoint is restricted to a rather small range of temperatures and densities.230

Therefore several attempts to develop a theory that accurately treats thecrossover from critical behaviour asymptotically close to the critical point toclassical behaviour sufficiently far from the critical point have been made by anumber of investigators; here we naturally focus on those applied to the SAFTfamily of equations. In particular, the approach of White,231–235 as imple-mented by Prausnitz and co-workers,236–239 and that of Kiselev,240–242 havebeen the most widely used and applied to many of the SAFT equationsdescribed in section 8.3. Although most crossover-SAFT studies to date havefocused on pure fluids, both approaches have been extended to mixtures usingthe so called iso-morphism assumption, which assumes that the thermodynamicpotential for a mixture has the same universal form as that for a pure fluid whenan appropriate isomorphic variable is chosen. While the density of the systemwould be a natural choice to replace the density of the pure fluid as the orderparameter, the isomorphism approximation requires the evaluation of thechemical potentials of each component as independent variables, rather thanthe mole fraction typically used in equations of state. A general procedure forincorporating the scaling laws into any classical equation of state for mixtureswith the mole fractions as independent variables was proposed by Kiselev andFriend242 and adopted by Cai and Prausnitz in the application of the method ofWhite to mixtures.243

White’s approach, based on the work of Wilson,244 incorporates densityfluctuations in the critical region using the phase-space cell approximation anduses a recursive procedure to modify the free energy for non-uniform fluids,thereby accounting for the fluctuations in density. The interaction potential isdivided into a reference contribution uref, due mainly to the repulsive interac-tions, and a perturbative contribution up, due mainly to the attractive interac-tions. Given the short-range nature of the repulsive term, the renormalization isonly applied to the attractive part of the potential, which is then divided intoshort-wavelength and long-wavelength contributions. The effect of the short-wavelength contributions less than a given cut off length L is calculated usingmean-field theory, while the contributions due to the long-wavelength fluctua-tions are taken into account through the phase-space cell approximation in a setof recursive relations that successively take into account the contribution oflonger and longer wavelength density fluctuations. The Helmholtz energy den-sity an(r) for a system at a density r can be described in a recursive manner as:

anðrÞ ¼ an�1ðrÞ þ danðrÞ ð8:30Þ


and accounts for the long-wavelength fluctuations through the following rela-tions:

danðrÞ ¼ �Kn lnOs

nðrÞOl

nðrÞ; 0 � r � rmax

2; ð8:31Þ

danðrÞ ¼ �0;rmax

2� r � rmax: ð8:32Þ

Here Os and Ol are the density fluctuations for the short-range and long-rangeattraction, respectively, rmax the maximum possible molecular density, and Kn isdefined by

Kn ¼ kBT23nL3: ð8:33Þ

The density fluctuations are calculated through evaluation of the followingintegral,

OsnðrÞ ¼

Zr0

exp�Gb

nðr; xÞKn

� �dx; ð8:34Þ

where

Gbnðr; xÞ ¼

�f bn ðrþ xÞ þ 2�f bn ðrÞ � �f bn ðr� xÞ2

ð8:35Þ

and b refers to both the short (s) and long (l) range attraction andGb depends onthe function �f as

�f lnðrÞ ¼ fn�1ðrÞ þ aðmrÞ2; ð8:36Þ

�f sn ðrÞ ¼ fn�1ðrÞ þ aðmrÞ2 fw2

22nþ1L2; ð8:37Þ

where f is an adjustable parameter, a is the interaction volume, and w representsthe range of the attractive potential, defined as

w2 ¼ � 1

3a

Zr2upðrÞdr: ð8:38Þ

Vega and co-workers have applied the method of White to the soft-SAFTequation68,71,74,245 to study the vapour-liquid equilibrium and derivativeproperties of the n-alkanes, carbon dioxide and 1-alkanols and their mixtures.The approach has also been combined with the PC-SAFT equation by Fu

230 Chapter 8

et al.158 and by Bymaster and co-workers246 to study the vapour-liquid equi-libria of the alkanes, and by Mi et al.247–252 who have considered several SAFT-like equations.247–252

The advantages of this method are that only a small number of additionalparameters are needed (between 2 and 3) and in principle the crossover treat-ment can be applied without altering the original SAFT molecular parameters;however, in practice such an approach is rarely taken, and the SAFT andcrossover parameters are more commonly simultaneously fitted to experimentaldata in order to obtain accurate agreement with experiment both near to andfar from the critical region. As discussed by Bymaster and co-workers,246 oneor both of the additional crossover parameters f and L can be fitted or fixeddepending upon the implementation of White’s method. Llovell and co-workers68 were the first to use both f and L as adjustable parameters in theirapplication of White’s method to the soft-SAFT equation and a new set ofparameters were proposed. Subsequently, Bymaster and co-workers246 foundthat good results could be obtained using the soft-SAFT equation withoutaltering the original SAFT parameters, while in the case of the PC-SAFTequation of state a soft additional fitted parameter (beyond f and L) is neededin order to describe the critical region of longer chain molecules precisely; aneffect most likely due to the fitted nature of the dispersion potential in the PC-SAFT equation. Bymaster et al.246 also noted that while Fu and co-workers158

fixed both f and L they made several changes to the implementation of White’smethod. A drawback of White’s method is that it can only be solved numeri-cally and so does not lead to a closed-form expression for the equation of state.The approach of Kiselev, based on the work of Sengers and co-workers253–255

and Kiselev and co-workers,240–242 utilizes a renormalized Landau expansionthat smoothly transforms the classical Helmholtz energy density into anequation that incorporates the fluctuation-induced singular scaling laws nearthe critical point, and reduces to the classical expression far from the criticalpoint. The Helmholtz energy density is separated into ideal and residual terms,and the crossover function applied to the critical part of the Helmholtz energyDa(DT, Dv), where Da(DT, Dv)¼ a(T, v)� abg(T, v) and the background con-tribution abg(T, v) is expressed as,

abgðT ; vÞ ¼ �Dv �P0ðTÞ þ ares0 ðT ; vÞ þ a0ðTÞ: ð8:39Þ

More expliclty,240–242

DaðDT ;DvÞ ¼ aresðDT ;DvÞ � ares0 ðDTÞ � lnðDvþ 1Þ þ Dv �P0ðDTÞ: ð8:40Þ

In eqs 8.39 and 8.40 DT¼T/T0c� 1 and Dv¼ v/v0c� 1 are dimensionless dis-tances from the calculated classical critical temperature (T0c) and classicalcritical molar volume (v0c), a0(T) is the dimensionless temperature-dependentideal-gas term, and �P0ðTÞ ¼ PðT ; v0;cÞv0;c=RT and ares0 ðT ; vÞ are the dimen-sionless pressure and residual Helmholtz energy along the critical isochore,respectively. The DT and Dv are then replaced with the renormalized values in


the critical part of the free energy, using

�t ¼ tY�a=2D1 þ ð1þ tÞDtcY2ð2�aÞ=3D1 ð8:41Þ

and

D�j ¼ DjY ðg�2bÞ=4D1 þ ð1þ DjÞDvcY ð2�aÞ=2D1 ; ð8:42Þ

so that DT ! �t and Dv! D�j in eq 8.40, where the superscripts correspond tothe critical exponents (see Table 8.1), t¼T/Tc� 1 is the dimensionless devia-tion of the temperature from the real critical temperature Tc, Dj¼Vm/Vm,c� 1is the dimensionless deviation of the molar volume from the real critical molarvolume Vm,c, and Dtc¼DTc/T0c¼ (Tc�T0c)/T0c and DVm,c¼DVm,c/Vm.0c

¼(Vm,c�Vm,0c

)/Vm,0care the dimensionless shifts of the critical parameters. The

crossover function Y(q) is given by,

YðqÞ ¼ q

1þ q

� �2D1

; ð8:43Þ

where q is a renormalized distance to the critical point and is obtained from thesolution of the crossover sine model,

q2 � tGi

� 1� p2

4b21� t

q2Gi

� ��

¼b2Dj 1þ v1Dj2 expð�10DjÞ þ d1t �

Gib

� �Y

1�2bD1

; ð8:44Þ

where v1, d1, and Gi are system-dependent parameters and p2 and b2 universalparameters.256–258 The term proportional to d1t in equation (8.44) corresponds

Table 8.1 Definitions and values of the major critical exponents.227,228 Thequantities KT and cV are the isothermal compressibility and con-stant volume specific heat capacity respectively. rliq and rvap are thedensities of the coexisting liquid (liq) and vapour (vap) phases, Tthe temperature, and Tc and Pc the critical temperature and pres-sure respectively.

Exponent Definition Classical Non-classical

a cVE(T�Tc)�a, r¼ rc, T-Tþc 0 0.110� 0.003

b rliq� rvapB(Tc�T)b, T-T�c 0.5 0.326� 0.002

gKT � �

1

V

@V

@P

��B T � Tcð Þ�g;T ! Tþc1 1.239� 0.002

d P� PcB rliq � rvap�� d;T ¼ Tc

3 4.8� 0.02

232 Chapter 8

to the rectilinear diameter of the coexistence curve, while the term proportionalto v1Dj

2exp(� 10Dj) ensures that Y¼ 1 at the triple point, hence, ensuring thecrossover is complete.While the approach of Kiselev requires more system-dependent parameters

(3-5) than the method of White, it does yield a closed-form analytical expres-sion. We also note that Kiselev recently proposed an analytical formulation forthe crossover sine model, which simplifies the calculation of the crossoverfunction and its first and second order derivatives.259 Crossover equations ofstate based on the approach of Kiselev have been developed for non-asso-ciating241,256,257 and associating fluids258 based on the SAFT expressions ofHuang and Radosz39,40 (SAFT-HR) and applied to study the phase behaviourof alkanes,241,259 refrigerants,256 water and ammonia.260 However, while thecrossover formulation improved the theoretical description in the vicinity of thecritical point compared to the classical equation of state alone, deviations fromexperimental data were in general observed at lower temperatures, which can beattributed to the use of the SAFT-HR equation. The need for an accurateunderlying classical equation of state was demonstrated by McCabe andKiselev102 with the development of the SAFT-VRX equation based on thecombination of Kiselev’s crossover technique and the SAFT-VR equation ofstate (see Figure 8.3). The SAFT-VRX approach has been shown to provide anexcellent description of the PVT and phase behaviour of both low and high

100

120

140

160

180

200

220

0

T/K

3010 20�/mol⋅dm-3

Figure 8.3 Temperature T as a function of coexisting liquid and gas densities r formethane. — SAFT-VRX;- - - -, SAFT-VR; – � –, SAFT-VR with rescaledparameters; � � � � � � , crossover SAFT-HR; &, experimental data. Rep-rinted with permission of Ind. Eng. Chem. Res.102


molar mass alkanes, carbon dioxide, water and their mixtures close to and farfrom the critical point.101,102,111,112

We conclude by noting that while it has been shown that crossover versionsof classical equations of state, including SAFT-like equations, typically provideaccurate predictions for vapour-liquid equilibria, PVT,112,261 and derivativeproperties74,245 of both pure fluids and their mixtures close to and far from thecritical region, such methods have yet to be applied to the study of liquid-liquidequilibria with the SAFT approach.

8.4.2 Polar Fluids

As described in section 8.3, the original form of the SAFT equation of state doesnot explicitly account for polar (permanent and induced) interactions. Inapplications of the SAFT equation of state to fluids with dipolar and quad-rupolar interactions, their effect is often taken into account implicitly throughfitted model parameters for the dispersion interaction, and sometimes throughthe use of association sites.262 For example, for a polar (1) and non-polar (2)mixture the presence of a 1-1 dipole interaction and absence of a 1-2 dipoleinteraction can be modelled by a 1-2 dispersion term that is much less than thegeometric mean average of the 1-1 and 2-2 values. A possible justification for thelatter approach is that at high temperature a Boltzmann averaging of the dipole–dipole interaction energy over all orientations leads to an angle-averaged (i.e.angle-independent) interaction varying as the sixth inverse power of inter-molecular distance, called the Keesom potential,263 which can be treated ascontributing to the overall van der Waals (dispersion) intermolecular interac-tion, while directional effects such as the formation of chains seen at low tem-perature in dipolar fluids are treated via association sites. Of course, directincorporation of dipolar interactions treats both of these regimes automatically.The absence of explicit polar terms in the model typically leads to large devia-tions from the usual geometric mean for the cross dispersion interaction and theneed for binary-interaction parameters in order to study mixtures of polar fluids,and so makes the approaches more dependent on experimental data and henceless predictive. Though we should note that the SAFT-VR approach92 (seesection 8.3.3), which accounts for the dispersion interactions through a variable-ranged potential, provides added flexibility over versions of SAFT based onfixed range potentials in terms of describing polar interactions; however, binary-interaction parameters fitted to experimental data are still typically needed inorder to study polar fluids with the SAFT-VR equation. In recent work Haslamet al.133 propose a predictive method to obtain unlike intermolecular interactionparameters in these systems with some degree of success (see section 8.5.3).Since polar interactions can have a significant effect on the phase behaviour

of both simple fluids and polymers, various approaches to explicitly incorpo-rate polar interactions have been proposed; in principle, these should provide amore precise description of the interactions in polar fluids and greater pre-dictive capabilities.

234 Chapter 8

Methods to describe polarity can be broadly classified into two groups: so-called molecule-based and segment-based approaches. Within the latter a dis-tinction can also be made between expressions in which the polar contributionis applied only at the level of the thermodynamics (the Helmholtz energy) of themonomer fluid and those in which the effect of polar interactions is also con-sidered at the level of the structure of the fluid.The first attempts to account for polar interactions within a SAFT-based

equation used perturbation theory to include the dipolar interaction approxi-mated as a third-order expansion in which the two- and three-body terms arecalculated explicitly, while the higher terms are approximated by the Padeapproximant following the work of Stell.264–266 Examples include the workof Walsh et al.267, who incorporated the expressions of Gubbins et al.268 forthe polar expansion together with a LJ version of SAFT, Kraska andGubbins269,270 who used the dipole-dipole interaction term of Gubbins andTwu271,272 to extend the Lennard-Jones-based SAFT equation (LJ-SAFT) ofMuller and Gubbins273 to study the phase behaviour of alkanes, alcohols, waterand their mixtures, and Li et al.274 who studied the critical micellar con-centrations of aqueous non-ionic surfactant solutions, again using the expres-sions of Gubbins and Twu to describe the dipole-dipole interactions in aLennard-Jones-based SAFT equation. We note that Kraska and Gubbins alsoattempted to account for the effects of polarizability through a state-dependenteffective dipole moment, since dipole-dipole induction leads to an increase inthe effective dipole moment in the liquid phase, following the work ofWertheim.275,276

A drawback of these molecule-level approaches is that the polar term isintroduced by assuming a single spherical segment, which is then used as thereference for the polar expansion, that effectively maps onto the chain; this issomewhat inconsistent with the description of dispersion interactions on asegment basis. As noted by Jog and Chapman,277 although this approach pre-serves the idea that the contribution to the fluid properties of a single dipole on amolecule becomes weaker as the size of the molecule increases, it is in pooragreement with molecular simulation results and underestimates the effect ofdipolar interactions for chain molecules. Additionally, application of the dipolarcontribution at the molecular level only allows for the description of a singlepolar group per molecule, which makes the extension of the approach topolymers with multiple polar groups in the repeat unit difficult. To overcomethese issues Jog and Chapman proposed a SAFT-HR-based approach fordipolar fluids in which the Pade approximant dipolar term of Rushbrook278 wasapplied at the segment, rather than molecular, level,279 with an additional modelparameter xp introduced to determine the fraction of dipolar segments in themolecule. For chains with a single dipolar site, xp should be equal to 1/m;however, for real fluids it is treated as an adjustable parameter. The authors alsoshowed that the contact value of dipolar hard-spheres can be approximated bythat of hard spheres in specific orientations277 and use the radial distributionfunction of a hard-sphere fluid in the chain term instead of that for a mixture ofhard and dipolar segments (i.e. the monomer fluid). The association and chain


terms remain unchanged from those in the original SAFT-HR equation and sothe effects of the dipole are not considered at the chain level of the theory. Thispolar-SAFT-HR approach was then applied to study the phase behaviour of(acetoneþ alkanes), with significant improvement over the original SAFT-HRequation seen, and to the study of cloud points in poly(methylpropenoate;ethene)þ butane, which further illustrated dipolar effects play an important rolein the phase behaviour.279 Note poly(methylpropenoate; ethene) is also knownas poly(ethylene-co-methyl acrylate).The Jog and Chapman term (referred to from hereon by JC) has been applied

by a number of authors to model polar fluids in combination with severalSAFT equations of state.280–283 In particular, Chapman and co-workers281,282

applied this approach to the PC-SAFT equation of state and evaluated theperformance of the JC dipolar term compared to a segment-level version of aterm proposed by Saager and Fischer.284,285 In the approach of Saager andFischer (SF) the polar contribution to the Helmholtz energy is based onempirical expressions fitted to simulation results of the vapour-liquid equilibriafor two-centre LJ plus point dipole (2CDLJ) molecules. These were thenmodified in the spirit of the JC approach in order to apply the expressions at thesegment level and take into account the non-spherical shape, and multiple-polar groups, of molecules; as in the original JC approach a model parameterxp was introduced. The addition of the dipolar contribution in both cases doesnot modify the remaining contributions to the free energy as given in the ori-ginal PC-SAFT equation. The two approaches (PC-SAFT-JC and PC-SAFT-SF) were used to study the phase equilibrium of ethers and esters and foundto yield very similar results; however, the parameters obtained from the PC-SAFT-JC equation were thought more physically meaningful and thereforeexpected by the authors to be superior to those obtained from the PC-SAFT-SFequation.281

Following the approach of Saager and Fischer of using molecular simulationdata to develop empirical expressions for the polar contributions to theHelmholtz energy, Gross and Vrabec (GV) have proposed a contribution fordipolar interactions in non-spherical molecules, again based on a fit to vapour-liquid equilibrium data for 2CDLJ molecules.286 The contribution, based onthird-order perturbation theory written in the Pade approximation, overcomesthe problems encountered when applying the SF term to strongly asymmetricsystems due to the empirical nature of the polar expressions. The proposed GVterm, has been implemented in the PC-SAFT equation of state, resulting in thePCP-SAFT equation, which has been used to study the phase behaviour of(dimethylsulfoxideþmethylbenzene) and (propanoneþ alkane) binary mix-tures.286 We note that the inclusion of the GV dipolar contribution does notintroduce an additional fitted model parameter (i.e. beyond xp), if experimentalvalues are used to determine the dipole moment as done in the original work ofGross and Vrabec.286 The PCP-SAFT and PC-SAFT-JC methods have beencompared in a number of ways: when the experimental value of the dipolemoment is used and xp is assumed equal to 1/m, and not treated as anadjustable model parameter, the PC-SAFT-JC equation incorrectly predicts

236 Chapter 8

liquid-liquid demixing at conditions where the corresponding experimentalsystems are fully miscible; when either xp or the dipole moment are fitted topure-component experimental data, the best results are obtained when theseparameters are zero (i.e. when the dipole term goes to zero); and finally, ifbinary mixture data are also used in the determination of the pure-componentparameters, as in the work of Dominik et al.,281 the PC-SAFT-GV model isshown to be in better overall agreement with experimental data for the systemsstudied than the PC-SAFT-JC model. The PC-SAFT equation with the SFdipolar contribution has also been studied, though minimal improvement overthe original PC-SAFT equation was seen for the systems studied.286

Kleiner and Gross subsequently considered the effect of molecular polariz-ability and induced dipole interactions through the combination of PCP-SAFTand the renormalized perturbation theory of Wertheim;275,276 however, for the36 polar fluids and their mixtures with hydrocarbons studied only slightimprovements in the agreement with experimental data was observed comparedto the results obtained from the PCP-SAFT equation alone.287

Using the perturbation theory proposed by Larsen et al.288 Karakatsani andEconomou289,290 have extended the PC-SAFT equation to account for dipole-dipole, dipole-quadrupole, quadrupole-quadrupole and dipole-induced dipoleinteractions. The exact second- and third-order perturbation terms in the workof Larsen are however rather complex and so a simplified version of the twoterms was also proposed based on the work of Nezbeda and co-workers.291,292

While simplifying the equation, in an effort to generate a more usable, engi-neering-type, approach, the simplification of the dipolar term in the modelintroduces an additional pure-component model parameter. We refer to theseas the PC-SAFT-KE and truncated PC-SAFT-KE (denoted PC-PSAFT andtPC-PSAFT by the authors respectively) approaches. In both KE approaches,the multipoles are assumed to be uniformly distributed over all segments in themolecule. The truncated and full PC-SAFT-KE approaches have been appliedto study a wide range of polar fluids and their mixtures,289,290,293,294 with thetruncated equation found to be as accurate as the full PC-SAFT-KE model,290

and the inclusion of polar interactions found to improve the theoretical pre-dictions in most cases.293

Recently, Al-Saifi and co-workers295 published an excellent comparison of theability of the PC-SAFT equation of state with the JC, GV and KE dipolar termsto describe the phase behaviour of 53 binary mixtures containing water, alcohol,or hydrocarbons. As an example, Figure 8.4 shows the results reported in ref 295for (methanolþ hexane). In general the PC-SAFT-JC equation was found toexhibit the best overall agreement with experimental data; although both PC-SAFT-JC and PC-SAFT-GV predict erroneous liquid-liquid behaviour for themethanol-hydrocarbon system at low temperatures. The agreement in the caseof the JC method was in part attributed to the magnitude of the dipolar con-tribution, which is larger than in the GV and KE terms; this results, for example,in better predictions for systems with two polar functional groups.As is clear from the discussion above, the majority of SAFT-based equations

for polar fluids focus on the dipole-dipole term, though some studies have also


considered dipole-quadrupole and quadrupole-quadrupole interactions. Theeffect of the inclusion of a quadrupolar term on the phase behaviour of(nitrogenþ hydrocarbon) was studied by Zhao et al.113 using the SAFT-VRequation and a quadrupolar term due to Benevides,296 which in turn is based inthe work of Larsen.288 Although the improvement in the prediction of thephase behaviour was small, the inclusion of a quadrupolar term reduced thenumber of binary-interaction parameters needed to study the nitrogen withalkane homologous series from two to one. Additionally, a quadrupolar termwas proposed by Gross297 in work similar to the development of the GVdipolar term; namely an expression for quadrupole-quadrupole interactionswas derived on the basis of third-order perturbation theory with model con-stants fitted to molecular simulation data for the two-centre LJ plus quadrupolefluid. The approach was tested through implementation in the PC-SAFTequation of state for mixtures of carbon dioxide and hydrocarbons, withimprovement over the original equation of state being observed. Recently,Vrabec and Gross have also proposed a term to describe the dipole-quadrupolecross interactions;298 however it has not yet been implemented and testedwithin the SAFT framework.Lucas and co-workers299 recently performed an extensive study to probe the

applicability and predictability of the PC-SAFT equation of state with both theGV dipolar term286 and a quadrupolar term due to Gross.297 We note thisequation is also termed the PCP-SAFT equation by Lucas and co-workers.

0.30

0.25

0.20

0.15

0.10

p/M

Pa

343 K

333 K

348K

x(CH3OH)

0.0 0.2 0.4 0.6 0.8 1.0

Figure 8.4 Pressure p as a function of methanol mole fraction x(CH3OH) for themethanolþ hexane vapour–liquid equilibrium at temperatures of (333,343 and 348) K. K, measurements; —, results predicted from the PC-SAFT equation with the JC polar term; – – –, results predicted from thePC-SAFT equation with the GV polar term; and – � – � –, results predictedfrom the PC-SAFT equation with the KE polar term. Reprinted withpermission of Fluid Phase Equilibria.295

238 Chapter 8

Gross and Vrabrec developed the polar terms used for systems consisting ofone multipolar molecule with either a dipole or a quadrupole and one non-polar molecule. Perhaps not surprisingly, therefore the theory was found to failfor systems in which one molecule has a significant dipole and the other asignificant quadrupole.299 Lucas and co-workers use multiple moments deter-mined from quantum mechanical calculations and so the experimental dipoleand quadrupole moments are not needed (see Section 8.5.2). In a subsequentpaper300 they compared their PCP-SAFT equation with a new approach inwhich the PC-SAFT equation is combined with polar terms derived fromperturbation theory with a spherical reference (based on a Pade approximationfor multipolar species with constants from one-centre LJ MC simulations)301

and an equation that combines the PC-SAFT equation with the GV term forthe dipolar interactions and spherical-reference perturbation theory for allother multipolar terms (dipole-quadrupole and quadrupole-quadrupole). Theequation based on the latter approach was found to out-perform both theiroriginal PCP-SAFT model and the new polar contribution.An alternative to perturbation expansions for inclusion of polar terms into

SAFT-based equations of state is via integral-equation theory. Whilst perhapsmore complex, integral-equation approaches do have the advantage of yieldinganalytic expressions for both the structure and the thermodynamics of the fluid,that are applicable (although not necessarily accurate) over wide ranges oftemperature, density, and interaction strength. In this spirit, Liu et al.302 haveproposed a Yukawa-based SAFT equation in which dipole-dipole interactionsare treated in a molecule-based (as opposed to segment-based) approach andexplicitly calculated with the analytical solution of the mean-sphericalapproximation (MSA) for the hard and dipolar Yukawa fluids due to Duh andMier-y-teran303 and Henderson et al.304 respectively; the corresponding termsfor the structure of the fluid are not considered in the chain term. Although thenew SAFT-based approach was found to provide a better representation ofboth polar and non-polar fluids than SAFT-HR, for non-polar and associatingfluids SAFT-LJ269 was in general in better overall agreement with experimentaldata.Zhao and McCabe have also used integral equation theory to propose a

SAFT-VR-based equation for polar fluids (SAFT-VRþD) in which dipolarinteractions are described through the solution of the generalized mean-sphe-rical approximation (GMSA) of Stell and co-workers.305 In the SAFT-VRþDequation the dipolar contribution is given at the segment level, with the dipolarsquare-well fluid (or a mixture of square-well and dipolar square-well fluids)describing the monomer fluid from which chains are formed. The effect of thedipolar interactions on the structure of the fluid is taken into account in thechain term through the use of a dipolar square-well radial distribution function.This explicit account of the effects of the dipolar interactions on both thethermodynamics and structure of the fluid is unique to the SAFT-VRþDequation. Following testing of the theory through comparison with molecularsimulation data,306,307 the SAFT-VRþD equation has been applied to studythe phase behaviour of hydrogen sulphide, water, and their binary mixtures,308


as well as (hydrogensulphideþ alkanes).309 Since experimental values for thedipole moment are used, no additional parameters are introduced into thetheory compared to the original SAFT-VR equation. In general, the SAFT-VRþD equation is found to improve the agreement with experimental dataover the original SAFT-VR approach; however, the improvement in some casesis slight, which could be due to the incorporation of a variable range potentialin SAFT-VR (see section 8.3.3).We conclude this section by noting that the general consensus from SAFT

studies involving polar contributions is that the explicit inclusion of a dipolarterm in general helps the predictive ability, as was shown most clearly in thework of Sauer and Chapman who compared polar versions of the PC-SAFTand original SAFT approach.283 However, the effect on the predictive capabilityof including polarizability and induced dipoles appears to be minimal and theadded complexity not justified.287 One problem to note is the use of fixed dipolemoments, be they experimental values or obtained from ab initio calculations,they will always limit the predictive capability of the approach since the effect ofchanges in temperature and state conditions on changes in the multipolemoments is not captured. This is clearly seen when one considers the bare dipolemoment of water, which has a value of 1.8D (1DE3.33564 � 10�30C �m),compared to condensed-phase estimates of between (2 and 3)D and the effect onthe solvation properties.310 While this is not an issue in the study of single-phasesystems or systems over narrow ranges of temperature and pressure, this willhave an effect when studying phase-equilibria properties and will hamper thedevelopment of a truly predictive approach.

8.4.3 Ion-Containing Fluids

The presence, or addition, of charged species, yielding an electrolyte solution,significantly affects the thermodynamic properties of fluids. Electrolyte solutionsare ubiquitous and of fundamental importance in both naturally occurring andindustrial processes (such as natural biological and batch biochemical processes,geochemistry, energy conversion, electrochemistry, corrosion and pollution).311–314

The development of a successful theory for charged fluids relies on the solution oftwo problems: firstly, an accurate intermolecular potential model must be pro-posed and secondly, the statistical mechanics relevant to the system must besolved with some precision. The long-range Coulombic interactions, includingpermanent and possibly induced multiple moments, make the solution of thestatistical mechanics of systems containing ions particularly difficult.Many models for electrolyte solutions have evolved from the Debye-Huckel

(DH) limiting law,315 which regards the ions as point charges immersed in adielectric continuum. In solution each ion of charge qi is surrounded by a co-sphere of oppositely charged ions, in a way that the ions interact not throughtheir bare Coulomb potential but through a screened potential, which is foundto take the form of a Yukawa-type potential (EA exp(� kr)/r where k is theinverse of the Debye screening length) and is obtained by solving the Poissonequation assuming a Boltzmann distribution for the charge distribution. The

240 Chapter 8

DHmodel yields a simple analytical expression for the Helmholtz energy, in theform

AElect:

NkBT¼ � k3

12pr; ð8:45Þ

where 1/k, the Debye screening length, is a measure of the diameter of the co-sphere, and is calculated from

k2 ¼ 4p�kBT

Xi

q2i ri: ð8:46Þ

In this equation, ri is the number density of ionic species i, e is the dielectricconstant of the solvent, and the sum is over all ionic species.The DH theory has served as the basis for many semi-empirical models for

electrolyte solutions, including the extended DH model,316 the models ofReilley and Wood317,318 and Scatchard et al.319 Perhaps the most widelyaccepted such theory is the ion-interaction model of Pitzer,320–323 which con-sists of an extended Debye-Huckel theory with virial-like coefficients to give aconcentration expansion of the excess Gibbs energy. The model is capable ofcorrelating electrolyte data very well over wide ranges of temperature and saltcomposition up to several molal. Variations of the model have been developedwhich take into account ion pairing324,325 and asymmetry in salt mixtures,326

and have been applied to near-critical and supercritical aqueous salt solu-tions.327–330 Reviews by Pitzer325,329 attest to the success of these models ininterpolating and facilitating the interpretation of experimental data. Anothersuccessful approach in this region is the extended corresponding-statesdescription of Gallagher and Levelt Sengers.331–333 A large class of models foractivity coefficients in electrolyte solutions334–347 can be classified as consistingof the Debye-Huckel model (usually in the extended form or Pitzer-modifiedform) to which a model for the non-electrostatic short-range contribution (suchas the non-random two liquid (NRTL) model of Renon and Prausnitz348 or theUNIQUAC model349) is added.A more fundamental approach is to attempt to model electrolyte solutions

using statistical mechanical methods, of which there are two kinds of models(reviewed extensively elsewhere350–355): Born-Oppenheimer (BO) level modelsin which the solvent species as well as the ionic species appear explicitly in themodel for the solution and McMillan-Mayer (MM) level models in which thesolvent species degrees of freedom are integrated out yielding a continuumsolvent approximation. Thus, for a BO level model, in addition to the interionicpair potentials one must specify the ion-solvent and solvent-solvent interactionsfor all of the ionic and solvent species. In this case, the interionic potentials donot contain the solvent dielectric constant in contrast to the MM-level models.Kusalik and Patey356,357 carefully discuss the distinction between these twoapproaches.


The DH model (eqs 8.45 and 8.46), can also be derived from statisticalmechanics as the solution of the mean spherical approximation (MSA) for anelectro-neutral mixture of point ions in a continuum solvent (i.e. it is anexample of a MM-level model). It represents a limiting behaviour of electrolytesolutions and breaks down quickly for concentrations higher than E0.01mol � dm�3; the extended DH model is accurate to E0.1mol � dm�3. For moreconcentrated solutions, it is natural to consider replacing the point ions withfinite-size ions; this leads to the consideration of so-called primitive models(PMs), which are MM models consisting of an electro-neutral mixture ofcharged hard spheres in a continuum solvent. The simplest PM is the restrictedprimitive model (RPM) consisting of an equimolar mixture of equal-diametercharged hard spheres in a dielectic continuum (i.e. r1¼ r�¼ r/2, s1¼ s�¼ s).To calculate the properties of the RPM, one approach is to solve the relation

between the direct correlation function and the pair correlation function givenby the Ornstein-Zernike (OZ) integral equation.358,359 The solution of the RPMin the MSA yields a simple analytical expression for the Helmholtz energy,given by,360

AElect:

NkBT¼ � 3x2 þ 6xþ 2� 2ð1þ 2xÞ3=2

12prs3; ð8:47Þ

where x¼ ks. The OZ equation has also been solved within the MSA for theunrestricted PM, consisting of an electro-neutral mixture of hard-sphere ions ofarbitrary size and diameter.361,362 The solution is again analytical, although aniterative method is required to solve for the scaling parameter, which is ageneralization of the parameter x in eq 8.47. Using the analytic solution of theunrestricted PM, Triolo et al.363–366 showed that by choosing an ionic-strength-dependent dielectric constant and taking into account hydration in the iondiameter, the unrestricted PM could fit the osmotic pressure and activitycoefficients of alkali halide electrolyte solutions up to 2mol � dm�3.Primitive models have been very useful to resolve many of the fundamental

questions related to ionic systems. The MSA in particular leads to relativelysimple analytical expressions for the Helmholtz energy and pair distributionfunctions; however, compared to experiment, a PM is limited in its ability tomodel electrolyte solutions at experimentally relevant conditions. Consider, forexample, that an aqueous solution of NaCl of concentration 6mol � dm3 (a highconcentration, close to the precipitation boundary for this solution) corre-sponds to a mole fraction of salt of just 0.1; i.e. such a solution is mostly water.Thus, we see that to estimate the density of such solutions accurately the sol-vent must be treated explicitly, and the same applies for many other thermo-dynamic properties, particularly those that are not excess properties. Thesuccess of the Triolo et al.363–366 approach can be attributed to the incor-poration of some of the solvent effect through state-dependent parameters, aswell as their focus on excess properties. Treating the solvent explicitly at somelevel is crucial to modelling real solutions; thus, we turn to BO-level models.

242 Chapter 8

The properties of BO-level models can be obtained from perturbation theory orintegral equations, and both have been used to develop versions of SAFTsuitable for modelling electrolyte solutions.We consider first obtaining the properties of BO models using perturbation

theories, where solvent-solvent, solvent-ion and ion-ion interactions can beeasily identified with different terms in the expansion. Chan367 combined theperturbation expansion of Stell and Lebowitz368 for the RPM with the Car-nahan-Starling33 equation of state for hard-spheres and noted that substitutingthe six-term expansion of Stell and Lebowitz by the much simpler DHexpression (eq 8.45) leads to little difference in comparison to experimentalmean activity coefficients, hence highlighting the key contribution of therepulsive term. In a second paper369 he tested the expansion of Hendersonet al.370 for the ion-dipole BO model but poor results were again obtained.Following the success of Chan’s simple proposition, and given the accuracy

of SAFT in modelling the properties of non-ionic fluids, it is a natural extensionto combine a treatment of the solvent and other non-Coulombic terms usingSAFT with a contribution to treat charge-charge interactions from a primitivemodel theory. The Helmholtz energy of the electrolyte solution is usuallytherefore written as

A

NkBT¼ AIdeal

NkBTþ AMono:

NkBTþ AChain

NkBTþ AAssoc:

NkBTþ AElect

NkBT; ð8:48Þ

where the first four terms in the right hand side correspond to those described insection 8.3 and AElect refers to the term or terms taking into account the elec-trostatic interactions. Most of the versions of SAFT described earlier have beentested in combination with Coulombic terms in the PM to treat a variety ofexperimental electrolyte systems. For the most part, research to date has con-centrated on strong (fully dissociated) electrolytes and single solvent systems(usually an aqueous solution). In the first attempt to model electrolyte fluidswith SAFT Liu et al.371 combined the original SAFT approach (with theexpression of Cotterman for the dispersion term)23 with the MSA in the PMapproximation and ion-dipole and dipole-dipole terms from the expansion ofHenderson et al.370 to regress the mean ionic activity coefficients (g� ) of single-salt and mixed salt aqueous electrolyte solutions at T¼ 298K. Using the dia-meter of the cation as an adjustable parameter for each electrolyte averageerrors for density and g� of the order ofo3% for concentrations up to 6 molalwere obtained. In later work the approach was used to correlate the criticalmicellar concentrations of charged surfactant solutions.372

Liu et al.373 have also considered the low density expansion of the non-pri-mitive MSA to treat the electrostatic interactions and discuss how the first threeleading terms can be identified with the solvent-solvent, solvent-ion and ion-ionelectrostatic contributions to the Helmholtz energy. Although expressions canbe obtained for each of these terms, for example, as provided in ref 373, it wasfound to be more accurate to use an expansion as presented in section 8.4.2 forsolvent-solvent interactions that include dipole-dipole terms and the usual


expressions of the MSA in the PM for the ion-ion interactions. It is interestingthat a term related to solvent-ion interactions is also identified which can berelated to the Born term for the contribution to the free energy due to ionsolvation at infinite dilution. In their work373 this term takes the form,

ASolvent-ion

NkBT¼ � z2ione

2

kBT�sion

�� 1

1þ sdipole=ðsionl� � ; ð8:49Þ

where l is a function related to the dielectric of the solvent and sion and sdipoleare the diameters of the ions and dipole, respectively. The expression abovereduces to the Born-term expression in the PM, when the diameter of thesolvent is neglected; i.e.

ABorn

NkBT¼ � z2ione

2

kBT�sionð�� 1Þ: ð8:50Þ

The dielectric constant in this model is concentration dependent and as suchcontributes to the chemical potential (activity coefficient) of the ions. It iscommon, however, not to take the derivative of the Born term with composi-tion as it is seen to introduce large discrepancies between the calculated valuesand the experimental ones. As shown by O’Connell,374 these problems typicallyarise through the inconsistent mixed use of MM and BO-level quantitieswithout paying due attention to the ensembles in which each property isderived.The SAFT-VR approach has been extended to treat electrolyte solutions

using both the primitive and non-primitive models within the DH model andMSA.375–378 In the first approach, SAFT-VRE,375 a MM level of theory is usedto describe the Coulombic interactions using either the solution of the DH orMSA in the RPM, and the vapour pressure of solutions in temperature rangesbetween (273 and 373)K (higher temperatures would require consideration ofion pairing) were modelled and the solution densities predicted. In contrastwith other approaches the ion diameters were not correlated, but instead takenfrom experimental Pauling radii. In the SAFT-VRE approach the solvent istreated implicitly; it is described as a non-polar solvent and so no dipole-dipoleor ion-dipole terms are included. The attractive interactions that lead to theformation of hydration shells are treated effectively through square-wellpotentials of variable range. An advantage of the SAFT-VRE approach is thatonly one adjustable parameter per ion is used; i.e. this is an ion-based salt-independent approach and so a parameter table per ion can be presented andthen used in a predictive fashion for salts not previously investigated, similar toa group contribution method. This idea has been tested in mixed salt solutionsand also applied successfully to predict the salting out of methane from aqu-eous solutions with added NaCl.379 Behzadi et al.377,380 later showed that theYukawa potential can also be used to model the ion-water and water-water

244 Chapter 8

interactions combined with the mean spherical approximation as extended byBlum361 for the non-restricted primitive model.In a second approach, the SAFT-VRþDE equation,378 a BO level model

with electrostatic properties obtained from integral equation solutions, ratherthan perturbation theory, is proposed. This approach draws heavily on thesolution of the OZ equation with the MSA closure for ionic and dipolar hardspheres due to Blum381 and on its extension to the generalized MSA (GMSA)and related approximations.305,382 In this modification of the SAFT-VRequation, a dipolar solvent is described explicitly and dipole-dipole and ion-dipole terms included. The importance of describing the size of the ions (i.e.using the PM over the DH approach) and an accurate description of thedielectric constant was clearly shown through comparison of the SAFT-VRþDE approach and other models with computer simulation data.378 Theapproach has been used to study simple electrolyte solutions and the effect ofthe solvent dipole moment on the system dielectric constant investigated asshown in Figure 8.5.383 The importance of using an explicit solvent over a

0

20

40

60

80

100

3.0

Water

Methanol

ε

0.0 0.5 1.0 1.5 2.0 2.5

�/D

Figure 8.5 Prediction of the water and methanol dielectric constant e as a function ofdipole moment m (1DE3.33564 � 10�30 C �m) from the SAFT-VRþDEequation of state at room temperature and pressure. K and ’, the the-oretical predictions;- - - - -, the experimental value of the dielectric con-stant for each solvent at room temperature and pressure. As would beexpected when the experimental bare dipole moment of water or methanolis used in the calculations the theory under predicts the dielectric constant,while values of the dipole moment for condensed phases yield predictionsin good agreement with experimental data. Taken from ref. 383.


dielectric continuum to describe the hydration of the ions was also demon-strated through a study of the Gibbs energy of hydration which was found to begood quantitative agreement with experimental data using the SAFT-VRþDEapproach, but not even in qualitative agreement when an implicit solvent modelis used.383

The ePC-SAFT384 (electrolyte PC-SAFT) equation follows similar ideas tothose used in the development of the SAFT-VRE equation, in that the DHapproach in the primitive model is used to account for the electrostatic inter-actions. The electrostatic term in references 384 and 385 is given in the generalform

AElect:

NkBT¼ � k3

12pr

Xi

wi; ð8:51Þ

where

wi ¼3

ðksiÞ3lnð1þ ksiÞ � ksi þ

k2s2i2

� �; ð8:52Þ

which leads to wiE1 by expanding the logarithm, so eq 8.45 is recovered. TheePC-SAFT has been successfully used to describe the vapour pressure, densityof solution and mean-ionic coefficient of 115 salts.385

Another strategy to fit ion-water solvation parameters is to develop salt-specific (or salt-dependent) models. The advantage here is that accuracy isusually higher,373,377 although also more reliant on experimental data. Tanet al.386,387 have proposed a hybrid method based on the SAFT1 equationwhere a hydrated diameter for the combined salt is used in addition to the usualionic volumes and ion-water dispersion (hydration) energies. Vapour pressures,activity coefficients and densities of the solutions are very well reproduced,which is difficult with ion-specific approaches. A key limitation, however, isthat the study of properties of solutions with mixed salts in this model requiresadded combining rules for the salt-specific parameters, and usually addedadjustable parameters that limit the predictive ability of the method.388,389

However, versions of the SAFT1 and SAFT2 (for divalent ions) approachesformulated as ion-specific are also available.146,390–392

In this area a key challenge remains the study of mixed-solvent systems. Thepreliminary works of Wu and Prausnitz393 and Patel et al.379 on salting out inaqueous solutions are promising, but are also simple cases in which the extremephase separation and difference in dielectric constant allows the electrolyte tobe considered only in one of the two phases; this greatly simplifies the treatmentof mixture dielectrics. The SAFT-VRþDE approach provides another pro-mising route for mixed solvent electrolyte systems, since the dielectric constantis a product of, and not a required input to, the calculations (see for exampleFigure 8.5). Non-aqueous solvents will also mean that ion pairing will need to

246 Chapter 8

be treated at some level. Initial work on combining a SAFT-like treatment witha chemical approach to treat ion-pairing has already been considered withencouraging results.394

8.4.4 Modelling Inhomogeneous Fluids

The use of SAFT within theories for inhomogeneous fluids395–397 such as thesquare-gradient theory398–400 (usually called density-gradient theory, DGT)and density-functional theory401,402 (DFT), provides a route towards thequantitative prediction of interfacial properties (interfacial thickness, inter-facial tension, adsorption, wetting and confinement effects) of complex fluids.Interfacial systems are of interest throughout industrial, technological andliving processes and involve increasingly complex molecules. Through thecombination of an underlying SAFT treatment of the fluid and DGT or DFTtreatments, non-spherical and associating or hydrogen bonding fluids can beconsidered in these frameworks.In DGT the thermodynamics of the system with an interface between

two fluid phases is described assuming that the density gradient betweenthe two phases is small compared to the reciprocal of the intermolecular dis-tance, so that the density and its derivatives can be treated as independentvariables. The Helmholtz energy of the inhomogeneous system is then obtainedas a Taylor series around the equilibrium state. One of the first attempts toexamine interfacial properties of real fluids with a SAFT approach and theDGT method was carried out by Kahl and Enders.403 The combinationof the accuracy provided by SAFT in terms of the bulk properties(liquidþ liquid coexistence densities) of the fluids and the DGT method,where one temperature-independent influence parameter is used, leads to veryaccurate correlations of the surface tension of real fluids. In the first paperof a series they investigated non-polar, and aromatic compounds, alcoholsand water and go on to treat mixtures using both the original version ofSAFT22,23 and PC-SAFT.404–406 The group of Fu and co-workers158,407–412

have also combined the PC-SAFT approach with the DGT method to correlatethe surface tension of a number of pure compounds and mixtures, and havestudied in detail the critical regions using the renormalization group approach(see section 8.4.1). The SAFT-VR102 and LJ413–415 versions of SAFT havealso been implemented in DGT formalisms to study interfacial properties ofreal fluids.While the DGT-based SAFT approach is easy to implement and can provide

a good representation of the surface tension of pure fluids and mixtures, ittypically requires the use of empirical adjustable parameters, the so-called‘‘influence’’ parameters, which limit the predictive ability of the method. Incontrast, DFT treatments, although more complex and numerically moredemanding, do not rely on adjustable parameters to provide information on theinterfacial properties. Chapman395 was the first to suggest the possibility ofincorporating a SAFT-like description of associating fluids within a DFT


approach, where the thermodynamic properties of the inhomogeneous fluid areexpressed as a function of a spatially varying density.401 Different levels ofapproximation are possible in DFT, such as the local-density approximation(LDA) in which the short-range correlations are neglected, weighted-densityapproximations (WDA) in which the short-range correlations are included, andthe fundamental measure theory (FMT) due to Rosenfeld416 in which a set ofgeometric ‘‘measures’’ define the weighted densities. Numerous works haveconcentrated on the behaviour of model associating systems in the presence ofwalls417–425 and in confinement,426,427 and have provided useful comparisonswith computer simulation data.Recently the DFT method combined with SAFT equations of state has been

used to predict the interfacial properties of real fluids. LDA methods areaccurate enough to treat liquid-liquid and liquid-liquid interfaces where thedensity profiles are usually smooth functions, and have been used in combi-nation with the SAFT-VR approach to predict the surface-tension of real fluidssuccessfully.428,429 The intermolecular model parameters required to treat realsubstances are determined by fitting to experimental vapour-pressure andsaturated liquid density data in the usual way (see section 8.5.1) and theresulting model is found to provide accurate predictions of the surface tension.A local DFT treatment has also been combined with the simpler SAFT-HSapproach,430,431 but in this case only qualitative agreement with experimentalsurface tension data is found due to the less accurate description of the bulkproperties provided by the SAFT-HS equation. Kahl and Winkelman432 havefollowed a perturbation approach similar to the one proposed with the SAFT-VR433 equation and have coupled a local DFT treatment with a Lennard-Jonesbased SAFT equation of state. They predict the surface tension of alkanes frommethane to decane and of cyclic and aromatic compounds in excellent agree-ment with experimental data.The most sophisticated DFT approaches incorporate weighted densities,

which depend on several weighting factors,401 often based on the FMT.416

The implementation of WDA in DFT provides accurate oscillatory profiles,such as those found in solid-fluid interfaces (near walls, in confined fluids)or in solid and other structured phases. The DFT for associating fluids ofSegura et al.418 combines Tarazona’s WDA DFT for hard-spheres withWertheim’s thermodynamic perturbation theory and has been used in anumber of studies of associating fluids in pores418,419,434 and with function-alized walls;435 in the limit of complete association a DFT for polymericfluids is obtained in this method.435–440 Based on these works, Chapman andco-workers have presented the interfacial-SAFT (iSAFT) equation,441 which isa DFT for polyatomic fluids formulated by considering the polyatomicsystem as a mixture of associating atomic fluids in the limit of completeassociation; this approach allows the study of the microstructure of chainfluids. Interfacial phenomena in complex mixtures with structured phases,including lipids near surfaces, model lipid bilayers, copolymer thin filmsand di-block copolymers,421,441,442 have all been studied with the iSAFTapproach.

248 Chapter 8

8.4.5 Dense Phases: Liquid Crystals and Solids

Although not strictly the main interest of this chapter, it is useful for com-pleteness to include briefly recent efforts that are considering applications ofWertheim’s perturbation theory to study structured phases. For example, Thiesand co-workers444,445 have combined the SAFT-HR equation with the classictheory of nematic formation of Maier and Saupe and study carbonaceouspitches (large polyaromatic compounds that can present liquid crystallinephase behaviour in toluene solution) with some success. A note of caution ishowever necessary here, as a correct treatment of mesogens (liquid crystalmolecules) requires the integration of the attractive energy as a function ofcontact distance, which is a function of orientation; failure to carry out thisintegration accurately can lead to predictions of the wrong type of phasebehaviour. In a more formal approach the association theory of Wertheim hasbeen used to study dimerizing liquid-crystal molecules combined with Onsa-ger’s theory for the isotropic-nematic phase transition.446 In a series of worksVega and co-wokers447–452 have shown that the fundamental proposition ofWertheim’s theory, in which the Helmholtz energy of a chain molecule isobtained through the use of the free energy and structure of a referencemonomer system can also be applied to solid phases. In this way global(solidþ liquidþ vapour) phase diagrams for pure model chain systems havebeen studied. This idea has been followed and used to correlate the solid-liquidequilibria of (argonþ krypton) and (argonþmethane).453

8.5 Parameter Estimation: Towards more Predictive

Approaches

The molecular model on which SAFT is based has the advantage of incor-porating important molecular details that allow the description of large chain-like molecules as well as treating molecular interactions such as hydrogenbonding. A price to pay is, however, the larger number of intermolecularparameters that need to be determined to characterize any given substance ormixture, compared to traditional cubic equations of state. The study of mix-tures adds additional complexity to the problem in terms of parameter esti-mation. In this section we discuss approaches concerned with minimizing theneed to fit parameters to experimental data.

8.5.1 Pure-component Parameter Estimation

SAFT intermolecular-potential model parameters are traditionally determinedby fitting to experimental data, as other equations of state; for pure compoundparameters vapour pressure and saturated liquid densities are typically used.These properties are chosen as the vapour pressure is of key interest in practicalapplications and depends strongly on the energy parameters, while the use ofliquid density data is important in determining size related parameters.


A least-squares objective function with the chosen residuals, given by

min yFObj ¼1

NP

XNp

i¼1

Pexp:sat;i � Pcalc:

sat;i ðyÞPexp:v;i

" #2

þ 1

Nr

XNr

j¼1

rexp:sat;j � rcalc:sat;j ðyÞrexp:sat;i

" #2;

ð8:53Þ

where y is the vector of intermolecular model parameters, NP is the number ofexperimental vapour pressure points, Nr the number of experimental saturatedliquid density points, Psat corresponds to the vapour pressure and rsat to thesaturated liquid density, is minimized. Gas-phase density data are rarelyincluded since they naturally yield parameters not well suited to modellingdense-phase properties. In addition, the fit is usually restricted to subcriticalconditions, since they are more accessible experimentally (and hence a lot moredata are available) and since the critical point is typically over predicted by thetheories (see section 8.4.1).It has been shown however that the traditional use of vapour-pressure and

saturated-liquid density data in SAFT approaches can lead to sub-optimalparameters125,136 due to a large degeneracy of models (i.e. different intermolecularparameter sets that lead to essentially the same error in comparison to theexperimental data considered). A technique that proposes to make the parameterspace discrete in order to carry out a global investigation of the parameter spaceillustrates the large degeneracy of models that result from the use of only twoproperties in determining parameters for associating compounds,125 and even innon-associating, relatively simple systems.454 In the case of water, for example,using the SAFT-VR approach it has been shown that four-site models withassociation energies between (1000 and 1400)K can lead to combined absoluteaverage deviations in vapour pressure and saturated liquid density of less than1 %. When the enthalpy of vaporization and the interfacial tension are includedin the fitting process, they are found not to facilitate the discrimination betweenmodels; however, the use of spectroscopic data, which provides information onthe fraction of associated (or free) molecules, was found to allow the differ-entiation between possible sets of parameters.267 Unfortunately, such data arerarely available over large temperature ranges. In section 8.5.2 the use of quantummechanical calculations to obtain hydrogen bond information is also discussed.Using the SAFT-Mie approach, Lafitte et al.136 have shown that if the equation

of state is precise enough at the level of the second-order derivative properties, acombination of vapour pressure, saturated-liquid density and single-phase density,together with speed of sound data leads to parameters that follow physical trendswithin a homologous series. This result suggests that the existence of many localminima that can lead to sub-optimal models when using local-optimizationmethods454 is resolved to a great extent when derivative properties are incorporated.A different approach is to reduce the number of parameters that need to be

determined by proposing transferable models. Towards this end, the use of

250 Chapter 8

parameter information transferable within a homologous series, usually fromsmaller members of the series where more experimental data are available tohigher-molar mass ones has been explored34,35,113,114,117,131,379,455–457 andtypically works well due to the physical basis of the SAFT model. While thismethod has not generally been very successful in the study of polymer systemsbeyond simple polymers such as polyethylene,100,457 due to the difficulty inobtaining parameters for an appropriate homologous series, it has been suc-cessfully applied to study poly(ethane-1,2-diol) (commonly known as poly-ethylene glycol) with the SAFT-VR approach.125

8.5.2 Use of Quantum Mechanics in SAFT Equations of State

The combination of a priori quantum mechanics (QM) calculations withequations of state is a very interesting proposition458 that is gathering interestwith more readily available computer power and equations of state thatimplement more physical models, which can lead to better, and even fullypredictive approaches. Specifically with the SAFT equation of state, Wolbachand Sandler were the first to propose the use of quantum mechanical calcula-tions to determine SAFT parameters.459–463 Starting with small pure-compo-nent hydrogen-bonding molecules, they relate an equilibrium constant ofassociation K obtained from molecular orbital calculations to the Da,b para-meter that characterizes the strength of association in SAFT. A relationbetween the SAFT size parameters (m and s) and the molecular volume cal-culated from QM was also derived in their original work.459 Calculations wereperformed with the Hartree-Fock method and density functional theory (DFT)using the B3LYP functional and 6-31þ g(2d,p) basis set to determine if a rig-orous level of theory was needed; the Hartree-Fock calculations were found tobe sufficiently accurate to model pure hydrogen-bonding fluids,459 such aswater and methanol, and binary mixtures of an associating fluid with a non-associating dilutant.460 In subsequent work the phase behaviour of binarymixtures in which cross association can occur were considered and a combiningrule for cross-association parameters in mixtures based on self-associating onesproposed.461 In general the combined SAFT-QM approach proposed byWolbach and Sandler was found to enable the description of mixture vapour-liquid equilibrium data for water, methanol and three acids with fewer adjus-table parameters and no loss of accuracy compared to the original SAFTequation. More recently, Yarrison and Chapman155 have used the originalSAFT equation23 and the association parameters developed by Wolbach andSandler to study the phase behaviour of (methanolþ alkanes) with goodresults, especially when the Hartree-Fock calculations were used.Similar to the work of Wolbach and Sandler for the molecular size para-

meters, Sheldon and co-workers464 derived values for m and s by mapping themolecular dimensions calculated from Hartree-Fock calculations onto asphero-cylinder; the remaining parameters were then determined by compar-ison to the usual vapour pressure and saturated liquid density data. Promising


results using this approach were obtained for the n-alkane series, nitrogen,carbon monoxide, carbon dioxide, benzene, cyclohexane, water and tworefrigerants.Although QM calculations can be expected to be more useful to determine

single-molecule, or dimer, properties (since they usually involve one or twomolecules, treated in vacuum), Lucas and co-workers465,299,300 have been ableto develop a successful framework to determine multipole moments and dis-persion interactions from QM. Using a fairly rigorous level of theory (MP2/aug-cc-Pvdz/B3LYP/TZVP) dipole and quadrupole moments, dipole polariz-ability and dispersion coefficients were determined and subsequently used withthe PCP-SAFT approach to study a number of pure fluids and their binarymixtures (see section 8.4.2).464

8.5.3 Unlike Binary Intermolecular Parameters

In the case of calculations of mixture properties and phase equilibria in mix-tures, mixture-specific parameters have to be calculated based on the pure-component (like, ii) and binary-interaction (unlike, ij) parameters with the useof mixing and combining rules. The simplest mixing rules for use in an equationof state are those presented by van der Waals,466 which use quadratic functionsof composition involving like and unlike parameters; e.g. in the one-fluidapproximation sx ¼

Pi

Pj

xixjsij , where the subscript x indicates a mixtureparameter466. Usually arithmetic and geometric (Lorentz-Berthelot) combiningrules are then used to determine the unlike size and energy parameters,respectively,

sij ¼sii þ sjj

2; ð8:54Þ

and

�ij ¼ 1� kij� � ffiffiffiffiffiffiffiffiffi

�ii�jjp

: ð8:55Þ

The quadratic form of the mixing rule has a theoretical basis in the compositiondependence of the second virial coefficient, but the Lorentz-Berthelot com-bining rules are well-known to fail in the case of highly non-ideal mix-tures.133,466–468 A correcting adjustable parameter kij that is mixture specificand typically determined by fitting to relevant experimental data for the mix-ture of interest is commonly introduced to improve agreement with experi-mental data.Given the failure of the Lorentz-Berthelot rules, in order to increase the

predictability of mixture models (be it equations of state or computer simula-tions) different combining rules have been proposed and investigated. Schnabelet al.469 have recently provided a quantitative assessment of eleven combiningrules in terms of their performance in describing binary mixtures classified bythe molecular shape and polarity of the components. Of particular interest is a

252 Chapter 8

new combining rule based on the original work of Hudson and McCoubrey forLJ mixtures,470 which can be used with potentials of general form and can beimplemented in SAFT-like models.133 The idea of the theory is to relate themodel potential to a more realistic compound potential (e.g. the Londonpotential for the case of dispersion interactions) so that an equation is obtainedfor one of the mixture parameters as a function of the others. For example, forthe case of molecules modelled as spherical segments interacting via SWpotentials, the unlike SW potential depth is obtained as

�ij ¼2s3iis

3jj

s6ij

l3ii � 1� �1=2

l3jj � 1� 1=2

l3ij � 1� 1=2

0B@

1CA IiIj

� �1=2Ii þ Ij

! ffiffiffiffiffiffiffiffiffi�ii�jjp

; ð8:56Þ

where, s, e, and l correspond to the usual SW parameters and Ii is the ioni-zation potential of pure compound i (in the case of chain molecules a corre-lation between the molecular ionization potentials and the model chain lengthm is proposed so that the approach can also be used in non-spherical models).It can be seen from this expression that the usual geometric (Berthelot) mean isrecovered in the special case of mixtures involving components of identicalionization potential, molecular size and potential range. In mixtures wherethese parameters are different for the two components values of kija0 shouldbe expected. The combining rule given in eq 8.56 has been shown to yield betteragreement with experimental data that the traditional geometric mean.For example, in the case of mixtures of alkanes and perfluoroalkanes useof the traditional Lorentz-Berthelot rules leads to the wrong type of phasebehaviour while the use of eq 8.56 to obtain kij leads to the correct behaviour(see for example Figure 8.6). The treatment has also been extended to dealwith mixtures involving polar and hydrogen-bonded molecules, although theresults in this case were found to be more varied. Huynh et al.471,472 havefollowed the work of Haslam et al.133 using the Hudson-McCoubrey method todevelop an approach to predict binary parameters based on the pseudo-ionization energy of functional groups in a group-contribution based method(see section 8.6).It is also worth noting that in the case of SAFT equations of state, the molecular

basis of the theory means that it is possible to transfer binary-interaction para-meters when studying mixtures within a homologous series very successfully. Thisidea has been used in numerous works to predict the phase behaviour of (alkanesþwater),34,75,379 (alkanesþ perfluoroalkanes),117 (alkanesþ carbon dioxide),114,130

(alkanesþ hydrogen chloride)106, and (alkanesþ nitrogen)113, and aqueous solu-tions of surfactants,123,455,473 to name but a few.

8.6 SAFT Group-Contribution Approaches

In group-contribution (GC) methods the properties of the system of interest(a pure compound or a mixture) are described in terms of the functional groups


making up the molecules; for example, a simple molecule like mono-ethanolamine (MEA) would be described in terms of one OH, one NH2, andtwo CH2 groups. The groups, or the contributions of the groups, are char-acterized by sets of inter- and intra-molecular group parameters, and theassumption is that combinations of these lead to the thermodynamic propertiesof the molecules or mixtures of interest. As such GC methods can turn intovery useful predictive tools, since from the description of a few groups thethermodynamic properties and phase equilibria of a large number ofpure compounds and mixtures can be accessed without additional fitting toexperimental data.The GC concept is not new; it has been widely implemented for example in

the calculation of pure component properties (an overview is provided in ref474), for the estimation of activity coefficients of liquids in the successfulUNIFAC approach,475 and more recently, in equations of state.476 The

Figure 8.6 Pressure p as a function of temperature T for the phase diagram of(C4H10þCF4). &, pure component critical point; J, pure componentcritical part;J, vapour pressure of the pure substances; ’, critical pointsof the mixture; —, predictions obtained from SAFT-VR equation with thepredicted value of kij¼ 0.063 determined from the use of eq 8.56 correctlyindicating 1PAlnQ phase behaviour; - - - - - , estimates obtained fromSAFT-VR with kij¼ 0 predicted an erroneous type of phase behaviour.Reprinted with permission of Fluid Phase Equilibria.133

254 Chapter 8

advantage of equation of state GC methods is that equations of state are notlimited, in principle, in their range of application, and more importantly, cantreat liquid and vapour phases continuously, and through thermodynamicrelations, can be used to calculate any thermodynamic property, which isimportant in process design applications.

8.6.1 Homonuclear Group-Contribution Models in SAFT

Based on the original homonuclear molecular SAFT model for chain molecules(see Figure 8.2), one of the first attempts to combine a group-contributionmethod with the SAFTmethodology was presented by Lora et al.59 who use theparameters of low-molecular-weight propanoates39 (homomorphs of theacrylate repeat groups) to calculate the size and energy parameters of poly-enoates.60 Following these ideas, Tobaly and co-workers have presented aseries of papers proposing a predictive implementation of the SAFT equationof state.477,478,479 The authors present two approaches for the predictive use ofthe equation of state within substances of the same chemical family. Inapproach 1, the parameters describing the members of a chemical family areobtained by fitting to experimental data in the usual way (i.e. by usingexperimental vapour-pressure and saturated-liquid density data for eachcompound) and then deriving relations between the molecular parameters andmolecular properties (e.g. molar mass); note that this approach amounts tousing different parameters for each molecule. In approach 2, all molecules of aseries (e.g. the alkane family) are assumed to be formed from identical seg-ments, and so only one set of segment parameter values is used. In the case ofalkanes, this means that the CH3, and CH2 chemical segments are described bythe same size and energy group parameters, and differences between membersof the same chemical family are treated simply by assigning different values ofthe chain length, m. Approach 2 was found to predict heavier n-alkane phasebehaviour more accurately than approach 1 and was therefore used to study thephase behaviour of other hydrocarbons, including alkenes, cyclic compoundsand mixtures of these,477,479 and alcohols,478 yielding good results. Unfortu-nately, the predictive ability of this technique is rather limited, since it is notfully formulated within the scope of contributions of functional groups, butmore an effort to model the behaviour of parameters within a certain homo-logous series by transferring parameters.A more sophisticated group-contribution approach, termed GC-SAFT, was

later introduced by the same authors.479 In this case, the molecules are decom-posed into functional groups and distinct group contributions are estimated. Theauthors implement the group-contribution method within the original SAFT23

and the SAFT-VR92,93 approaches. The concept of groups building up tomolecules is introduced explicitly in this approach, but the underlying molecularmodel is still a homonuclear one in which group parameters need to be averagedto predict pure and mixture properties. This method has been used to studyalkanes, a-olefins, 1-alkanols, alkyl-benzenes and alkyl-cyclohexanes and their


mixtures,479,480 and later alkyl-esters and formates,481,482 and their mixtures,using the segment-based polar contribution of Jog and Chapman.277 Thephase equilibria of mixtures of esters with alkanes, cyclohexanes, alkyl-benzenesand 1,4-dimethylbenzene have also been studied in detail and the perfor-mance of the method was found to lie within an uncertainty of a few percentin a fully predictive manner (i.e. no binary interaction parameters wereused).Group-contribution based approaches have also been studied within the PC-

SAFT framework. As discussed in section 8.3.4, the basis of the perturbed-chain SAFT approach (PC-SAFT)147 is slightly different to other versions ofSAFT in that the Helmholtz energy of a homonuclear hard chain fluid (asopposed to that of the monomer fluid) is first considered and the dispersioninteractions are described by a perturbation expansion with the hard-chainfluid as reference. As such, it is, in principle, not the best suited to be recast as aGC approach, though relations between molecular model parameters andmolecular properties (e.g. molar mass) can be derived so as to identify thecontribution of different chemical groups to the properties of molecules. Forexample, the intermolecular model parameters for a series of hydrofluoroetherswhere experimental data (saturated liquid densities and pressures) are availablehave been determined and used to calculate the contribution of each functionalgroup (CH3, CH2 CF3 CF2 and O groups are identified).483 In a later study theester series was considered484 and the phase-behaviour predictions of esters notincluded in the regression database were found to compare well using the GC-SAFT approach, including dipole-dipole interactions using the perturbationexpansion of Twu and Gubbins,271 and the original SAFT, SAFT-VR and PC-SAFT equations; however for large esters the deviation from experimental datais rather significant (e.g. 50 % standard deviation was found for methyl tet-racosanoate). The same method, with a term also describing a quadrupolar-quadrupolar contribution,272 has also been used for the prediction of equili-brium properties of polycyclic aromatic hydrocarbons and their mixtures.485

The vapour pressures of 19 chemical families including hydrocarbons, cyclicand aromatic hydrocarbons, alcohols, amines, nitriles, esters, ketones, ethers,and others have also been considered.486 The recent work of Tihic et al.,487 inwhich the PC-SAFT equation is combined with the group contribution ofConstantinou et al.488 to incorporate first and second-order groups so thatisomers and proximity effects can be considered, is interesting and has beenused to study a wide range of polymer systems.218,487,489 Unfortunately, as inprevious approaches, the underlying homonuclear model does not enable thedifferentiation of groups at the molecular level.

8.6.2 Heteronuclear Group Contribution Models in SAFT

Following the original first-order thermodynamic perturbation theory ofWertheim heteronuclear models can be proposed where the segments in a givenmodel molecule are arbitrarily different. The earliest works in this direction

256 Chapter 8

considered molecules formed from heteronuclear hard-spherical segments withattractive interactions treated at the van der Waals mean-field level;490–493

expressions for heteronuclear SW chains139,494–496 and LJ chains64,497,498 havealso been presented.A heteronuclear model, which allows the free energy of the fluid to be

written explicitly in terms of group parameters that are different, is ideallysuited to developing a detailed molecular-level group-contribution modeland if the heteronuclear nature of the model is retained in both themonomer and chain fluid, the detailed connectivity of the segments can bepreserved and hence some isomers differentiated. Although Wertheim’s theoryis based on tangentially bonded segments, a tangential model with integervalues of m to represent each functional group is not capable of describingmultifunctional molecules accurately.499 This is also seen in homonuclearversions of SAFT when modelling chain fluids; for example the commonly usedrelationship for the number of segments in the model chain to describe alkanesas a function of the number of carbon atoms C, m¼ 1/3(Cþ 1)� 1, prescribesm values of 1.33 for ethane, 1.67 for propane and 2 for butane. Therefore,butane is modelled by two tangentially bonded segments, while ethane andpropane are essentially modelled by fused segments. The implementation of aheteronuclear-based GC SAFT approach has recently been proposed in theSAFT-g499,500 and GC-SAFT-VR501 approaches, where the difficulties asso-ciated with the underlying tangential models in SAFT using generalizations ofthe SAFT-VR equation of state to model heteronuclear chain molecules areaddressed.In SAFT-g499,500 a model of fused segments is proposed; each group is

described by an integer number of segments nk and a parameter reflecting thecontribution of each group k, the so-called shape factor Sk, to the overallmolecular properties is introduced to describe molecules as fused segments. Theshape factor Sk,

500 describes the contribution that a given segment k of dia-meter skk makes to the overall molecular geometry and contributes to the meanradius of curvature, the surface area and the volume of the molecule. A givengroup k is then fully described by the number of segments in the group nk, theshape factor Sk, a diameter skk, a dispersive energy ekk and range lkk. In casesof associating groups, two additional parameters are introduced for each site-site a-b interaction, namely the energy �HB

kk;ab and range rckk;ab, and for unlikegroup interactions, ekl, �HB

kl;ab and rckl;ab also need to be determined. In the GC-SAFT-VR501 approach non-integer values of m are considered for each func-tional group and therefore, although still based on the tangentially bondedapproach of Wertheim, the model can also be considered as describing fusedchains. In this approach a given group k is described by its chain length mk,diameter skk, dispersive energy ekk and range lkk, as well as the same additionalparameters as in the SAFT-g approach when considering associating groups. Incomparing the two approaches, using

nkSk ¼ mk; ð8:57Þ


where nk is the number of segments in a group of type k following the notationin,500 mk is the chain length as defined in ref 501 and

xs;k ¼XNC

i¼1xs;ki; ð8:58Þ

where NC is the number of components in the mixture, to change betweenthe definition of the fraction of segments of type k in the mixture used inref 500 and the fraction of segments of type k in a molecule of type i used inref 501. It can be seen that the two theories are fundamentally the same withrespect to the ideal and monomer terms. In SAFT-g the original mappingfor the effective density92 is used to obtain the mean attractive Helmholtzenergy a1 (see section 8.3.3), while in GC-SAFT-VR a mapping suitable forlonger square-well ranges503 is used, but the two lead to essentially identicalresults for the shorter ranges. A second small difference is in the use of a dif-ferent mixing rule for the effective diameter of the mixture sx in the A1 term; theoriginal (m1b) mixing rule93 does not lead to the correct limit when thediameter of one segment goes to zero, however this limit is not encountered inapplications to model real fluids, and so is not an issue, in the group-con-tribution approaches.The key difference between the two approaches lies in the treatment of the

chain term, AChain. In SAFT-g the SAFT-VR chain term is used; through theuse of effective parameters, the contribution to the Helmholtz energy due tochain formation is a function of the number of segment-segment contacts inthe chain and the contact radial distribution function of an effective fluid.In the case of the GC-SAFT-VR the heterogeneity of the segments is explicit inAChain, through a generalization of the Wertheim expression that reduces to theoriginal Wertheim term (as used in SAFT-VR) in the limit of homonuclearchains and/or an integer number of chain segments.In both the GC-SAFT-VR and SAFT-g approaches, compared in Figure 8.7,

the group parameters are determined by fitting to the experimental vapourpressures and saturated-liquid densities of the smaller members of chosenchemical families (i.e. alkanes, branched alkanes, 1-alkenes, alkylbenzenes,ketones, alkyl acetates and methyl esters, among others). The predictive cap-ability of the methods is then tested by assessing the description of the fluidphase behaviour of larger molar mass compounds that were not included in thedetermination of the group parameters.A key additional advantage of both SAFT-g and GC-SAFT-VR is that mix-

tures can be treated in a fully predictive manner without the need to proposecombining rules and adjustable parameters. In this context both methods havebeen shown to provide a good description of the pressure and compositionof mixtures including binary mixtures of alkanes, alkenes, alkanols, namely(alkanesþ alkylbenzenes), (alkaneþ ketones), (alkaneþ esters), (alkaneþ acids),(alkanesþ amines) and even cases with highly non-ideal behaviour; includingliquid-liquid equilibrium (LLE) and polymer systems.500–502

258 Chapter 8

370

380

390

400

410

420

430

0.0

x(propyl butanoate)

x(hexane)

p/kP

a T/K

1.00.80.60.40.2

Figure 8.7 TOP: Temperature T as a function of propyl butanoate mole fractionx(propyl butanoate) for the mixtures (heptaneþ propyl butanoate) and(nonaneþ propyl butanoate). K, measurements of (heptaneþ propylbutanoate);506 ’, measurement of (nonaneþ propyl butanoate);506 ——,predictions from the GC-SAFT-VR equation using CH3, CH2, CO andOCH2 groups without recourse to a fit against any measurements thatfurther demonstrates the versatility of the GC approach. Simply theaddition of two CH2 segments as the alkane chain is increased in length issufficient to capture the dramatic change in phase behaviour. Reprintedwith permission of Fluid Phase Equilib.501 BOTTOM: Pressure p as afunction of hexane mole fraction x(hexane) for the mixtures (hexaneheptaneþ propan-2-one). J, measurments at T¼ 283.15K;506 K; mea-surments at T¼ 313.15K;506 &, measurments at T¼ 338.15K;506 ——,predictions obtained from the SAFT-g equation with CH3, CH2, and COgroups without recourse to a fit against any measurements that furtherdemonstrates the versatility of the approach. Reprinted with permission ofFluid Phase Equilibria.502


These are promising approaches, which incorporate a detailed molecularmodel in which groups can be differentiated. In this way, they benefit fromadvantages of the successful UNIFAC approach, and overcome the difficultiesassociated with its underlying lattice model. They are accurate over largepressure ranges and can be used consistently for liquid and vapour phases. Inaddition, as mentioned above, their formulation as continuum fluid theoriesmeans that the binary interaction parameters can be determined from purecomponent data.

8.7 Concluding Remarks

SAFT approaches in any of their many variants provide a means towardsmodelling complex fluids from detailed molecular models. They all share acommon basis stemming from the thermodynamic perturbation theory ofWertheim, which provides the key expressions to obtain the free energy ofassociating chain fluids from knowledge of the thermodynamics and structureof a monomer reference fluid; the choice and level of detail incorporated intreating this reference fluid lead to the differentiation of the many versions. Afundamental strength of the method is its close link to the proposed molecularmodel; this often allows direct comparison with simulation data, and helps thecontinuous improvement of the approach. We have presented an overview ofthe most popular SAFT-based approaches and their application to modellingthe thermodynamics and phase behaviour of fluids, and provided a survey ofrecent efforts to extend the applicability of SAFT-based equations to morecomplex systems (such as polar fluids, electrolyte solutions, and critical andinhomogeneous fluids). We have specifically tried to focus on methodologicaladvances since the comprehensive reviews of Economou504 and Muller andGubbins,505 rather than aim at a comprehensive review of all systems studied.

Acknowledgements

The authors gratefully acknowledge M. Carolina dos Ramos for her carefulproof reading and useful comments. CMC acknowledges support from theNational Science Foundation through grant numbers CBET-0453641, CBET-0452688 and CBET-0829062. AG acknowledges support from the Engineeringand Physical Sciences Research Council (EPSRC) of the UK for funding underthe Molecular Systems Engineering grant (EP/E016340).

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CHAPTER 9

Polydisperse Fluids

DIETER BROWARZIK

Institute of Physical Chemistry, Martin-Luther-University Halle-Wittenberg,D-06099 Halle, Germany

9.1 Introduction

Many industrial operations require quantitative predictions of thermodynamicproperties, particularly, of phase equilibria of polydisperse fluids. Such fluidsconsist of a huge number of components most of which are similar to oneanother. Examples are polymer systems, petroleum fluids, asphaltenes, naturalgases, waxes and vegetable oils. Often it is impractical or even impossible todetermine the concentration of each individual component. In this situation,the description of the composition needs approximations. There are twoprincipal approaches, and these are namely the pseudo-component method andcontinuous thermodynamics. In the pseudo-component method, the real mix-ture is replaced by a mixture of few pseudo-components. If the pseudo-com-ponents are chosen in a suitable way, this approach can lead to reasonableresults. Whereas in the pseudo-component method the number of componentsis reduced, continuous thermodynamics is based on continuous distributionfunctions corresponding to an infinite number of components. These functionsdepend on one or more identifying quantities, such as molar mass, normalboiling-point temperature or number of carbon atoms. Using analytical func-tions, in some cases the integrals of continuous thermodynamics possess ana-lytical solutions and an essential mathematical simplification is achieved.In section 9.2, the polydispersity is proven to influence strongly the phase

equilibrium. As an example the liquidþ liquid equilibrium of a polymer solu-tion is discussed. In section 9.3, the approaches to polydispersity are treated in




280

detail especially the calculation of phase equilibria based on continuous ther-modynamics. In this, both the application of Gibbs-energy models and ofequations of state are discussed. Section 9.4 presents numerous examples ofphase equilibria of polymer systems, petroleum fluids, asphaltenes, waxes andother polydisperse systems, particularly considering papers published duringthe last decade.

9.2 Influence of Polydispersity on the Liquid+Liquid

Equilibrium of a Polymer Solution

Different thermodynamic properties depend on polydispersity to a differentdegree. The liquidþ liquid equilibrium (LLE) of a polymer solution is a par-ticular good example to demonstrate the polydispersity influence. The nature ofthis influence is not only a quantitative one but rather a qualitative one too. Letus consider a polymer solution and the corresponding phase diagram byplotting the temperature T or the pressure p against the mass fraction oB of thepolymer. In the monodisperse case there is only one binodal curve with thecritical point in the maximum (or minimum). In the polydisperse case thesituation is quite different. Figure 9.1 shows the phase diagram of {ethe-neþ poly(but-3-enoic acid; ethene) [ethylene-co-vinylacetate]} at T¼ 433.15K.In Figure 9.1, the pressure is plotted as a function of the total mass fraction ofthe copolymer. The molar mass averages of the copolymer are Mn ¼ 33400 g �mol�1 for the number average and Mw ¼ 137700 g �mol�1 for the mass aver-age. In terms of the quantity U ¼Mw=Mn � 1, that is a measure of poly-dispersity, the sample is highly polydisperse (U¼ 3.12). The average massfraction of but-3-enoic acid in the copolymer is 0.175. Additionally to themolar-mass polydispersity copolymers show a polydispersity with respect tochemical composition. The curves of Figure 9.1 are based on the Sako-Wu-Prausnitz equation of state (SWP-EoS) and on continuous thermodynamics(considering both types of polydispersity).1,2 Figure 9.1 shows a single binodalcurve, a cloud-point curve, a shadow curve and an infinite number of coex-istence curves with only two pairs of shown. The experimental data of Niesz-porek3 is also shown in Figure 9.1. To understand these curves we start at themass fraction 0.133 of the copolymer at high pressure in the homogeneousregion and lower the pressure until the cloud-point is reached where the firstdroplets of a second phase are formed. However, the overall polymer content ofthe second phase does not correspond to a point on the cloud-point curve butto the corresponding point on the shadow curve. With further pressurereduction the two coexisting phases do not change their overall mass fraction ofthe polymer according to the cloud-point curve or to the shadow curve butaccording to the coexistence curve. This coexistence curve is divided into twobranches beginning at corresponding points on the cloud-point curve and onthe shadow curve. Each feed mass fraction of the copolymer belongs to one pairof coexistence curves. Only in the case of the critical composition does coex-istence curve close with the critical point at a maximum. Also at this point the

281Polydisperse Fluids

cloud-point curve and the shadow curve intersect. In contrast to the mono-disperse case the critical point is not located at the maximum of the cloud-pointcurve or of the shadow curve. In the limit of vanishing polydispersity the cloud-point curve, the shadow curve and all coexistence curves become identical. Thiscomplicated behaviour originates from the polydispersity of the polymer.Polymers in coexisting phases have different molar-mass distributions which

are also different from that of the initial homogeneous system. Obviously, theinfluence of polydispersity on the LLE is not only of a quantitative nature butof a qualitative nature as well. This demixing behaviour is important for somepractical problems, for example, in the high-pressure synthesis of low-densitypolyethene [polyethylene] or of poly(but-3-enoic acid; ethene) [poly(ethylene-co-vinylacetate)]. The polyethene is obtained as a solute in supercritical ethene

Figure 9.1 Pressure p as a function of the total mass fraction oB of the copolymer for{etheneþ poly(but-3-enoic acid; ethene)[ethylene-co-vinylacetate]} atT¼ 433.15K where molar mass averages of the copolymer are Mn ¼33400 g �mol�1 and Mn ¼ 137700 g �mol�1. The calculated coexistencecurves with a starting feed mass fraction of 0.133 for the copolymer arecompared with the experimental data of Nieszporek.3 m, polymer-richphase;3 ., polymer-lean phase;3 –––––, Calculated cloud-point curve; – – –shadow curve; and – � – coexistence curves.

282 Chapter 9

and precipitated by pressure reduction where there is a polymer-rich phase anda polymer-lean phase. The molar-mass distributions of the polymer in bothphases (and thus the quality of the polyethene produced) depend on the tem-perature and on the precipitation pressure. Engineering models for this pro-blem require the combination of an equation of state and a method foraccounting for the effect of polydispersity.

9.3 Approaches to Polydispersity

9.3.1 The Pseudo-component Method

In many papers phase equilibria for ill-defined polydisperse fluids are calculatedon the basis of usual thermodynamics by introducing pseudo-components.Pseudo-components may be some of the real components of the mixture but,often, these are hypothetical components the properties of which are calculatedas average values according to a specific algorithm. If the pseudo-componentsare chosen in a suitable way this approach can lead to reasonable results asshown for instance by Pedersen.4 Pedersen investigated vapour-liquid equili-brium (VLE) of North-Sea oils and of gas condensates. The real systemsconsisted of more than 80 components. To calculate bubble points and dewpoints and to perform flash calculations the real system was replaced by amixture of 6 pseudo-components. Based on the Soave-Redlich-Kwong equa-tion of state (described in Chapter 4) a precise description of the experimentalresults can be achieved. The pseudo-components were chosen in such a waythat they all have approximately the same mass fractions and that the criticalproperties are mass-mean average values. The papers of Pedersen et al.5–8 havebeen concerned with very similar problems. Jensen and Fredenslund appliedthe pseudo-component concept to flash calculations based on the Peng-Robinson equation of state.9 Gani and Fredenslund generalized all theexperience about equation-of-state applications to equilibrium calculations ofpetroleum fractions developing an expert tuning system.10 This includes analgorithm to choose the pseudo-components for this type of system. VonBergen et al.11 calculated densities and vapour-liquid compositions for crudeoils and natural gases using the Sanchez-Lacombe equation of state. Fractionsof the heavy ends were approximated as pseudo-components defined in termsof their equivalent alkanes.Marano and Holder have calculated the VLE of the Fischer–Tropsch sys-

tem.12 The pseudo-components were defined with the aid of an analyticalmolar-mass distribution function (Anderson-Schulz-Flory distribution). Theproperties of a pseudo-component were based on a hypothetical model com-ponent in each carbon-number cut.Gonzalez et al.13 modeled the asphaltenes precipitation in live oils with the

(Perturbed Chain-Statistical Associating Fluid Theory (PC-SAFT); see Chap-ter 8 for additional material. It is not an easy task to apply a complicated modelsuch as PC-SAFT to systems consisting of a very large number of chemical


components not all of which may have been identified. The model also requiresdata for each pure substance. In this case, pseudo-components are inevitablyused to model a complex and polydisperse system. In ref 13 the oil wasrepresented by up to 10 pseudo-components, four of which describe the gaseoussubstances including nitrogen and methane, two the saturated hydrocarbonsand the aromatics and the remaining four the asphaltenes and resins. Even thisis a rough approximation of a compositionally very complex fluid the modelwas able to predict satisfactorily, as determined by the agreement withexperiment, asphaltenes precipitation arising from changes in compositioncould be modeled in good agreement.Nichita et al.14 calculated the wax precipitation from hydrocarbon mixtures

using a cubic equation of state (see Chapter 4) to describe the vapour and theliquid lumping into pseudo-components to simplify the phase equilibriumcalculation. However, the information lost in this procedure effected thelocation of the predicted solid phase transition. This issue was avoided by aninverse lumping procedure, in which the equilibrium constants of the originalsystem are related to some quantities evaluated from lumped fluid flash results.The method was tested for two synthetic and one real mixture yielding goodagreement between calculated and experimental results.Many authors15–19 have applied the pseudo-component concept to polymer

systems and computed the LLE. Because of the qualitative influence of poly-dispersity, described above, the model must also include polydispersity. Insome cases, if the molar-mass distribution is wide, the number of necessarypseudo-components becomes very large, for example, Krenz and Heidemannneed 100 pseudo-components to describe the phase equilibrium of polyetheneblends in hydrocarbons with the Sanchez-Lacombe equation of state.19 How-ever, Tork et al.15 introduced a method that needs only few pseudo-componentsthat relies on the phase behaviour being mainly influenced by a limited numberof moments of the molar-mass distribution function. If the pseudo-componentsare chosen in such a way that they represent a certain number of these momentsthe influence of polydispersity on the phase equilibrium is accommodated. Inprinciple, the moments may be calculated by integration of the experimentalmolar-mass distribution provided by gel-permeation chromatography. How-ever, in literature this information is rarely given. Therefore, Tork et al.15 usedthe 3-parameter Hosemann-Schramek distribution function for which only thefirst three moments �Mn; �Mw; �Mzð Þ have to be known and all other momentscalculated from the distribution function by numerical integration. A pseudo-component is described by its mole fraction and its molar mass. Thus, npseudo-components can represent 2n moments of the molar-mass distribution.For example, to describe ten moments only five pseudo-components arenecessary. A disadvantage of this approach is the Hosemann-Schramek dis-tribution function falsifies the higher moments. Such deviations have littleinfluence on the cloud-point curve but can change the location of the shadowcurve, the coexistence curves and the critical point.However, the few cases discussed show that pseudo-components can be a

successful applied to describe polydispersity and calculate phase equilibrium by

284 Chapter 9

convenient choice of pseudo-components that is in the end more an art than apredictable procedure. The selection depends both on the type of system and onthe phase equilibrium to be calculated. Most procedures are empirical and theirdevelopment requires much experience. In the next section as a secondapproach to handle polydispersity, continuous thermodynamics is presented.Here, the real system is replaced in a definite way by a continuous or semi-continuous mixture consisting of an infinite number of components. Never-theless, continuous thermodynamics is also an approximation for which thereare also difficulties.

9.3.2 Continuous Thermodynamics

9.3.2.1 History and Fundamentals

The history and fundamentals of continuous thermodynamics will be brieflypresented here and has been discussed in detail elsewhere.20 Before the 1980’smany authors applied continuous distribution functions to specific cases ofnon-equilibrium thermodynamics, statistical thermodynamics, the VLE ofpetroleum fractions and the LLE of polydisperse polymer systems.21–39 Startingin 1980 a consistent version of chemical thermodynamics directly based oncontinuous distribution functions was developed and called continuous ther-modynamics. The work of Kehlen and Ratzsch,40,41 Gualtieri et al.,42 Salacuseand Stell,43 Briano and Glandt,44 are to be mentioned as sources of informa-tion. In the following years several groups applied continuous thermodynamicsto nearly all important types of polydisperse systems.45–51 Cotterman andPrausnitz52 reviewed the literature up until about 1990. In the 1980’s con-tinuous modelling of phase equilibria was mostly focused on polymer systems,petroleum fractions and natural gases. In the last ten years, this has beenexpanded to also include problems with asphaltene precipitation from crudeoils and wax precipitation from hydrocarbon mixtures. In section 9.4 the morerecent papers are discussed.The most important fundamentals of continuous thermodynamics will be

outlined below. Firstly, let us consider a continuous ensemble consisting of avery large number of chemically similar species only differing in the char-acterization variable M. For example, this can be a polymer, a petroleumfraction or a wax. As characterization variable, the molar mass, the normalboiling-point temperature or the number of carbon atoms may be considered.The extensive distribution function w(M) is defined by

nM 0;M 0 0 ¼ZM 0 0M 0

wðMÞdM; ð9:1Þ

where the quantity on the left hand side represents the amount of substancewith molar masses between M0and M00. If the integration is performed for the


total range of molar mass the overall amount of substance n is obtained. Themolar distribution function W(M) is defined by

WðMÞ ¼ wðMÞ=n; ð9:2Þ

being the analogous quantity to the mole fractions of thermodynamics. Inte-grating eq 9.2 for the total range of M the value 1 is obtained similarly to thesummation of all mole fractions. Generally, in continuous thermodynamicsintegrals occur instead of the sums of the usual thermodynamics.Considering a thermodynamic extensive quantity, for example, the Gibbs

energy G, that in thermodynamics depends on the temperature T, the pressure pand the amounts of substances n1, n2, n3, � � � In continuous thermodynamicsG is only a function of T and p and includes a functional with respect to thedistribution function w. The functional is a mapping that assigns a number toeach function of a given class of functions. Practically, in continuous thermo-dynamics this functional is always a definite integral with fixed limits and thesolution of which is a number. A value G is assigned to each function w (atgiven T and p) depending on the total course of the distribution function. Toexpress this particularity G¼G(T, p; w) is written with a semicolon before w.In thermodynamics, the differentials (variations) dG of the extensive quan-

tities G play an important role. In this chapter, they are signified by the symbold to distinguish them from differentials belonging to integrations. With respectto T and p this differential is formed at a given T and p from the limit:

dG ¼ limt!0

@GðT ; p;wþ tdwÞ@t

: ð9:3Þ

In equation 9.3, the well-known rules of differentiation have been applied. SincedG is the first approximation of the change dw(M) of the distribution functiondG may be written as

dG ¼ZM

mðMÞdwðMÞdM and dG ¼Xi

midni: ð9:4Þ

In the first equation of eq 9.4, m(M) is the continuous version of the partialmolar chemical potential and corresponds to the usual thermodynamicexpression provided as the second equation of eq 9.4. The integral relates to thetotal range of the characterization variable M. In principle, knowing G andcalculating dG according to eq 9.3 the comparison of eqs 9.3 and 9.4 leads tothe partial molar quantities m(M). Based on the previously outlined principlesthe well-known fundamental equations of usual thermodynamics may betranslated into continuous thermodynamics to give the basic equations

G ¼ZM

wðMÞmðMÞdM and Gm ¼ZM

WðMÞmðMÞdM; ð9:5Þ

286 Chapter 9

where the subscript m indicates the molar quantity. Gm and m(M) are functionsof T and p and also functionals with respect to the molar distribution functionW(M). Furthermore, the Gibbs-Duhem equation is given:

ZM

wðMÞdmðMÞdM ¼ 0 or

ZM

WðMÞdmðMÞdM ¼ 0: ð9:6Þ

More details may be found in the literature.1,41,47

Considering the chemical potential the assumption of an infinite number ofcomponents in continuous thermodynamics leads to difficulties. In usualthermodynamics the chemical potential of component i of a liquid mixture maybe written as

mi ¼ m�i ðT ; pÞ þ RT ln xi þ RT ln gi ð9:7Þ

The first term of eq 9.7 relates to the pure liquid component i. In equation 9.7,R is the universal gas constant, xi the mole fraction and gi the activity coefficientof component i measuring the deviations from the ideal mixture. To obtain acontinuous version of eq 9.7 would strictly require replacement of xi byW(M)dM and result in

mðMÞ ¼ m�ðM;T ; pÞ þ RT ln WðMÞdM½ � þ RT ln gðMÞ: ð9:8Þ

Because of the infinite number of components dM-0 would yield m(M)-�N. At this point the assumption of an infinite number of components cannotbe applied and dM must be replaced by the finite quantity DM(M) that can beincorporated into the pure-component contribution with mo (M, T, p)¼ m* (M,T, p)þRT ln [DM (M)] so that eq 9.8 can be replaced written as

mðMÞ ¼ moðM;T ; pÞ þ RT lnWðMÞ þ RT ln gðMÞ: ð9:9Þ

For phase equilibrium the first term on the right hand side of eq. 9.9 is the samefor both phases and, therefore, does not matter. Thus, for practical purposes,the use of chemical potential in continuous thermodynamics is not a seriousone.If an equation of state is to be used for the calculation, it is often better to

introduce the fugacity coefficient fi. This is particularly important if there is noexpression for the Gibbs energy as in the case of empirical cubic equations ofstate like Soave-Redlich-Kwong and Peng-Robinson. In usual thermodynamicsthe fugacity coefficient may be calculated from

RT lnfi ¼ZNV

@p

@ni

� �T ;V ;nj 6¼i

�RT

V

( )dV � RT lnZ; ð9:10Þ


where V is the volume and Z¼ pV/(nRT) is the compressibility factor.45 Thecontinuous form of eq 9.10 reads

RT lnf Mð Þ ¼ZNV

@p

@w Mð Þ �RT

V

� �dV � RT lnZ: ð9:11Þ

The quantity @p/w(M), which is the continuous version of @p/@ni, can bedetermined from

limt!0

@p T ;V ;wþ tdwð Þ@t

¼ZM

@p

@wðMÞ dwðMÞdM ð9:12Þ

The integral on the left hand side of eq 9.12 is obtained from an equation ofstate p¼ p(T, V; w). The terms before dw(M)dM are equivalent to the derivative@p/w(M).In this treatment, presented previously, all chemical species of the mixtures are

considered to be similar. Thus, they all are described by one distribution function.In a generalized version the occurrence of several ensembles of very similar species(e.g. paraffinic and aromatic hydrocarbons or polymer blends) in the mixture maybe accounted for by describing each ensemble with its own distribution function.Furthermore, some individual components may also be present (e.g. in a polymersolution), that need to be included into the formalism too. Polydisperse mixturesof this kind are often called semicontinuous mixtures. There are no fundamentaldifficulties in generalizing the simple version of continuous thermodynamicsdiscussed previously, but the equations become somewhat more complex. Here,the principles of generalization are to be presented only by some examples (for amore detailed treatment the reader should refer to ref. 48 for instance).In the generalized version of continuous thermodynamics a mixture is con-

sidered to contain the I individual components A 0, A00, . . . , A(I) and the Densembles (distributions) of very similar species B0, B00, . . . , B(D). The amountsof the individual components may be specified as usual by the amounts ofsubstances nA with A0, A00, . . . , A(I). For the ensembles of very similar speciesthe extensive distribution functions wB(M), with B0, B00, . . . , B(D), are used.Furthermore, there are the mole fractions of the individual components, xA,and of all species of an ensemble B, xB, and the molar distribution functionsWB(M) defined as previously outlined. Hence, the following normalizationconditions must be fulfilled

1 ¼Zt

WBðMÞdt; B ¼ B0;B 0 0; . . . ;BðDÞ ð9:13Þ

1 ¼XA

xA þXB

xB;XA

¼XAðIÞA¼A0

; andXB

¼XBðDÞB¼B 0

:

288 Chapter 9

The extensive Gibbs energy G now is a function of T, p and {nA} and a func-tional with respect to {wB}, where fnAg ¼ nA 0 ; nA 0 0 ; :::nAðIÞ means the ensemble ofthe amounts of discrete substances and fwBg ¼ wB 0 ;wB 0 0 ; :::wBðDÞ means theensemble of the extensive distribution functions. As with the generalization ofeqs 9.3 and 9.4 the differential dG (at constant values of T and p) becomes

dG ¼ limt!0

@

@tG T ; p; nA þ tdnAf g; wB þ tdwBf gð Þ

¼XA

mAdnA þXB

mBðMÞdwBðMÞdMð9:14Þ

Analogous equations are valid for other extensive quantities. The general-ization of eq 9.5 results in

G ¼XA

nAmA þXB

ZM

wBðMÞmBðMÞdM;

Gm ¼XA

xAmA þXB

ZM

WBðMÞmBðMÞdM:

ð9:15Þ

Generalizing eq 9.6 for the Gibbs-Duhem equation provides:

XA

xAdmA þXB

ZM

xBWBðMÞdmBðMÞdM ¼ 0: ð9:16Þ

Instead of eq 9.11 the fugacity coefficients are now given by

RT lnfA ¼ZNV

@p

@nA� RT

V

� �dV � RT lnZ; A ¼ A0;A0 0; :::;AðIÞ ð9:17Þ

RT lnfB Mð Þ ¼ZNV

@p

@wB Mð Þ �RT

V

� �dV � RT lnZ; B ¼ B0;B0 0; :::;BðDÞ

In equation 9.17, the partial derivatives @p/@nA and @p/@wB (M) are at constanttemperature T, volume V, amounts nA of all discrete components (except theconsidered) and of all distribution functions wB (except the considered). Thesepartial derivatives may be calculated applying eq 9.12 if there w is replaced bywB with B¼B0, B00, . . . ,B(I).In some cases a polydisperse mixture cannot be adequately described by a

single distribution variable and multivariate distribution functions have to beapplied. In principal, there are no limitations on the number of variablesinvolved in the distribution function; however, in practice, the limit is two.Briano and Glandt44 were the first to discuss the need to introduce bivariantdistribution functions. Cotterman et al.45 have showed how the chemical


potential may be calculated in the case of multivariate distribution functions.Copolymer systems are important types of polydisperse systems that requirebivariant distribution functions. In this case, a distribution function depends onmolar mass and on chemical composition has to be considered. Distributionfunctions of this type have been applied to calculate the LLE of copolymersolutions and of copolymer blends.53–56

9.3.2.2 Phase Equilibrium

Let us start with the LLE of a solution of a polydisperse polymer B in a solventA. Because of the large differences in molecular size in polymer thermo-dynamics it is usual to describe the composition of the system on the basis ofamounts of segments rini instead of amounts of substance ni. Choosing astandard segment one can calculate segment numbers ri for all the molecules ofthe system. As intensive quantities, corresponding to the amounts of segments,segment fractions are used. So, for example, in the monodisperse case thesegment fraction of the polymer is cB¼ rBnB/(rAnAþ rBnB). The segmentfractions possess the magnitude of the mass fraction or the volume fraction.Accordingly it is useful to chose the segment number r as a characterizationvariable and to replace the molar distribution function WB(r) by the segmentrelated distribution function

Ws;BðrÞ ¼WBðrÞr=�rB: ð9:18Þ

In eq 9.18, �rB is the number average of the segment number defined by

�rB ¼Zr

WBðrÞdr or 1=�rB ¼Zr

Ws;BðrÞ1

rdr: ð9:19Þ

The integration in eq 9.19 is performed for the total domain of definition of r.Practically, in the most cases the limits of the integral are 0 and N. Further-more, instead of the usual chemical potential the segment related quantityms,B(r)¼ mB(r)/r is defined by:

ms;BðrÞ ¼ mos;BðrÞ þ RT1

rln cBWs;BðrÞ½ � þ 1

r� 1

�rB

� �þ RT ln gs;B; ð9:20Þ

where cB is the total segment fraction of the polymer. The second term of eq9.20 is the Flory-Huggins contribution of the ideal athermic solution takinginto account the entropic effects originating from the size differences of themolecules. gs,B of eq 9.20 is the segment-molar activity coefficient and may becalculated from the segment-molar excess Gibbs energy. The segment-molaractivity coefficients do not represent the deviations from the ideal mixture butrather the deviations from the ideal athermic solution. It is usual to assume gs,Bis independent of the segment number.57 If there is an equilibrium between two

290 Chapter 9

liquid phases I and II the equality of the segment-molar chemical potentials inthe frame of continuous thermodynamics is given by

1� cIIB ¼ 1� cI

B

� �exp rArAð Þ ð9:21Þ

where

cIIBW

IIs;BðrÞ ¼ cI

BWIs;BðrÞ exp rrBð Þ: ð9:22Þ

The abbreviations rA and rB are given by

rA ¼1

rAcIB � cII

B

� �þ cII

B

�rIIB� cI

B

�rIBþ ln gIA � ln gIIA and ð9:23aÞ

rB ¼1

rAcIB � cII

B

� �þ cII

B

�rIIB� cI

B

�rIBþ ln gIB � ln gIIB ð9:23bÞ

Usually, cIB and the distribution functionWI

s,B (r) are given and cIIB and T (or p)

at constant p (or T) are the unknowns to be calculated. For this purpose, eq9.22 has to be changed into scalar equations by integrating it to give cII

B andcIIB =�rIIB :

cIIB ¼ cI

B

Zr

WIs;BðrÞ exp rrBð Þdr ð9:24Þ

and

cIIB =�r

IIB ¼ cI

B

Zr

1

rWI

s;BðrÞ exp rrBð Þdr ð9:25Þ

The unknowns cIIB, T (or p) and �rIIB may be calculated by solving eqs 9.21, 9.24

and 9.25. The function T(cIB) at constant p or p(cI

B) at constant T is the cloud-point curve. The function T(cII

B) at constant p or p(cIIB) at constant T is the

shadow curve. Equation 9.22 permits the calculation of the distribution func-tion WII

s,B (r) of the incipient phase II. Assuming a Schulz-Flory distributionfor WI

s,B (r) and all integrals possess analytical solutions.57 If a copolymersolution is considered in eqs 9.24 and 9.25 the single integrals have to bereplaced by double integrals with respect to the segment number and to thechemical composition.53–55 Using a Stockmayer distribution the double inte-grals possess analytical solutions.57 If the system consists of several polymersand of several solvents the procedure discussed here may be generalized instraight forward way.57

To calculate the coexistence curves the mass balance relationships have to betaken into account. Usually, the segment fraction cF

B of a feed phase F and the


corresponding distribution function WFs,B (r) are given (at constant p or T).

Furthermore, the relative amount F ¼ nIIs =nFs of segments in phase II as given

and using eqs 9.23a and 9.23b the unknowns cIIB, T (or p) and �rIIB may be

calculated from

1� cIIB ¼

1� cFB

Fþ ð1� FÞ exp �rArAð Þ ð9:26Þ

cIIB ¼

Zr

cFBW

Fs;BðrÞ

Fþ ð1� FÞ expð�rrBÞdr ð9:27Þ

and

cIIB =�rIIB ¼

Zr

1

r

cFBW

Fs;BðrÞ

Fþ ð1� fÞ expð�rrBÞdr: ð9:28Þ

The quantity cIB (occurring in rA and rB) may be expressed by cII

B and F usingthe balance equation

cFB ¼ ð1� FÞcI

B þ FcIIB : ð9:29Þ

By analogy �rIB may be calculated from the balance equation:

cFB=�rFB ¼ ð1� FÞcI

B=�rIB þ FcII

B =�rIIB : ð9:30Þ

Equations 9.26 to 9.28 contain integrals have to be solved iteratively becausethere are no analytical solutions independent of the distribution function of thefeed phase. For F¼ 0, eqs 9.26 to 9.28 are equivalent to eqs 9.21, 9.24 and 9.25.The treatment may be easily generalized to mixtures of several polymers andseveral solvents.57 Equations 9.26 to 9.30 play an important role in polymerfractionation. In each fractionation step a feed phase F splits into two coex-isting phases I and II.A very simple method to do so is called the Successive Precipitation Frac-

tionation (SPF).58 The initial solution of the polydisperse polymer B in a sui-table solvent A forms the feed phase F. A change of temperature results in aphase separation. The polymer-rich phase II is removed and forms fraction 1.The polymer-lean phase I serves directly as the feed phase for step 2. A furtherchange of temperature results in a second phase separation (step 2). Again, thepolymer-rich phase II is removed forming fraction 2 whereas the polymer-leanphase I is the feed phase of the next step. This procedure is continued. The feedphase of each step equals the polymer-lean phase I of the preceding step. At theend of this process instead of the widely distributed original polymer there areseveral fractions (1, 2,� � �) with narrow molar-mass distributions.

292 Chapter 9

Another type of fractionation is called the Successive Solution Fractionation(SSF).58,59 In the SSF after phase separation the polymer-lean phase is removedand forms fraction 1. The polymer-rich phase is diluted by addition of solventup to the initial volume of the feed phase and forms now the feed phase forseparation step 2 etc. Continuous thermodynamics has also been applied toBaker-Williams fractionation60,61 where the polymer is fractionated in a col-umn using a solvent and a non-solvent. The superposition of a solvent and non-solvent gradient and a temperature gradient leads to a very high separationefficiency.Continuous thermodynamics has also been applied to derive equations for

spinodal, critical point and multiple critical points.62–69 To do so with con-tinuous thermodynamics is much easier than in usual thermodynamics. Spi-nodal and critical points may be calculated for very complex systems or forcases in which the segment-molar excess Gibbs energy and depends on somemoments of the distribution function. In simple cases (for example, a solutionof a polymer in a solvent, where the segment-molar excess Gibbs energy isindependent of the distribution function) the equations of the spinodal and thecritical point are known from the usual thermodynamic treatment. However,for more complex systems continuous thermodynamics has achieved realprogress, for example, for polydisperse copolymer blends, the polydispersity isdescribed by bivariant distribution functions.68

For phase-equilibrium calculations of polymer systems at high pressures theapproach based on the segment-molar excess Gibbs energy is only suitable ifthe pressure dependence of the quantity is known. In this case, a direct appli-cation of equations of state is often more useful. The fugacity coefficient iscalculated from eq 9.17. If a solution of a polymer B in a solvent A is consideredthe fugacity coefficients may be written as

lnfA ¼ rABA � lnpVs�r

M

RT

� �and lnfBðrÞ ¼ rBB � ln

pVs�rM

RT

� �ð9:31Þ

In eq 9.31 Vs is the segment-molar volume and �rM is the number average of thesegment number of the mixture. In eq 9.31, BA and BB are functions of T, Vs

and cB that may be derived from eq 9.17 for a given equation of state. If anequilibrium between the phases I and II is considered the fugacities of bothphases are equal. Then, the equilibrium condition for the solvent is given by

1

rAln

1� cIB

� �VII

s

1� cIIB

� �VI

s

" #þ BI

A � BIIA ¼ 0: ð9:32Þ

The equilibrium condition for the polymer species is given by

WIIs;BðrÞ ¼

cIBV

IIs

cIIB V

Is

exp r BIB � BII

B

� �� WI

s;BðrÞ: ð9:33Þ


Integrating this equation for the total domain by definition of r, a scalarequation with the left side taking the value 1 is obtained. Multiplication of eq9.33 by r and integrating provides a further scalar equation for �rIIB . If T (or p)and cI

B are given both these scalar equations when combined with eq 9.32 serveto calculate the unknowns p (or T), cII

B and �rIIB . The segment-molar volumes VIs

and VIIs have to be computed by solving the equation of state. The distribution

function of the incipient phase II may be calculated from eq 9.33. The functionp (cI

B) at constant T or the function T (cIB) at constant p is the cloud-point

curve. The function p (cIIB) or T (cII

B) is the shadow curve. The calculation ofcloud-point curves and shadow curves involve a complicated numerical pro-cedure. However, ifWI

s,B (r) is a Schulz-Flory distribution (see section 9.3.2.3) aconsiderable simplification is achieved70 because all integrals possess analyticalsolutions and �rIIB may be explicitly calculated. Furthermore, the distributionfunction of the incipient phase II proves to also be a Schulz-Flory distribution.Obtaining the spinodal and critical point of a polydisperse system with an

equation of state has been illusive but the problem has been solved with con-tinuous thermodynamics;71–74 an analytical solution of the determinants oftraditional thermodynamics has been shown to be possible.72,73 Phase equili-brium calculations including polydisperse systems have been performed70,75,76

by treating the non-linear part DAs of the segment-molar Helmholtz energy.For a solution of a polymer B in a solvent A this quantity is given by70

DAs ¼RTZr

cB

rWs;BðrÞ ln cBWs;BðrÞ½ �drþ 1� cB

rAln 1� cBð Þ

8<:

9=;

� RT

�rMlnVs þ Js:

ð9:34Þ

In equation 9.34, the quantity Js includes the equation of state and is given by:

Js ¼ZNVs

p� RT

�rMVs

� �dVs ð9:35Þ

To find the spinodal condition, the first step requires the determination of thesecond differential d2As of the segment-molar Helmholtz energy with respect todVs, dcB and d cBWs;BðrÞ½ �. The second step is to find that variation functiond cBWs;BðrÞ½ �� minimizing the second differential d2As. In this the conditionRrd cBWs;BðrÞ½ �dr ¼ dcB has to be taken into account by using Lagrange’smethod of undetermined multipliers as minimization procedure. Settingd cBWs;BðrÞ½ �� into d2As the quantity d2As dVs; dcB; d cBWs;BðrÞ½ ��f g is found.The expression has the form of a usual second order differential with respect todVs and dcB. Finally one has to find that variation dc�B that minimises d2As.This procedure is similar to that of traditional thermodynamics and withd2As dVs; dc

�B; d cBWs;BðrÞ½ ��

�¼ 0 the spinodal condition follows.70 The first

294 Chapter 9

differential of the spinodal condition together with d cBWs;BðrÞ½ ��results in thecritical condition.70 Knowing phase equilibrium and spinodal the system maybe evaluated with respect to its stability (stable, metastable or unstable). Fur-thermore, knowledge of both spinodal and critical point are useful to findsuitable starting values for phase-equilibrium calculations.For petroleum fractions or similar systems the treatment of phase equi-

libria will now be discussed briefly. The basic principles are the same asthose outlined for polymer systems without recourse to segment-molarquantities. For petroleum fractions the phase-equilibrium problem of impor-tance is the so-called flash calculation that is analogous to the calculationof coexistence curves for a polydisperse polymer solution and in the simplestcase a single distribution function is required. For example, the system maycontain many alkanes characterized by their normal boiling-point tempera-tures Tb that in this work will be denoted by t. At moderate pressures theequilibrium condition is given by the continuous thermodynamics form ofRaoult’s law:

WVðtÞp ¼WLðtÞp � ðt;TÞ: ð9:36Þ

In equation 9.36, the superscripts V and L indicate the vapour phase and theliquid phase, respectively. In eq 9.36 p*(t, T) is the vapour pressure of a purespecies at t and is dependent on temperature. Considering a feed phase Fsplitting in liquid phase L and a vapour phase V in addition to the equilibriumcondition there is also a requirement for material balance given by:

WFðtÞ ¼ ð1� xÞWLðtÞ þ xWVðtÞ: ð9:37Þ

In equation 9.37, x¼ nV/nF is the ratio of the amount of substance in the vapourphase nV to that of the feed phase nF. This quantity is analogous to the quantityF used for the LLE of polymer systems. Combination of eqs 9.36 and 9.37 andaccounting for the normalization condition of W(t) the integration yields theflash equation:

Zt

p� p � ðt;TÞð1� xÞpþ xp � ðt;TÞW

FðtÞdt ¼ 0: ð9:38Þ

Specifying p (or T) eq 9.38 permits the calculation of the equilibrium tem-perature (or p) as function of x. The curve T(x) at constant p or p(x) at constantT is called Equilibrium-Flash-Vaporization (EFV) curve. Knowing T, p and xwith eqs 9.36 and 9.37 the distribution functions for the phases L and Vmay becalculated.77 An analytical solution of the integral of eq 9.38 is only possible forx¼ 0, that is when the feed and liquid phases are equal, when the followingassumptions hold: (1), the distribution function of the given liquid phase has tobe Gaussian; and (2), the vapour pressure function p*(t, T) has to be calculatedwith a combination of Clausius-Clapeyron’s equation and Trouton’s rule.


These two assumptions permit the vapour pressure and the distribution func-tion of the vapour phase to be calculated.78 Generalizing eq 9.38 to systemsconsisting of several individual components and several distribution functionsgives:

XA

p� gAp�AðTÞ

ð1� xÞpþ xgAp�AðTÞxFA

þXB

Zt

p� gBðtÞp�Bðt;TÞð1� xÞpþ xgBðtÞp�Bðt;TÞ

xFBWFB ðtÞdt ¼ 0

ð9:39Þ

In eq 9.39, xFA denotes the mole fractions of the individual components in thefeed phase and xFB the total mole fractions of the ensembles. Equations tocalculate the mole fractions and distribution functions of the phases L and Vhaves also been derived.77,79 Equation 9.39 plays an important role for thecalculation of distillation columns.80,81 For equations of state the same treat-ment requires fugacity coefficients as detailed by Browarzik and Kehlen20

where the flash is given by

XA

xFA fVA � fL

A

� �ð1� xÞfV

A þ xfLA

þXB

Zt

xFBWFB ðtÞ f

VB ðtÞ � fL

BðtÞ�

ð1� xÞfVB ðtÞ þ xfL

BðtÞdt ¼ 0: ð9:40Þ

If the pressure or the temperature is specified the Equilibrium-Flash-Vapor-ization (EFV) curve T(x) or p(x) is obtained.

9.3.2.3 Distribution Functions

In many cases distribution functions are determined experimentally: the char-acterization of petroleum fractions by true-boiling-point distillation or gas-chromatographically simulated distillation, and the characterization of poly-mers by gel-permeation chromatography. In principle, the integrals of con-tinuous thermodynamics may be directly solved based on these experimentallydetermined distribution functions. However, this approach delicate numericalanalyses and the assumption the complete distribution function has beenobtained by experiment; clearly this is no the case, for example, for somepolymers only molar-mass averages are determined. Thus, there are numerouscases where smoothed or analytical distribution function provides more reliablephase equilibrium calculation than those obtained by use of the experimentallydetermined distribution function. When the integrals of continuous thermo-dynamics possess analytical solutions considerably numerical simplification isafforded and this is one motive for the desire to have analytical expressions forthe distribution function.The best-known analytical distribution is the Gaussian distribution that has

been applied to petroleum fractions with the normal boiling-point temperature

296 Chapter 9

t as the characterization variable given by:

WðtÞ ¼ 1

sffiffiffiffiffiffi2pp exp � t� yð Þ2

2s2

" #; ð9:41Þ

where W(t) is the molar distribution function and the quantities y and s arethe mean value of t and the corresponding standard deviation, respectively,given by

y ¼Zt

W tð Þtdt and s2 ¼Zt

W tð Þ t� yð Þ2dt: ð9:42Þ

The Gaussian distribution is symmetrical with respect to the mean value y and,consequently, application to asymmetrical molar-mass distributions of poly-disperse polymers is not possible. However, the composition of petroleumfractions the Gaussian distribution is acceptable and in many practical casesthe lower integration limit is �N and this complete symmetry results in otherdifficulties.These issues can be overcome by use of the Gamma distribution given by

WðtÞ ¼ kk

GðkÞ1

y� t0

t� t0y� t0

�k�1exp �k t� t0

y� t0

�; ð9:43Þ

where, G is the gamma function. In eq 9.43, the quantity t0 is the lower limit ofthe t-range so that W(t) is normalized within the limits t0 and þN and thequantity k is given by k¼ (y� t0)

2/s2, where the mean value y and the standarddeviation s obey eq 9.42. The gamma distribution is very suitable for char-acterization of petroleum residues, as shown by Whitson.82 It may be appliedboth to petroleum fractions (and similar polydisperse fluids) and to polymersolutions. Applying eq 9.43 to polymer solutions t0¼ 0 is assumed and insteadof the boiling-point temperature t the molar massM or the segment number r isused. This results in the replacement of y by the number average �Mn of themolar mass or by the corresponding number average �r of the segment number.For the former the molar distribution function W(r) is used while the latter thesegment-molar distribution function Ws(r). For case of Ws(r) eq 9.43 becomes

WsðrÞ ¼kk

�rGðkÞr

�r

� �kexp �k r

�r

� �: ð9:44Þ

In eq 9.44 the distribution function is called generalized Schulz-Flory dis-tribution (or Schulz-Zimm distribution) because of its introduction by Schulz in1935,83 and by Flory in 1936.84 To evaluate the polydispersity of polymersU ¼ �Mw= �Mn � 1 is an important quantity. The quantity k of the Schulz-Florydistribution is given by k¼ 1/U. The two parameters �r and k of the


Schulz-Flory distribution may be determined from the experimental values of�Mn and �Mw. The calculation of the cloud-point curve and of the shadow curveof polymer systems is essentially simplified applying the Schulz-Flory dis-tribution because all occurring integrals possess analytical solutions.For widely distributed polymers such as radically polymerized polyethene the

Schulz-Flory distribution function is unable to describe the high degree ofasymmetry in the distribution. In this case, the Wesslau distribution (loga-rithmic normal distribution) is used and given by:

WsðrÞ ¼1

bffiffiffipp

rexp � lnðr=r�Þ½ �2

b2

( ): ð9:45Þ

In eq 9.45 the parameter b is related to U by U¼ exp(b2/2)� 1 and the para-meter r* when expressed in terms of �r and U related by r� ¼ �r

ffiffiffiffiffiffiffiffiffiffiffiffiffi1þUp

.The three-parameter Hosemann-Schramek distribution given by

WsðrÞ ¼ adðkþ1Þ=a

G ðkþ 1Þ=að Þ rk exp �drað Þ; ð9:46Þ

can accommodate very different degrees of asymmetry. In eq 9.46 the threeparameters a, k and d are equated to experimental values of �Mn, �Mw and the so-called z-average �Mz. For the case that a¼ 1 and d ¼ k=�r eq 9.46 the Schulz-Flory distribution is recovered. A summary of distribution functions forpolymers has been given by Berger.85

The bivariant Stockmayer distribution has been used for copolymers.86 Inthis case, the product of a Schulz-Flory distribution with respect to molar mass(or segment number) and of a Gaussian distribution with respect to chemicalcomposition y is formed. If the copolymer consists of two kinds of monomersthe chemical composition is defined as the segment fraction of one of themonomers within the copolymer. For the segment number and chemicalcomposition the Stockmayer distribution is given by:

Wsðr; yÞ ¼kk

�rGðkÞr

�r

� �kexp �k r

�r

� ��ffiffiffiffiffiffiffir

2p�

rexp � r y� �yð Þ2

2�

" #; ð9:47Þ

where �y is the average of the chemical composition and e describes the width ofthe chemical part of the distribution. The quantity e may be calculated from �yand kinetic parameters that are specified for the copolymer considered. Forrandom copolymers the chemical part of the distribution function is narrow.Nevertheless, chemical polydispersity can still strongly influence theliquidþ liquid equilibrium of copolymer and the use of the Stockmayer dis-tribution enables an analytical solution of the double integrals of continuousthermodynamics to be solved and both cloud-point and shadow curvescalculated.

298 Chapter 9

Willman and Teja49,87 have described the composition of fossil-fuel mixtures(including gas condensates, absorber oils, crude oils, coal liquids) with abivariant log-normal distribution that depends on the boiling-point tempera-ture and the specific gravity. This scheme was used to determine dew-pointswith the Patel-Teja equation of state.In some cases the polydisperse fluids cannot be described by a single dis-

tribution function and these require a sum of a finite number of simple dis-tribution functions.48,88

Analytical solutions of the integrals of continuous thermodynamics often donot exist and either approximations have to be applied,20 or a numericalintegration is performed where the integral is replaced by a summation overdiscrete points and in this case the continuously distributed components arereplaced by a collection of pseudo-components. When the method of Gaussianquadrature is used these pseudo-components are not arbitrarily chosen20,46,50

because the number of quadrature points (corresponding to ‘‘effective pseudo-components’’) are selected to provide the most precise representation of thevalue of the integral. Thus the choice of quadrature points permits the optimaldiscrete representation of the continuous mixture for a specified number ofpoints. It has been shown that very small number of quadrature points (e.g. 4)proves to be sufficient to calculate the integrals of continuous thermodynamicswith sufficient precision.20 There are of course different quadraturemethods50,89,90 and these alternatives have been applied by Schlijper,91 and byYing et al.92 and discussed elsewhere.20

9.4 Application to Real Systems

9.4.1 Polymer Systems

The spinodal curve and the critical points (including multiple critical points) onlydepend on fewmoments of the molar-mass distribution of the polydisperse systemwhile the cloud-point curve the shadow curve and the coexistence curves arestrongly influenced by the whole curvature of the distribution function. Themethods used that include the real molar-mass distribution or an assumed ana-lytical distribution replaced by several hundred discrete components have beenreviewed by Kamide.93 In the 1980s continuous thermodynamics was applied, forexample, by Ratzsch and Kehlen to calculate the phase equilibrium of a solutionof polyethene in supercritical ethene as a function of pressures at T¼ 403.15K.94

The Flory’s model95 was used with an equation of state to describe the poly-dispersity of polyethene with a a Wesslau distribution. Ratzsch and Wohlfarthapplied continuous thermodynamics to the high-pressure phase equilibrium of{ethene [ethylene]þ poly(but-3-enoic acid; ethene) [poly(ethylene-co-vinylace-tate)]} and to the corresponding quasiternary system including ethenyl ethanoate[vinylacetate].96 In addition to Flory’s equation of state Ratzsch and Wohlfarthalso tested the Schotte model as a suitable means to describe the phase equili-brium97 neglecting the polydispersity with respect to chemical composition of the


copolymer. Most applications of continuous thermodynamics in the 1980s andpart of the 1990s were restricted to normal pressure and were based on relativelysimple Gibbs energy models. Ratzsch et al.98 calculated the cloud-point curve andcoexistence curves for solutions of poly(ethenylbenzene) [polystyrene] in cyclo-hexane. They applied a modified Flory-Huggins model and the generalizedSchulz-Flory distribution to describe the polydispersity of the poly(-ethenylbenzene) samples. Choi et al.99,100 performed similar calculations with theextended Flory-Huggins model describing the polydispersity of the polymer witha Wesslau distribution providing calculations of the cloud-point curve and theshadow curve. A generalized Schulz-Flory distribution was applied for calcula-tions with poly(ethenylbenzene)[polystyrene]þ poly(methoxyethene) [poly(vinylmethyl ether)] blends101 and for poly(ethenylbenzene)[polystyrene]þ cetena-poly[(methylphenylsilicon)-m-oxo][poly(methylphenylsiloxane)].102 Enders et al.103

studied experimentally and theoretically the polymer blend {poly(di-methylsiloxane)þ poly(ethylmethylsiloxane)}. The calculations were based oncontinuous thermodynamics describing the polydispersity of the polymers by ageneralized Schulz-Flory distribution and studied also the influence of molar-mass distribution on the compatibility of polymers104 In these calculations amonodisperse polymer and a polydisperse polymer were considered. Thepolydispersity was also described by a generalized Schulz-Flory distribution.The influence of the polydispersity on phase equilibrium was studied by var-iation of the quantity U ¼ �Mw= �Mn � 1 at constant mass average �Mw. Endersand co-workers represented the polydisperse polymer by 100 components andcalculated the phase equilibrium by direct minimization of the Gibbs energy ofmixing. The results quantitatively agree with those obtained from computa-tions on the basis of continuous thermodynamics. Eckelt et al.105 studied bothexperimentally and theoretically, with calculations based on continuous ther-modynamics, the phase equilibrium of systems consisting of two polydispersepolymers and a solvent. The continuous thermodynamics was also appliedsolely to the polymer fractionation.58–61 Stockmayer’s bivariant distributionfunction was used to calculate the phase equilibria of the copolymer systemstaking into account the double polydispersity of the copolymers. In thiscalculation, the activity coefficients were expressed with the w-parameter ofFlory and Huggins theory that is considered to be a quadratic polynomial withrespect to the average of the chemical composition �y.54–56,106,107

Beginning in the 1990s calculations of high-pressure phase equilibria of poly-disperse polymer systems were performed. For example, Enders and de Loos108

calculated cloud-point and spinodal curves in the high-pressure range for{methylcyclohexaneþ poly(ethenylbenzene)} and compared their results withexperimental data. Enders and de Loos108 used a Gibbs-energy model withpressure dependent parameters and models that include an equation of state, suchas the lattice fluid model introduced by Hu et al.109,110 for the monodisperse andpolydisperse case by Hu et al.111,112 Enders and de Loos108 performed computersimulations showing the influence of the polydispersity on the cloud-point curve,the shadow curve, the spinodal curve and the critical point for polymer solutionsat different pressures. For {propan-2-oneþ poly(ethenylbenzene)} the calculated

300 Chapter 9

upper critical solution temperature (UCST) and lower critical solution tem-perature (LCST) formed an hour-glass phase diagram. Browarzik and Kowa-lewski75 calculated the high-pressure phase equilibrium and spinodal curves forthe system {poly(ethenylbenzene)þ cyclohexaneþ carbon dioxide} describing thepolydispersity of poly(ethenylbenzene) by a generalized Schulz-Flory distributionand continuous thermodynamics with the Sako-Wu-Prausnitz equation ofstate113 that is a generalization of the Soave-Redlich-Kwong equation of state.Browarzik and Kowalewski70,76 performed calculations of the phase equilibria of{methylcyclohexaneþ poly(ethenylbenzene)}with an equation of state containingone additional parameter that accommodates the flexibility of the polymer chain.The pressure concentration diagram for {methylcyclohexaneþ poly(ethenyl-benzene)} shows both a curve with a lower critical point (LCP) and a curve withan upper critical point (UCP) in a limited temperature region. At low tempera-tures there is an hour-glass shaped two-phase region. For nearly monodispersepoly(ethenylbenzene) there are experimental cloud-point and spinodal data forboth types of phase diagram. The model parameters were fitted to these dataresulting in good predictions of the cloud-point and spinodal curves. The authorsperformed computer simulations showing the transition of the LCP/UCP beha-vior into the hour-glass behavior for the polydisperse case and was demonstratedto be quite different from the monodisperse case. Figures 9.2a through 9.2dillustrate the transition and the results of the calculations.

Figure 9.2a Pressure p as a function of the total mass fractionoB for methylcyclohexane(A) with polydisperse poly(ethenylbenzene) (B) ( �Mn ¼ 16 500 g �mol�1,U¼ 1) illustrating the Lower Critical Point and Upper Critical Pointbehaviour at a temperature of 296.65K. –––––, cloud-point curves;- - - -,shadow curves; � � � � � � , spinodal curves; and ’, critical points.70,76


In Figure 9.2a shows the cloud-point, shadow and spinodal curves pressurefor a solution of a polydisperse poly(ethenylbenzene) in methylcyclohexane as afunction of polymer mass fraction at a temperature of 296.65K. At this tem-perature all curves show LCP/UCP behavior. The critical points are intersec-tion points of the corresponding cloud-point and shadow curves. In the criticalpoints the spinodal curves possess the same slope as the respective cloud-pointcurves. In the monodisperse case the transition of the LCP/UCP behaviour intothe hour-glass-behavior takes place simultaneously for the cloud-point and thespinodal curves. For the polydisperse case, shown in Figure 9.2b, the cloud-point and the shadow curves coincide at their extremes forming a cross whilethe spinodal curves still show the LCP and UCP behaviour. Lowering thetemperature further, as shown in Figure 9.2c, the cloud-point and the shadowcurves are hour-glass shaped and only the cloud-point curve 2 and the corre-sponding shadow curve 2* cross in the critical points. Lowering the tempera-ture further results in the spinodal curves merging to form first a cross ofstraight lines and then hour-glass behaviour (not illustrated in Figure 9.2).Further temperature reductions, shown in Figure 9.2d, both critical pointsapproach each other and finally merging to form a double critical point. If thetemperature is further lowered (but not illustrated in Figure 9.2) the doublecritical point vanishes and the cloud-point curve 2 and the shadow curve 2*separate. Browarzik and Kowalewski70,76 found the calculated cloud-point and

Figure 9.2b Pressure p as a function of the total mass fractionoB for methylcyclohexane(A) with polydisperse poly(ethenylbenzene) (B) ( �Mn ¼ 16 500 g �mol�1,U¼ 1) illustrating the coincidence of the cloud-point curves at a a tem-perature of 296.623 K. –––––, cloud-point curves; - - - -, shadow curves;� � � � � � , spinodal curves; and ’, critical points.70,76

302 Chapter 9

spinodal curves were in good agreement with experimental values for a realsolution of a widely distributed poly(ethenylbenzene) in methylcyclohexane.Browarzik also studied the transition of the LCST and UCST behaviour intothe hour-glass shaped curves for {poly(ethenylbenzene)þ poly(pentyl 2-methylpropenoate) [poly(n-pentyl methacrylate)]} at normal pressure in thetemperature concentration diagram.114 The calculations were based on thelattice model of Flory and Huggins assuming the w-parameter was a quadraticpolynomial with respect to temperature. The parameters were fitted toexperimental data for the nearly monodisperse case. Using these parameters theexperimental data were described reasonably well.The Sako-Wu-Prausnitz equation of state was also applied to high-pressure

phase equilibria of polyolefin systems by Tork et al.15 The calculations werebased on the pseudo-component method where the number of pseudo-compo-nents used were between 2 and 8. The small number of pseudo-components is aresult of the very efficient estimation method used to adjust the pseudo-compo-nents to the moments of the distribution function (described in section 9.3.1). Inso doing Tork et al.15 were able to provide a good description of the experimentaldata and show, perhaps not surprisingly, the agreement between calculated andexperimental data improved with increasing number of pseudo-components.Browarzik et al.1,115 used the Sako-Wu-Prausnitz equation of state to study

the high-pressure phase equilibrium of the {etheneþ poly(but-3-enoic acid;

Figure 9.2c Pressure p as a function of the total mass fractionoB for methylcyclohexane(A) with polydisperse poly(ethenylbenzene) (B) ( �Mn ¼ 16 500 g �mol�1,U¼ 1) illustrating the hour-glass behaviour of the cloud-point and theshadow curves at a a temperature of 296.50K. - - - - -, cloud-point curves;––––, shadow curves; � � � � � � � � � � , spinodal curves; and ’, criticalpoints.70,76


ethene)} [EVA-copolymer] and {etheneþ poly(but-3-enoic acid; ethene)þethenyl ethanoate} including polydispersity; this approach differs from thattaken by Ratzsch and Wohlfarth.96 Browarzik et al.1,115 was able to show withcomputer simulation the importance of polydispersity. For the quasibinary{etheneþ poly(but-3-enoic acid; ethene)} the only mixing parameter kAB wasfitted to the experimental data of Wagner.116 The calculated results agree rea-sonably well with these experimental data. Thus, the kAB-parameter was knownfor 11 copolymer samples differing in the average ethenyl ethanoate segmentfraction �y at five temperatures. Therefore, kAB could be expressed as a functionof �y and T. Using this function the phase equilibrium for systems of this type,that were not involved in the fit to determine the parameter, could be used topredict any values of �y and T. As a test the authors predicted the cloud-pointcurve for three systems studied experimentally by Nieszporek and Folie.3,117

Figure 9.3 shows the agreement between the measured and calculated pressure asa function of copolymer mass fraction for {etheneþ poly(but-3-enoic acid;ethene) [EVA3-1]} (for which �Mn ¼ 13 000 g �mol�1, U¼ 0.7 and �y ¼ 0:325)and {ethyleneþ poly(but-3-enoic acid; ethene) [EVA3-7]} (for which�Mn ¼ 83 600 g �mol�1, U¼ 1.76 and �y ¼ 0:26).More recent articles have are focussed on the application of either lattice-

fluid models or SAFT in the framework of continuous thermodynamics or ofthe pseudo-component concept. Koak et al.16 has calculated cloud-point and

Figure 9.2d Pressure p as a function of the total mass fraction oB for methylcyclohexane(A) with polydisperse poly(ethenylbenzene) (B) ( �Mn ¼ 16 500 g �mol�1,U¼ 1) illustrating the formation of a double critical point at a a temperatureof 296.208K. –––––, cloud-point curves; - - - -, shadow curves; � � � � � � ,spinodal curves; and ’, critical point.70,76

304 Chapter 9

shadow curves as well as spinodals and critical points at high pressures for(ethyleneþ polyethylene) and (1-buteneþ polybutene) comparing the resultswith their own experimental data. The calculations included both the Sanchez-Lacombe equation of state and SAFT.118–122 The polydispersity of the polymersamples was described by pseudo-components and required between 13 and 36pseudo-components. The calculations provided values that differed unsa-tisfactorily from experimental data and alternative strategies to estimate thepolymer parameters have been suggested.Phoenix and Heidemann developed a new computer algorithm, based on the

work of Michelsen and Koak,124,125 to calculate the cloud-point and the sha-dow curve of polydisperse polymer solutions in the framework of continuousthermodynamics using the Sanchez-Lacombe equation of state.123 The methodwas tested with (hexaneþ polyethene) and (etheneþ polyethene). To describe

Figure 9.3 Pressure p as a function of the mass fraction o of poly(but-3-enoic acid;ethene) in etheneþ poly(but-3-enoic acid; ethene) [EVA3-1] (for which�Mn ¼ 13 000 g �mol�1, U ¼ 0:7 and �y ¼ 0:325) and etheneþ poly(but-3-enoic acid; ethene) [EVA3-7] (for which �Mn ¼ 83 600 g �mol�1, U¼ 1.76and �y ¼ 0:26) at a temperature of 433.15K. ––––––, calculated cloud-point curves;1,115 - - - -, calculated shadow curves;E, measurements foretheneþ poly(but-3-enoic acid; ethene) [EVA3-1] reported by Niesz-porek;3,117 ’, measurements for etheneþ poly(but-3-enoic acid; ethene)[EVA3-7] reported by Nieszporek.3,117


the polydispersity the generalized Schulz-Flory distribution and the Wesslaudistribution (logarithmic normal distribution) were used. Applying a modifiedSanchez-Lacombe equation of state the cloud-point was calculated for iso-therms of polyetheneþ hydrocarbon.126 These isotherms can exhibit abruptchanges in slope when the polyethene has a broad molar-mass distribution. Thebehaviour can be described by showing the origin is three-phase equilibriumand other examples provided where there is four-phase equilibria. The poly-dispersity of the polymer was represented by 100 pseudo-components.Heidemann et al.127 also presented a discontinuous method to calculate

spinodal curves and critical points using two different versions of the Sanchez-Lacombe equation of state and PC-SAFT.128 Moreover, Krenz and Heide-mann applied the modified Sanchez-Lacombe equation of state to calculate thephase behaviour of polydisperse polymer blends in hydrocarbons.129 In thisanalysis the polymer samples were represented by 100 pseudo-components.Taimoori and Panayiotou130 developed a lattice-fluid model incorporating theclassical quasi-chemical approach and applied the model in the framework ofcontinuous thermodynamics to polydisperse polymer solutions and mixtures.The polydispersity of the polymers was expressed by the Wesslau distribution.Recently, PC-SAFT has been increasingly applied to mostly nearly mono-

disperse, polymer systems and in some cases to polydisperse polymer andcopolymer systems with either two or three pseudo-components.131–133

Using the pseudo-component concept with multimodal molar-mass dis-tributions, requires relatively high numbers of pseudo-components and in thiscase the numerical effort required for phase-equilibrium calculations is enor-mous. To overcome this deficiency, Behme et al.134 proposed a multicomponentflash algorithm known by the acronym POLYMIX that is intended for poly-disperse polymer for which the computing time is independent of the number ofassumed pseudo-components. Behme et al.134 applied POLYMIX to polymermixtures containing polyethene and polypropene. Applying SAFT in the fra-mework of the pseudo-component concept Ghosh et al.18 proposed an algo-rithm that simplifies the calculation of the stability and of the phase equilibriumof polydisperse polymer systems.Recently, phase behaviour of mixtures consisting of a polydisperse polymer

(polystyrene) and nematic liquid crystals (p-ethoxy-benzylidene-p-n-butylani-line) was calculated and determined experimentally.135 The former used a semi-empirical model based on the extended Flory-Huggins model in the frameworkof continuous thermodynamics and predicted the nematic-isotropic transition.The model was improved with a modified double-lattice model including Maier-Saupe theory for anisotropic ordering and able to describe isotropic mixing.136

9.4.2 Petroleum Fluids, Asphaltenes, Waxes and Other

Applications

Most applications concerning phase equilibria of the systems the title of thissection includes are based on equations of state with both pseudo-component

306 Chapter 9

method and continuous thermodynamics. In the 1980s industrial processesas supercritical-fluid extraction were discussed.137–139 In the 1980s and 1990scubic equation of states such as the Soave-Redlich-Kwong equation ofstate45,46,137,140,141 and the Peng-Robinson equation of state have been used inthe calculations of phase equilibria for polydisperse systems.142–146Von Bergenet al.11 applied the Sanchez-Lacombe equation of state to phase equilibria ofnatural gases and crude oils using the pseudo-component mehod. Koch et al.147

calculated EFV-curves for a mixture of a light (relatively low molar mass) oiland propan-2-ol within the framework of continuous thermodynamics with theaid of the mean field lattice-gas model of Kleintjens and Koningsveld.148

Reviews of the work have been given by Cotterman and Praunitz52,149 and byBrowarzik and Kehlen.20 In few cases cubic equations of state are still appliedincluding the work of Pederson et al.8 whom studied the PVT behaviour ofreservoir fluids using the Soave-Redlich-Kwong equation of state and thePeng-Robinson equation of state in the framework of the pseudo-componentmethod.Holderman et al.150 calculated the differential distillation and crystallization

of multicomponent alkane mixtures with a 94 component mixture representedby a function decaying exponentially with increasing number of carbon atoms.A 12-component quadrature representation (Laguerre-Gauss quadrature) ofthe same mixture Holderman et al.150 authors assumed a solid solution beingno ideal in nature and obtained results for the differential distillation processthat did not differ significantly from that obtained with complete descriptionwhile the crystallization process results are essentially different. The compu-tations suggest that the strong non-ideality of the solid solution can render itspseudo-component description inadequate, leading to unrealistic results whenphase equilibrium is considered. Datta and Singh151 used continuous thermo-dynamics to propose a new approach to find suitable quadrature points forthe calculation of vapour-liquid equilibrium of petroleum mixtures. Lage152

proposed a quadrature method to solve the integrals occurring in thephase-equilibrium treatment of continuous thermodynamics by choosingquadrature points so that the moments of the distribution are representedcorrectly. This treatment is analogous to the pseudo-component methodreported by Tork et al.15 for LLE calculations on polydisperse polymer solu-tions. In some cases simple methods are used to choose the pseudo-componentswith success, for example, Marano and Holder12 have calculated the vapour-liquid equilibrium of Fischer–Tropsch liquids using carbon-number basedpseudo-components where the properties of a pseudo-component were basedon a hypothetical model component in each carbon-number cut.Vakili-Nezhaad et al.153 applied continuous thermodynamics to the calcu-

lation of the vapour pressure of petroleum fluids improving a method reportedby Kehlen and Ratzsch41 that used an analytical expression for the vapourpressure of a continuous ensemble starting with a continuous version ofRaoult’s law. Assuming a combination of Trouton’s rule and the Clausius-Clapeyron equation for the vapour pressure of the pure species and a Gaussiandistribution to represent the polydispersity of the system to solve the integral


Vakili-Nezhaad et al.153 found relatively good agreement between calculatedand experimental data neglecting the intermolecular interactions. The agree-ment between the calculated and experimental vapour pressure was improvedwhen the intermolecular interactions were taken into account by the intro-duction of the UNIFAC model.Nichita et al.14 applied the pseudo-component method to the wax pre-

cipitation from hydrocarbon mixtures. To do so a general form of a two-parameter equation of state was used for vapour and liquid phases. The heavycomponents were assumed to precipitate in a single solid solution. Becauselumping in pseudo-components often results in difficulties in solid-liquidequilibrium calculations the authors proposed a delumping procedure (men-tioned in section 9.3.1). Lira-Galeana et al.154 calculated wax precipitation inpetroleum mixtures by assuming the wax consisted of several solid phases eachdescribed as a pure component or pseudo-component immiscible with othersolid phases.Browarzik and Matthai155 applied continuous thermodynamics to the sol-

vent-deoiling process of waxes. In this work, the wax was considered to consistof n-alkanes and non-n-alkanes including all the other types of paraffinic spe-cies. Each alkanes type were described by separate Gaussian distributions withrespect to the number of carbon atoms. Because of the use of the Gaussiandistribution, the equilibrium conditions were integrated analytically; the inte-grals of the material balances do not possess analytical solutions and theproposal of Cotterman et al.45 was adopted to simplify the solution byrestricting the distribution functions to some moments. Browarzik and Mat-thai155 assumed complete miscibility in the solid phase and described the non-ideality of the liquid phase by the Flory-Huggins theory. To evaluate theaccuracy of the calculations three slack waxes were deoiled in laboratory underconditions comparable with that of the industrial process. The composition ofthe feed wax, the deoiled wax and the oily wax were determined by gas chro-matography. The agreement between the calculated and the experimental datawas considered to be relatively good considering the complexity of the systemand the simplicity of the model.Much effort has been directed at asphaltenes flocculation and precipitation in

petroleum crude mixtures that is a significant problem in oil production,transmission and processing facilities. Pressure, temperature, the chemicalcomposition of the oil and the amount of dissolved gases affect this undesiredphenomenon. In general, the prediction of asphaltenes precipitation is verydifficult and suffers from the definition of an asphaltene. Usually, asphaltenesare defined as the part of the crude oil that is soluble in methylbenzene andbenzene but insoluble in pentane or heptane. Asphaltenes consist of manythousands of species, differing in size and chemical structure. The aromaticcharacter of the asphaltenes and their content of heteroatoms influence theirsolubility in different solvents and the tendency to flocculate. In most of thecalculations of the phase equilibria the pseudo-component method has beenused. For example, Victorov and Smirnova156,157 developed a model forasphaltenes precipitation that is based on the assumption of asphaltenes-resin

308 Chapter 9

micelles. To describe the dependence of aggregation equilibrium on pressure,the fugacities of the monomeric species were calculated with the Peng-Robin-son equation of state. Akbarzadeh et al.158 adopted an alternative strategydividing the crude oil into saturates, aromatics, resins and asphaltenes wherethe asphaltenes were further divided into several fractions with the generalizedSchultz-Flory distribution with respect to molar mass. These fractions wereconsidered to be pseudo-components. The asphaltenes were treated as polymer-like compounds assuming the high molar mass asphaltenes to be aggregates ofmonodisperse asphaltenes monomers. Akbarzadeh et al.158 applied the Soave-Redlich-Kwong equation of state modified by a group-contribution method tocalculate the LLE between the oil phase and an asphaltene-rich phase. In otherwork the Peng-Robinson equation of state has been used,159 and in others ageneralized regular solution model has been presented.160,161 Recently, Gon-zalez et al.13 applied PC-SAFT to asphaltene precipitation using the pseudo-component method (see section 9.3.1).Monteagudo et al.162 characterized the asphaltenes as a continuous ensemble

for which the distribution function was taken from the fractal aggregationtheory. The asphaltene family was discretized in pseudo-components by theGauss-Laguerre quadrature. Only the asphaltene polydispersity was taken intoaccount. All other components were represented by as ‘‘solvent’’ whose prop-erties (molar volume and solubility parameter) were calculated form a cubicequations of state. Aggregation of asphaltenes was considered to be a reversibleprocess. And it was assumed the phase equilibrium was between a liquid phaseand a pseudo-liquid phase containing only asphaltenes.Very recently, Manshad and Edalat163 developed an algorithm based on the

Scott-Magat theory of polydisperse polymer solutions and continuous ther-modynamics. Three commonly used continuous distribution functions wereexamined for characterization of asphaltenes. The best results were obtainedusing the fractal molar mass distribution function. The authors calculated thesolubility parameter and the volume fraction of asphaltenes as well as theamount of asphaltenes precipitated through minimization of Gibbs energyusing the phase equilibrium condition.Browarzik et al.164 calculated asphaltenes flocculation at high pressures for

methaneþ crude oilþ 2,2,4-trimethylpentane [i-octane] using continuousthermodynamics where 2,2,4-trimethylpentane acts as a precipitant. Theasphaltene flocculation was considered to be a liquidþ liquid equilibrium.Browarzik et al.164 applied the van der Waals equation of state. The poly-dispersity of the crude oil was considered to be described by the solubilityparameter of the Scatchard-Hildebrand theory. Within this distribution theasphaltenes represent the species with the highest solubility parameters. Thecalculated results were compared to experimental data.165 For oils with a verylow content of asphaltenes the model describes the experimental flocculationdata reasonably well. However, on contrary to the experimental results, themodel predicts the asphaltenes to show a higher flocculation tendency withincreasing asphaltenes content of the crude oil. Based on these comparisonsfurther work was undertaken by Browarzik et al.166 and the associates formed


by ‘‘asphaltene monomers’’ were taken into account. Using continuous ther-modynamics the mass action law was applied to the association equilibriumand an analytical expression for the size distribution function of the associatedasphaltenes was derived. This distribution depends on temperature, pressureand concentration. The concept was applied to the (methaneþ crudeoilþ 2,2,4-trimethylpentane), but in this case the crude oil was represented by asemicontinuous mixture of a discrete maltene, which represents all componentsof the crude oil except the asphaltenes (or matter soluble in heptane and hex-ane), and a continuous ensemble of asphaltenes associates. The liquid feedphase was assumed to be in equilibrium with a vapour phase containing onlymethane and a pseudo-liquid phase containing only asphaltenes. For themodelling Browarzik et al.166 applied the Sako-Wu-Prausnitz equation ofstate.113 The aim of the investigation was the calculation of the volume of theprecipitant (2,2,4-trimethylpentane) required to provoke flocculation and thedependence on pressure. The results were compared to experimental data andwhereas the previous model predicted a very strong increase of flocculationtendency with increasing asphaltenes content of the oil,164 the new modelpredicted only a very slight dependence on the asphaltenes content.166 Fur-thermore, the old model predicted flocculation for higher asphaltene contentswithout the presence of the precipitant 2,2,4-trimethylpentane contrary to theexperimental observations. However, as shown in Figure 9.4 the more recent

0 5 10 15 20 25 300.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Vp

/Voi

l

p /MPa

Figure 9.4 Ratio Vp/Voil of the volume of the precipitant 2,2,4-trimethylpentane Vp

to the volume of oil Voil as a function of pressure at the onset of floccu-lation for a crude oil with asphaltene mass fraction of 0.0923. –––––,calculated at T¼ 373.15K; - - - -, calculated at T¼ 328.15K; J, experi-mental data at T¼ 373.15K; and n, measurements at T¼ 328.15K.165,166

310 Chapter 9

model was able to describe reasonably well the experimental observations evenfor a crude oil with mass fraction of asphaltenes of 0.0923. Only at the highertemperature and higher pressures did the calculated results deviate from theexperimental values. It is plausible these differences arose because in thistemperature and pressure range the assumption of a vapour phase consistingsolely of methane is rather to naıve. The superiority of the predictions obtainedfrom ref 166 is a result of the consideration of asphaltene association.There are other examples of the calculation of phase equilibrium that are

of practical interest and worthy of further discussion. For example, Rodrigueset al.167 studied the phase equilibrium for (rice bran oilþ fatty acid-sþ ethanolþwaterþ g-oryzanol þ tocols). g-oryzanol is a mixture of severalferulic acid [(E)-3-(4-hydroxy-3-methoxy-phenyl)prop-2-enoic acid] esters ofsterol and triterpene alcohols. Tocols consist of tocopherols and tocotrienols.g-oryzanol. Tocols are minor components of rice bran oil possessing anti-oxidant and other beneficial physiological properties. Rodrigues et al.167

applied the pseudo-component method with the NRTL and the UNIQUACmodel to calculate the LLE of this compositionally very complex system. Forparameter estimation model fatty systems were investigated.Abdel-Quader and Hallett investigated the part played by internal mixing in

the evaporation of droplets of polydisperse mixtures.168 Continuous thermo-dynamics was applied to develop the formulations of the liquid phase transportequations and diffusivities. The mixtures were described by single distributionfunction as well as with two widely separated distributions, composed of a verylight and a very heavy fraction.Continuous thermodynamics may be applied to association and to aggregation

processes too. One example is the asphaltene association outlined above.166 How-ever, also in mixtures of simple molecules as alcohols or amines a huge number ofchain associates of differing size are formed. Applying mass action law and con-tinuous thermodynamics to the association equilibrium an analytical expression forthe size distribution of the associates was derived by Browarzik.169–171 Unlike themolar-mass distribution function of a polymer this function depends on tempera-ture and on the total concentration of the associated component. Using the size-distribution function of the associates, simple expressions for the excess functionswere found. Flory-Huggins theory was applied to describe the entropic effectoriginating from the size differences of the species and to take the intermole-cular interactions (by the w-parameter) into account. For several binary systemscontaining methanol, ethanol, aniline and water the excess Gibbs energy wascalculated with excess enthalpy, VLE and LLE (if there is one) and the results werefound to be in good agreement with the experimental data.169–171 This improvedmodel is called continuous thermodynamics of associated systems (CONTAS).171

In other systems, aggregation phenomena play an important role, forexample, micellization in aqueous surfactant solution. Browarzik and Bro-warzik172,173 have calculated the LLE for several aqueous solutions of CiEj

surfactants CH3(CH2)i�1 (OCH2CH2)jOH with tail length i varying from 4 to12 and head length j varying from 1 to 6. To perform these calculations con-tinuous thermodynamics was applied in manner similar to that described for


association above and thus micellization and the self association of water wastaken into account. For all studied systems the LLE calculated was in goodagreement with the experimental data. The model was also extended to{waterþCiEjþ alkane}174 and because of the numerous experimental data for{waterþC4E1þ dodecane} this system was investigated in detail. For {water-þC4E1þ dodecane} at temperatures below 306K there is an equilibriumbetween an alkane-in-water microemulsion and an alkane-excess phase.Increasing the temperature a three-phase equilibrium between a bicontinuousmicroemulsion phase, an alkane-excess phase and a water-excess phase isformed. The upper corner of the three-phase triangle that characterizes themiddle phase shifts from the water-rich side to the alkane-rich side with furthertemperature increments. At temperature above 345K the three-phase equili-brium vanishes. At higher temperatures a water-in-alkane microemulsion is inequilibrium with a water-excess phase. The authors calculate the phase equi-librium at several temperatures showing the development of the three-phaseequilibrium. The temperature dependence of the three-phase equilibrium wasconveniently presented as vertical intersection through the phase prism byplotting the temperature versus the surfactant mass fraction. This intersectionis usually performed at constant alkane to water mass ratio. The resulting phaseboundaries resemble the shape of a fish. Figure 9.5 shows the calculated ‘‘fish’’

0.0 0.2 0.4 0.6 0.8

300

320

340

360

T/K

�C4E1

Figure 9.5 Temperature T as a funciton mass fraction oC4E1of C4E1 for {water-

þC4E1þ dodecane}. ––––––, calculated fish-shaped phase diagram forrelated to equal mass fractions of water and dodecane;174 ’, experimentaldata.

312 Chapter 9

at equal mass fractions of water and alkane that is in relatively good agreementwith the experimental data of Burauer et al.175 The ‘‘fish body’’ (closed part ofcurve) shown in Figure 9.5 includes the three-phase region. The ‘‘fish tail’’separates the one-phase region (shown on the right hand side of Figure 9.5)from the two-phase region (shown on the left hand side of Figure 9.5). In theregion above the fish a water-in-alkane microemulsion is in equilibrium with anexcess water phase. In the region below the fish an alkane-in-water micro-emulsion equilibrates with an alkane-excess phase.There are also other models that incorporate the size distribution of the

aggregates in surfactant systems. For example, Nagarajan and Ruckenstein176

developed a thermodynamic treatment for aqueous solutions of non-ionicsurfactants, taking into account the geometrical type of micelles. Enders andHantzschel177 improved this model and applied it to aqueous carbohydratesolutions. Recently, Enders and Kahl applied a similar treatment to CxGy

surfactantsþwaterþ alcohol mixtures.178 The CxGy surfactant moleculesconsist of a hydrophobic alkyl chain with x carbon atoms and of a hydrophilichead group with y glucose groups. In this work it was assumed the mixedmicelles consisted of surfactant and alcohol molecules and the principle ofmultiple-chemical equilibrium was applied. In this way bivariant micellar dis-tribution functions depending on the aggregation number and on the chemicalcomposition of the mixed micelles were obtained. The critical micelle con-centration and the surface tension of the systems was calculated with this modeland found to be in good agreement with experimental data.

9.5 Conclusions

This Chapter has shown the thermodynamic treatment of polydisperse systemshas gained in importance during the last decade. This is particularly so for theestimation of liquidþ liquid equilibria of polymer systems and the precipitationof asphaltenes from crude oils and of waxes from petroleum fractions with boththe pseudo-component and continuous thermodynamics methods. The choiceof method depends strongly on the problem to be studied. In some casescontinuous thermodynamics results in essential mathematical and numericalsimplifications, particularly, if the occurring integrals possess analytical solu-tions. However, such advantages have lost importance because of the highpower of modern computers. In many cases the integrals of continuous ther-modynamics do not possess analytical solutions and then quadrature methodsare well suited to solve the integrals and ‘‘effective pseudo-components’’ aredetermined. In such cases, continuous thermodynamics is not different from thepseudo-component method, but it is the method to choose the most suitablepseudo-components. If the pseudo-components are chosen by fitting to themoments of a continuous distribution function the methods are similar. In thisreport many powerful algorithms for phase equilibrium calculations of poly-disperse systems have been discussed. If there is enough experimental infor-mation about the distribution of the species polydispersity may be adequately


described. A further improvement of these methods or the development of newapproaches to treat polydispersity is not the main task for future work insteadthe effort should be expended to improve the models describing the inter-molecular interactions. This is because of the growing importance of high-pressure phase equilibria for which precise equations of state are of increasinginterest. Recently, SAFT and its modifications have been preferred. In mostcases the models and equations of state contain pure-component parametersthat have to be fitted to experimental data. For polydisperse systems there isoften no suitable experimental data, for example, in the case of asphaltenes. Inother cases there is experimental data but not for the preferred quantities and,unfortunately, the modelling results depend strongly on the choice of measuredproperties used in the fit. This is today more of an art than a science and toovercome these matters is a challenge for future work.

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320 Chapter 9

CHAPTER 10

Thermodynamic Behaviour ofFluids near Critical Points

HASSAN BEHNEJAD*, JAN V. SENGERS AND MIKHAILA. ANISIMOV

Institute for Physical Science and Technology and Department of Chemicaland Biomolecular Engineering, University of Maryland, College Park, MD20742, U.S.A.

10.1 Introduction

Critical phenomena in fluid and fluid mixtures have been the subject of manytheoretical and experimental studies during the past decades as has been eluci-dated in various reviews.1–15 The most striking result of these studies has beenthe discovery of critical-point universality. Universality of critical behaviourresults from the presence of large fluctuations in the order parameter associatedwith the critical phase transition (density in one component fluids and/or con-centration in fluid mixtures): the range of these fluctuations becomes much largerthan any microscopic scale. As a consequence, the thermodynamic behaviourcan be characterized by scaling laws with universal critical exponents and uni-versal scaling functions of three-dimensional (3D) Ising-like systems. That is, the3D Ising model, which can be reformulated as a lattice-gas model,16–18 is theprototype model for the critical behaviour of fluids.The nonanalytic critical behaviour of fluids and fluid mixtures is universal in

terms of so-called scaling fields that are analytic functions of the physical fields.The variety of critical phenomena that are actually observed in fluids and fluid




*On leave of absence from Department of Chemistry, University of Tehran, Tehran, Iran.

321

mixtures follow from the different relationships between the scaling fields and thephysical fields appropriate for the individual critical phase-separation phenom-ena, a principle known as isomorphic critical behaviour of fluid mixtures.6,14,19–23

An (interesting) new development in the theory of critical phenomena of fluidsand fluid mixtures is the concept of complete scaling proposed by Fisher andcoworkers.24–26 According to this concept, the scaling fields are not only func-tions of the thermodynamically independent fields as had been assumed untilrecently, but they are functions of all physical fields. While the Ising model issymmetric in terms of the order parameter, critical phase separation in fluids andfluid mixtures is not symmetric. The concept of complete scaling turns out toaccount for the asymmetric nature of fluid critical behaviour.27–31 In the presentchapter we shall not repeat a discussion of some features of critical behaviour offluids covered in previous reviews.12–15 Instead, we shall focus our attention ondevelopments in formulating appropriate scaling laws for fluid critical behaviour.We shall proceed as follows. In Section 10.2 we provide the general

theoretical frame work for the asymptotic thermodynamic critical behaviour ofIsing-like systems. In Section 10.3 we consider the application of the theory toone-component fluids near the vapour-liquid critical point. In Section 10.4 wediscuss the extension of the theory to binary fluid mixtures. In Section 10.5we address the problem of nonasymptotic crossover behaviour of fluids.Concluding remarks are presented in Section 10.6.

10.2 General Theory of Critical Behaviour

10.2.1 Scaling Fields, Critical Exponents, and Critical

Amplitudes

Fluids and fluid mixtures belong to the universality class of Ising-like systems,whose critical behaviour is characterized by two independent scaling fields, h1 andh2, and one dependent scaling field, h3, which are analytic functions of the physicalfields. For instance, in one-component fluids the physical fields are temperature,chemical potential and pressure, as further discussed in Section 10.3. Asympto-tically close to the critical point the scaling fields become a linear combination ofthe physical fields, with their critical values subtracted, so that h1¼ h2¼ h3¼ 0 atthe critical point. Throughout this chapter all physical quantities, and hence thescaling fields, are being made dimensionless in terms of appropriate combinationsof the critical parameters. The theory of critical phenomena predicts that close tothe critical point the dependent field h3 becomes a generalized homogeneousfunction of the two independent scaling fields h1 and h2 of the form

2,32

h3ðh1; h2ÞE h2j j2�af �h1

h2j j2�a�b

!; ð10:1Þ

where a and b are two universal critical exponents and where f� is a scalingfunction with the superscript � referring to h2>0 and h2o0, respectively. In

322 Chapter 10

eq 10.1, h1 is a so-called ‘‘strong’’ scaling field (also called ordering field) and h2 isa ‘‘weak’’ scaling field (also called thermal field). Associated with these scalingfields are two conjugate scaling densities, a strongly fluctuating scaling density j1

(order parameter) and a weakly fluctuating scaling density j2, such that

dh3 ¼ j1dh1 þ j2dh2; ð10:2Þ

with

j1 ¼@h3@h1

� �h2

; j2 ¼@h3@h2

� �h1

: ð10:3Þ

In addition one may define three susceptibilities, a ‘‘strong’’ susceptibility w1, a‘‘weak’’ susceptibility w2, and a ‘‘cross’’ susceptibility w12:

w1 ¼@2h3@h21

� �h2

¼ @j1

@h1

� �h2

; w2 ¼@2h3@h22

� �h1

¼ @j2

@h2

� �h1

w12 ¼@j1

@h2

� �h1

¼ @j2

@h1

� �h2

: ð10:4Þ

A schematic representation of the phase diagram as a function of h2 and theorder parameter j1 is shown in Figure 10.1 with the critical point (C.P.) locatedat the origin. For positive values of h2 the system is homogeneous for all valuesof j1; for negative values of h2 there exists a region of two-phase equilibriumbounded by a coexistence curve where j1¼ �jcxc. For h2>0, h1¼ 0 only atj1¼ 0; for h2o0, h1¼ 0 everywhere in the two-phase region including at thephase boundary j1¼ �jcxc. The scaling law, eq 10.1, implies that varies ther-modynamic properties exhibit critical power-law behaviour along specific ther-modynamic paths, as indicated in Table 10.1. Note that the superscripts � referto h2>0, and h2o0, while the prefactors � refer to j1>0 and j1o0, respec-tively. The critical exponents g and d are related to a and b by g¼ 2� a� 2b andbd¼ 2� a� b, and A�0 , B0, G

�0 , D0 are system-dependent critical amplitudes.

For future reference in Section 10.3.5 we have included in Table 10.1 a correc-tion term proportional to |h2|D1. The values of the universal critical exponentsfor 3D Ising-like systems are presented in Table 10.2. The scaling law, given byeq 10.1, is universal except for two system-dependent critical amplitudes, towhich all other asymptotic critical amplitudes are related through universalratios listed in Table 10.3.32–35 The correction amplitudes A�1 , B1, and G�1 alsosatisfy universal amplitude ratios.36–38 The singular thermodynamic behaviour iscaused by large fluctuations in the order parameter whose spatial extent can becharacterized by a correlation length x which at h1¼ 0 diverges as:18,39

xExþ0 h2j j�n at h2 > 0; j1 ¼ 0 and xEx�0 h2j j�n at h2o0;j1 ¼ �jcxc;

ð10:5Þ

323Thermodynamic Behaviour of Fluids near Critical Points

Figure 10.1 Phase diagram in terms of h2 and j1. For h2>0, h1¼ 0 only at j1¼ 0; forh2o0, h1¼ 0 everywhere in the two-phase region, i.e., for h2o0,|j1|rjcxc. The variables r and y of the parametric linear model are alsoindicated.

Table 10.1 Universal critical power laws.

Critical power law Thermodynamic path

j1E� B0 h2j jb 1þ B1 h2j jD1þ . . .h i

h2o0, j1¼ �jcxc

j2EA�01� a

h2 h2j j�a 1þ A�1 h2j jD1þ . . .h i

j1¼ 0

w1EGþ0 h2j j�g 1þGþ1 h2j jD1þ . . .h i

h2>0, j1¼ 0

w1EG�0 h2j j�g 1þ G�1 h2j jD1þ . . .h i

h2o0, j1¼ �jcxc

h1E�D0 j1j jd 1þ . . .½ � h2¼ 0

324 Chapter 10

where the exponent n is related to a by the hyperscaling relation n¼ (2–a)/3,while the amplitudes x�1 are related to the thermodynamic amplitudes by two-scale-factor universality, which states that the free energy in a volume x3 (dividedby the volume v0 of a unit cell) should be universal.18,34,35

In the mean-field approximation, when the effects of the critical fluctuationsare neglected, eq 10.1 reduces to an asymptotic Landau expansion:6,40,41

�h3E1

2a0h2j2

1 þ1

4!u0j4

1 � h1j1; ð10:6Þ

where a0 and u0 are the two mean-field system-dependent amplitudes. In themean-field approximation the critical exponents attain the classical valuesa¼ 0, b¼ 1/2, g¼ 1, d¼ 3, and v¼ 1/2.

10.2.2 Parametric Equation of State

It is not possible to write the scaling law eq 10.1 for h3 as an explicit function ofh1 and h2 without creating some incorrect singular behaviour in the one-phaseregion. This problem is solved by replacing the two independent scaling fields,h1 and h2, with two parametric variables: a variable r which measures a

Table 10.2 Critical exponents of Ising-like systems and for the classicaltheory.

Critical exponent 3D Ising system Classical value

a 0.110� 0.003 0b 0.326� 0.002 1/2g 1.239� 0.002 1d 4.80� 0.02 3n 0.630� 0.002 1/2D1 0.52� 0.02 1

Table 10.3 Universal critical-amplitude relations.

Ratio 3D Ising system Restricted linear model Crossover Landau model

Aþ0 /A�0 0.523� 0.01 0.523 0.50Gþ0 /G�0 4.95� 0.15 4.85 5.0aAþ0 Gþ0 /B2

0 0.058� 0.001 0.056 0.052

Gþ0 D0Bd� 10 1.57� 0.23 1.68 1.73

xþ0 /x�0 1.96� 0.01 – –aAþ0 (xþ0 )3/v0 0.0188� 0.0002 – –Aþ1 /B1 1.10� 0.25 0.21 0.83B1/G

þ1 0.90� 0.21 1.6 0.87

B1/G�1 0.29� 0.08 – –

Aþ1 /A�1 1.12� 0.29 – –


‘‘distance’’ from the critical point and an angular variable y, which indicatesthe location on a contour of constant r:

h1 ¼ r2�a�bH yð Þ; h2 ¼ rT yð Þ: ð10:7Þ

The idea is that the distance variable r accounts for the asymptotic singularthermodynamic behaviour near the critical point, while H(y) and T(y) areanalytic functions of y. It then follows from eq 10.1 that the order parameter j1

has the form:

j1 ¼ rbM yð Þ; ð10:8Þ

where M(y) is another analytic function of y. The choice of the functions H(y),T(y), and M(y) is not unique.42 The parametric variables r and y frequentlyadopted are defined by a transformation of the form

h1 ¼ ar2�a�by 1� y2� �

; h2 ¼ r 1� b2y2� �

: ð10:9Þ

To define an equation of state one needs to introduce an approximation for thefunction M(y) in the expression eq 10.8 for the order parameter j1. The mostcommon choices are the ‘‘linear model’’ in which43 M(y)¼ ky and the ‘‘cubicmodel’’ in which44 M(y)¼ ky(1þ cy2). In these equations b2 and c are universalconstants, while a and k represent the two system-dependent coefficients thatare related to the critical amplitudes. In this chapter we specify the relevantequations only for the simplest parametric equation which is the linear modelwith order parameter

j1 ¼ krby: ð10:10Þ

The meaning of the parametric variables is indicated in Figure 10.1. Thevariable y varies from y¼ 0 at j1¼ 0 at h2>0 to y¼ � 1/b at h2¼ 0 to y¼ � 1at j1¼ �jcxc at negative h2. A limitation of the linear model is that it does notallow for extrapolation into the two-phase region for |y|>1. For a repre-sentation of interfacial density and concentration profiles other approaches areneeded.45–47

The linear-model equations for the scaling fields, the scaling densities andscaling susceptibilities are listed in Table 10.4.18,48,49 We note that we havegeneralized the traditional linear-model expression for h3 to:

14,49

h3 ¼ �akr2�a f yð Þ � y2 1� y2� ��

� 1

2Bcrr

2 1� b2y2� �2

; ð10:11Þ

where we have added an analytic term with another system-dependent coeffi-cient Bcr to account for an analytic fluctuation contribution to the (caloric)properties also predicted by the theory.41,49 The universal constant b2 is

326 Chapter 10

Table

10.4

Linear-model

representationofthermodynamic

functions.

Scalingfields

h1¼

ar2�a�

by1�y2

��

h2¼

r1�b2y2

��

Singularpart

ofthermodynamic

potential

h3¼�akr2�afyðÞ�

y21�y2

��

��

1 2Bcrr2

1�b2y2

�� 2

with

fðyÞ¼

f 0þf 2y2þf 4y4

f 0¼�

g�2b

ðÞ�

b2ag

2b4ð2�aÞð1�aÞa

f 2¼

g�2b

ðÞ�

b2að1�2bÞ

2b2ð1�aÞa

f 4¼�1�2b

2a

Scalingdensities

conjugate

tothescalingfields

j1¼

@h3

@h1

�� h

2

¼krby

j2¼

@h3

@h2

�� h

1

¼akr1�asðyÞ�Bcrr1�b2y2

��

with

sðyÞ¼

L0s 0þs 2y2

�� ;

L0¼

1� 2b

41�a

ðÞa;

s 0¼

g�2b

ðÞ�

b2ag;

s 2¼

a�1

ðÞg�2b

ðÞb

2

Scalingsusceptibilities

w 1¼

kr�

g q1ðyÞ=a

w 2¼

akr�

a q2ðyÞ�

Bcr

w 12¼

krb�1q12ðyÞ

with

q1ðyÞ¼

1�b2y2þ2bb

2y2

�� q 0

;q2ðyÞ;

q2¼ð1�aÞð1�3y2ÞsðyÞ�

bþg

ðÞ2s 2y2ð1�y2Þ

�� q 0

q12ðyÞ¼

y�gþðg�2bÞy2

��

�� q 0

;q0ðyÞ¼ð1�3y2Þð1�b2y2Þþ

2b2bþg

ðÞy

2ð1�y2Þ


commonly approximated by (‘‘restricted’’ linear model):6,18

b2 � b20 ¼g� 2bð Þ

g 1� 2bð Þ ’ 1:36: ð10:12Þ

In the restricted linear model the expression for the isomorphic ‘‘heat capacity’’(@j2/@h2)j1

becomes independent of y:48,50

@j2

@h2

� �j1

¼ w2 �w212w1¼ akg g� 1ð Þ

2ab20r�a � Bcr: ð10:13Þ

The linear-model expressions for the critical amplitudes are presented inTable 10.5. The values of the universal critical-amplitude ratios implied by therestricted linear model are included in Table 10.3. For a corresponding set ofexpressions for the cubic model the reader is referred to the literature.18,51 Moresophisticated parametric equations8,46,52,53 have also been considered in theliterature, that are not discussed here.

10.3. One-Component Fluids

10.3.1 Simple Scaling

Let T be the temperature, P the pressure, m the chemical potential, r the molardensity, w¼ (@r/@m)T¼ r(@r/@P)T the (isothermal) susceptibility, Sm the molarentropy, Am the Helmholtz energy per mole, CV,m the isochoric molar heatcapacity, and CP,m the isobaric molar heat capacity. The scaling law given by eq10.1 represents the singular part of the thermodynamic potential. To account foranalytic background contributions it is convenient to introduce an analytic‘‘caloric’’ background function m0(T ) and an analytic ‘‘mechanical’’ background

Table 10.5 Critical amplitudes for the linear model.

Linear model Restricted linear model

Gþ0 ¼k

a

k/a

G�0 ¼ ðb2 � 1Þg ka

1� b2ð1� 2bÞ2ðb2 � 1Þ

0.206 k/a

Aþ0 ¼ �akð2� aÞð1� aÞf0 0.984 ak

A�0 ¼ �akð2� aÞð1� aÞ f0 þ f2 þ f4

ðb2 � 1Þ2�a1.859 ak

B0 ¼k

ðb2 � 1Þb1.394 k

D0 ¼a

kdðb2 � 1Þbd�3 0.477 a/kd

328 Chapter 10

function P0(T ), such that m0(Tc)¼ mc and P0(Tc)¼Pc. In this chapter we adoptthe convention that a subscript c indicates the value of a property at the criticalpoint. To formulate the scaling laws for one-component fluids near the vapour-liquid critical point, one also introduces deviation functions:41

DT ¼ T � Tc; DP ¼ P� Pc; Dr ¼ r� rc; and Dm ¼ m� m0ðTÞ: ð10:14Þ

In order to normalize the scaling fields, the thermodynamic properties are madedimensionless with the aid of the critical parameters Tc and rc:

T ¼Tr ¼T

Tc; r ¼ rr ¼

rrc; P ¼ Pr ¼

P

rcRTc; m ¼ mr ¼

mRTc

;

s ¼sr ¼Sm

R; a ¼ ar ¼

Am

RTc; and CV ¼

CV ;m

R;

ð10:15Þ

where R is the universal gas constant. Equation 10.15 defines the reducedthermodynamic properties and provides the IUPAC symbols used in otherchapters. The notation ^ indicating a reduced property is adopted here to beconsistent with the literature and because different reduced properties, to beindicated by ~, will also be used later in this chapter. All other thermodynamicproperties are made dimensionless consistent with eq 10.15. We also introduce adimensionless entropy density s¼ rS, Helmholtz-energy density a¼ rA, andisochoric heat-capacity density cV¼ rCV. Strictly speaking, the appropriatetemperature variable in the scaling laws is not T, but the inverse temperature 1/T.14,54 While the asymptotic critical behaviour remains the same, the inversetemperature should be used when considering nonasymptotic critical behaviourto be discussed in Section 10.3.5.The simplest theoretical prototype of the critical vapour-liquid transition is

the lattice gas which is a reformulation of the 3D Ising model in terms of fluidvariables.16–18 For the lattice gas the scaling fields are:

h1 ¼ Dm; h2 ¼ DT ; and h3 ¼ DP: ð10:16Þ

As a consequence, the pressure is given by

P ¼ h3ðDm; DTÞ þ P rðDm; DTÞ; ð10:17Þ

where h3(Dm, DT ) satisfies the scaling eq 10.1 and where P(Dm, DT ) is a regularbackground contribution that is approximated by

PrðDm;DTÞ ¼ P0ðDTÞ þ Dm: ð10:18Þ

Equation 10.17 represents a fundamental equation for the pressure as afunction of chemical potential and temperature from which all other


thermodynamic properties can be derived by differentiation:

dP ¼ sdT þ rdm: ð10:19Þ

Rather than using the chemical potential and the temperature as character-istic variables, it is in practice more convenient to use density and temperatureas characteristic variables. Such a fundamental equation is obtained byapplying a Legendre transformation a¼ mr–P, so that

a ¼ m0 r� P0 � ðh3 � h1j1Þ ð10:20Þ

with

j1 ¼ r� 1: ð10:21Þ

For the isothermal susceptibility, the entropy density, and the isochoric heatcapacity one obtains

w ¼ w1; ð10:22Þ

s ¼ dP0

dT� r

dm0dTþ j2; ð10:23Þ

cV

T¼ d2P0

dT2� r

d2m0dT2

þ w2 �w212w1: ð10:24Þ

The various thermodynamic properties satisfy asymptotic power laws listed inTable 10.6 with critical amplitudes A�0 , B0, G

�0 , D0 that are identical with the

Ising amplitudes, A�0 , B0, G�0 , D0 defined in Section 10.2.1.

Substitution of the linear-model expressions for the scaling fields, the scalingdensities, and scaling susceptibilities from Table 10.4 in the equations aboveyields the linear-model expressions for the thermodynamic properties of a one-component fluid near the vapour-liquid critical point. In practical applicationsof the linear model the fluctuation-induced analytic contributions to the caloric

Table 10.6 Critical power laws for thermodynamic properties.

Path Power law

TrTc, r¼ rcxc Drcxc ¼ �B0 DT b

T¼Tc Dm ¼ D0ðDrÞ Drj jd�1TZTc, r¼ rc w ¼ Gþ0 DT

�gTrTc, r¼ rcxc, one-phase w ¼ G�0 DT

�gTZTc, r¼ rc cV

�T ¼ Aþ0 DT

�a�Bcr

TrTc, r¼ rc two-phase cV�T ¼ A�0 DT

�a�Bcr

330 Chapter 10

properties proportional to Bcr have usually been neglected. In Table 10.7 wegive a list of the substance-dependent linear-model parameters a and k, gath-ered from the literature, for a number of fluids. These parameter values shouldonly be treated as informed estimates, since the values may be somewhataffected by the types of nonasymptotic corrections used in the fits to theexperimental data. In fitting the linear model to experimental data the back-ground functions P0(DT ) and m0(DT ) are in practice represented by truncatedTaylor expansions

P0ðDTÞ ¼ Pc þXi¼1

PiðDTÞi and m0ðDTÞ ¼ mc þXi¼1

miðDTÞi ð10:25Þ

with adjustable coefficients Pi and mi. In the expansion for m0(DT ) the first twocoefficients mc and ml depend on the choice of zero energy and entropy and donot appear in the expressions for any of the physically observable thermo-dynamic properties.

Table 10.7 Estimated critical parameters for a number of fluids.

Critical-point parameters Restricted linear modelCorrelationlength

Tc/K Pc/MPa rc/(kg �m�3) a k x0/nm3Hea 3.31 0.114 41.45 5.12 0.77 0.26Neb 44.479 2.72 484 18.5 1.02 0.13Arc 150.663 4.860 535.13 16.3 1.00 0.16Krb 209.29 5.493 908 18.5 1.02 0.16Xeb 289.72 5.840 1110 18.5 1.02 0.19N2

b 126.21 3.398 314 22.6 1.06 0.15HDb 35.957 1.484 48.1 16.4 0.92 0.16D2O

d 643.847 21.671 356 23.3 1.42 0.13H2O

d 647.096 22.064 322 23.3 1.42 0.13CO2

e 304.107 7.3721 467.69 23.4 1.22 0.15NH3

f 405.367 11.336 235 25.4 1.27 0.14SF6

g 318.717 3.7545 742 25.0 1.23 0.19CH4

h 190.564 4.5992 162.380 16.4 1.05 0.18C2H4

b 282.35 5.040 214.165 19.2 1.11 0.18C2H6

h 305.322 4.8718 206.581 18.5 1.13 0.19Iso-C4H10

b 407.84 3.629 225.5 22.0 1.19 0.22n-C5H12

i 469.610 3.372 231.168 29.6 1.27 0.23n-C7H16

i 539.860 2.727 227.604 35.1 1.32 0.22CFCl2CF2Cl

i 486.968 3.39 566.981 29.2 1.32 0.21

aRef. 55.bRef. 5.cRef. 56.dRef. 57.eRef. 58.fRef. 59.gRef. 60.hRef. 61.iRef. 30.


From the identification of h1 with Dm¼ m–m0(T ) it follows that in the simple-scaling approximation the chemical potential is an analytic function in fieldh1¼ 0 (i.e., at y¼ 0, y¼ � 1). Furthermore, at constant temperature, m(r,T ) isan anti-symmetric function around the critical isochore r¼ rc.

18,62 As a con-sequence, the susceptibility w(r,T ) is a symmetric function of r around rc inthis approximation.63 In contrast to the temperature dependence of the che-mical potential, the expansion of the temperature expansion of the pressure infield h1¼ 0 has a non-analytic term proportional to |DT |2�a.64 Of particularinterest is the asymptotic temperature dependence of the coexisting vapour andliquid densities, r0 and r0 0:

r0 0 � r0

2� DrcxcEB0 DT

b and r0 0 þ r2� 1 � Drd ¼ 0: ð10:26Þ

Thus in the simple-scaling approximation the two-phase boundary is com-pletely symmetric.

10.3.2 Revised Scaling

Guided by some theoretical models that predicted a singular behaviour of thecoexistence-curve diameter rd as a function of temperature, a revised-scalingapproximation has been proposed in which the scaling fields are defined by:65

h1 ¼ Dm; h2 ¼ DT þ b2Dm; and h3 ¼ DP: ð10:27Þ

The coefficient b2 in the definition of the weak scaling field h2 has been fre-quently referred to as ‘‘mixing’’ parameter. An additional regular asymmetriccontribution is obtained by adding a term proportional to DmDT in theexpansion eq 10.18 for the background function Pr(Dm, DT ), so that66

P ¼ h3ðh1; h2Þ þ P0ðDTÞ þ Dmþ P11DmDT : ð10:28Þ

In revised scaling b2 and P11 are two additional system-dependent coefficientsthat account for a lack of vapour-liquid symmetry of the phase transition.Equation 10.28 is a fundamental equation for the pressure as a function ofchemical potential and temperature.In the revised-scaling approximation one obtains for the various thermo-

dynamic properties

r ¼ 1þ P11DT þ j1 þ b2j2; ð10:29Þ

a ¼ m0r� P0ðDTÞ � h3 � h1 j1 þ b2j2ð Þ½ �; ð10:30Þ

332 Chapter 10

s ¼ dP0

dT� r

dm0dTþ j2; ð10:31Þ

w ¼ w1 þ 2b2w12 þ b22w2; ð10:32Þ

cv

T¼ d2P0

dT2� r

d2m0dT2

þ w2w1 � w212w1 þ 2b2w12 þ b22w2

: ð10:33Þ

The thermodynamic properties continue to satisfy the asymptotic power lawsdefined in Table 10.6 with the same amplitudes as in simple scaling. In addition,the chemical potential at h1¼ 0 is still an analytic function of temperature givenby m0(DT ). Substitution of the linear-model expressions for the scaling fields,scaling densities, and scaling susceptibilities from Table 10.3 in the equationsabove yields a parametric equation of state in the revised-scaling approxima-tion. In practice this revised linear model has been used in conjunction with anextension to include a symmetric nonasymptotic correction to the scaling law(given by eq 10.1) to be discussed in Section 10.3.5. It follows from eq 10.29 thatthe coexisting vapour and liquid densities, r0 and r0 0 at y¼ � 1, are nowrepresented by

r00 � r0

2� DrcxcEB0 DT

b ð10:34aÞ

and

r00 þ r0

2� 1 � DrdE� b2

A�01� a

DT 1�a�Bcr DT

� �� P11 DT

: ð10:34bÞ

We conclude that in this approximation the coexistence-curve diameter has asingular contribution proportional to |DT |1�a, a fluctuation-induced con-tribution and a regular contribution proportional to |DT |.

10.3.3 Complete Scaling

As was recently pointed out by Fisher and coworkers,24–26 to completelyaccount for all asymmetric features of the critical phase transition one needs torelate the scaling fields to all the physical fields. For one-component fluids onethus should write the scaling fields in linear approximation as

h1 ¼ a1Dmþ a2DT þ a3DP; h2 ¼ b1DT þ b2Dmþ b3DP; and

h3 ¼ c1DPþ c2Dmþ c3DT :ð10:35Þ

Consistent with the linear approximation we shall in this subsection approx-imate Dm as Dm ’ m� mc, neglecting terms of higher order in DT. On comparingeq 10.35 with eqs 10.16 and 10.27, we note that by making the thermodynamic


fields dimensionless in terms of the critical-point parameters, we can normalizethe scaling fields such that a1¼ b1¼ c1¼ 1. In addition, c2¼ � rc¼ � 1 andc3¼ � sc, so that:

h1 ¼ Dmþ a2DT þ a3DP; h2 ¼ DT þ b2Dmþ b3DP; and

h3 ¼ DP� Dm� scDT :ð10:36Þ

Anisimov and coworkers have shown that the expressions for the scaling fieldscan be further simplified by taking sc¼ (@P/@T )cxc,c, such that30,31

c3 ¼ sc ¼ @P=@T� �

cxc;cand a2 ¼ a3c3 ¼ �a3sc ¼ �a3 @P=@T

� �cxc;c

; ð10:37Þ

where (@P/@T)cxc,c is the slope of the saturation pressure at the critical pointwhich is a unique direction in the phase diagram. Hence, in complete scalingthere are three system-dependent coefficients, a3, b2, and b3, in the scaling fieldsof eq 10.35 which are related to asymmetry as compared to one such a coeffi-cient b2 in the revised-scaling approximation of eq 10.27.Introducing the short-hand notation

t0 ¼ 1� b3c3 ¼ 1þ b3sc; ð10:38Þ

one obtains for the thermodynamic properties:26

Dm ¼ h1 � a3h3

ð1þ a3Þ; ð10:39Þ

DT ¼ � b2 þ b3ð Þh1 þ 1þ a3ð Þh2 þ a3b2 � b3ð Þh3t0ð1þ a3Þ

; ð10:40Þ

DP ¼ 1þ b2c3ð Þh1 � c3 1þ a3ð Þh2 þ 1� a3b2c3ð Þh3t0ð1þ a3Þ

; ð10:41Þ

r ¼ @P

@m

!T

¼ 1þ f1 þ b2f2

1� a3f1 � b3f2

E1þ ð1þ a3Þf1 þ a3ð1þ a3Þf21 þ ðb2 þ b3Þf2 þ . . . ;

ð10:42Þ

s ¼ @P

@T

!m

¼ sc þ a2f1 þ f2

1� a3f1 � b3f2

Esc þ t0f2 þ :::; ð10:43Þ

334 Chapter 10

w ¼ @r@m

� �T

E 1þ a3ð Þ2 1þ 3a3f1ð Þw1 þ b2 þ b3ð Þ2w2

þ 2 1þ a3ð Þ b2 þ b3ð Þw12 þ ::::ð10:44Þ

The isochoric specific heat capacity is related to P, T, and m by

cV

T¼ @2P

@T2

!m

� @2P

@m@T

!2,@2P

@m2

!T

: ð10:45Þ

An alternative equation is the so-called Yang-Yang relation67

cV

T¼ @2P

@T2

!r

�r @2m

@T2

� �r: ð10:46Þ

Substitution of the parametric equations in Table 10.4 for the scaling fieldsand scaling densities into eqs 10.39 to 10.42 yields expansions of Dm, DT, DPand Dr at h1¼ 0 (y¼ 0 or y¼ � 1) h1¼ 0 in terms of powers of the variable r.Upon inverting the expansion of DT one obtains expansions of Dm, DP, and Drin terms of DT. Using the relationships between the parametric coefficientsa and k and the Ising amplitudes A�0 , B0, and G�0 , one finds that thethermodynamic properties satisfy the asymptotic power laws specified inTable 10.6 with

A�0 ¼ A�0 t0j j2�a; Bcr ¼ Bcr t0 2; G�0 ¼ G�0 t0j j�g 1þ a3ð Þ2 and

B0 ¼ B0 t0j jb 1þ a3ð Þ:ð10:47Þ

We note that the asymmetry coefficients a3, b3, and c3 affect the criticalamplitudes but not the amplitude ratios which continue to have the sameuniversal values listed in Table 10.3. Substitution of the parametric expres-sions for the scaling fields, scaling densities, and scaling susceptibilities fromTable 10.4 into eqs 10.39 to 10.44 yields a linear-model equation of stateconsistent with complete scaling.

10.3.4 Vapour-Liquid Equilibrium

The temperature expansions for the saturation chemical potential ms and thesaturation pressure Ps are:

Dms ¼ �a3

1þ a3

A�0 DT 2�a

2� að Þ 1� að Þ �1

2Bcr DT 2" #

þ ::: ; ð10:48Þ


and

DPs ¼ �c3DT þ1

1þ a3

A�0 DT 2�a

2� að Þ 1� að Þ �1

2Bcr DT 2" #

þ ::: : ð10:49Þ

From eq 10.46 it follows that the isochoric specific heat capacity in the two-phase region is related to the temperature derivates of ms and Ps by:

cV T ; r� �T

¼ d2Ps

dT2� r

d2msdT2

: ð10:50Þ

Hence, when the specific heat capacity in the two-phase region is plotted as afunction of density at constant temperature, the data fall on a straight line,referred to as Yang-Yang plots. We note that complete scaling predictsthat both d2Ps/dT

2 and d2ms/dT2 contribute to the divergence of the isochoric

heat capacity. Since

d2msdT2

¼ �a3d2Ps

dT2; ð10:51Þ

the divergence of the slope d2ms/dT2 of the Yang-Yang plots, called Yang-Yang

anomaly,25,26 is completely determined by the asymmetry coefficient a3. Forsymmetric fluids like helium the Yang-Yang anomaly is expected to be negli-gibly small.30,55,68 Theoretical evidence for the existence of a Yang-Yanganomaly has been found from computer simulations.27,28,69 Experimental evi-dence for a Yang-Yang anomaly has been reported for propane and carbondioxide;25 however, the experimental evidence is ambiguous, since Yang-Yangplots are strongly affected by small impurities.70 More convincing experimentalevidence for complete scaling has been obtained from analyses of coexistence-curve data to be discussed below.29–31

The difference between the coexisting vapour and liquid densities is still givenby the simple power law

r00 � r0

2� DrcxcEB0 DT

b; ð10:52aÞ

but for the sum of the coexisting densities one now obtains:30,31

r0 0 þ r2� 1 � DrdE

a3

1þ a3B20 DT 2b

� b2 þ b3ð Þt0

A�01� að Þ DT

1�a�Bcr DT !

:

ð10:52bÞ

336 Chapter 10

Comparing this result with eq 10.34b, we see that complete scaling intro-duces a new singular contribution to the coexistence-curve diameter propor-tional to |DT |2b replacing the classical background term P11|DT | in eq 10.34b,which is no longer present in complete scaling. Equation 10.52b showsthat the coexistence-curve diameter depends on the asymptotic behaviourof the specific heat capacity in Table 10.6 and on two asymmetry coefficients,namely a3 and beff¼ (b2þ b3)/t0. An analysis of experimental coexistence-curve in terms of eqs 10.52a and 10.52b has been reported by Wangand Anisimov30 yielding values of A�0 , Bcr and the asymmetry coefficients a3and beff for a number of fluids. As an example we show in Figure 10.2 coex-istence-curve diameters Drd of ethane and nitrogen. Since the asymmetrycoefficients a3 and beff can be either positive or negative, the actual temperaturedependence of the coexistence-curve diameters can appear rather different indifferent fluids.Classical theories satisfy complete scaling in the mean-field approxi-

mation.30,72 Classical equations of state are analytic at the critical point,so that the a, m, and P can be expanded in terms of powers of DT andDr.18 The classical power series are recovered when the relations of thescaling fields and densities with the physical fields and densities, inaccordance with complete scaling, are introduced in the classical Landauexpansion of eq 10.6 with appropriate values for the coefficient a0 and u0and for the asymmetry coefficients a3, b2, and b3 in the scaling fields.Wang and Anisimov30 have collected values for a0 and u0 and for theasymmetry coefficients a3 and beff, associated with several classical equationsof state.

Figure 10.2 Diameter of the vapour-liquid coexistence curve for nitrogen (a) andethane (b). J, experimental data;71 ––––, fit to eq 10.65b; - - - -, repre-sents the 2b term; and ......, indicates both the 1� a and the linear term.Reprinted with permission of Phys. Rev. E.30


10.3.5 Symmetric Corrections to Scaling

The equations presented thus far describe the behaviour of the thermodynamicproperties of fluids asymptotically close to the critical point. The actual tem-perature range of asymptotic scaling behaviour depends on the magnitude ofcorrection-to-scaling terms. They are incorporated by extending eq 10.1 to

h3 h1; h2ð ÞE h2j j2�af �h1

h2j j2�a�b

!1þ h2j jD1 f �1

h1

h2j j2�a�b

!" #; ð10:53Þ

where D1 ’ 0:52 is a universal correction-to-scaling exponent and where f�1 is acorrection-to-scaling function that is universal except for one system-dependentamplitude.32 Equation 10.53 represents the first two terms of a so-calledWegner expansion.73 It implies corrections to the asymptotic power laws asshown in Table 10.5 with correction-to-scaling amplitudes that satisfy universalamplitude ratios given in Table 10.3. In zero field these corrections are sym-metric with respect to the sign of the order parameter j1.To obtain a simple extended linear-model equation of state consistent with

eq (10.53) in terms of the parametric variables r and y, defined by eq (10.9), onehas proposed to extend eq 10.10 for the order parameter to

j1 ¼ rby kþ rD1k1� �

; ð10:54Þ

where k1 is an additional system-dependent constant.18,66 As can be seen from thenumerical data in Table 10.3, the linear-model parametric equation of state givesan excellent representation of the amplitudes of the asymptotic scaling laws, butonly an order-of-magnitude estimate of the correction-to-scaling amplitudes.18,38

A second correction to the asymptotic power laws defined in Table 10.5arises from the observation that the actual field variables in the scaling lawsshould be 1/T rather than T, P/T rather than P, and m/T rather than m. Hence,it is more appropriate to introduce slightly revised dimensionless thermo-dynamic variables:18

~r ¼ rrc; ~T ¼ �Tc

T; ~P ¼ PTc

TPc; and ~m ¼ m rcTc

TPc: ð10:55Þ

The reduced pressure P satisfies a differential relation of the form

d ~P ¼ ~ud ~T þ ~rd~m: ð10:56Þ

Upon comparing with eq 10.19 we note that in terms of these modified vari-ables the role of the reduce entropy density s is taken over by that of thereduced energy density u. The extended linear model, defined by eqs 10.9 and10.54, has been used to represent the thermodynamic properties of a number offluids in the critical region in the revised-scaling approximation, including light

338 Chapter 10

steam,74 heavy steam,75 isobutane,76 carbon dioxide,58 methane77 and ethy-lene.78 A complete list of the thermodynamic properties for the revised andextended linear model has been presented by Levelt Sengers et al.74,76,79 Gen-erally, the revised and extended linear model yields an accurate representationof the thermodynamic properties of one-component fluids in the criticalregion in a range of densities bounded by18 � 0.006rDTrþ 0.06 and� 3.0rDrrþ 0.03 Note that the range of validity for temperatures below thecritical temperature is rather small; this problem can been remedied by theadoption of complete scaling as discussed in Section 10.3.4.

10.4 Binary Fluid Mixtures

10.4.1 Isomorphic Critical Behaviour of Mixtures

In fluid mixtures one encounters a variety of vapour-liquid, gas-gas, andliquid-liquid critical phenomena.12,80–82 The subject received renewed interestafter Van Konynenburg and Scott identified six types of phase diagrams thatare possible on the basis of a Van der Waals equation of state for binary fluidmixtures.83,84 In retrospect, these types of phase diagrams were already knownat the beginning of the 20th century.85 In the past decades additional possiblephase diagrams have been identified from mathematical analyses of binary fluidmodels. Other interesting phenomena are closed solubility loops and re-entrantcritical behaviour.86 A recommended systematic nomenclature specifying typesof phase diagrams has been proposed by IUPAC in 1998.87

In the simplest case the vapour-liquid critical points of the two componentsmay be connected by a continuous locus of vapour-liquid critical points ofmixtures at various concentrations with or without additional liquid-liquidimmiscibility. In other cases the locus of vapour-liquid critical points is inter-rupted and starting from one of the components may wonder off to highertemperatures or may crossover to an upper or a lower consolute point. Theprinciple of isomorphic critical behaviour19–21 asserts that the thermodynamicbehaviour associated with the critical behaviour in mixtures can still bedescribed by the scaling-law expression of eq 10.1 in terms of two independentscaling fields, h1 and h2, and a dependent scaling field h3. The different types ofcritical phenomena observed experimentally are caused by different relation-ship of these scaling fields with the actual physical fields.22,23

In binary mixtures, we need to consider four physical fields, namely, the(dimensionless) temperature T¼T/Tc, pressure P¼P/rcRTc, chemical poten-tial of the solvent m1¼ m1/RTc, chemical potential of the solute m2¼ m2/RTc,or chemical-potential difference m21¼ m2–m1, and the deviation variablesDT¼ T–1, DP¼ P–Pc, Dm1¼ m1–m1c, and Dm2¼ m2–m2c, or Dm21¼ m21–m21c.Complete scaling asserts that the scaling fields depend on all four field vari-ables. In linear approximation one obtains instead of eq 10.35:31

h1 ¼ a1Dm1 þ a2DT þ a3DPþ a4Dm21; ð10:57aÞ


h2 ¼ b1DT þ b2Dm1 þ b3DPþ b4Dm21; ð10:57bÞ

and

h3 ¼ c1DPþ c2Dm1 þ c3DT þ c4Dm21: ð10:57cÞ

It should be pointed out that all system-dependent parameters, namely thecoefficients ai, bi, and ci in the expressions for the scaling fields and the criticalparameters Tc, Pc, m1c, and m21c now depend parametrically on the actualposition on the critical locus that may be specified by any of the four criticalparameters. The two theoretical scaling fields f1 and f2 continue to be definedas j1¼ (@h3/@h1)h2 and j2¼ (@h3/@h2)h1. Since

dP ¼ sdT þ rdm1 þ rxdm21; ð10:58Þ

where x is the mole fraction of solute, it follows that31

x ¼ xc þ a4j1 þ b4j2

1þ a1j1 þ b2j2

; ð10:59Þ

r ¼ 1þ a1j1 þ b2j2

1� a3j1 � b3j2

; ð10:60Þ

and

s ¼ sc þ a2j1 þ b1j2

1þ a1j1 þ b2j2

: ð10:61Þ

Just as for a one-component fluid, c1¼ 1 and c2¼ � 1, while c3¼ � sc andc4¼ � xc. In the one-component limit the coefficients a4, b4 and c4 vanish as xc.It has been shown that the principle of isomorphic critical behaviour

accounts not only for the thermodynamic behaviour of mixtures near vapour-liquid critical points and near critical liquid-liquid mixing critical points, butalso near special critical points, like azeotropic critical points, critical pointswhere the critical temperature exhibits a maximum or a minimum as a functionof temperature, re-entrant critical points and critical double points, dependingon the values of the coefficients ai, bi, and ci in the expressions for the scalingfields.22 In this chapter we restrict ourselves to some more common cases ofcritical phase behaviour in mixtures.

10.4.2 Incompressible Liquid Mixtures

In this and the next section we consider liquid-liquid phase separation in liquidmixtures terminating in either an upper or a lower critical solution point. Since thepressure does not affect concentration fluctuations we neglect in first approx-imation the contribution of the pressure to the independent scaling fields, h1 and

340 Chapter 10

h2, which induce critical fluctuations. The Ising model can also be translated into amodel for phase separation in symmetric solid and liquid mixtures.88 A symmetricliquid mixture satisfies simple scaling such that in terms of dimensionless variablesh1¼Dm21, h2¼DT, and h3¼DP so that a4¼ 1, b1¼ 1, and c1¼ 1. Hence, for anasymmetric incompressible liquid mixture the scaling fields become31

h1 ¼ Dm21 þ a1Dm1 þ a2DT ; h2 ¼ DT þ b2Dm1 þ b4Dm21; and

h3 ¼ DP� Dm1 � scDT � xcDm21:ð10:62Þ

On comparing eq 10.62 with eq 10.36 we see that for an incompressible liquidmixture at constant pressure DP the scaling fields h1, h2, and h3 are analogous tothe fields h1, h2, and –h3 of a one-component fluid with m21 and m1 now playing therole of m and � P and with the coefficients a1, a2, b4, and b2 playing the sameroles as the asymmetry coefficients � a3, a2, b2, and –b3 in eq 10.36. Just as inSection 10.3.3, the coefficient a2 can be eliminated by selecting

�c3 ¼ sc ¼ � @m1=@T� �

h1¼0;cand

a2 ¼ �a1a3 ¼ a1sc ¼ �a1 @m1=@T� �

h1¼0;c:

ð10:63Þ

Equation 10.62 yields for the mole fraction x of an incompressible liquid mixture:

x ¼ xc þ j1 þ b4j2

1þ a1j1 þ b2j2

’ xc þ 1� xca1ð Þj1

� a1 1� xca1ð Þj21 þ b4 � xcb2ð Þj2 þ :::

ð10:64Þ

Except for a trivial factor xc, eq 10.64 has exactly the same form as the expressionof eq 10.42 for the density of a one-component fluid. Hence, in analogy witheq 10.52, we can immediately conclude that the mole fractions x0 and x0 0 of thesolute along the two sides of the phase boundary will vary with temperature as:

x00 � x0

2xc� DxcxcEB0 DT

b; ð10:65aÞ

and

x00 þ x0

2xc� 1 � Dxd ¼ �

xca1

1� xca1ð Þ B20 DT 2b

� b4 � xcb2ð Þxc 1� b2scð Þ

A�01� a

DT 1�a�Bcr DT

!:

ð10:65bÞ

In eq 10.65b A�0 and Bcr now refer to the critical behaviour of the isomorphicspecific heat capacity which for liquid mixtures is the isobaric specific heatcapacity.


Figure 10.3 Mole fraction distance Dxcxc and diameter Dxd of the liquid-liquidcoexistence curves for solutions of pentane (1), heptane (2), octane (3),and decane in nitrobenzene as a function of |DT | .––––, obtained fromeq 10.65; and n, &, J and $ experimental data. Reprinted with per-mission of Phys. Rev. E.31

342 Chapter 10

Figure 10.4 Closed solubility loops in 2,5 lutidineþwater and 2,6 lutidineþwater.––––, represent values calculated from eq 10.66; J and K, experimentaldata.92 Reprinted with permission of Z. Phys. Chem.91


Just as eq 10.52b for Drd, we see that the coexistence diameter Dxd forthe mole fraction of a liquid mixture depends on the asymptotic behaviourof the specific heat capacity and on two asymmetry coefficients, namelyaeff¼ � xca1/(1–xca1) and beff¼ (b4–xcb2)/xc(1–b2sc).Equations 10.65a and 10.65b yield an excellent description of the mole

fractions of the coexisting liquid phases as a function of temperature.31 As anexample we show in Figure 10.3 Dxcxc and Dxd of four hydrocarbon solutionsin nitrobenzene. Wang et al.31 have found empirical evidence that the asym-metry coefficients aeff and beff are related to the ratio of the molecular volumesof the two components.In some liquid mixtures one may encounter a closed solubility loop between

an upper critical solution point with temperature TU and concentration xU anda lower critical solution point with temperature TL and concentration xL. Onecan obtain a quantitative representation of such closed solubility loops if thetemperature variable |DT| is replaced by86,89,90 DTUL�(TU–T )(T–TL)/TUTL.This procedure has been applied successfully in the revised-scaling approx-imation (i.e., without a contribution proportional to |DTUL|

2,b), but with theaddition of a correction-to-scaling contribution proportional to |DTUL|

D1 asdiscussed in Section 10.3.5:91

x ¼xL � B00 DTUL

b 1þ B01 DTUL

D1n o

þ B0a DTUL

1�aþ xU � xL

TU � TLT � TLð Þ;

ð10:66Þ

where B00, B01, and B0a are effective amplitudes. As an example we show in

Figure 10.4 closed solubility loops in 2,5 lutidineþwater and 2,6 lutidineþwater. While the role of the pressure can be neglected in dealing with con-centration fluctuations, it cannot be neglected in dealing with other propertieslike the density r¼ (@P/@m1)T,m21. Hence, for a more complete treatment ofnearly incompressible liquid mixtures we must incorporate some effect of thepressure on the fluctuations as elucidated below.

10.4.3 Weakly Compressible Liquid Mixtures

In practice, liquid mixtures are weakly compressible. That is, in liquid mixturesthe pressure does not induce fluctuations directly but indirectly, since the cri-tical parameters in Dm21, Dm1, and DT in eq 10.61 depend on the pressure:

T � Tc P� �’ T � Tc P0

� �� dTc

dPP� P0� �

; ð10:67Þ

with similar expressions for m21–m21c(P) and m1–m1c(P), and where P0 is theactual experimental reference pressure. Substitution of these expansions intothe expressions for the scaling fields h1 and h2 in eq 10.57 yields for weakly

344 Chapter 10

compressible liquid mixtures94

h1 ¼ Dm21 þ a1Dm1 þ a2DT þ a3DP; ð10:68aÞ

and

h2 ¼ DT þ b2Dm1 þ b4Dm21 þ b3DP; ð10:68bÞ

where now

DT ¼T � TcðP0Þ; Dm1 ¼ m1 � m1cðP0Þ; Dm12 ¼ m12 � m12cðP0Þ;and DP ¼ P� P0:

The coefficients a3 and b3 are related to the other coefficients by

a3 ¼ �dm21cdPþ a1

dm1cdPþ a2

dTc

dP

!; ð10:69Þ

and

b3 ¼ �dTc

dPþ b2

dm1cdPþ b4

dm21cdP

!: ð10:70Þ

We note that the total derivatives are taken along the critical locus.From eq 10.59, we see that the relationship of the mole fraction x with

the scaling densities j1 and j2 is independent of either a3 or b3. Hence, thetheoretical expressions for the temperature dependence of the mole fractionalong the two phase boundaries, developed in the previous section, remainequally valid for weakly compressible liquid mixtures. This is the physicalreason why eq 10.65 yields an excellent representation of the behaviourof the mole fraction x for liquid-liquid equilibria. As an example we showin Figure 10.5 closed solubility loops in 2-butanolþwater.91,93 As we explainedearlier, closed solubility loops can be represented by the expansion of eq 10.65provided that |DT | is replaced by |DTUL| in accordance with eq 10.66. Theclosed solubility loops collapse into a double critical point at P¼ 85.6MPa andT¼ 340 K. The implications of the theory for the behaviour near such adouble critical point have been elucidated by Wang et al.91 Both near the uppercritical solution temperature TU and near the lower critical solutiontemperature TL, Dxcxc varies as |DTUL|

b in accordance with eq 10.65a. Nearthe double critical point both TU and TL approach the temperature TD

of the double critical point. Hence, near the double critical point

DTUL

b’ j T � TD

� �2=T2

Djb ¼ T � TD

� ��TD

2b. This is the phenomenon of


exponent doubling when the double critical point is approached at constantpressure.22,86

From eq 10.60, we note that, in contrast to the mole fraction, the massdensity does depend on a3 and b3. Hence, the temperature dependence of thecoexisting densities in liquid-liquid equilibria is affected by the pressuredependence of the critical parameters. Specifically,

r ¼ 1þ a1 þ a3ð Þj1 þ a3 a1 þ a3ð Þj21 þ b2 þ b3ð Þj2: ð10:71Þ

We conclude that for a non-vanishing value of a3 the expansion for Dr containsa term proportional to j2

1. Hence, the pressure dependence of the criticalparameters causes a singular term proportional to |DT |2b in the temperatureexpansion of the density diameter Drd. Such a singular term has been detectedfrom experimental density data for liquid mixtures.94 Traditionally the presenceof a term proportional to |DT |2b was considered an artifact because thedensity was not a ‘‘correct’’ order parameter for liquid mixtures.1 We now seethat this term is a consequence of the pressure dependence of the criticalparameters.

Figure 10.5 Closed solubility loops in 2-butanolþwater at various pressures. ––––,values calculated from eq 10.66; J, experimental data.93 Reprinted withpermission of Z. Phys. Chem.91

346 Chapter 10

10.4.4 Compressible Fluid Mixtures

In this section, we consider vapour-liquid equilibrium in binary fluid mixtures.A locus of vapour-liquid critical points may emanate from the critical point ofeither component. In the simplest case a single continuous locus of vapour-liquid critical points may connect the critical points of the two components. It isimportant to consider the thermodynamic behaviour of the mixture at constantchemical potential m21. On comparing eq 10.57 with eq 10.35, we see that, atconstant m21, the scaling fields become identical to those of a one-componentfluid. Hence, the thermodynamic behaviour of mixtures at constant m21 can bedescribed by exactly the same equations as for one-component fluids near thevapour-liquid critical point, except that the critical parameters and the system-dependent coefficients will depend parametrically on the ‘‘hidden’’ field m21. Useof m21 as the hidden field is not convenient, since it diverges in the two one-component limits. This problem is avoided by adopting an alternative hiddenfield proposed by Leung and Griffiths:95

z ¼ 1

1þ e�m21: ð10:72Þ

The advantage of the choice of eq 10.72 for the hidden field variable is that zruns from 0 to 1, when the mole fraction x varies from 0 to 1. The definition ofeq 10.72 of z is not unique due to the arbitrary choice of zeroes of the chemicalpotentials.95 To simplify the calculations one adopts, following a proposal ofMoldover and Gallagher,96 as an additional constraint that z¼ x everywhereon the critical locus. This additional constraint has been referred to as thecritical-line condition.97,98 The procedure of extending the thermodynamicequations of one-component fluids to describe the thermodynamic propertiesof mixtures near the vapour-liquid critical locus by allowing the system-dependent coefficients to be analytic functions of z has been generally adopted.A comprehensive review of vapour-liquid equilibrium in binary fluid mixturesand their correlation in terms of a scaled equation of state has been presentedby Rainwater,7 where all appropriate references prior to 2000 can be found. Itmust be emphasized that the thermodynamic behaviour of mixtures is iso-morphic with that of one-component fluids at constant z but not at constant x.When the critical point is approached along a path at constant x, one-com-ponent-like power-law behaviour crosses over to a modified asymptotic power-law,6,14,22,99 a phenomenon known as exponent renormalization as originallypredicted by Fisher.100

10.4.5 Dilute Solutions

As shown in Section 10.4.3, in weakly compressible liquid mixtures the tem-perature and the chemical potentials of solvent and solute contribute to thecritical fluctuations directly, while the pressure could be treated as a non-ordering field,101,102 whose influence only manifests itself through the


dependence of the critical parameters on the pressure. Similarly, we assume thatin dilute solutions m21 can be treated as a non-ordering field. That is, we assumethat in eq 10.57 the coefficients ai, bi, and ci for i¼ 1, 2, 3 are the same as for thepure solvent, while the coefficients a4, b4, and c4 originate from the dependenceof critical parameters on m21. Thus the scaling fields h1 and h2, given by eq 10.36for the pure solvent, become for dilute solutions:103

h1 ¼ Dm1 þ a2DT þ a3DPþ a4Dm21; ð10:73aÞ

and

h2 ¼ DT þ b2Dm1 þ b3DPþ a4Dm21; ð10:73bÞ

where, in analogy with eqs 10.69 and 10.70,

a4 ¼ �dm1cdm21

þ a2dTc

dm21þ a3

dPc

dm21

!; ð10:74Þ

and

b4 ¼ �dTc

dm21þ b2

dm1cdm21

þ b3dPc

dm21

!: ð10:75Þ

We note that for a dilute mixture on the critical locus

m21 ¼ ln xc; ð10:76Þ

while the coefficient a2 continues to be related to a3 and (@P/@T)cxc,c of the puresolvent in accordance with eq 10.37. Transforming the derivatives with respectto m21 to derivatives with respect to x and using the Gibbs-Duhem relation eq10.58, one can rewrite expressions 10.74 and 10.75 for a4 and b4 as:

103

a4 ¼ �xc 1þ a3ð ÞK � 1� �

; ð10:77Þ

and

b4 ¼ �xcdTc

dxþ b2 K � 1

� �þ b3

dPc

dx

" #; ð10:78Þ

where K¼K/rcRTc is the dimensionless version of what has become known asthe Krichevskii parameter104

K � dPc

dx� @P

@T

� �cxc;c

dTc

dx: ð10:79Þ

348 Chapter 10

Values for the Krichevskii parameters of dilute solutions have been reportedby many investigators, primarily for dilute aqueous solutions,63,105–108 but alsofor other solutions, such as dilute solutions in carbon dioxide.109,110 However,because of cancellations when one evaluates the difference in slopes in eq 10.79,there is actually a considerable spread in the values reported for K in the lit-erature.63 From eq 10.78 we see that in the revised-scaling approximation(a3¼ b3¼ 0) the thermodynamic behaviour of near-critical dilute mixtures iscompletely determined by the concentration derivative dTc of the criticaltemperature and the Krichevskii parameter K. Complete scaling (a3a0, b3a0)causes the thermodynamic behaviour of near-critical dilute solutions to dependon the Krichevskii parameter K and on both dTc/dx and dPc/dx. Anotherinteresting feature of near-critical dilute solutions is that the limiting values ofthe partial molar properties of the solvent depend on the path along which thesolvent’s critical point is approached, while the partial molar properties of thesolute diverge, as elucidated by Levelt Sengers and coworkers.9,111–114

10.5 Crossover Critical Behaviour

10.5.1 Crossover from Ising-like to Mean-Field Critical

Behaviour

The general theory of critical phenomena presented in Section 10.2 concerns theasymptotic (Ising-like) thermodynamic critical behaviour. It is possible toextend the theory to include a crossover from fluctuation-induced Ising-likecritical behaviour to classical mean-field critical behaviour. For this purpose itis more convenient to replace the potential h3 in Section 10.2 with anotherpotential F with characteristic variables j1 and h2:

F ¼ � h3 � h1j1ð Þ: ð10:80Þ

From eq 10.20, we note that for the lattice gas F is to be identified with thecritical part of the Helmholtz-energy density. From eq 10.11, we see that in theclassical mean-field approximation Fcl has an asymptotic Landau expansion ofthe form:

FclE1

2a0h2j2

1 þ1

4!u0j4

1: ð10:81Þ

Introducing rescaled variables defined by53

t ¼ cth2; M ¼ crj1 such that a0 ¼ ctc2r; and u0 ¼ u��uLc4r ð10:82Þ

we rewrite the expansion for Fcl as:

FclE1

2tM2 þ 1

4!u��uLM4: ð10:83Þ


In eq 10.82, u� ’ 0:472 is a universal coupling constant, u a scaled system-dependent coupling constant, and L a dimensionless cutoff wave numberrelated to the microscopic (molecular) length scale xD¼ v1/30 /pL; for the latticegas v0 is the volume of the unit cell and for fluids v0 is the molecular volume. Asa measure of a distance from the critical point we consider a parameter k,related to the inverse correlation length, which in the classical limit is pro-portional to the square root of the inverse susceptibility w1 and which has theform:

k2cl ¼@2Fcl

@M2

� �t

¼ tþ 1

2�uu�LM2: ð10:84Þ

The renormalized potential Fr becomes

Fr ¼1

2tM2TDþ 1

4!�uu�LM4D2U� 1

2t2K; ð10:85Þ

which is obtained by replacing t in the classical two-term Landau expansion ofeq 10.83 with tTU�1=2, by replacing M with MD1=2U1=4, and by adding a(caloric) fluctuation-induced contribution � 1=2ð Þt2K . The correspondingexpression for the distance variable k becomes

k2 ¼ tT þ 1

2�uu�LM2DU: ð10:86Þ

For the rescaling functions T; D; U; and K we have adopted approximantsoriginally proposed by Chen et al.115,116 and referred to as crossover model IIby Tang et al.:38

T ¼ Y 2n�1ð Þ=D1 ; D ¼ Y g�2nð Þ=D1 ; U ¼ Yn=D1 ;

K ¼ n=a�uLð Þ Y�a=D1 � 1 � ð10:87Þ

in terms of a crossover function Y defined by

1� 1� �uð ÞY ¼ �u 1þ L=kð Þ2h i1=2

Yn=D1 : ð10:88Þ

Equations 10.85 to 10.88 define what has been called a two-term crossoverLandau model (CLM). In the classical limit (L/k{1) the crossover function Yapproaches unity and one recovers from eq 10.85 the classical expansion of eq10.83. In the critical region (L/kc1) the crossover function approaches zero asYE k=�uLð ÞD1=n and one recovers from eq 10.85 the power-law expansionsspecified in Table 10.5 with expressions for the critical amplitudes listed inTable 10.8.53 The values for the critical-amplitude ratios implied by thecrossover Landau model are included in Table 10.3. The nonasymptotic criticalbehaviour is governed by u and L/c1/2t or, equivalently by u and by NG, known

350 Chapter 10

as the Ginzburg number:41,53

NG ¼ n0 �uL=ffiffiffiffictpð Þ2 with n0 ’ 0:0314: ð10:89Þ

The coupling constant u controls the magnitude of the corrections to theasymptotic power-law behaviour as can be seen from Table 10.8. The Ginzburgnumber NG is a measure of the value of the temperature variable t, where thecrossover from Ising-like to mean-field critical behaviour occurs.Another theoretical formulation of crossover from Ising-like to classical

mean-field critical behaviour has been derived by Bagnuls and Bervillier.117 Ithas been further developed and applied to one-component fluids by Garrabosand coworkers.118–122

10.5.2 Effective Critical Exponents

As an example let us consider the susceptibility w�1¼ c2r(@2F/@M2)t. From eq

10.85 it follows that in field zero in the one-phase region:123–125

w�1 ¼ c2r tYg�1ð Þ=D1 1þ u�n

2D12k2

L21þ L2

k2

�nD1þ 1� �uð ÞY1� 1� �uð ÞY

�� 2n � 1

D1

� ��1 !:

ð10:90Þ

A convenient quantitative description of the nonasymptotic criticalbehaviour of the susceptibility is obtained by defining an effective criticalexponent126 geff �� (d logw/d logt). In Figure 10.6 we show the valuespredicted by the crossover Landau model for geff as a function of u andt=NG ¼ t=ctNG ¼ t=n0 �uLð Þ2. For uo1, geff exhibits a smooth and gradualcrossover from the asymptotic Ising value 1.24 to the mean-field value of unity.For u>1 (strong interactions) the crossover becomes steep and

Table 10.8 Leading and correction-to-scaling amplitudes for the crossoverLandau model (CLM)a.

Gþ0 ¼ 0:871gg�1a�10

G�0 ¼ 0:174gy�1a�10

Aþ0 ¼ 2:27ga a20�u0

� �A�0 ¼ 4:55ga a20

�u0

� �B0 ¼ 2:05g1=2�b a0=u0ð Þ1=2

D0 ¼ 0:129g 3�dð Þ=2a0 u0=a0ð Þ d�1ð Þ=2

Aþ1 ¼ 0:439g�D1 1� �uð ÞGþ1 ¼ 0:439g�D1 1� �uð ÞB1 ¼ 0:531g�D1 1� �uð Þa

g ¼ �uLð Þ2.ct


nonmonotonic.123 We note that the crossover behaviour occurs at a tempera-ture t¼ txE10NG.Luijten and coworkers127–129 have reported numerical studies of the 3D Ising

model as a function of temperature (t¼ h2) for a variety of interaction rangesand, hence, for various values of u. For the Ising model the cutoff equals the cellsize (L¼ 1), so that the crossover behaviour should only depend on u and,hence, on t/tx. Figure 10.7 shows the effective susceptibility exponent geff as afunction of t/tx for the 3-dimensional Ising model: the symbols represent thevalues deduced by numerical differentiation of the computer simulations andthe curve represents the values predicted from eq 10.90.130

The susceptibility can be determined experimentally by measuring the intensityof scattered light as a function of temperature. Light scattering yields the iso-morphic susceptibility associated with the actual order parameter regardless ofits physical origin. Hence, eq 10.90 can be used to represent experimental light-scattering data in fluids and liquid mixtures in field h1¼ 0 in the one-phase regionwhere asymmetry corrections are very small; u and L=

ffiffiffiffictp

are then used as thetwo adjustable crossover parameters. As an example we show in Figure 10.8the effective susceptibility exponent geff of xenon (Xe) near the vapour-liquidcritical point and of the liquid mixtures 3-methylpentaneþ nitroethane(3MPNE), isobutyric acidþwater (IBAW), and of tetra-n-butyl ammonium

Figure 10.6 Effective susceptibility exponent geff as a function of u and log(t/NG).Reprinted with permission of J. Chem. Phys.125

352 Chapter 10

picrateþ 1,4-butanediol/1-dodecanol (TPDB) near the critical point of mix-ing.123 In Xe, geff decreases monotonically from its Ising limiting value 1.24; in3MPNE the behaviour remains asymptotic over a sizable temperature range; inthe aqueous ionic solution IBAW and the nonaqueous ionic solution TPDB thecrossover behaviour is nonmonotic as predicted by the theory for systems withvery strong short-range interactions.It is also interesting to define an effective exponent beff for the temperature

dependence of the order parameter at phase coexistence.130 Calculation ofphase coexistence from the crossover equations is a bit more complicated, sinceit requires the application of phase-equilibrium conditions. Hence, we do nothave an analytic solution for j1 in field zero along the phase boundary. As anapproximate solution one may use125

j001 � j

01

2

� �� Djcxc ¼

1

cr

6t

u��uL

� �1=2

Y 2b�1ð Þ=2D1 : ð10:91Þ

The theory of crossover critical behaviour, presented in Section 10.5.1, notonly accounts for crossover from asymptotic Ising-like critical behaviour toclassical mean field critical behaviour away from Tc (in the limit u-0), but also

Figure 10.7 Effective susceptibility exponent geff as a function of t/tx. K, ’, n, ,,v, x, c, b, �, & andB, represent numerical simulation data; ––––,values obtained from eq 10.90. Reprinted with permission of J. Stat.Phys.130


crossover from Ising-like critical behaviour to tricritical behaviour (in the limitL-0), as observed in polymer solutions of polymers with increasing largerdegrees of polymerization near the theta temperature.124,131,132

10.5.3 Global Crossover Behaviour of Fluids

The equations presented in the preceding sections deal with the thermodynamicbehaviour of fluids in the near-critical region. For many applications one needsa global equation of state that not only incorporates the effects of criticalfluctuations in the near-critical region, but also yields a representation of thethermodynamic properties of fluids over large ranges of temperatures and

Figure 10.8 Effective susceptibility exponent as a function of logt for xenon, liquidmixtures of 3-methylpentaneþ nitroethane (3MPNE), isobutyricacidþwater (IBAW), and of tetra-n-butyl ammonium picrateþ 1,4-butanediol/1-dodecanol (TPDB). J, B, &, �, n, þ and *, indicatevalues deduced numerically from experimental light-scattering data;––––,represent values calculated from eq 10.90. Reprinted with permission ofPhys. Rev. Lett.123

354 Chapter 10

densities. We discuss here five different approaches that have become availableto deal with this problem.

10.5.3.1 Switching from Scaled Equations to Classical Equations

A conceptually simple approach would be to use the sum of a scaled equationand a classical analytic equation with weights determined by a switchingfunction chosen so that the analytic equation is suppressed in the vicinity of thecritical point and the scaled equation is suppressed far away from the criticalpoint.133 However, it appears to be impossible to interpolate between twodifferent equations without introducing spurious behaviour of the derivatives inthe switching region.134 Hence, this procedure requires already a rather perfectmatching of the two equations before applying any switching function. This hasbeen done thus far successfully in one case only, namely for H2O.135

10.5.3.2 Crossover from Ising-like to Nonasymptotic Mean-FieldCritical Behaviour

The theory described in Section 10.5.1 accounts for the crossover from Ising-like critical behaviour to asymptotic mean-field critical behaviour. To extendthe range of applicability one may consider crossover from Ising-like criticalbehaviour to nonasymptotic mean-field critical behaviour by including higher-order terms in the classical Landau expansion of eq 10.83:116

FclE1

2tM2 þ �uu�L

4!M4 þ a05

5!M5 þ a06

6!M6 þ a14

4!tM4 þ a22

2!2!t2M2: ð10:92Þ

Equation 10.92 contains an asymmetric term pM5 that is not present in the(symmetric) Ising model or lattice gas. We note that in the mean-fieldapproximation M2 is of order t. Equation 10.85 for the renormalized potentialFr now becomes:

Fr ¼1

2tM2TDþ �uu�L

4!M4D2Uþ a05

5!M5D5=2VUþ a06

6!M6D3U3=2

þ a14

4!tM4TD2U1=2 þ a22

2!2!t2M2T2DU�1=2 � 1

2t2K:

ð10:93Þ

In eq 10.93, T;D;U;K continue to be the rescaling functions defined by eq10.87, while V is an additional rescaling function

V ¼ Y 2Da�nð Þ=2D1 ; ð10:94Þ

with a critical exponent Da ’ 1:323 associated with a new asymmetric confluentsingularity induced by the M5 term in the Landau expansion 10.92.137,138 For


the distance parameter k one either continues to use eq 10.86 for easierapplication41,52,116,138,139 or a generalized expression more consistent with thehigher-order terms in the expansion of eq 10.92:10,14,57,60,140–143

k2 ¼ tTþ �uu�L2

M2DUþ a05

3!M3D3=2VUþ a06

4!M4D2U3=2

þ a14

2!tM2TDUþ a22

2!t2T2U�1=2:

ð10:95Þ

The six-term crossover Landau model given by eq 10.93 has been applied to one-component fluids in terms of the fluid variables defined by eq 10.55 in the revised-scaling approximation, i.e., with one asymmetry coefficient in the mixing of thescaling fields and with analytic background contributions of eq 10.25, and withd1¼ P11 in eq 10.29 as a second empirical asymmetry coefficient.10,41,116 A para-metric version of the six-term crossover Landau model is also available.52 Theprocedure has been applied to represent thermodynamic properties in the extendedcritical region for a variety of fluids including carbon dioxide,144 methane andethane,145 some refrigerants,138,140,141 and H2O and D2O.57,142 The six-termcrossover Landau model has also been applied to represent the thermodynamicproperties of mixtures of carbon dioxide and ethane,144 mixtures of methane andethane,145 and mixtures of H2O and D2O

143,146 by allowing the system-dependentquantities to be analytic functions of the hidden field z as explained in Section10.4.4. Attempts have also been made to use the crossover procedure to accountfor the effects of critical fluctuations on closed-form classical equations ofstate.147,148

The confluent singularity resulting form the asymmetric M5 term in theexpansions of eqs 10.92 and 10.94 was thought to be responsible for thedeviations from symmetry of the coexistence-curve diameter.149 However, wenow know that this asymmetry is well represented by a symmetric Hamiltonianprovided that one relates the scaling fields to all physical fields according to theprinciple of complete scaling discussed in Section 10.3.3. Hence, rather thanusing a six-term crossover Landau model in the revised-scaling approximation,it would be of interest to revisit the two-term crossover Landau model butmade consistent with complete scaling. This has not yet been done.

10.5.3.3 Phenomenological Parametric Crossover Equations

A somewhat less fundamental but more practical approach has been developedby Kiselev150 who has formulated a phenomenological extension of the linear-model parametric equation described in Section 10.2.2 with variables

DT ¼ r 1� b2y2� �

; ð10:96Þ

and

Dr ¼ kR�bþ1=2 qð Þrbyþ d1DT : ð10:97Þ

356 Chapter 10

The Helmholtz-energy density has the form

rA ¼ a ¼ Daþ rm0 DT� �

� P0 DT� �

ð10:98Þ

with

Da r; yð Þ ¼ kr2�aRa aC0 yð Þ þX4i¼1

cirDiR�Di qð ÞCi yð Þ

" #: ð10:99Þ

In this expression Ci are and C1 are universal functions61 with C0 and C1

representing an asymmetric and a first correction-to-scaling function, respec-tively, while Di are universal exponents with D2¼ 2D1 and D3¼D4¼ gþ b� 1.In eqs 10.97 and 10.99 R(q) plays the role of a crossover function definedas150,151

R qð Þ ¼ 1þ q2

1þ q

� �2

; ð10:100Þ

where the variable q is related to the parametric variable r by

q ¼ grð Þ1=2: ð10:101Þ

In these equations k, a, ci, d1 (¼ P11 in eq 10.29), and g are system-dependentcoefficients with g being related to the inverse of the Ginzburg number NG.Slightly different versions for the crossover function R(q) have also beenused.14,61,152–157 In the critical limit q-0 one recovers the linear-model para-metric equation in Section 10.2.2 with coefficients a and k. In the classical limitq-p, Da becomes an analytic function of DT and Dr. For a comparison ofthis phenomenological parametric crossover equation with the crossoverLandau models the reader is referred to some previous publications.14,41,61

The phenomenological parametric crossover equation of Kiselev has beenused to represent the thermodynamic properties of many fluids including car-bon dioxide,152 hydrocarbons61,150,152 and H2O and D2O.155,156 A detailedcomparison of the application of the six-term crossover Landau model and thephenomenological parametric crossover equation to H2O and D2O has beenmade by Kostrowycka Wyczalkowska et al.57 Most importantly, Kiselev andcoworkers have successfully applied the procedure to represent the thermo-dynamic properties of a large number of mixtures150,152,156 by combining theprinciple of isomorphism described in Section 10.4.4 with the principle ofcorresponding states.153,154,158 Kiselev and Friend159 have applied the methodto account for critical fluctuations in cubic engineering equations of state.Kiselev and Ely160–162 have modified the procedure by replacing the asymptoticlinear-model parametric equation of state with trigonometric parametricequations of state. Unlike the linear-model, trigonometric models allow for anextrapolation into the metastable region.46 Another interesting development is


the application of the method by Kiselev and coworkers,163–170 also adopted byother researchers,171,172 to account for critical fluctuations in the statisticalassociating fluid theory (SAFT). Sun et al.173 have extended the procedure todevelop a multiparameter equation of state applicable to a very wide range oftemperatures and densities. Other examples of parametric crossover equationshave been proposed by Kim and coworkers.174,175

10.5.3.4 Numerical Implementation of RenormalizationTransformation

The theoretical approaches discussed so far lead to universal mathematicalequations for the effects of critical fluctuations on the thermodynamic prop-erties in which the system-dependent quantities are to be determined from fitsof the equations to experimental data. An alternative approach is to numeri-cally implement the renormalization-group theory for the effects of criticalfluctuations. The most practical version of such an approach has been devel-oped by White and coworkers176–180 in which a mesoscopic Helmholtz-energyexpression with a hard-core contribution and an attractive term is renormalizedby successive numerical iteration in a so-called phase-space cell approxima-tion.181 The method has been applied to fluids of molecules interacting with asquare-well potential182–185 and those interacting with a Lennard-Jonespotential.186–188 Prausnitz and coworkers have applied the method to describethe thermodynamic properties of n-alkanes185,189 and have also extended themethod to binary mixtures184,190,191 and even to multicomponent mixtures.192

Tang and coworkers186,193 have combined this numerical approach with amean-spherical approximation outside the critical region. The most interestingdevelopment in our opinion is the successful merger of this approach withvarious versions of the SAFT equation of state, discussed in Chapter 8.193–198

The procedure is especially useful for a description of volumetric thermo-dynamic properties.179,180 It yields a less accurate representation of caloricproperties like the isochoric heat capacity.199 In contrast to the mathematicalcrossover equations, the critical exponents incorporated in this method aresubjected to the restriction g¼ 2v implied by the numerical application scheme.However, unlike the mathematical crossover equations, the method is able toobtain estimates for system-dependent properties.

10.5.3.5 Hierarchical Reference Theory

A more fundamental but much more complex numerical approach is providedby the hierarchical reference theory (HRT).200–205 In the HRT the numeralimplication of the renormalization transformation is applied on a microscopicmodel of the fluid. One starts from a reference system with short-rangerepulsive interactions and then formulates a hierarchy of integral equationsaccounting for successively longer-range fluctuations. The theory has also beenextended to fluid mixtures.204–208 The HRT provides estimates for both

358 Chapter 10

universal and nonuniversal properties,209,211 such as the critical locus of binarymixtures as a function of concentration.206,210,212 The values obtained for thecritical exponents are approximate209,213,214 and subject to the restrictiong¼ 2v, as is also case the for the phase-space cell approximation discussed inSection 10.5.3.4. An improved smooth cutoff reformulation of the HRT hasbeen proposed very recently.215–217 Attempts have been made to unify the HRTwith the so-called self-consistent Ornstein-Zernike approximation,218–220

commonly referred to as SCOZA,221 which is not further discussed here.

10.6 Discussion

In this chapter, we have presented a survey of the major theoretical approachesthat are available for dealing with the effects of critical fluctuations on thethermodynamic properties of fluids and fluid mixtures. Special attention hasbeen devoted to our current insight in the nature of the scaling densities andhow proper relationships between scaling fields and physical fields account forasymmetric features of critical behaviour in fluids and fluid mixtures. We havediscussed the application of the theory to vapour-liquid critical phenomena inone-component fluids and in binary fluid mixtures and to liquid-liquid phaseseparation in weakly compressible liquid mixtures. Because of space limitationsthis review is not exhaustive. In particular for the interesting critical behaviourof electrolyte solutions we refer the reader to the relevant literature.15,125,222–234

Acknowledgements

We have benefited from valuable discussions with Claudio A. Cerdeirina andMichael E. Fisher. Mikhail A. Anisimov acknowledges support from ThePetroleum Institute, Abu Dhabi Oil and Gas Company, UAE. Hassan Beh-nejad thanks the Research Council of the University of Tehran for financialsupport of his sabbatical leave; he has also appreciated the hospitality of theInstitute for Physical Science and Technology at the University of Maryland.

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CHAPTER 11

Phase Behaviour of Ionic LiquidSystems

MAAIKE C. KROONa AND COR J. PETERSa, b

aDepartment of Process & Energy, Delft University of Technology,Leeghwaterstraat 44, 2628 CA Delft, The Netherlands; b ChemicalEngineering Program, The Petroleum Institute, P.O. Box 2533, Abu Dhabi,United Arab Emirates

11.1 Introduction

In the last two decades, ionic liquids have received much attention for use asnovel environmentally benign solvents.1–3 Ionic liquids are molten salts that areliquid at temperatures below 373K. They solely consist of ions. Commonlyused cations (with different functional groups R, which are usually alkyl chains)and anions are depicted in Figure 11.1.The most remarkable property of ionic liquids is that their vapour pressure at

room temperature is negligibly small, although some ionic liquids have adetectable albeit low vapour pressure at higher temperatures.4 This non-vola-tility is the reason why ionic liquids are considered to be environmentallybenign solvents, even though a lot of ionic liquids contain halogen atoms or aretoxic. Because it is possible to tune the physical and chemical properties of ionicliquids by varying the nature of the anions and cations, they can be designed tobe ‘green’. It is estimated that there are approximately 1018 accessible ionicliquids.5

Applications include the use of ionic liquids as electrolytes in electrochemicaldevices, as solvents in chemical synthesis and catalysis, separation technology,as lubricants and heat-transfer fluids.6 For the design of separation processes




368

that include ionic liquids, such as gas absorption,7 liquid-liquid extraction,8,9

and membrane separation,10,11 the phase behaviour of ionic liquid mixture is ofgreat importance. The phase behaviour of binary, ternary and multi-compo-nent ionic liquid systems has therefore been widely studied.

11.2 Phase Behaviour of Binary Ionic Liquid Systems

The phase behaviour of binary ionic liquid systems can roughly be divided intothree classes: (i) ionic liquids with dissolved gases, (ii) ionic liquids with waterand (iii) ionic liquids with organic solvents. The first class involves vapour-liquid equilibrium (VLE) data, whereas the last two classes mainly involveliquid-liquid equilibrium (LLE) data.

11.2.1 Phase Behaviour of (Ionic Liquid+Gas Mixtures)

The most investigated binary ionic liquid systems are mixtures with carbondioxide (CO2). In 1999, Blanchard et al.12 showed that the solubility of CO2 inimidazolium-based ionic liquids is very high, however, CO2 is not able to dis-solve any ionic liquid. Therefore, it was found possible to extract a solute froman ionic liquid using supercritical CO2 without any contamination by the ionicliquid.13

Common cations:

N+

R1

R3 R2R4

N+

R2R1

NN+

R1 R2

Ammonium Imidazolium Pyrrolidinium

P+

R1

R3 R2R4

N+

R

C+

N

N NR6

R5

R2R1

R4

R3

Pyridinium Phosphonium Guanidinium

Common anions:

Halides: Cl−, Br−, I−

Sulfates: CH3OSO3−

Sulfonates: CF3SO3−

Acetates: CF3CO2−

Phosphates: PF6−

Borates: BF4−

Imides: N(CF3SO2)2−

Cyanates: N(CN)2−

Figure 11.1 Most common cations and anions for ionic liquids.

369Phase Behaviour of Ionic Liquid Systems

The phase behaviour of many (ionic liquidþCO2) systems was subsequentlystudied, including CO2 solubility in imidazolium-based ionic liquids with tetra-fluoroborate anions,14–22 hexafluorophosphate anions,16,17,21–28 bis(trifluoro-methylsulfonyl)amide anions,17,18,21,28–30 and other ionic liquids.17,18,22,28,30–33

All these systems show similar phase behaviour as depicted in Figure 11.2.From Figure 11.2 it can be concluded that the CO2 solubility in an ionic

liquid is high at lower pressures, but a nearly infinite bubble-point slope ispresent at a specific maximum concentration of CO2, beyond which increasingthe external pressure hardly increases the CO2 solubility in the ionic liquid.According to Huang et al.34, the reason for this sharp pressure increase at acertain maximum CO2 concentration is that at this point all cavities in the ionicliquid phase are occupied by CO2, so that further insertion of CO2 wouldrequire ‘‘breaking’’ the cohesive structure of the ionic liquid.It was found that the anion predominantly determines the CO2 solubility in

the ionic liquid.17,18 Ionic liquids with anions containing fluoroalkyl groups, forexample, bis(trifluoromethylsulfonyl)amide, show the highest CO2 solubi-lity.17,28–30 It was also observed that an increase in the alkyl chain length on thecation increases the CO2 solubility in the ionic liquid.25,26

The extremely low solubility of imidazolium-based ionic liquids in CO2,as indicated by the straight dew point line at a CO2 mole fraction of 1 inFigure 11.2, resulted in the use of (ionic liquidþCO2) for biphasic catalysis,where the CO2 was used to transport the reactants to and the productsfrom the reactor and the reaction took place in the ionic liquid phase.35–40

However, some ionic liquids, such as trihexyltetradecylphosphonium chloride,were found soluble in CO2 up to a mass fraction of 0.07,41 indicating thatsingle-phase (ionic liquidþCO2) solvent for chemical synthesis are alsoconceivable.

0

20

40

60

80

100

0 0.2 0.4 0.6 0.8 1

x (CO2) or y (CO2)

p/M

Pa

0

20

40

60

80

100

Figure 11.2 (p,x) section of the phase behaviour of binary ionic liquidþCO2 systems.———, x; - - - - - -, y.

370 Chapter 11

The phase behaviour of binary ionic liquid systems with other gases has alsobeen determined. The gases considered include oxygen,15,18,30,42 carbon mon-oxide,18,30,42 nitrogen,42,43 nitrous oxide,18,30 hydrogen,42–46 hydrogen sul-fide,47,48 methane,42,43 ethane,18,30,42,46,49 ethene,18,28,30 benzene18,30 andtrifluoromethane.25,26,50,51 Although their solubilities depend on the specificionic liquid considered, some general trends can be observed. Benzene andtrifluoromethane are more soluble in ionic liquids than CO2.

18,26,30 After CO2,the gases nitrous oxide and hydrogen sulfide have the highest solubilities andstrongest interactions with the ionic liquid,18,30,47 followed by ethene, ethaneand methane.28,42,49–51 Oxygen, nitrogen and carbon monoxide are less solu-ble.15,18,30,42 Hydrogen is the least soluble of all gaseous solutes studied.42–46

All gas solubilities increase with pressure.18,42 However, the effect of tem-perature is ambiguous. As temperature increases, the solubilities of CO2 andhydrocarbons in ionic liquids decrease, but the solubility of hydrogen andnitrogen increases with increasing temperature.42,43

Although CO2 is not able to dissolve any imidazolium-based ionic liquid,12

other gases do.25,26 Ionic liquids are especially soluble in hydrocarbons thathave a strong molecular interaction with the ionic liquid, such as benzene andtrifluoromethane (CHF3).

18,25,26,30 Figure 11.3 shows the general phase beha-viour of (ionic liquidþCHF3) systems.50,51 This phase diagram is completelydifferent from the phase diagram of (ionic liquidþCO2) systems shown inFigure 11.2. The (ionic liquidþCHF3) shows a closed phase envelope,including the occurrence of a critical point,50,51 whereas (ionic liquidþCO2)system with the same ionic liquid has an immiscibility gap between the CO2-phase and the ionic liquid-phase, even up to very high pressures. This has beenattributed to the stronger molecular interactions between CHF3 (with its strong

0

10

20

30

40

50

0 0.2 0.4 0.6 0.8 1

x (CHF3) or y (CHF3)

p/M

Pa

0

10

20

30

40

50

V

L

L + V

Figure 11.3 (p, x) section of the phase behaviour of (ionic liquidþCHF3).E, criticalpoint.


permanent dipole moment) and the ionic liquid compared to those betweenCO2 (no dipole moment) and the ionic liquid.26

11.2.2 Phase Behaviour of (Ionic Liquid+Water)

Most ionic liquids are immiscible with water at room temperature.52 However,at higher temperatures the mutual solubilities between ionic liquid and waterincrease, eventually ending in upper critical solution temperature (UCST)behaviour52–62 as shown in Figure 11.4. Examples of ionic liquids that showUCST behaviour with water are imidazolium-based ionic liquids with tetra-fluoroborate anions,53–58 hexafluorophosphate anions54,57–60 and bis(trifluoro-methylsulfonyl)amide anions,55–57,60 and other ionic liquids.55,57,61,62

It can be noticed from Figure 11.4 that the mutual solubilities of ionic liquidsand water are not symmetric. Generally, a much higher mole fraction of wateris present in the ionic liquid phase than ionic liquid present in the water phase atthe same temperature.52–62 The UCST is thus found at very low fractions of theionic liquid.60 This asymmetry is also observed in (polymerþwater) systems.52

Therefore, there are analogies between the phase diagrams of ionic liquidsolutions and polymeric ones, although this may be due to differentmechanisms.The mutual solubilities of ionic liquids and water are primarily defined by the

type of anion and the cation alkyl side chain length of the ionic liquid.52,57 Anincrease in the alkyl chain length of the cation decreases the mutual solubility.Ionic liquids with bis(trifluoromethylsulfonyl)amide anions and hexa-fluorophosphate anions are more hydrophobic than ionic liquids with tetra-fluoroborate anions.52,57 However, even the most hydrophobic ionic liquids still

300

400

500

600

700

800

0 0.2 0.4 0.6 0.8 1

x (IL)

T/K

300

400

500

600

700

800

L2

L

L1 + L2

L1

Figure 11.4 (T, x) section of the phase behaviour of {ionic liquid (IL)þwater}.E,critical point.

372 Chapter 11

show some mutual solubility with water. Therefore, if an ionic liquid is used asa solvent to extract solutes from water, the dissolution of the ionic liquid in theaqueous phase could represent a waste water treatment challenge.54 Anotherproblem is the water vapour uptake. Even hydrophobic ionic liquids areslightly hygroscopic.54 Activity coefficients at infinite dilution for water in manyionic liquids were measured to be slight positive deviations from Raoult’slaw.63–66

Some (ionic liquidþwater) systems do not exhibit the usual UCST beha-viour, but show a lower critical solution temperature (LCST) behaviourinstead. Here, the phase separation takes place at higher temperatures andmutual solubilities increase upon cooling until a homogeneous solution isreached.67 Only a small change in ionic liquid structure can already change thetype of phase behaviour (UCST or LCST). For example, the tetrabutyl-phos-phonium-based ionic liquid with trans 2-butenedione (fumarate) as anionshows UCST behaviour, whereas the similar ionic liquid with the cis 2-bute-nedione (maleate) shows LCST behaviour after mixing with water.68

11.2.3 Phase Behaviour of (Ionic Liquid+Organic)

The binary phase behaviour of ionic liquids with different organic sol-vents has been measured. These organics include alkanes,55,61,63,64,66,69–79

alkenes,63,64,66,72–79 alkynes,74,78,79 cyclic hydrocarbons,55,61,63,64,66,69,71,72,74,78,79

aromatic hydrocarbons,55,61,63,64,66,69,71–83 alcohols,55–58,60–66,71,75–78,82–94 alde-hydes,75–77,95 ketones,56,61,63–66,78,94,95 ethers,61,63,65,66,78,94 esters,63,66,75–77,94 andchloroalkanes.78,89,94,96

Most (ionic liquidþ organic) show immiscibility in the liquid phase with anupper critical solution temperature (UCST) found at low mole fractions of theionic liquid.55–58,60–62,69–71,84–86,88–91 Therefore, the phase behaviour of (ionicliquidþ organic) is similar to that of (ionic liquidþwater) (Figure 11.4).However, unlike the system with water, the UCST of (ionic liquidþ organic)decreases with increasing alkyl chain length of the cation.55–58,60,84,89–92

It was also found that a decrease in the alkyl chain length of the organicresults in a decrease in the UCST.56,86,87,89–91 This means that the UCST ofsmall organics, like methanol, is often found at temperatures below roomtemperature, resulting in complete miscibility at room temperature.86 For largeorganics the observation of the UCST is limited by the boiling temperature ofthe organic.71 Ionic liquids are more soluble in aromatic hydrocarbons com-pared to alkanes and cycloalkanes with the same number of carbon atoms.61

Branching of the organic results in a higher solubility of the organic in the ionicliquid phase, and thus lowering the UCST.56,89–91 Increasing hydrogen bondingopportunities (i.e., the presence of polar groups on the organic,75 or the pre-sence of acidic hydrogen on the cation56) also increases the mutual solubility.Finally, the choice of anion of the ionic liquid has a large impact on the

UCST of the system. Organics are much more soluble in ionic liquids with the


bis(trifluoromethylsulfonyl)amide anion compared to ionic liquids with thetetrafluoroborate anion.56,57,69,89–91

Common UCST behaviour has also been observed for mixtures consisting oftwo ionic liquids with the same anion and a different cation.97 Less commonlower critical solution temperature (LCST) behaviour has been observed forbinary ionic liquid mixtures with benzene,78–80 chloroalkanes78,89,94,96 andpolymers.98 Interestingly, the LCST behaviour of benzene with an imidazo-lium-based ionic liquid changed via ‘hour-glass’ to UCST behaviour when theimidazolium alkyl chain length was increased.80 Moreover, imidazolium-basedionic liquids were found to form clathrate structures with benzene at lowtemperatures,81 whereas the solid-liquid equilibria (SLE) of other organics orwater with ionic liquids are simple eutectic systems.55,71,86

11.3 Phase Behaviour of Ternary Ionic Liquid Systems

The phase behaviour of four classes of ternary ionic liquid systems has beendetermined: (i) ionic liquidþ carbon dioxideþ organic, (ii) ionicliquidþ aliphaticþ aromatic, (iii) ionic liquidsþwaterþ alcohol and (iv) ionicliquid systems with azeotropic organic mixtures.

11.3.1 Phase Behaviour of (Ionic Liquid+Carbon

Dioxide+Organic)

Figure 11.5 shows the general phase behaviour of {ionic liquid(l)þCO2(v)þ organic (l)}. When the ionic liquid and the organic compoundare completely miscible at ambient conditions (liquidþ vapour), it is possible toinduce the formation of a second liquid phase by placing a pressure of CO2

upon the mixture (liquidþ liquidþ vapour).99–106 The most dense phase is rich

Figure 11.5 Pressure P as a function of temperature T illustrating the phase beha-viour of ternary {ionic liquid (l)þCO2(v)þ organic (l)}.

374 Chapter 11

in ionic liquid, the newly formed liquid phase is rich in organics and the gasphase mostly contains CO2 with some organics. Further pressurization leads toexpansion of the organic-rich phase with increased CO2 pressure, while theionic liquid-rich phase expands relative little. Eventually this will lead to thedisappearance of the vapour phase at the point, where the organic-rich phasemerges with the gas phase.100–106At this moment the last traces of ionic liquidthat remained in the organic-rich liquid phase are expelled, and the resulting(CO2þ organic) phase contains no detectable ionic liquid. Eventually, when thepressure is increased even further, one homogeneous liquid region isreached.107,108

Interestingly, as Figure 11.6 shows, it is thus possible to induce ternary (ionicliquidþCO2) to undergo a ‘two-phase’ – ‘three-phase’ – ‘two-phase’ – ‘one-phase’ transition by only changing the CO2 pressure. Although the simplephase transition from two to three phases by addition of CO2 was alreadyknown to occur in ternary CO2 systems without an ionic liquid, only recently itwas also discovered to occur in ternary CO2 systems in presence of an ionicliquid.99 Initially this phenomenon was wrongly identified as Lower CriticalEnd Point (LCEP).99 Thereafter, the transition from three to two phases atfurther CO2 pressure increase was discovered, and also wrongly identified as K-point.100 After all, both transitions are normal phase transitions without anycriticality involved.109 Only recently, the formation of a homogeneous liquidphase at even higher CO2 pressures was found.

107 The location of this homo-geneous liquid phase is hard to find, because it occurs in a relatively narrowrange of CO2 mole fractions.108

The conditions at which the different phase transitions occur depends on thetype of organic, the type of ionic liquid and the concentrations.101,110–114

Stronger interaction between the ionic liquid and the organic makes it moredifficult for CO2 to induce the formation of a second liquid phase.110–114 Thisdifference in affinity was used for selective extraction of specific organics fromionic liquids by using CO2.

106 CO2 at low concentrations was found to work asco-solvent (increasing the solubility of organics into the ionic liquid phase),

p(CO2)

CO2-rich phase(+ organic)

Ionic liquid-rich phase(+ organic + CO2)

CO2-rich phase(+ organic)

organic-rich phase(+ ionic liquid + CO2)


CO2 + organic


V

V

L L1

L2

L1

L2Homogeneous phase

(ionic liquid +organic + CO2)

L1

Figure 11.6 CO2 induced ‘two-phase’ – ‘three-phase’ – ‘two-phase’ – ‘one-phase’transition in the ternary {[bmim][PF6]þmethanolþCO2}.

7


while CO2 at higher concentrations worked as anti-solvent (decreasing thesolubility of organics in the ionic liquid phase).107,113–115 The same type ofphase behaviour was also observed for systems in which the organic is a solidinstead of a liquid.115,116 Therefore, it is possible to crystallize an organic out ofan ionic liquid using CO2 as anti-solvent.

115

When an organic reaction is carried out in an (ionic liquidþCO2) system, thereaction rate depends strongly on the number of phases present.104 The highestreactions rates are obtained in the homogeneous liquid phase.107 By switchingback to the multiphase regime, the product can be recovered from the phasethat does not contain any ionic liquid.107,108

Furthermore, it is possible to separate hydrophilic ionic liquids from waterwith CO2.

117 CO2 can cause liquid-liquid separation in hydrophilic ionicliquidþwater mixtures.117–119 So (ionic liquidþCO2þwater) show similaritieswith (ionic liquidþCO2þ organic) systems.

11.3.2 Phase Behaviour of (Ionic Liquid+Aliphatic+Aromatic)

The general (ionic liquidþ aliphaticþ aromatic) liquid–liquid equilibrium,including tie lines, is shown in Figure 11.7. It can be noticed that aromaticsshow stronger attractive interactions (smaller immiscibility gaps) and highersolubilities in ionic liquids than aliphatic hydrocarbons. This observed highersolubility of aromatics over aliphatic hydrocarbons in ionic liquids was used forthe separation by liquid–liquid extraction.120–132

Figure 11.7 Liquid–liquid equilibrium of (ionic liquidþ aliphaticþ aromatic) atconstant temperature.

376 Chapter 11

The effectiveness of the extraction of aromatics from alkanes by ionic liquidsis determined by the ratio of the solubilities in the two phases. This quantity,known as the selectivity, was found to increase with increasing carbon numberof the alkane.120,122,125 Moreover, the selective extraction of aromatics fromalkanes by ionic liquids increases with decreasing aromatic content in the feedand at lower temperatures.123–125 However, the distribution ratio, which is ameasure of the extractive capacity, decreases with decreasing aromatic contentand temperature.125 Therefore, the choice of extraction conditions is a trade-offbetween selectivity and capacity.The choice of ionic liquid also has a large influence on the selectivity of the

extraction. Selectivities are higher for cations with shorter alkyl side chainlengths.12,129,131 Ionic liquids containing a pyridinium-based cation have amore aromatic character than imidazolium-based ionic liquids, resulting in ahigher selectivity.124,125,128 Quaternary phosphonium-based and ammonium-based ionic liquids show lowest selectivity.125 The selectivities for ionic liquidswith hexafluorophosphate anions are higher than those with tetrafluoroborateanions.121,126 Generally, separation factors with ionic liquids as extractant are afactor two higher than those obtained with sulfolane, which is a conventionalsolvent for the extraction of aromatic hydrocarbons from a mixed close-boiling(aromaticþ aliphatic) hydrocarbon stream.125,130,131

Other investigated separations with ionic liquids as extractant include the(aromaticþ cyclic hydrocarbon) extraction,133–136 the (alkeneþ alkane) extrac-tion,136 and the (sulfur-containing aromaticþ aliphatic) extraction.137–144 Theseextractions follow the same trends observed for general (aromaticþ aliphatic)extractions. However, it is more difficult to extract aromatics from cyclichydrocarbons compared to linear aliphatic hydrocarbons by using ionicliquids.133–136

11.3.3 Phase Behaviour of (Ionic Liquid+Water+Alcohol)

Ionic liquids were found to be suitable entrainers for the separation of azeotropic(waterþ alcohol) mixtures by means of extractive distillation or solvent ex-traction. The alcohols investigated include ethanol,62,145–152 -1-propanol,153,154

-2-propanol,153,155–158 2-methyl-2-propanol,159 -1-butanol160,161 and severalpolyols.162–165

Ionic liquids have stronger interactions with water than with alcohol.Therefore, the relative volatility of the alcohol is increased by addition ofthe ionic liquid. A higher ionic liquid concentration results in a higherrelative volatility of the alcohol, eventually resulting in breaking of theazeotrope.146–148,150,154,156,158,159 The general vapour-liquid equilibrium of(ionic liquidþwaterþ alcohol), including the influence of ionic liquid con-centration, is shown in Figure 11.8.Generally, most hydrophilic ionic liquids are the best entrainers.146

A decrease in alkyl chain length of the cation enhances the relative volatility ofthe alcohol.146,148,153,159 The choice of anion also has a large influence. Ionic


liquids with chloride and acetate anions are much better entrainers compared toionic liquids with dicyanamide and tetrafluoroborate anions.146,151,157

11.3.4 Phase Behaviour of Ionic Liquid Systems with Azeotropic

Organic Mixtures

Azeotropic organic mixtures can also be separated by using ionic liquids asentrainers in extractive distillations. The most commonly investigated azeo-tropic systems are (alcoholþ organic), where the organic is an alkane,166–170

chloroalkane,171,172 alkene,173 ketone,155,174,175 ether176 or ester.177–181 In allcases, the interaction of the ionic liquid with the alcohol is stronger than theinteraction with the other organic compound. The relative volatility of theother organic compound can thus be increased by addition of an ionic liquid,and the azeotrope can be broken. Most polar ionic liquids are the bestentrainers for azeotropic (alcoholþ organic).166–169,177,180,181 Compared toconventional liquid entrainers, ionic liquids have two advantages: (i) their ioniccharacter results in stronger interactions with the alcohol and therefore agreater separating effect, and (ii) their negligible vapour pressure allows thealcohol to be recovered without any ionic liquid contamination.174

Other azeotropic organic mixtures that can be separated by using ionic liquidsas entrainers include (alkaneþ ester),170,182 (cycloalkaneþ ketone)183 and(benzeneþ hexafluorobenzene).184 Especially the effect of the ionic liquid on thelast-mentioned mixture is interesting. The binary (benzeneþ hexafluorobenzene)is known to have double azeotropes that is, a minimum pressure and a

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1x (OH)

y (O

H)

0

0.2

0.4

0.6

0.8

1

Figure 11.8 Vapour–liquid equilibrium y(x)p, where y and x are the vapour andliquid alcohol mole fractions, respectively, for (ionic liquidþwaterþ alcohol) at constant pressure as a function of the mole fractionsof ionic liquid x(IL). ———, x(IL)¼ 0; ————, x(IL)¼ 0.1; – - – - – - –,x(IL)¼ 0.2; - - - - - -, indicates the one-to-one line.

378 Chapter 11

maximum pressure azeotrope. The addition of an ionic liquid allows thebreaking of both the minimum pressure azeotrope (vapour-liquid equilibrium)and the maximum pressure azeotrope (liquid-liquid equilibrium).184

11.4 Modeling of the Phase Behaviour of Ionic Liquid

Systems

Because measurement of the phase behaviour of ionic liquid systems is timeconsuming, it is desirable to develop predictive methods to estimate this. Dif-ferent approaches were proposed for modeling the phase behaviour of ionicliquid systems: (i) molecular simulations, (ii) excess Gibbs-energy methods, (iii)equation of state modeling and (iv) quantum chemical methods.

11.4.1 Molecular Simulations

Molecular simulations have been used to obtain thermodynamic properties andphase equilibria data of ionic liquid systems: (i) Monte Carlo simulationtechniques were employed to predict the solubility of gases and water in ionicliquids185–190 and (ii) molecular dynamics simulations were used to investigatethe solvation dynamics of water and various organics in ionic liquids.191–195

The reliability of these molecular simulations depends both on the quality ofthe force field and on the use of a proper simulation method.196

The most common approach to develop a classical force field for an ionicliquid is to represent the bonding and energetics with simplified analyticalpotential functions, which are inexpensive to evaluate numerically. Such a forcefield is typically based on an OPLS-AA/AMBER framework with two types oftorsions (proper and improper) and has the following form for the totalpotential energy Utot:

197–206

Utot ¼Xbonds

kb l � l0ð Þ2þXangles

ky y� y0ð Þ2 þX

torsions

kf 1þ cos nf� gð Þ½ �

þX

impropers

kc c� c0ð Þ þXC�1i¼1

XCj¼iþ1

4�ijsijrij

� �12

� sijrij

� �6" #

þ qiqj

rij

( ):

ð11:1Þ

The first four terms represent the bonded interactions that is, bonds, angles,proper torsions and improper torsions. In eq 11.1, kb is the bond force constant,(l� l0) is the distance from equilibrium bond length that the atom has moved, kyis the angle force constant, (y� y0) is the angle from equilibrium between threebonded atoms, kf is the dihedral force constant, n is the multiplicity of thedihedral function, f is the dihedral angle, g is the phase shift, kc is the Eulerangle force constant and (c�c0) is the out of plane angle. The non-bondedinteractions are described in the last term, including the Lennard-Jones inter-actions and the Coulombic interactions of atom-centered point charges.196–206


In eq 11.1, eij is the Lennard-Jones well depth, sij is the distance at which theinterparticle potential is zero, rij is the distance between atoms i and j, and qi isthe charge on atom i.Methods to determine the force-field parameters vary. Generally, bond

lengths, angles and point charges are calculated by ab initio calculations, whilethe Lennard-Jones parameters and geometric force constants are taken fromestablished sources such as OPLS, AMBER or CHARMM.197–199,202–205 Insome cases, geometric force constants were also obtained from quantum cal-culations.201 Force-field parameters may be adjusted to available experimentaldata, although this was rarely found to be necessary.196

Once the force field is chosen, a proper simulation method needs to beselected. Molecular dynamics simulations are applied to determine the solva-tion behaviour of ionic liquids by means of solving the Newtonian equations ofmotion for all molecules in the presence of a gradient in potential energy.191–195

Ionic liquid phase equilibria are determined by using Monte Carlo simulationsin the isothermal isobaric Gibbs ensemble,185–188 grand canonical ensemble orosmotic ensemble189,190 with clever sampling schemes.

11.4.2 Excess Gibbs-energy Methods

Excess Gibbs-energy methods are by far the most commonly used to correlatethe phase behaviour (pressure and temperature versus phases present andcomposition) of ionic liquid systems. For example, the Wilson equation hasbeen used for modeling of binary and ternary (ionic liquidþ alcohol) sys-tems,56,155 the Margules equation has been used to correlate binary (ionicliquidþ chloroalkanes) systems51 and regular solution theory appears to beable to describe the solubilities of several gases28,190 and organics78 in ionicliquids. However, the most often applied excess Gibbs-energy methods are thenonrandom two-liquid (NRTL) equation and the universal quasi-chemical(UNIQUAC) equation.Althought originally developed for systems involving nonelectrolytes, the

NRTL model has successfully been applied to correlate the UCST behavi-our of binary ionic liquid with water56,57,60–62,207,208 and organics.51,56,57,60–62,69,72,77,82,83,86,88,91,95,207,208 Also, the NRTL equation is able to model (ionicliquidþ aliphaticþ aromatic) systems,120,121,123,124,127–129,134,135,137–144,207,208

(ionic liquidþwaterþ alcohol) systems62,147–150,153,155,257,159,207,208 and ter-nary azeotropic organic systems.155,166,167,169,170,175–178,181–183,207,208 TheUNIQUAC equation, also developed for nonelectrolyte mixtures, successfullydescribes binary and ternary ionic liquid systems with water and organ-ics.65,72,86,88,140,141,155,179,182,183,207,210 Reason for the success of these none-lectrolyte models is the fact that ionic liquids under most circumstances can beconsidered as neutral ion pairs.209 However, in combination with strongly polarcompounds such as water or alcohols, ionic liquids start to dissociate into itsconstituent ions, especially at low ionic liquid concentrations.209 The electrolytenonrandom two-liquid (eNRTL) model, which takes the ionic character of

380 Chapter 11

ionic liquids into account, was used to model the phase behaviour of binary andternary ionic liquid systems with water and alcohols.57,150,154,172,174,178,207,210,211

Parameterization for all excess Gibbs-energy methods commonly takes placeby minimizing an objective function based on the squared differences betweencalculated and experimental compositions. Binary systems are modeled withtwo parameters, whereas multi-component systems need two parameters foreach possible binary pair. This means that six parameters are used to fit ternarydata.207 Each time when new experimental data become available, a new esti-mation for the parameters of excess Gibbs-energy methods is made. A majordisadvantage of excess Gibss energy methods is that they cannot predict thephase behaviour of ionic liquid systems prior to making extensive measure-ments. Moreover, it is impossible to correlate the unusual phase behaviour of(ionic liquidþ carbon dioxide) systems with excess Gibbs-energy methods.190

11.4.3 Equation of State Modeling

Different types of equations of state have been used to model the phasebehaviour of ionic liquid systems. Cubic equations of state such as the Peng-Robinson equation50,102 and the Redlich-Kwong equation16,31,184,212 have beenused to describe the solubility of carbon dioxide, trifluoromethane and organicsin ionic liquids. Because cubic equations of state require the critical parametersof ionic liquids, which are unknown, these have to be estimated by using group-contribution methods.16 Thus estimates obtained from cubic equations of statefor ionic liquid systems are unreliable. Moreover, cubic equations of state canonly describe the carbon dioxide solubility in ionic liquids at low concentra-tions, but cannot predict the dramatic increase in bubble point pressure athigher carbon dioxide concentrations.50

Corresponding-states correlations, which were used for describing the phasebehaviour of (ionic liquidþ organics) systems,58,70,85,213 also suffer from thesame problem that the critical parameters of ionic liquids are required yetunknown.More reliable phase behaviour predictions for binary ionic liquid systems

with carbon dioxide or organics come from group-contribution equations ofstate, such as the universal functional activity coefficient (UNIFAC)method,66,136 the group-contribution nonrandom lattice fluid equation ofstate21,214 and the group-contribution equation of state of Skjold-Jørgensen.215

In group-contribution methods, molecules are decomposed into groups whichhave their own parameters. Generally, ionic liquids are decomposed into a largegroup, consisting of the anion and the methylated (aromatic) ring of the cation,and a CH3 group and various CH2 groups that form the alkyl chain of thecation.21,215 For example, Figure 11.9 shows how the ionic liquid 1-butyl-3-methylimidazolium tetrafluoroborate ([bmim1][BF4

�]) is decomposed into oneCH3 group, three CH2 groups, and one [mim1][BF4�] group. Pure groupparameters are regressed from liquid density data.21 Binary interaction para-meters are fitted from infinite dilution activity coefficients of alkanes in ionic


liquids and vapour-liquid equilibrium data of binary ionic liquidþ carbondioxide or organics systems.215 In this way, the unknown critical parametersand vapour pressures of ionic liquids are not needed to determine group-con-tribution equation of state parameters. Once the parameters for one binaryionic liquidþ carbon dioxide system were determined, the solubility of carbondioxide in other ionic liquids of the same homologous series could predictedwith high accuracy.21,215

Statistical-mechanics based equations of state are most predictive, becausethey account explicitly for the microscopic characteristics of ionic liquids.The statistical association fluid theory models tPC-PSAFT216–218 (truncatedperturbed chain polar statistical associating fluid theory) and soft-SAFT219

(soft statistical associating fluid theory) have successfullly been used tomodel the phase behaviour of binary and ternary ionic liquid systems withcarbon dioxide, organics and/or water over a wide pressure range. These sta-tistical-mechanics based equations of state consider the ionic liquids to beasymmetrical neutral ion pairs, either with a dipole moment to account for thecharge distribution of the ion pair (for tPC-PSAFT)216–218 or with an asso-ciating site mimicking the interactions between the cation and anion as a pair(in case of soft-SAFT).219 Also, the associating interactions between ionicliquids and carbon dioxide, organics and water are accounted for.216–219 Allpure-component parameters for ionic liquids are calculated from availablephysicochemical data of the constituent ions, such as size, polarizability andnumber of electrons.216 This means that all parameters are physically mean-ingful. Only one binary interaction parameter for each possible binary pair isadjusted in order to fit the model to experimental vapour-liquid equilibriumdata.216–219 Statistical associating fluid theory models predict the phase beha-viour of ionic liquid systems with carbon dioxide or nonpolar organics withhigh accuracy.216–219 However, it is more difficult to predict the phase equilibriaof ionic liquid systems with strongly polar compounds (water and alcohols),218

because ionic liquid dissociation into its constituent ions is not taken intoaccount.

Figure 11.9 Chemical structure of cation and anion of 1-butyl-3-methylimidazoliumtetrafluoroborate. The dashed lines illustrate how the ionic liquid 1-butyl-3-methylimidazolium tetrafluoroborate ([bmim1][BF4

�]) isdecomposed into one CH3 group, three CH2 groups, and one[mim1][BF4�] group.

382 Chapter 11

11.4.4 Quantum Chemical Methods

The conductor-like screening model for real solvents (COSMO-RS)220 is anunimolecular quantum chemical method that has been used to predict ionicliquid phase equilibria. In COSMO-RS, molecules are treated as a collection ofsurface segments, each with a screening charge density as if they are imbeddedin a conductor. An expression for the chemical potential of segments in thecondensed phase is derived, in which interaction energies between segments arecalculated. The chemical potential of each molecule is obtained by summing thecontributions of the segments. COSMO-RS uses a single radius and one dis-persion constant per element and a total number of eight COSMO-RS inherentparameters, which have been optimized by using thermodynamic property dataof over 200 small neutral organic compounds.220

The phase behaviours of binary and ternary ionic liquid mixtures with car-bon dioxide,221 organics66,84,173,222,223 and water222,224–226 have been deter-mined using COSMO-RS. In the COSMO-RS framework, ionic liquids areconsidered to be completely dissociated into cations and anions.227,228 Ionicliquids are thus taken as an equimolar mixture of two distinct ions, whichcontribute as two different compounds. Because ionic liquids only dissociate inthe presence of strongly polar substances,209 the COSMO-RS prediction of thephase behaviour of ionic liquid systems with polar compounds (water andalcohols) is more accurate than that of ionic liquid systems with nonpolarcompounds (carbon dioxide and organics).223–228 Especially the COSMO-RSprediciton of the solubility of (relatively nonpolar) carbon dioxide in ionicliquids shows considerable deviations (E15 %) from experimental values.223

IUPAC Technical Reports document the measurements of the thermo-dynamic and thermophysical properties of 1-hexyl-3-methylimidazolium bis[(trifluoromethyl)sulfonyl]amide and the recommended values.229,230

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CHAPTER 12

Multi-parameter Equations ofState for Pure Fluids andMixtures

ERIC W. LEMMONa AND ROLAND SPANb

a Thermophysical Properties Division, National Institute of Standards andTechnology, Boulder, CO 80305, U.S.A.; bRuhr-Universitat Bochum,Lehrstuhl fur Thermodynamik, 44780 Bochum, Germany

12.1 Introduction

Thermodynamic fluid properties with low uncertainties are needed for thedevelopment of a variety of industrial and scientific applications. Althoughsignificant improvements have been made in predicting properties from theo-retical methods, the need for more accurate empirical equations of state forapplications in engineering system design and analysis continues to grow. Asenergy costs rise and the need for lower uncertainties to reduce resource con-sumption continues, precise property calculations for natural gas, petroleum,and biofuels will enable better conservation and enhanced usage of renewablesources for future generations.A typical thermodynamic property formulation is based on an equation of

state that allows the calculation of all thermodynamic properties of the fluid,including properties such as entropy that cannot be measured directly. In thiscase the term ‘‘equation of state’’ is used to refer to an empirical modeldeveloped for calculating fluid properties such as those reported by Jacobsenet al.1 and Span et al.2 The equation of state is based on one of four funda-mental relations: internal energy as a function of volume and entropy; enthalpy




394

as a function of pressure and entropy; Gibbs energy as a function of pressureand temperature; or Helmholtz energy as a function of density and tempera-ture. Modern equations of state for pure fluid properties are usually funda-mental equations explicit in the Helmholtz energy as a function of densityand temperature. Thermodynamic relationships can then be used to calculateother properties from the equation of state once a state point has been specifiedby an appropriate number of independent properties. For equations basedon one of the four fundamental properties, such calculations require onlydifferentiation of the equation. Equations of state of the type reported hereare valid for gas and liquid states (or for an equilibrium of liquid and vapour),and include supercritical states above the critical point. All thermodynamicproperties can be calculated without additional ancillary equations forsaturation properties through the use of the Maxwell criterion (equal pressuresand Gibbs energies at constant temperature during phase changes forpure fluids). The range of validity of the equation ends on the melting line; theproperties of the fluid phase can be calculated along this line, but not theproperties of the solid.The development of an accurate property formulation requires analysis of

the available data and correlation with a suitable functional form. The processof determining the optimum correlation often involves considerable judgmentin addition to more objective protocols, and experience plays a significant partin the determination of the final result. The objective of the correlator is toascertain the uncertainty of the available experimental data for the particularfluid or system under investigation, and to develop a mathematical modelcapable of representing the data within the reported or estimated experimentaluncertainty. The practical models of today are empirical or semi-empirical innature, although virtually all are based upon sound theoretical principles. Thelimitations of the model selected must be mutually understood by the correlatorand the user for effective system optimization and related work.

12.2 The Development of a Thermodynamic Property

Formulation

The importance of experimental data of known uncertainty in the developmentof thermodynamic property correlations is well established. Systematic errorsin experimental data sets will be reflected in the quality of the correlation basedupon such data. If data are not available for a particular region of the surface, itis often helpful to use methods that allow for smooth interpolation or extra-polation with regard to temperature, pressure, or density in the fitting process.The calculations can be verified or discounted later when experimental data areavailable.There are several fixed points, reference state properties, and molecular data

that should be available for each pure fluid for which a thermodynamicproperty correlation is developed. These include the temperature, density, and

395Multi-parameter Equations of State for Pure Fluids and Mixtures

pressure at the critical point and triple point, the molar mass, and the enthalpyand entropy reference values as outlined in Span3 and Lemmon et al.4

The following experimental data are generally considered essential indeveloping an accurate equation of state: ideal gas heat capacities Cpg

p,m

expressed as functions of temperature T, vapour pressure psat, and density rdata in all regions of the thermodynamic surface. Precise speed of sound w datain both the liquid and vapour phases have recently become important for thedevelopment of equations of state. The precision of calculated energies can beimproved if the following data are also available: CV,m(r, T) (isochoric heatcapacity measurements), Cp,m(p, T) (isobaric heat capacity measurements),DHm(p, T) (enthalpy differences), and Joule-Thomson coefficients.Correct behaviour of the equation of state in the critical region is a concern

of users of property formulations. Table 12.1 lists the constraints that must beplaced on equations of state to obtain an accurate representation of the thermalproperties. Classical equations cannot represent the theoretically expected non-analytical behaviour at the critical point. However, state-of-the-art multi-parameter equations of state represent the experimental data with sufficientprecision in the critical region to satisfy most data needs (although they shouldnot be used as a basis for theoretical calculations regarding the limitingbehaviour at the critical point because they do not include cross over to thecritical behaviour). Older and usually less precise multi-parameter equations ofstate may show significant shortcomings with regard to the representation ofproperties in the critical region.Accuracy and thermodynamic consistency in a wide-range equation of state

for a pure fluid imply the following:

1. All thermodynamic properties can be calculated within the limits ofexperimental uncertainty by differentiating or integrating the equationof state, that is, arbitrary corrections to derived properties are notrequired;

2. The equation of state reduces to the ideal gas equation of state in thelimit as r-0;

Table 12.1 Typical constraints imposed on the equation of state so that theestimated values are equal to the physical property of derivatives.

Constraint Property or condition

Pressure at the critical point pcDensity at the critical point rcTemperature at the critical point Tc

Isochoric derivative at the critical point @p=@Tð Þrc¼ dp=dT jsa

First derivative of pressure with respect to densityat the critical point

@p=@rð ÞTc¼ 0

Second and subsequent derivatives of pressure withrespect to density at the critical point

@2p=@2r� �

Tc¼ @3p=@3r� �

Tc¼ 0

afrom the vapour pressure equation at the critical point.

396 Chapter 12

3. The equation of state obeys the Maxwell criterion (equal Gibbs ener-gies for saturated liquid and saturated vapour states at a givensaturation temperature and the corresponding vapour pressure);

4. The behaviour in the critical region (� 0.25 � rc and � 0.05 �Tc) isreasonably consistent with experimental measurements and theoreticalconsiderations except at and very near the critical point (approximately� 0.15 � rc and � 0.002 �Tc);

5. The behaviour of calculated constant property lines is consistent withavailable experimental data and with theoretical predictions (forexample, isotherms should not intersect at high pressures).

An important attribute of an equation of state for engineering applications isrelative simplicity, resulting in minimal computation time for calculation of thenecessary properties. Although the use of developed software has becomecommon in engineering applications, situations that require user specific soft-ware are still frequent. If requirements of either precise estimate or demandswith regard to the range of validity can be relaxed, then equations of state withfewer terms for specified levels of precision may be developed to reduce com-putation times.In studies to determine equations of state for fluids, a number of empirical

polynomial and exponential forms have been suggested for use in fittingexperimental measurements. Generally, experimental data are represented by aconvenient functional form with linear or nonlinear least-squares curve fitting.The methods used by correlators in developing accurate multi-parameterequations of state vary widely, depending upon the intended use of the equationdeveloped. The methods discussed here are generally applicable to the deter-mination of an optimal functional form that represents a large number of datapoints. Statistical analysis of the results of the least-squares fits to determine athermodynamic equation of state is useful as a guide. Such analysis should beconfirmed by the calculation of properties from the formulation for directcomparison to experimental data otherwise not included in the regressionanalyses.

12.3 Fitting an Equation of State to Experimental

Data

Fitting an equation of state to experimental data basically involves severalsteps. The first step is to assign an experimental uncertainty sj to each datapoint j to which the equation of state is fitted. This step requires analysis of theavailable data. Because information on the uncertainty of published data is notalways reliable and often incomplete, assigning a suitable uncertainty alsorequires information on the uncertainty of the measured independent variables,such as temperature T and pressure p. The correlator has to assess the availabledata set by detailed comparisons with measurements of properties obtainedfrom methods that suffer from quite different sources of systematic error. Often


experimental data for different properties have to be considered simulta-neously; data are included and excluded from the data set and the impact on therepresentation of other properties is evaluated. This technique of assigninguncertainties to the available data, which is used to weight the data in ananalysis, is inevitably an iterative procedure and is usually one of the most timeconsuming steps in developing an equation of state.If the uncertainty of a data point is known, a weighted residual can be cal-

culated. If the experimental data point is, for example, a pressure measured as afunction of temperature and density [a typical p(r, T) data point] the relativeresidual zj becomes

Bj ¼pj calc; T measð Þ;r measð Þ; nf g � pj meas; T measð Þ;r measð Þf g

sp;j: ð12:1Þ

The parameter zj for the point j is unity when the deviation between the value,in this case pressure, calculated from the equation of state pj{calc, T (meas), r(meas), n} and the experimental value pj{meas, T (meas), r (meas)} is equal tothe experimental uncertainty sp,j. Typically, for a reference equation of state,the zj should be less than unity for almost all data (typically495 % of thepoints if s is considered to be equal to two times the standard deviation asappropriate for an expanded uncertainty at a confidence interval of 0.95). In eq12.1 the calculated pressure depends on the parameter vector n, thus on thecoefficients of the equation of state that are fitted. In practice, the dimensionlesscompression factor Z is commonly used instead of the pressure. Thus, theresidual becomes

Bj ¼Zj calc;T measð Þ;r measð Þmeas; n� �

� Zj meas;T measð Þ;r measð Þf gsZ;j

¼ pj calc;T measð Þ; r measð Þ; nf g � pj meas;T measð Þ;r measð Þf gT measð Þr measð ÞRð ÞsZ;j

:

ð12:2Þ

Fitting the equation of state means that the parameter vector n is determined insuch a way that the sum of squares w2 over all M considered data pointsbecomes a minimum, for example, for M data points the sum of squares for thecompression factor is

w2 nð Þ ¼XMj¼1

Bj nð Þ2

¼XMj¼1

Zj calc;T measð Þ;r measð Þ; nf g � Zj meas;T measð Þ;r measð Þf gsZ;j

� �2

:

ð12:3Þ

In multi-property fits, which are common, the sum of squares containsdimensionless residuals of experimental data for different thermodynamic

398 Chapter 12

properties. As long as the residuals are reduced properly with the experimentaluncertainty of the respective data point, there is no limit to the number ofdifferent properties that can be considered in the sum of squares. The mathe-matical steps required to do so are described in detail by Span.3 This processrequires derivatives of the residua with respect to each adjustable parameter ni.The combination of dependent and independent variables commonly used

for multi-parameter equations of state today is an empirical description of thereduced Helmholtz energy Am as a function of the reduced density d and theinverse reduced temperature t,

Amðr;TÞRT

¼ aðd; tÞ ¼ apgðd; tÞ þ arðd; tÞ; ð12:4Þ

with t¼Tr/T and d¼ r/rr; the critical temperature and critical density of thefluid are commonly used as reducing parameters, Tr and rr. The ideal part ofthe reduced Helmholtz energy, apg(d,t), can be derived from a two-fold inte-gration of a function describing the ideal gas heat capacity, which is a functionof temperature only. The constants introduced by this two-fold integration arecommonly used to adjust the reference states of enthalpy and entropy.The residual part of the reduced Helmholtz energy, ar(d,t), has to be

described empirically. The simplest form of an empirical equation for thereduced Helmholtz energy, which uses a simple polynomial series with I termsfor the residual part, becomes

a d; tð Þ ¼ apg d; tð Þ þXIi¼1

nidditti : ð12:5Þ

Because the Helmholtz energy is a function of temperature and density that isone form of the axiom of the second law of thermodynamics at constantamount of substance, all thermodynamic properties can be calculated bycombinations of itself and its derivatives (see also Sec. 12.5.7). For example, thecompressibility factor becomes

Z rn;Tð Þ ¼ rnRT

@Am

@rn

� �T

¼ rnRT

@Apgm

@rn

� �T

þ @Arm

@rn

� �T

�

¼1þ d@ar

@d

� �t¼ 1þ

XIi¼1

nididditti ;

ð12:6Þ

when expressed in molar quantities. From eq 12.6, the weighted residual of thecompression factor becomes

Bj ¼1þ

PIi¼1

nididdij t

tij � Zj meas;T measð Þ;r measð Þf g

sZ;j: ð12:7Þ


The derivative of the residual with respect to the ith coefficient ni, which isrequired to fit the parameter vector n, becomes

@Bj@ni

� �t;d;nk6¼i

¼did

dij t

tij

sz;j: ð12:8Þ

If di and ti are considered given parameters of the functional form of theequation of state, this relation contains neither ni itself nor any other adjustableparameters. Because the derivative of the residual can be calculated straight-forwardly, a comparably simple linear fitting algorithm can be used. Thecoefficients ni are linearly adjustable parameters, and experimental data for thecompression factor Z(T, rn) are thus often called linear data.The situation becomes different if di and ti are considered adjustable para-

meters. In this case, starting values for di and ti are required to initially calculatethe derivative of the residual with respect to ni. These starting values have to beupdated continuously throughout the fitting process, which becomes an itera-tive process requiring a ‘‘non-linear fitting algorithm’’. Thus, di and ti arenonlinearly adjustable parameters.In a similar way, data can be non-linear if they are measured, for example, as

a function of temperature and pressure instead of temperature and density or iftheir relation to the reduced Helmholtz energy results in residual that containsnonlinear combinations of the derivatives of ar. In the first case, the densityr(expt) has to be replaced by a density r{calc, T(expt), p(expt), n}, which has tobe continuously updated throughout the fitting process. This is the case forexample for enthalpy Hm,

Hm T ; rn calcð Þ½ �RT

¼ 1þ t apgt þ art� �

þ dard; ð12:9Þ

which result in linear residua, but are usually experimentally determined as afunction of temperature and pressure. In the second case, the derivative of theresiduals with respect to ni contains other coefficients, which have to be updatedcontinuously. This is the case for experimental data for the isobaric heatcapacity Cp,m,

Cp;m T ; rn calcð Þ½ �R

¼ �t2 apgtt þ artt� �

þ1þ dard � dtardt� �21þ 2dard þ d2ardd

; ð12:10Þ

or for the speed of sound. So far we have given molar quantities Hm and Cp,m

but these can be clearly changed to specific or massic quantities. Due to the waythe derivatives of the Helmholtz energy, which are functions of t, d, and n, haveto be combined to calculate these properties, any derivative of the residual withrespect to a coefficient ni will contain all elements of the parameter vector n.Thus, starting values and iterative procedures are again required.

400 Chapter 12

Algorithms have been developed that make nonlinear data accessible forlinear fitting by linearization. However, these algorithms always result initerative strategies because values calculated from linearization [for example,densities r{calc, T(expt), p(expt), n}] have to be recalculated with the latestequation of state until no further improvement seems possible; only nonlinearalgorithms allow for a direct fit to nonlinear data. The theory required for linear,linearized, and nonlinear multi-property fits was established during the 1970’s,see Wagner,5,6 Bender,7 McCarty8 and Ahrendts and Baehr.9,10 Fitting proce-dures were widely applied in the 1980’s and resulted in state-of-the-art referenceequations of state, such as Schmidt and Wagner11 and Jacobsen et al.12

The residual part of a modern reference equation of state has the generalform

ar d; tð Þ ¼XIi¼1

nidditti þ

XIþJi¼Iþ1

nidditti exp �deið Þ

þXIþJþK

i¼IþJþ1nif

ci t; d; nci� �

;

ð12:11Þ

where the functions fc correspond to mathematically complex terms designed toimprove the description of properties in the critical region. Terms like this wereused for a number of fluids with an experimentally well-described critical regionby Setzmann and Wagner13 and Span and Wagner.14 The number of poly-nomial terms is given by I, the number of exponential terms is given by J, andthe number of critical region terms is given by K. In addition to the coefficientsni, this equation involves a large number of parameters such as the temperatureand density exponents, ti, di, and ei, and the parameters in the critical regionterms. These parameters are generally referred to as the ‘‘functional form’’ ofan equation of state. The functional form has been determined by trial anderror. In general, the resulting equations of state had more terms than necessaryfor a certain level of accuracy and showed strong inter-correlations betweendifferent terms. These inter-correlations frequently resulted in poor extra-polation behaviour and sometimes even in unreasonable behaviour of derivedproperties, such as heat capacities and speeds of sound.This problem was addressed by Wagner15, who introduced a modified

regression analysis that was used to optimize the functional form of vapourpressure equations. The same algorithm was later applied to equations of stateby de Reuck and Armstrong.16 A far more advanced optimization algorithmwas the one by Ewers and Wagner17 who adapted evolutionary principles tooptimize the functional form of an equation of state; see also Schmidt andWagner.11 Most optimization algorithms that are in use today are basedon the algorithm by Setzmann and Wagner,18 which combines evolutionaryand deterministic elements. The use of optimized functional forms led tohighly accurate equations of state with reduced inter-correlations between thedifferent terms and improved extrapolation behaviour.


Although the complexity of the various optimization programs is quite dif-ferent, they all have one basic feature in common: the optimization processstarts with the definition of a ‘‘bank of terms’’. The correlator selects parametercombinations (for example, values for ti, di, and ei in an exponential term) thatare in principle considered promising, resulting in a large number of terms. Forthese terms, which form the bank of terms, for the selected set of linear or lin-earized experimental data, the derivatives of the residual required for a linear fitare calculated and stored in a matrix. From the bank of terms, the optimizationalgorithm selects the combination of terms, which for a given number of selectedterms results in the lowest sum of squares. The sum of squares is calculated withthe information stored in the matrix, mentioned above, by algorithms that areequivalent to a linear fit. The number of terms selected for the equation of statecan be optimized in parallel by application of statistical tests for the equation asa whole and for the least significant term in the equation of state.A reasonable bank of terms should be restricted to about 100 for the

regression analysis by Wagner,15 and can contain more then 1 000 for theoptimization algorithm by Setzmann and Wagner.18 This can result in moreadvanced optimization algorithms primarily by allowing a smaller step width inthe temperature exponents ti (for a proper expansion in the gas phase, thedensity related exponents di and ei must be integers) and in the parameters ofthe critical region. While older equations of state typically use integers for thetemperature exponents, the use of advanced optimization algorithms allowedtemperature exponents with step widths of 1

4, 18, or even 1

16, which is a significant

advantage when attempting to fit highly precise experimental measurementswith as few terms and as small inter-correlations between terms as possible.The principle disadvantage of the described optimization algorithms is that

they are restricted to linear and linearized data. As long as precise (p, r, T) dataare the dominant source of experimental information in homogeneous states,and precise ancillary equations are available for vapour pressure and saturatedliquid and saturated vapour densities to calculate linearized phase equilibriumdata,5,6 this does not result in a significant disadvantage. A subsequent linear fitof the coefficients ni is sufficient to make full use of the available linear data.However, in the 1990’s, when highly precise speed of sound data over broadtemperature and pressure ranges became an important source of experimentalinformation, linear optimization algorithms showed some limitations withregard to the representation of speeds of sound. To overcome these limitations,Tegeler et al.19 developed a ‘‘quasi nonlinear’’ optimization algorithm, wheredecisions on the functional form were based on nonlinear fits. Nevertheless, thisalgorithm still used a bank of terms set up with predefined parameter combi-nations and both linear and linearized experimental data; this approach doesnot fully overcome the limitations of linear algorithms.

12.3.1 Recent Nonlinear Fitting Methods

The next evolution in the development of equations of state was the jump fromthe quasi-nonlinear algorithm to fully-nonlinear methods. Nonlinear fitting has

402 Chapter 12

many advantages over linear fitting, such as the ability to directly fit any type ofexperimental data. Shock-wave measurements of the Hugoniot curve are anexample where nonlinear fitting can use (p, r, h) measurements, even when thetemperature for any given point is unknown. Another advantage in nonlinearfitting is the ability to use ‘‘greater than’’ or ‘‘less than’’ operators for con-trolling the extrapolation behaviour of properties such as heat capacities orpressures at low or high temperatures. Curves can be controlled by ensuringthat a calculated value along a constant property path is always greater (or less)than a previous value; thus only the shape is specified, not the magnitude. Thenonlinear fitter then determines the best magnitude for the properties based onother information in a specific region.A reasonable preliminary equation is required as a starting point for non-

linear fitting. The exponents for density and temperature, along with thecoefficients and exponents in critical region terms, are determined simulta-neously with the coefficients of the equation. In addition, the terms in the idealgas heat capacity equation and the reducing parameters (critical temperatureand density) of the equation of state can also be fitted. Thus, with an 18-termequation, there are at times up to 90 values being fitted simultaneously to derivethe equation.The nonlinear algorithm adjusts the parameters of the equation of state to

minimize the overall sum of squares of the deviations of calculated propertiesfrom the input data, where the residual sum of squares is represented as

s ¼X

WrnF2rnþX

WpF2p þ

XWCV ;mF

2CV ;mþ � � � ; ð12:12Þ

where W¼ 1/s2 is the weight assigned to each data point and F is thefunction used to minimize the deviations. The equation of state is fitted to(p, r, T) data with either deviations in pressure F(p)¼ {p(expt)� p(calc)}/p(expt) for vapour-phase and critical-region data, or deviations in density,F(r)¼ {r(expt)� r(calc)}/r(expt), for liquid phase data. Because the calculationof density requires an iterative solution that greatly extends calculation timeduring the fitting process, the nearly equivalent, non-iterative form

Fr ¼p exptð Þ � p calcð Þf g

r exptð Þ@r@p

� �T

; ð12:13Þ

is used instead, where p(calc) and the derivative of density with respect topressure are calculated at the r and T of the data point. Other experimentaldata are fitted in a like manner, for example, F(w)¼ {w(expt)�w(calc)}/w(expt)for the speed of sound. The equation of state is constrained to the criticalparameters by adding the values of the first and second derivatives of pressurewith respect to density at the critical point, multiplied by some arbitrary weight,to the sum of squares. In this manner, the calculated values of these derivativesare nearly zero at the selected critical point.


Other fitting techniques and criteria that can be used include proper handlingof the second and third virial coefficients, elimination of the curvature of lowtemperature isotherms in the vapour phase, control of the two-phase loops andthe number of false two-phase solutions, convergence of the extremely hightemperature isotherms to a single line, and proper control of the ideal curveswith, for example, the Joule inversion curve that will be discussed later. Thework of Lemmon and Jacobsen20 for pentafluoroethane and Lemmon et al.4

for propane describe the properties that can be added to the sum of squares sothat the equation of state meets the criteria.When a nonlinear fit of the temperature exponents ti and of the para-

meters in the critical region terms is used, the restriction of finite step widths ina bank of terms becomes obsolete. Thus, the functional form can be optimizedmore precisely than with linear optimization algorithms, provided the availableexperimental data are accurate enough to distinguish between very similarsolutions and a good starting solution is available for the functional form.Otherwise, purely deterministic fitting algorithms are likely to end up in localminima.

12.4 Pressure-Explicit Equations of State

Although modern empirical equations of state are usually formulated in termsof the reduced Helmholtz energy, the most common form used in the 20th

century for multi-parameter equations of state in technical applications was thepressure-explicit form. The discussion that follows highlights several practicalforms of multi-parameter pressure-explicit equations of state and indicatesapplications for each. The virial equation of state is not detailed in this work,but can be found in Chapter 3 of this book.

12.4.1 Cubic Equations

The simplest form of the pressure explicit equation of state (with independentvariables of density and temperature) are those based on a cubic expression ofthe fluid volume such as the Peng-Robinson and Soave-Redlich-Kwong andthese equations are discussed in Chapter 4. When temperature and pressureinputs are available, the cubic equation can be solved non-iteratively for den-sity. Thus, the calculation speed of cubic equations of state is rapid whencompared to other methods explained that are provided below, and the use ofthese equations is quite popular in many industrial applications. Unfortu-nately, the advantage of speed of calculation is offset by the disadvantage ofhigher uncertainties.

12.4.2 The Benedict-Webb-Rubin Equation of State

An equation of state that can be extended to reasonably high densities wasdeveloped by Benedict et al.21 in 1940. The Benedict-Webb-Rubin equation of

404 Chapter 12

state is given as

p ¼rRT þ B0RT � A0 �C0

T2

� �r2 þ bRT � að Þr3

þ aar6 þ cr3

T2

� �1þ gr2� �

exp �gr2� �

:

ð12:14Þ

This equation has eight empirical constants: A0, B0, C0, a, b, c, a and g. Valuesfor these constants were reported by Benedict et al.22 for 12 hydrocarbons. TheBenedict-Webb-Rubin equation was the first equation of state that used anexponential term in density to extend the capabilities of simple polynomialexpansions and is thus considered the ancestor of almost all modern equationsof state.In 1962, Strobridge23 extended the BWR equation of state using the form

p ¼rRT þ n1RT þ n2 þn3

Tþ n4

T2þ n5

T4

�r2 þ n6RT þ n7ð Þr3 þ n8Tr4

þ r3n9

T2þ n10

T3þ n11

T4

�exp �n16r2� �

þ r5n12

T2þ n13

T3þ n14

T4

�exp �n16r2� �

þ n15r6;

ð12:15Þ

with 16 adjustable parameters ni. In 1973, Starling24 published a further wellknown modification of the Benedict-Webb-Rubin equation with 11 adjustablecoefficients.The Benedict-Webb-Rubin equation and its simple modifications yield suf-

ficiently precise representation of the thermodynamic surface for typical tech-nical applications in the gas phase and at supercritical states with low andmedium densities. However, estimates of energy at liquid or liquid-like super-critical states may be in error by more than � 10 %. These simple equations areunable to provide precise estimates for what are called both scientific andadvanced technical applications.

12.4.3 The Bender Equation of State

Bender25 developed an equation of state with a modified Benedict-Webb-Rubinform given by:

p ¼ rT Rþ Brþ Cr2 þDr3 þ Er4 þ Fr5 þ GþHr2� �

r2 exp �a20r2� ��

;

ð12:16Þ

where the coefficients B, C, D, E, F, G, and H are each a polynomial in T thatrequire an additional 19 adjustable parameters for a total of 26 coefficients.Bender published coefficients for several cryogenic fluids including argon,nitrogen, oxygen, carbon dioxide and methane. The Bender equation was used


by Maurer and co-workers during the 1980’s to describe a broad variety oftechnically relevant substances. The works of Polt26,27 and Platzer andMaurer28 contain Bender equations for more than 50 substances.The Bender equation was one of the first modifications of the Benedict-Webb-

Rubin equation that was specifically intended to describe vapour-liquid phaseequilibria as well as energetic properties in the liquid phase with results that repre-sent the measured values with an uncertainty suitable for technical applications.

12.4.4 The Jacobsen-Stewart Equation of State

In 1973, Jacobsen and Stewart29 developed what is termed an advanced form ofa modified Benedict-Webb-Rubin equation of state that has been given theacronym mBWR and is given by

p ¼X9n¼1

anrn þ exp �gr2� �X15

n¼10anr2n�17; ð12:17Þ

where g¼ 1/r2c and each an is a function of temperature as follows:

a1 ¼ RTa2 ¼ b1T þ b2T

1=2 þ b3 þ b4T�1 þ b5T

�2 a9 ¼ b19T�2

a3 ¼ b6T þ b7 þ b8T�1 þ b9T

�2 a10 ¼ b20T�2 þ b21T

�3

a4 ¼ b10T þ b11 þ b12T�1 a11 ¼ b22T

�2 þ b23T�4

a5 ¼ b13 a12 ¼ b24T�2 þ b25T

�3

a6 ¼ b14T�1 þ b15T

�2 a13 ¼ b26T�2 þ b27T

�4

a7 ¼ b16T�1 a14 ¼ b28T

�2 þ b29T�3

a8 ¼ b17T�1 þ b18T

�2 a15 ¼ b30T�2 þ b31T

�3 þ b32T�4:

ð12:18Þ

In the original work of Jacobsen and Stewart, the functional form given byeqs 12.17 and 12.18 were used to describe the thermodynamics properties ofnitrogen and has been used subsequently by others including Younglove andMcLinden30 and Outcalt and McLinden.31 The mBWR form has been used forreference equations for the properties of a variety of fluids, including refrig-erants and cryogens.

12.4.5 Thermodynamic Properties from Pressure-Explicit

Equations of State

The values of entropy Sm, enthalpy Hm, internal energy Um, and heat capacity(Cp,m, CV,m, or Csat) at various state points are calculated with the pressureexplicit equation of state and an ancillary equation that represents the ideal gasheat capacity. For a pure fluid, the equations that represent the vapour pressureand melting curve are used to identify the temperatures of the phase changes fromliquid to vapour and solid to liquid, respectively. Properties are evaluated through

406 Chapter 12

the two-phase region by numerical or analytical integration to calculate propertiesin the liquid. This procedure is valid if the equation of state was developed withprocedures to include the conditions for two-phase equilibrium in the least-squares determination of the coefficients in the equation of state. The relations forthe calculation of thermodynamic properties are summarized below. Functionsfor the integrals and derivatives of the equation of state required to perform thesecalculations are based on the specific pressure explicit equation of state used.The entropy S of any thermodynamic state is calculated from

Sm T ;rnð Þ ¼S�Jm T�J� �þZTT�J

Cpgp;m Tð ÞT

� dT � R ln RTrnð Þ

þZrn0

R

rn� 1

r2n

� �@p

@T

� �rn

" #T

dr:

ð12:19Þ

An ancillary equation is used to evaluate the ideal gas specific heat, Cpgp,m. The

reference entropy of the ideal gas at T�J and p

�J is taken from a suitable sourcefor the fluid under investigation. The enthalpy of any state may be calculated from

Hm T ;rnð Þ ¼H�J T�J� �þ T

Zrn0

p

Tr2n

� �� 1

r2n

� �@p

@T

� �rn

" #T

drn

þ p� rnRTrn

� �þZTT0

Cpgp;m Tð ÞdT :

ð12:20Þ

It is convenient to replace the first integral term in eq 12.20 with

T

Zrn0

p

Tr2n

� �� 1

r2n

� �@p

@T

� �rn

" #T

drn ¼TZrn0

R

rn

� �� 1

r2n

� �@p

@T

� �rn

" #T

drn

þZrn0

p

r2n

� �� RT

rn

� �� T

drn:

ð12:21ÞThe resulting expression for enthalpy Hm is

Hm T ;rnð Þ ¼H�Jm T�J� �þ T

Zrn0

R

rn

� �� 1

r2n

� �@p

@T

� �rn

" #T

drn

þZrn0

p

r2n

� �� RT

rn

� �� T

drn þp

rn� RT

� �þZTT�J

Cpgp;m Tð Þ dT :

ð12:22Þ


The reference enthalpy of the ideal gas at T�J and p

�J is taken from a separatesource. The internal energy Um of a fluid state is given by

Um T ;rð Þ ¼ Hm T ; rð Þ � p

r: ð12:23Þ

The heat capacity at constant volume CV,m is determined from

CV ;m T ; rnð Þ ¼ Cpgp;m Tð Þ � R

h i�Zrn0

T

r2n

� �@2p

@T2

� �rn

" #drn: ð12:24Þ

The heat capacity at constant pressure Cp,m is obtained from the relationship

Cp;m T ;rnð Þ ¼ CV ;m T ;rnð Þ þ T

r2n

� �@p

@T

� �2

rn

@rn@p

� �T

" #: ð12:25Þ

12.5 Fundamental Equations

Fundamental equations contain calorimetric and reference state information sothat absolute values of specified properties may be calculated directly by dif-ferentiation without the need for integration. For simple pure-fluids, there arefour fundamental relations, given as

Um ¼ Um Sm;Vmð Þ; ð12:26Þ

Hm ¼ Hm Sm; pð Þ; ð12:27Þ

Am ¼ Am T ;Vmð Þ ð12:28Þ

and

Gm ¼ Gm T ; pð Þ: ð12:29Þ

The energy Um and enthalpy Hm are generally not used for correlation workbecause the independent variable entropy Sm is not measurable. The Gibbsenergy Gm can be used only to represent the liquid surface or the vapour sur-face, but not both, due to the discontinuity in slope in the Gibbs energy at thephase boundaries. Two independent formulations would be required for thevapour and liquid phases, and matching these formulations at supercriticalconditions would be difficult. Thus, the formulation for the Helmholtz energyAm with independent variables temperature T and volume Vm (or density rn),or their non-dimensional equivalents, d and t, is the only fundamental relationsuitable for the development of equations of state that describe the whole fluidregion.

408 Chapter 12

The following sections present descriptions of Helmholtz energy equations ofstate used to represent the thermodynamic properties of a number of fluids overwide ranges of temperature and pressure. Although the selection of examples isarbitrary, those given here are selected to illustrate both the theoretical basisand the empirical nature of each.

12.5.1 The Equation of Keenan, Keyes, Hill, and Moore

In 1940, Benedict, Webb, and Rubin21 published their formulation both interms of pressure and in the residual Helmholtz energy. However, the for-mulation in pressure was considered the original formulation, and the Helm-holtz energy formulation was rarely used because the ideal gas contribution wasmissing. The first equation of state that was formulated exclusively in terms ofthe Helmholtz energy a was published in 1969 by Keenan et al.32 for water andsteam. The equation is given in specific quantities by

a ¼ apgðTÞ þ RT ln rþ RTrQ r; tð Þ; ð12:30Þ

where t¼ 1000K/T. The function a0(T) represents the ideal gas behaviour ofthe fluid and is

apgðTÞ ¼X6i¼1

Ci

ti�1þ C7 lnT þ

C8 lnT

t: ð12:31Þ

The empirical function Q(r, t) of eq 12.30 describes the residual contribution tothe Helmholtz energy and is expressed as

Q r; tð Þ ¼ t� tcð ÞX7j¼1

t� taj� �j�2 X8

i¼1Aij r� raj� �i�1þe�ErX10

i¼9Aijri�9

" #:

ð12:32Þ

Today, this equation of state has fifty adjustable parameters, Aij, to representthe residual of the Helmholtz energy and in this respect is interesting only froma historical perspective.

12.5.2 The Equations of Haar, Gallagher, and Kell

In 1978, Haar and Gallagher33 developed an equation of state for ammonia,and in 1984 Haar, Gallagher and Kell34 published an equation of state forwater. Both equations became accepted as standards, and although they havebeen superseded by more recent formulations by Tillner-Roth et al.35 forammonia and Wagner and Pruß36 for water, they are still frequently used inindustrial applications.


The equation by Haar and Gallagher for ammonia is given in specificquantities by

a r;Tð Þ ¼ apg Tð Þ þ ar r;Tð Þ; ð12:33Þ

where apg(T) is the contribution of the ideal gas and ar(r,T) is the residualcontribution. The ideal gas contribution is given as

apgðTÞ ¼ RT a1 ln T=Kð Þ þX11i¼2

ai T=Kð Þi�3 � 1þ ln 4:818 T=Kð Þ½ �( )

: ð12:34Þ

The residual is given by

arðr;TÞ ¼ RT ln rð Þ þ rX9i¼1

X6j¼1

aijri�1ðt� tcÞj�1 !" #

: ð12:35Þ

The equation of state for water includes a so called base function ab as well asthe residual component and the ideal gas contribution,

a r;Tð Þ ¼ apg Tð Þ þ ab r;Tð Þ þ ar r;Tð Þ: ð12:36Þ

The ideal gas contribution to the Helmholtz energy is given by

apgðTÞ ¼ �RT 1þ C1

TRþ C2

� �lnTR þ

X18i¼3

CiTi�6R

" #: ð12:37Þ

The base function is used to represent the dilute-gas region and was derivedfrom the Ursell-Mayer virial theory,

ab r;Tð Þ ¼ RT � lnð1� yÞ � b� 1

1� yþ aþ bþ 1

2ð1� yÞ2þ 4y

B

b� g

� �� a� bþ 3

2þ ln

rRTp0

" #;

ð12:38Þ

where a, b, g and p0 are constants and both B and b are temperature-dependentmolecular parameters that contain in total eight adjustable coefficients. Theresidual part consists of 40 adjustable parameters and has the form:

ar r;Tð Þ ¼X36i¼1

ai

ki

T0

T

� �li

1� e�rð ÞkiþX40i¼37

aidlii exp �aid

kii � bit

2i

�: ð12:39Þ

410 Chapter 12

12.5.3 The Equation of Schmidt and Wagner

In the mid 1980’s, Schmidt and Wagner11 and Jacobsen et al.12 developedequations of state that can be regarded as the origin of most of the recent socalled reference equations of state. In this approach, optimization algorithmswere used for the first time to determine the functional form of equations ofstate. In the equation by Schmidt and Wagner, the reduced Helmholtz energy isgiven by:

Amðrn;TÞRT

¼ aðd; tÞ ¼ a0ðd; tÞ þ arðd; tÞ; ð12:40Þ

where t¼Tc/T, d¼ r/rc, Tc is the critical temperature, and rc is the criticaldensity. In general, the ideal gas contribution is given by

apgðd; tÞ ¼ Hpg;rm tRTc

� Spg;rm

R� 1þ ln

dt0d0t� tR

Ztt0

Cpgp;m

t2dtþ 1

R

Ztt0

Cpgp;m

tdt; ð12:41Þ

where the superscript r refers to an arbitrary reference condition and Cpgp,m is

obtained from statistical mechanics or low density heat capacity or speed ofsound measurements. The evolutionary optimization algorithm by Ewers andWagner17 was used to establish the functional form of the final Schmidt-Wagner equation based on the bank of terms created from the equation

arðd; tÞ ¼X9i¼1

X12j¼�1

aijditj=2 þ expð�d2Þ

X10i¼0

X17j¼4

aijditj=2 þ expð�d4Þ

X5i¼2

X23j¼10

aijditj

ð12:42Þ

containing 336 terms. The final optimized equation is

arðd; tÞ ¼X13i¼1

aidditti þ expð�d2Þ

X24i¼14

aidditti þ expð�d4Þ

X32i¼25

aidditti ; ð12:43Þ

with 32 adjustable parameters. This equation introduced exponential terms ofthe type exp(� d4) that resulted in substantial improvements with regard to therepresentation of properties in the critical region.

12.5.4 Reference Equations of Wagner

In combination with a suitable optimization algorithm, the functional formsintroduced in the banks of terms by Schmidt and Wagner11 and Jacobsen etal.12 are sufficient to match the uncertainty of experimental data of the bestobtainable uncertainty with an acceptable number of terms. More complexfunctional forms are required to describe precisely the properties in the


extended critical region, and an increased number of terms was necessary whenthe equation was required to better represent the measured values for certainsubstances. New Gaussian bell-shaped terms were added in the work performedat the Ruhr-University Bochum under the leadership of Wagner in the 1990’sand 2000’s. To do so, the functional form of ar was modified to give:

ar d; tð Þ ¼Xk1k¼1

nkddkttk þ

Xk2k¼k1þ1

nkddkttk exp �dlk

� �

þXk3

k¼k2þ1nkd

dkttk exp �Zk d� �kð Þ2�bk t� gkð Þ2h i

;

ð12:44Þ

which differs from the previous functions in the third term on the right handside of eq 12.44 where Z and b are adjustable parameters. These new terms in eq12.44 were used in the reference equations of state for methane by Setzmannand Wagner,13 for carbon dioxide as reported by Span and Wagner,14 for waterreported by Wagner and Pruß,36 for nitrogen by Span et al.,37 for argon in thework of Tegeler et al.,38 Smukala et al.39 for ethene, Bucker andWagner40,41 forethane, butane, and methylpropane, and Lemmon et al.4 for propane. Exampledeviations of experimental data from some of these equations are given inSection 12.6.

12.5.5 Technical Equations of Span and of Lemmon

There is also a substantial need for technical equations of state suited foradvanced technical applications, such as process calculations requiring ener-getic properties, where state-of-the-art measurements are not available for thesefluids and for which very low uncertainties are not required. Unlike equationsthat claim to represent precise measurements over a wide range of both tem-perature and pressure, which often requires 20 to 50 adjustable parameters todescribe densities with an uncertainty between (0.01 to 0.1) %, technicalequations have fewer terms (which are often fixed) and thus a well-establishedfunctional form to characterize the fluid properties. Span and Wagner42–44

developed two 12-term fundamental equations with fixed functional forms: onefor non-polar or slightly polar substances and one for polar fluids. Precise datasets were used to develop the functional forms and to assess the uncertainty ofthe resulting equations of state. Thus, Span and Wagner42–44 concentrated onrepresenting fluid properties for substances already described with highly-pre-cise equations of state. Equations with fewer parameters permit substantiallyreduced computation times by between (2 to 10) times over those required forequations with 50 parameters. This difference depends on the number of termsin the alternative equations and on the form the terms, particularly those in thecritical region.The technical equations of the type reported by Span and Wagner42–44 were

developed with the same constraints required of the reference equations to

412 Chapter 12

ensure proper behaviour. These so called technical equations can be extra-polated to lower temperatures (as demonstrated by the curvature of the isobaricand isochoric heat capacities and the speed of sound) and to higher tempera-tures (as demonstrated by the ideal curves). In addition, the number of terms inthe equations was limited to 12 and this also decreased the correlation betweenterms and the possibility of over-fitting. Although the smaller number of termsdecreased the flexibility of the equation and thus its ability to represent pre-cisely the properties of a fluid, the fixed and rigid form has the benefit that it ismore suited for substances for which there are limited experimental data andthe form of the equations permits interpolation to temperatures, pressures, anddensities for which there are no measured values in a manner that is perhapsmore acceptable than the other formulations discussed so far. This has been akey feature in the ability to represent the properties of some of the fluids thatare listed in Table 12.2. The objective of the work of Span and Wagner42–44 inthe development of technical equations was to provide the best functional form(or forms) that represented the data for all chosen substances that were fittedsimultaneously with the procedure described by Span et al.45; the determinationof a functional form that best fits the data was not in this case the primaryobjective. Two forms of the equation were ultimately developed, one thatrepresented polar and the other non-polar fluids, and for each coefficients couldbe fitted for each category of fluid. In the work of Wagner and Span,42–44 theform of the equation chosen for non-polar fluids was used to describe theproperties of the alkanes from methane to octane as well as argon, oxygen,nitrogen, 2-methylpropane, cyclohexane, and sulfur hexafluoride. Ethene wasalso included in this scheme to ensure that the non-polar form was also capableof representing weakly polar fluids. The data sets for nitrogen and pentane werenot included in the development of the functional form but were used to test thetransferability of the scheme to other fluids. The polar form was used for therefrigerants trichlorofluoromethane, dichlorodifluoromethane, chlorodi-fluoromethane, difluoromethane, 1,1,2-trichloro-1,2,2-trifluoroethane, 2,2-dichloro-1,1,1-trifluoroethane, pentafluoroethane, 1,1,1,2-tetrafluoroethane,1,1,1-trifluoroethane, and 1,1-difluoroethane along with the fluids ethene,carbon dioxide, and ammonia. The data sets for trichlorofluoromethane and1,1,2-trichloro-1,2,2-trifluoroethane were not included in the development ofthe functional form but were again used to test the transferability to other polarfluids. The new form was able to represent the properties of the associating fluidammonia. However, attempts to describe water as well with these equationswere unsuccessful, one plausible cause for this arises from hydrogen bonding.In general, the polar form is not considered suitable for associating fluids.The work of Lemmon and Span46 continued that of Span andWagner42–44 to

substantially increase the number of fluids described by technical equations ofstate for industrial applications and to show that the coefficients could be fittedfor most fluids with these generalized forms. For several fluids, the formula-tions were the first to describe the thermodynamic properties with multi-parameter equations. For other fluids, this new formulation replaced equationspresented by Polt26 with the modified BWR form as well as extended


Table

12.2

Wide-rangethermodynamic

property

form

ulationsforscientificandengineeringapplicationsincludingthemini-

mum

Tminandmaxim

um

Tmaxtemperaturesofvalidityalongwiththemaxim

um

pressure

pmax.

Fluid

Authors

YearPublished

Tmin/K

toTmax/K

apmax/M

Pa

Gradeb

air(asapseudo-pure

fluid)

Lem

monet

al.56

2000

59.75to

2000

2000

Bacetone

Lem

monandSpan46

2006

178.5

to550

700

Cammonia

Tillner-R

oth

etal.35

1993

195.495to

700

1000

Bargon

Tegeler

etal.38

1999

83.8058to

2000

1000

Abenzene

Poltet

al.27

1992

278.7

to635

78

Cbutane

Bucker

andWagner

41

2006

134.895to

575

69

Bbutene

Lem

monandIhmels5

72005

87.8

to525

70

Bcarbondioxide

SpanandWagner

14

1996

216.592to

2000

800

Acarbonmonoxide

Lem

monandSpan46

2006

68.16to

500

100

Ccarbonylsulfide

Lem

monandSpan46

2006

134.3

to650

50

Ccis-butene

Lem

monandIhmels5

72005

134.3

to525

50

Ccyclohexane

Penoncelloet

al.58

1995

279.47to

700

80

Ccyclopropane

Poltet

al.27

1992

273to

473

28

Cdecane

Lem

monandSpan46

2006

243.5

to675

800

Cdeuterium

McC

arty59

1989

18.71to

423

320

Ddim

ethylether

IhmelsandLem

mon60

2007

131.65to

525

40

Cdodecane

Lem

monandHuber

61

2004

263.6

to700

700

Cethane

Bucker

andWagner

40

2006

90.368to

675

900

Aethanol

DillonandPenoncello62

2004

250to

650

280

Cethylene

Smukala

etal.39

2000

103.986to

450

300

Afluorine

deReuck

63

1990

53.4811to

300

20

CD

2O

Hillet

al.64

1982

276.97to

800

100

Chelium

McC

artyandArp

65

1990

2.1768to

1500

100

Cheptane

SpanandWagner

43

2003

182.55to

600

100

Chexane

SpanandWagner

43

2003

177.83to

600

100

Chydrogen

(norm

al)

Leachmanet

al.66

2009

13.957to

1000

2000

Bhydrogen

sulfide

Lem

monandSpan46

2006

187.7

to760

170

Cmethylpropane

Bucker

andWagner

41

2006

113.73to

575

35

B

414 Chapter 12

2-m

ethylprop-1-ene

Lem

monandIhmels5

72005

132.4

to550

50

C2-m

ethylpentane

Lem

monandSpan46

2006

119.6

to550

1000

C2-m

ethylbutane

Lem

monandSpan46

2006

112.65to

500

1000

Ckrypton

Lem

monandSpan46

2006

115.775to

750

200

Cmethane

Setzm

annandWagner

13

1991

90.6941to

625

1000

Amethanol

deReuck

andCraven

67

1993

175.61to

620

800

Cneon

Kattiet

al.68

1986

24.556to

700

700

C2,2-dim

ethylpropane

Lem

monandSpan46

2006

256.6

to550

200

Cnitrogen

Spanet

al.37

2000

63.151to

2000

2200

Anitrogen

trifluoride

Younglove6

91982

85to

500

50

Cnitrousoxide

Lem

monandSpan46

2006

182.33to

525

50

Cnonane

Lem

monandSpan46

2006

219.7

to600

800

Coctane

SpanandWagner

43

2003

216.37to

600

100

Coxygen

SchmidtandWagner

11

1985

54.361to

2000

82

Bparahydrogen

Leachmanet

al.66

2009

13.8033to

1000

2000

Bpentane

SpanandWagner

43

2003

143.47to

600

100

C1,1,1,2,2,3,3,4,4,4-decafluorobutane

ECSd

189to

500

30

C1,1,1,2,2,3,3,4,4,5,5,5-

dodecafluoropentane

ECSd

148.363to

500

30

C

propane

Lem

monet

al.4

2009

85.525to

625

1000

Apropene

Overhoff70

2006

87.953to

575

1000

Bprop-1-yne

Poltet

al.27

1992

273to

474

32

Ctrichlorofluoromethane(R

-11)

Jacobsenet

al.71

1992

162.68to

625

30

C1,1,2-trichloro-1,2,2-trifluoroethane

(R-113)

Marx

etal.72

1992

236.93to

525

200

C

1,2-dichloro-1,1,2,2-tetra-

fluoroethane(R

-114)

Platzer

etal.73

1990

273.15to

507

21

C

1-chloro-1,1,2,2,2-pentafluoroethane

(R-115)

Lem

monc

173.75to

550

60

C

1,1,1,2,2,2-hexafluoroethane(R

-116)

Lem

monandSpan46

2006

173.1

to425

50

Cdichlorodifluoromethane(R

-12)

Marx

etal.72

1992

116.099to

525

200

C2,2-dichloro-1,1,1-trifluoroethane

(R-123)

Youngloveand

McL

inden

30

1994

166to

600

40

C


Table

12.2

(continued

)

Fluid

Authors

YearPublished

Tmin/K

toTmax/K

apmax/M

Pa

Gradeb

1-chloro-1,2,2,2-tetrafluoroethane

(R-124)

deVries

etal.74

1995

120to

470

40

C

1,1,1,2,2-pentafluoroethane(R

-125)

Lem

monandJacobsen20

2005

172.52to

500

60

Bchlorotrifluoromethane(R

-13)

Magee

etal.75

2000

92to

403

35

C1,1,1,2-tetrafluoroethane(R

-134a)

Tillner-R

oth

&Baehr7

61994

169.85to

455

70

Btrifluoroiodomethane(R

-13I1)

Lem

monc

120.to

420

20

Ctetrafluoromethane(R

-14)

Platzer

etal.73

1990

120to

623

51

C1,1-dichloro-1-fluoroethane(R

-141b)

Lem

monandSpan46

2006

169.68to

500

400

C1-chloro-1,1-difluoroethane(R

-142b)

Lem

monandSpan46

2006

142.72to

470

60

C1,1,1-trifluoroethane(R

-143a)

Lem

monandJacobsen77

2000

161.34to

650

100

B1,1-difluoroethane(R

-152a)

OutcaltandMcL

inden

31

1996

154.56to

500

60

Cdichlorofluoromethane(R

-21)

Platzer

etal.73

1990

200to

473

138

C1,1,1,2,2,3,3,3-octafluoropropane(R

-218)

Lem

monandSpan46

2006

125.45to

440

20

C

chlorodifluoromethane(R

-22)

Kamei

etal.78

1995

115.73to

550

60

C1,1,1,2,3,3,3-heptafluoropropane(R

-227ea)

Lem

monc

146.35to

475

60

C

trifluoromethane(R

-23)

Penoncelloet

al.79

2003

118.02to

475

120

C1,1,1,2,3,3-hexafluoropropane(R

-236ea)

ECSd

242to

500

60

D

1,1,1,3,3,3-hexafluoropropane(R

-236fa)

OutcaltandMcL

inden

80

1995

179.52to

500

40

C

ECSd

200to

500

60

D

416 Chapter 12

1,1,2,2,3-pentafluoropropane(R

-245ca)

1,1,1,3,3-pentafluoropropane(R

-245fa)

Lem

monandSpan46

2006

171.05to

440

200

C

difluoromethane(R

-32)

Tillner-R

oth

&Yokozeki81

1997

136.340to

435

70

B1,1,1,3,3-pentafluorobutane(R

-365mfc)

Lem

monc

239to

500

35

C

fluoromethane(R

-41)

Lem

monandSpan46

2006

129.82to

425

70

C1,1,2,2,3,3,4,4-octafluorocyclobutane

(R-C

318)

Platzer

etal.73

1990

233.35to

623

60

C

sulfurdioxide

Lem

monandSpan46

2006

197.7

to525

35

Csulfurhexafluoride

Guder

andWagner

82

2009

223.555to

625

150

Amethylbenzene

Lem

monandSpan46

2006

178to

700

500

Btrans-butene

Lem

monandIhmels5

72005

167.6

to525

50

Cwater

Wagner

andPruß36

2002

273.16to

2000

1000

Axenon

Lem

monandSpan46

2006

161.405to

750

700

C

aThetemperature

scale

forequationspublished

in1990orbefore

are

basedonIPTS-68.

bGrade:A

isassigned

toareference

equationofstate,Bto

amoderately

precise

equationofstate,Cto

aso

called

technicalequationofstate,andD

andbelowan

equationofstate

thathaseither

abadfunctionalform

orinsignificantdata.

cUnpublished

equation,coeffi

cients

are

given

inREFPROPbyLem

monet

al.83

dUsesextended

correspondingstatesasoutlined

inHuber

andEly

47.


corresponding states such as presented by Huber and Ely.47 Many of the BWRand corresponding states equations replaced had deficiencies providingunphysical behaviour, for example, negative heat capacities at low tempera-tures and in some instances physically impossible behaviour in accessible single-phase regions of the fluid surface. Additional details as well as figures are givenby Lemmon and Jacobsen.20 The unphysical behaviour can be particularlyproblematic for models used for mixtures that are based on the equation ofstate for a pure fluid. The 20 equations of state presented in the work ofLemmon and Span46 are listed in Section 12.7.

12.5.6 Recent Equations of State

Recently, there have been a number of advances in the methods used to cor-relate experimental measurements for fluids with an equation of state. Thepublication of Lemmon and Jacobsen20 for 1,1,1,2,2-pentafluoroethane givesdetails of these techniques that reduced the number of terms, while maintaininga precise representation of the measurements, and reduced the magnitude of thetemperature exponents. The temperature exponent t could be equal to 50 andconsequently would produce extremely large values of pressure within the two-phase region. Because of the large curvature of isotherms near the triple point,values in the single phase vapour region do not match those obtained with thetheoretically correct virial equation. Other matters addressed by Lemmon andJacobsen20 were the elimination of false solutions for the saturation conditions,and proper extrapolation at extremely high temperatures, pressures and den-sities; overall the work of Lemmon and Jacobsen20 simplified the terms requiredto represent fluid properties. Although the data for 1,1,1,2,2-pentafluoroethaneare quite extensive and comprehensive, the uncertainties in the data are notquite low enough to label this formulation as a reference equation of state.The work of Lemmon et al.4 presents a new reference formulation for the

properties of propane. This new equation has features similar to those of thefunctional form used to represent the properties of 1,1,1,2,2-pentafluoroethane,and in addition represents both measurements of the speed of sound to within0.03 % as well as recent measurements of the density to within 0.02 %, similarto that of other high quality reference equations. This equation also extra-polates to temperatures as low as about 40K, resulting in a reduced functionalform that is valid at reduced temperatures as low as 0.1 �Tc, making this fluid aperfect choice for use in corresponding states because no other equationextends this low.Upon completion of the equation of state for propane4 (in 2007), work began

on an equation for propene that resulted in improved constraints and methodsto control the derivatives of the equation of state and that permitted extra-polation of the equation to temperatures that tend toward 0.1K without anyadverse behaviour. In order to control the derivatives of the equation of state,new code was written so that the correlator could specify what property was tobe controlled and how it should behave. For example, the first input to the

418 Chapter 12

fitting routines could specify that the shape of the isochoric heat capacity wouldbe modified, the second input would specify that CV,m should be calculatedalong the saturated liquid line, and the third input would specify that thederivatives of CV,m with respect to the temperature along the saturation lineshould all be positive. The derivatives include the first, second, third, and fourthwith respect to temperature. Lemmon et al.4 report the resulting shape of CV,m

along the saturation line has no apparent abnormalities. Other propertiescalculated from the equation of state can be controlled by forcing the first andthird derivates, for example of the virial coefficients, to be positive and thesecond and fourth derivatives to be negative.Current work is focused on the expansion of this concept with additional

constraints for various properties with the intent of permitting equations to befitted when the experimental data are extremely limited. This strategy impliesthe resulting equation of state will probably have the correct behaviour eventhough there might be only a few vapour pressures and saturated liquid den-sities for the regression. When additional measurements are made the resultscan be compared to values obtained from the equation of state to confirm ordeny the predicted values. But as is the case with so many fluids where corre-lations have not been developed, these techniques provide methods to obtainequations of state for the vast number of fluids where equations were hithertounavailable.

12.5.7 Thermodynamic Properties from Helmholtz Energy

Equations of State

The following equations give the relations among the common thermodynamicproperties including those frequently measured for fluids of engineeringimportance. These relations are derived from the fundamental equation.Properties can be expressed in either molar or mass units depending on thevalue used for the gas constant R.

p ¼ r2n@Am

@rn

� �T

¼ rnRT 1þ d@ar

@d

� �t

� ; ð12:45Þ

Um

RT¼ Am þ TSm

RT¼ t

@apg

@t

� �dþ @ar

@t

� �d

� ; ð12:46Þ

Sm

R¼ � 1

R

@Am

@T

� �rn

¼ t@apg

@t

� �dþ @ar

@t

� �d

� � apg � ar; ð12:47Þ

Hm

RT¼ Um þ pVm

RT¼ t

@apg

@t

� �dþ @ar

@t

� �d

� þ d

@ar

@d

� �tþ1; ð12:48Þ

Gm

RT¼ Hm � TSm

RT¼ 1þ apg þ ar þ d

@ar

@d

� �t; ð12:49Þ


CV ;m

R¼ 1

R

@Um

@T

� �rn

¼ �t2 @2apg

@t2

� �dþ @2ar

@t2

� �d

� ; ð12:50Þ

Cp;m

R¼ 1

R

@Hm

@T

� �p

¼ CV ;m

Rþ

1þ d@ar

@d

� �t�dt @2ar

@d@t

� �� 2

1þ 2d@ar

@d

� �tþd2 @2ar

@d2

� �t

� ; ð12:51Þ

andw2M

RT¼ M

RT

@p

@r

� �s

¼ 1þ 2d@ar

@d

� �tþd2 @2ar

@d2

� �t

�1þ d

@ar

@d

� �t�dt @2ar

@d@t

� �� 2

t2@2apg

@t2

� �dþ @2ar

@t2

� �d

� :

ð12:52Þ

In eq 12.52 r is the mass density. Equations 12.45 through 12.52 are used forthe calculation of thermodynamic properties. Other properties can also becalculated by differentiation of the fundamental equation.Problems that arise from calculating properties with fundamental equations

usually result from the choice of independent variables. Any combination of vari-ables that are different from T and rn, for example, (Tsat, T and p, Hm and Sm)requires one or two-dimensional iterations to determine the corresponding values oftemperature and density. Details for algorithms suitable to achieve the transfor-mation have been given by Span;3 in general, the recommendation is to use avail-able programs and so avoid the inherent problems with these iterative procedures.

12.6 Comparisons of Property Formulations

Regardless of whether the property formulation for a particular fluid is explicitin pressure, Helmholtz energy, Gibbs energy, or another property, the usermust be given an assessment of the uncertainty of the predicted properties sothat the equation can be considered practical. The quality of a thermodynamicproperty formulation is best determined by its ability to model the physicalbehaviour to represent the measured properties of the fluid. Statistics anddeviation plots are used to show how thermodynamic properties calculatedfrom equations of state compare to experimental data.Group statistics typically used are based on the per cent deviation of a

particular property X given by:

DX ¼ 100 � X exptð Þ � X calcð ÞX exptð Þ

� : ð12:53Þ

For X�r Figure 12.1 shows for nitrogen selected experimental data50–53 asdeviations from the values calculated with the equation of state of Span et al.37

420 Chapter 12

Another quantitative measure of the ability of an equation of state to representexperimental data for a thermodynamic property is obtained from the averageabsolute relative deviation (AARD) that is given by:

AARD ¼ 1

n

Xni¼1

DXij j; ð12:54Þ

where n is the number of data points being considered. A relatively high valueof the AARD indicates either a systematic or a large random difference betweenthe data and the equation of state. The approach is shown in Figure 12.2,48,49

which compares the experimental speed of sound data for propane with valuesobtained from the equation of state of Lemmon et al.4 These graphical com-parisons illustrate the capability of the equation of state to predict the mea-surements of a property.Constant-property lines obtained from the formulation are used to qualita-

tively examine the equation’s of state behaviour and are particularly useful forthe critical region or the liquid region at low reduced temperatures that aredifficult to represent. Details of this procedure have been given by Span andWagner54 and Lemmon and Jacobsen20 and are shown in Figures 12.3 and 12.4for the propane equation of state reported by Lemmon et al.4 In general, the

100.{�

(expt)-�(calc)}/

�(expt)

Figure 12.1 Fractional deviations Dr¼ r(expt)� r(calc) of the measured densityr(expt) for nitrogen50–53 from the values estimated with the equation ofstate reported by Span et al.37


calculation of derived properties such as heat capacities at constant volume andconstant pressure and speed of sound is a far more sensitive test of the equationof state than properties such as pressure and enthalpy.The extrapolation behaviour of empirical multi-parameter equations of state

has been summarized by Span and Wagner.54 Aside from the representation ofshock tube data for the Hugoniot curve at very high temperatures and pres-sures, an assessment of the extrapolation behaviour of an equation of state canalso be based on the so called ‘‘ideal curves’’ that were first discussed byBrown.55 While reference equations of state generally result in reasonableestimates for the Boyle, ideal, and Joule-Thomson inversion curves, the pre-diction of reasonable Joule inversion curves is still a challenge. Equations mayresult in unreasonable estimates of Boyle, ideal and Joule-Thomson plotsespecially when the equations are based on limited experimental data.Most reference equations of state provide reasonable estimates when extra-

polated out of the range of the experimental data and in some case the extra-polation can extend to the limits of chemical stability of the substance.However, in general, multi-parameter equations of state should not be extra-polated beyond the range of validity cited. When extrapolation of the equationis necessary for the intended application, the reliability of the results might bereported by the authors.

Figure 12.2 Fractional deviations Dw¼w(expt)�w(calc) of the measured liquid phasespeed of sound w(expt) for propane48,49 from values estimated with theequation of state of Lemmon et al.4

422 Chapter 12

12.7 Recommended Multi-Parameter Equations of

State

Table 12.2 lists sources of multi-parameter equations of state for a set of substancesthat are considered by the authors as providing the most precise representation ofthe thermodynamic properties that can be used for system design and analysis, aswell as in scientific applications. However, the uncertainty of an equation for asubstance depends on the available measurements as well as the method used tocorrelate them and the precision of the equations listed in Table 12.2 could varysignificantly. The assessment of the suitability of an equation for an applicationmight be reported by the authors of the particular formulation.The equations of state listed in Table 12.2 have been classed into four groups

based on the uncertainty of properties predicted and the capability of thefunctional form to provide estimates when extrapolated outside the range forwhich there is experimental data. Equations with the lowest uncertainty thatalso exhibit exceptional extrapolation capabilities are labelled as referenceequations of state and assigned Grade A. These equations typically have

CV,m/J. K

-1. m

ol-1

Figure 12.3 Isochoric heat capacities CV,m for propane as a function of temperatureT obtained from the equation of state of Lemmon et al.4 Isobars areshown at pressures of (0, 1, 2, 3, 4, 5, 6, 8, 10, 20, 50, 100, 200, 500, 1000and 2000)MPa. The melting line is shown intersecting the liquid phaseisotherms. State points below the melting line are extrapolations of theliquid phase to very low temperatures.


uncertainties in density less than 0.03% and are suitable to predict properties tocalibrate instruments used for scientific research. Equations that were devel-oped with densities of uncertainty of 0.1 %, vapour pressures to 0.1 %, speed ofsound to 0.5 %, and heat capacities to 1 % are given a grade of B and areconsidered to be equations of state with moderate uncertainty. Those equationsgiven a grade C are referred to as technical equations of state and were oftendeveloped with either limited experimental data sets or with measurements ofhigher uncertainties than grade B. Typically, equations with grade C should notbe used for calibration of a scientific apparatus or scientific research. Equationsfor which either the physical behaviour shows discrepancies or the parameterswere determined by regression to inferior measurements are given a Grade Dclassification and are those than need to eventually be replaced.

12.8 Equations of State for Mixtures

There are many practical models for calculating properties of mixtures of twoor more fluids. A mixture equation of state should provide an accurate

Figure 12.4 Sound speeds w for propane as a function of temperature T obtainedfrom the equation of state of Lemmon et al.4 Isobars are shown atpressures of (0, 1, 2, 3, 4, 5, 6, 8, 10, 20, 50, 100, 200, 500, 1000 and2000)MPa. The melting line is shown intersecting the liquid phase iso-therms. State points below the melting line are extrapolations of theliquid phase to very low temperatures.

424 Chapter 12

representation of the thermodynamic properties of the mixture over a widerange of compositions, including liquid and vapour properties. Virial equationsof state derivable from statistical mechanics can be used to express the devia-tions from the perfect gas equation as a power series in density or volume. Thetwo methods listed below describe the current state of the art in mixturemodels.

12.8.1 Extended Corresponding States Methods

Huber and Ely47 expanded the extended corresponding states models reportedby Leach84 and Ely85 to predict the thermodynamic properties of mixturesassuming the mixture behaves as a hypothetical equivalent pure substance. Todetermine mixture properties, the states or properties of the mixture, identifiedby the subscript x, and those of a reference fluid, designated by subscript ref,must be in correspondence:

arx dx; txð Þ ¼ arref dref ; trefð Þ ¼ arref dxjx; txyxð Þ ð12:55Þ

Zx dx; txð Þ ¼ Zref dref ; trefð Þ ¼ Zref dxjx; txyxð Þ ð12:56Þ

Corresponding states are found with shape factors, fx and yx, that relate thereduced properties of the mixture to those of the reference fluid. The thermo-dynamic properties for the mixture are reduced by the pseudo-critical para-meters rc,x and Tc,x defined by

rc;x ¼Xp

Xq

xpxq

rc;p;q

!" #�1ð12:57Þ

and

Tc;x ¼

Pp

Pq

xpxqTc;p;q=rc;p;qPp

Pq

xpxq=rc;p;q; ð12:58Þ

where the terms rc,p,q and Tc,p,q are reduced temperatures and densities that aredefined in the work of Huber and Ely.47 Exact shape factors are determined bysimultaneous solution of the equations for the Helmholtz energy and com-pressibility factor with equations for either methane, nitrogen, or 1,1,1,2-tet-rafluoroethane (R-134a) used as the reference fluid; other equations could beused for the reference fluid provided the saturation lines are relatively long andthe equations of state are considered precise representations of extensivemeasurements.


12.8.2 Mixture Properties from Helmholtz Energy Equations of

State

Lemmon,86 Lemmon and Jacobsen,87 Tillner-Roth et al.88 and Lemmon andTillner-Roth89 have developed generalized mixture models that are based on theequations of state for one of the pure fluids in the mixture and an excess functionto account for the interaction between different species. The work of Lemmonand Jacobsen90 documents the equations currently in use for mixtures ofdifluoromethane (R-32), pentafluoroethane (R-125), 1,1,1,2-tetrafluoroethane(R-134a), 1,1,1-trifluoroethane (R-143a) and 1,1-difluoroethane (R-152a). Kunzet al.91 expanded the methods previously reported by Lemmon86,87,89 includingadditional coefficients for both the reducing parameters and the equation ofstate as well as providing revised coefficients for mixtures including methanethrough butane or with nitrogen and carbon dioxide, and new coefficients formixtures with alkanes of higher molar mass and with the intent of preciselyrepresenting the thermodynamic properties of natural gas systems with up to 21components including dilutants such as hydrogen, helium, and gaseous water.The equation can also be used to calculate the properties of moist air.The Helmholtz energy for mixtures of fluids can be calculated with the

equation

Am ¼ Aidm þ AE

m; ð12:59Þ

where the Helmholtz energy for the ideal mixture is

Aidm ¼

XCi¼1

xi Apgm;i rn;Tð Þ þ Ar

m;i d; tð Þ þ RT ln xi

h i: ð12:60Þ

In this equation, C is the number of components in the mixture, Apgm,i is the

ideal gas Helmholtz energy for component i, and Ari is the pure fluid residual

Helmholtz energy of component i evaluated at a reduced density and tem-perature defined below.The excess contribution to the Helmholtz energy from mixing is

AEm

RT¼ aE ¼

XC�1i¼1

XCj¼iþ1

xixjFij

�XKpol

k¼1Nkd

dkttk þXKpolþKexp

k¼Kpolþ1Nkd

dkttk exp �Zk d� �kð Þ2�bk d� gkð Þ �2

435;

ð12:61Þ

where the coefficients and exponents were obtained from nonlinear regressionof experimental mixture data. The parameter Fij is used in a generalization to

426 Chapter 12

relate the excess properties of one binary mixture to those of another. With thisparameter, the same set of mixture coefficients can be used for several binarymixtures in the model. Several binary pairs do not use the generalized para-meter and instead have binary specific excess functions for the coefficients andexponents and these binary mixtures include the following: (methaneþ nitro-gen), (methaneþ carbon dioxide), (methaneþ ethane), (methaneþ propane),(methaneþ hydrogen), (nitrogenþ carbon dioxide) and (nitrogenþ ethane).There were sufficient experimental data for these binary mixtures to permit thefit of individual equations.All single-phase thermodynamic properties can be calculated from the

Helmholtz energy as described in Sec. 12.5 with the relations

a0 ¼XCi¼1

xiA0

i rn;Tð ÞRT

þ ln xi

� ð12:62Þ

and

ar ¼XCi¼1

xiari d; tð Þ þ aE d; t;xð Þ; ð12:63Þ

where the derivatives are taken at constant composition. The reduced values ofdensity and temperature for the mixture are

d ¼ r=rr xð Þ ð12:64Þ

and

t ¼ Tr xð Þ=T ; ð12:65Þ

where r and T are the mixture density and temperature, and rr(x) and Tr(x) arethe reducing values given by:

1

rr xð Þ¼XCi¼1

XCj¼1

xixjbv;ijgv;ijxi þ xj

b2v;ijxi þ xj

1

8

1

r1=3c;i

þ 1

r1=3c;j

!3

ð12:66Þ

and

Tr xð Þ ¼XCi¼1

XCj¼1

xixjbT ;ijgT ;ijxi þ xj

b2T ;ijxi þ xj

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiTc;iTc;j

p: ð12:67Þ

The parameters b and g are used to define the shapes of the reducing tem-perature lines and reducing density lines, respectively. These reducing para-meters are not the same as the critical parameters of the mixture and are


determined simultaneously in the nonlinear fit of experimental data with theother parameters of the mixture model.

12.9 Software for Calculating Thermodynamic

Properties

There are a wide variety of software packages available that have implementedthe equations of state presented in this work. Two of these programs areREFPROP, available from NIST (www.nist.gov/srd/nist23.htm), and Ther-moFluids (www.FirstGmbH.de), developed by Wagner and co-workers at theRuhr University in Bochum, Germany. Both programs calculate thermo-dynamic properties from equations of state and include Dynamic LinkLibraries (DLL) for the calculation of properties in user-defined applications,for example, Microsoft Excel. The programs also provide figures, for example,of pressure as a function of enthalpy, and can be generated based on inputsprovided by the user for the iso-property lines.Although the use of readily programmed algorithms is recommended in most

cases, it can also lead to new problems. In the archival literature the valuesobtained from programs are frequently referred to as the data source. Becausethe equation of state used to generate data for certain fluids may be updated inlater versions of the program, such a reference is ambiguous and may result inirreproducible scientific results. Even if commercially available software is used,it is important to quote the equation of state that is used to calculate propertydata and associated reference. It is therefore imperative that software productsprovide the original references for the user.

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428 Chapter 12

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430 Chapter 12

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65. R. D. McCarty and V. D. Arp, Adv. Cryo. Eng., 1990, 35, 1465–1475.66. J. W. Leachman, R. T Jacobsen, S. G. Penoncello and E. W. Lemmon,

J. Phys. Chem. Ref. Data, 2009, 38, 721–748.67. K. M. de Reuck and R. J. B. Craven, International Thermodynamic

Tables of the Fluid State-12 Methanol, International Union of Pure andApplied Chemistry, Blackwell Scientific Publications, London, 1993.

68. R. S. Katti, R. T Jacobsen, R. B. Stewart and M. Jahangiri, Adv. Cryo.Eng., 1986, 31, 1189–1197.

69. B. A. Younglove, J. Phys. Chem. Ref. Data, 1982, 11.70. U. Overhoff, Development of a New Equation of State for the Fluid Region

of Propene for Temperatures from the Melting Line to 575K with Pressuresto 1000MPa as Well as Software for the Computation of ThermodynamicProperties of Fluids, Ph. D. Dissertation, Ruhr University, Bochum,Germany, 2006.

71. R. T Jacobsen, S. G. Penoncello and E. W. Lemmon, Fluid Phase Equilib.,1992, 80, 45–56.

72. V. Marx, A. Pruß, and W. Wagner, New Equation of State for R 12, R 22, R11 and R 113, Fortschr. -Ber. VDI, Dusseldorf: VDI-Verlag, Volume 19,Number 57, 1992.

73. B. Platzer, A. Polt and G. Maurer, Thermophysical Properties of Refrig-erants, Springer-Verlag Berlin Heidelberg, Germany, 1990.

74. B. de Vries, R. Tillner-Roth, and H. D. Baehr, Thermodynamic Propertiesof HCFC 124, 19th International Congress of Refrigeration, The Hague,The Netherlands, International Institute of Refrigeration, IVa:582-589,1995.

75. J. W. Magee, S. L. Outcalt and J. F. Ely, Int. J. Thermophys., 2000, 21,1097–1121.

76. R. Tillner-Roth and H. D. Baehr, J. Phys. Chem. Ref. Data, 1994, 23,657–729.

77. E. W. Lemmon and R. T Jacobsen, J. Phys. Chem. Ref. Data, 2000, 29,521–552.

78. A. Kamei, S. W. Beyerlein and R. T Jacobsen, Int. J. Thermophys., 1995,16, 1155–1164.

79. S. G. Penoncello, E. W. Lemmon, R. T Jacobsen and Z. Shan, J. Phys.Chem. Ref. Data, 2003, 32, 1473–1499.


80. S. L. Outcalt and M. O. McLinden, An Equation of State for the Thermo-dynamic Properties of R236fa, NIST report to sponsor (U. S. Navy, DavidTaylor Model Basin) under contract N61533-94-F-0152, 1995.

81. R. Tillner-Roth and A. Yokozeki, J. Phys. Chem. Ref. Data, 1997, 26,1273–1328.

82. C. Guder and W. Wagner, J. Phys. Chem. Ref. Data, 2009, 38, 33–94.83. E. W. Lemmon, M. L. Huber, and M. O. McLinden, NIST Standard

Reference Database 23: Reference Fluid Thermodynamic and TransportProperties-REFPROP, Version 8.0, National Institute of Standards andTechnology, Standard Reference Data Program, Gaithersburg, 2007.

84. J. W. Leach, Molecular Structure Corrections for Application of the Theoryof Corresponding States to Non-Spherical Pure Fluids and Mixtures, Ph. D.Dissertation, Rice University, Houston, Texas, 1967.

85. J. F. Ely, Adv. Cryog. Eng., 1990, 35, 1511–1520.86. E. W. Lemmon, A Generalized Model for the Prediction of the Thermo-

dynamic Properties of Mixtures Including Vapour-Liquid Equilibrium, Ph.D. Dissertation, University of Idaho, Moscow, 1996.

87. E. W. Lemmon and R. T Jacobsen, Int. J. Thermophys., 1999, 20, 825–835.88. R. Tillner-Roth, J. Li, A. Yokozeki, H. Sato, and K. Watanabe, Ther-

modynamic Properties of Pure and Blended Hydrofluorocarbon (HFC)Refrigerants, Japan Society of Refrigerating and Air Conditioning Engi-neers, Tokyo, 1998.

89. E. W. Lemmon and R. Tillner-Roth, Fluid Phase Equilib., 1999, 165, 1–21.90. E. W. Lemmon and R. T Jacobsen, J. Phys. Chem. Ref. Data, 2004, 33,

593–620.91. O. Kunz, R. Klimeck, W. Wagner, and M. Jaeschke, The GERG-2004

Wide-Range Equation of State for Natural Gases and Other Mixtures,GERG TM15, Fortschritt-Berichte VDI, Volume 6, Number 557, 2007.

432 Chapter 12

CHAPTER 13

Equations of State in ChemicalReacting Systems

SELVA PEREDA, ESTEBAN BRIGNOLE AND SUSANABOTTINI

Planta Piloto de Ingenierıa Quımica (PLAPIQUI) – CONICET, UniversidadNacional del Sur, Camino La Carrindanga Km 7 – C.C: 717, Argentina

13.1 Introduction

Phase and chemical equilibrium calculations are essential for the design ofprocesses involving chemical transformations. Even in the case of reactions thatcannot reach chemical equilibrium, the solution of this problem gives infor-mation on the expected behaviour of the system and the potential thermo-dynamic limitations. There are several problems in which the simultaneouscalculation of chemical and phase behaviour is mandatory. This is the case, forexample, of reactive distillations where phase separation is used to shift che-mical equilibrium. Also, the calculation of gas and solid solubility in liquids ofhigh dielectric constants requires at times the resolution of chemical equili-brium between the different species that are formed in the liquid phase. Severalalgorithms have been proposed in the literature to solve the complex non-linearproblem; however, proper thermodynamic model selection has not receivedmuch attention.In recent times, the use of supercritical solvents has emerged as an important

technique to improve rates and selectivities in diffusion-controlled reactions.Phase behaviour near the critical point of mixtures is very sensitive to processoperating conditions and mixture compositions. The selection and designof the appropriate phase conditions to exploit process potential require




433

thermodynamic models able to deal with highly asymmetric mixtures involvingpermanent gases, supercritical solvents and non-volatile substrates.This chapter presents a phase equilibrium engineering approach to analyze

the phase behaviour of chemical reacting systems. The use of group con-tribution equations of states in these systems is discussed. The main advantageof these models is that they have predictive capability for compounds that werenot included in the parameterization process. The lack of equilibrium data inreactive mixtures is quite common; therefore, group contribution methodsallow designers to gain knowledge on the changes in phase behaviour as thereaction proceeds.

13.2 The Chemical Equilibrium Problem

The general criterion for equilibrium conditions is that the Gibbs energy of asystem reaches a minimum value for a given temperature and pressure. For asingle-phase reacting system, this condition is conveniently expressed in termsof the chemical potentials mi and the stoichiometric number ni of each species i isgiven by:

Xi

@G

@ni

� �T ;p;nj 6¼i

ni ¼Xi

mini ¼ 0; ð13:1Þ

where ni has a positive sign for products and negative for reactants. In a non-ideal solution the chemical potential is related to the activity ai of the corre-sponding compound:

mi ¼ G�Ji þ RT ln ai: ð13:2Þ

Placing this equation into the equilibrium condition gives:

DrG�Jm ¼ �RT ln K ; ð13:3Þ

where DrG�Jm is the standard molar Gibbs energy change of the reaction and the

equilibrium constant K is related to the mixture activities by:

K ¼Yi

aið Þvi ð13:4Þ

By definition the activity of component i is equal to the ratio between thefugacity coefficients fi in the solution and in the standard state f�Ji by:

ai ¼fiðp;T;xÞf�Ji ðp;TÞ

: ð13:5Þ

434 Chapter 13

The composition of the reactive system at equilibrium can be obtained bycalculating fugacities with equations of state.Relevant examples of the effect of non-ideality on chemical equilibria can be

found in the synthesis of ammonia and in the production of methanol fromcarbon monoxide and hydrogen. Graaf et al.1 studied the chemical equilibriumin the synthesis of methanol at low pressures, coupled with the water-gas shiftreaction. They compare experimental data with values obtained from thermo-dynamic calculations using different equations of state and taking into accountthe effect of pressure, temperature and composition. The best results wereachieved with the Soave-Redlich-Kwong equation of state,2 even though theuse of the ideal gas assumption gave reasonable values at pressures in the rangeof (1 to 8) MPa and at temperatures between (470 and 530) K. Bertucco et al.(1995) also studied the homogeneous chemical equilibrium in the synthesis ofmethanol and ammonia and in the water-gas shift reaction using a modifiedSoave-Redlich-Kwong equation of state. Liu et al.3 analyzed the effect of sol-vents under supercritical conditions in the synthesis of methanol, at tempera-tures and pressures similar to those studied by Graaf et al.1 The effect ofsolvents such as hexane, heptane and nitrogen on the equilibrium conversionwere evaluated with the Soave-Redlich-Kwong equation of state and conven-tional mixing rules. The results were insensitive to the values of the binaryinteraction coefficient, kij; hence, all of these parameters were set equal to zero.Saim and Subramanian4 addressed the effect of supercritical CO2 as solvent

media in the homogeneous chemical equilibria of different reacting systems,including the isomerisation of hexane and 1-hexene and the oxidation of 2-methylpropane. The Peng-Robinson equation of state5 was used to calculate thecritical loci of CO2 with each hydrocarbon; the information was used to avoidthe two-phase region in the computation of conversion for each system studied.The problem of simultaneous chemical and phase equilibrium calculations

are mathematically rather complex and several authors have proposed differentalgorithms to look for the final equilibrium state. A review paper by Smith6 anda textbook by Smith and Missen7 present the general formulation of thisproblem. Algorithms can be significantly simplified in ideal systems. The rig-orous solution implies the minimization of the Gibbs energy, counting not onlyover all the components NC present in the system but also over all NP phases:

ðGÞmin ¼XNC

i

XNP

j

nijGij

!; ð13:6Þ

subject to the atomic mass balance equations:

A � n ¼ b; ð13:7Þ

where A is the matrix with the number Aik of atoms of element k in molecule i.Vectors b and n contain, respectively, the total number of atoms of element k inthe system and the number of moles of component i.

435Equations of State in Chemical Reacting Systems

A major problem in finding the correct solution is that the number of phasespresent at equilibrium is not known a priori. The algorithm proposed byMichelsen8 based on the sequential addition of a new phase followed by astability test, is a robust method to circumvent this problem.Castier et al.9 and Gupta et al.10,11 studied the simultaneous solution of

multiple-phase and chemical equilibria taking into account the non-ideality ofthe mixture using an equation of state to calculate fugacities. The Gibbs energyminimization procedure was used for both as was the Michelsen’s phase sta-bility analysis. Castier et al.9 applied the Soave-Redlich-Kwong equation ofstate to study the high-pressure synthesis of methanol and proved that con-densation may occur at T¼ 473K and pressures about 30 MPa. Gupta et al.11

showed similar results for the same reaction. Burgos- Solorzano et al.12 vali-dated Castier et al.9 solutions to the phase and chemical equilibrium problemusing a deterministic interval analysis, which guarantees that the global mini-mum in the Gibbs energy is found.Phoenix and Heidemann13 on the other hand, adapted Michelsen 0s

approach14,15 for chemical equilibrium calculation in multiphase reacting sys-tems, applying iterative corrections to ideality with the Soave-Redlich-Kwongequation of state. Phoenix and Heidemann16 applied this procedure with thePeng-Robinson equation of state to study the phase behaviour of natural gasescontaining elemental sulfur, which is known to exist as a number of species upto S8.

16

Quite frequently, papers that deal with simultaneous phase and chemicalequilibrium place the emphasis on the computational algorithms required tomake the calculations, but disregard the importance of the thermodynamicmodel used. It is common to find the use of inappropriate equations of statewith conventional mixing rules for highly non-ideal systems, which invalidatesthe numerical results of the complex algorithms proposed.

13.3 Reactions under Near-Critical Conditions

The use of supercritical fluids can greatly enhance the performance of reactionslimited by the partial miscibility of reactants. These reactions are generallydiffusion-controlled. The use of supercritical fluids reduces this controlling step,by eliminating the interface and increasing the reactants diffusivity. Therefore,reaction rates are increased.17,18 In addition, better selectivities can be achievedat supercritical conditions due to the possibility of uncoupling process vari-ables. For example, while gasþ liquid hydrogenation reactions require hightemperatures to increase hydrogen solubility, in the supercritical process thetemperature can be modified with no effects on compositions. This allows theselection of an operating temperature that improves selectivity without redu-cing conversion. Consequently, isomerisation reactions favoured by the lack ofhydrogen at the catalyst surface can be avoided.19,20

In recent years, many authors21–24 also propose the use of supercritical fluidsas reaction media, but working under a two-phase region. In these supercritical

436 Chapter 13

multiphase reactions the use of CO2 has received special emphasis. In this casethe only requirement is that the supercritical fluid dissolves in large amounts inthe liquid phase21 but it is not required that the substrates have good solubilityin the supercritical phase. For this reason, the use of CO2 has received specialattention as the main limitation of this solvent (that is, its poor solvent capacityfor many substrates) is circumvented.Gas-liquid reactions are one of the areas where the use of equations of state is

particularly attractive. In these reacting systems equations of state can handlesubcritical and supercritical components under a wide range of conditions.Multiphase and supercritical conditions can be described with a proper equa-tion of state, going from the heterogeneous to the critical and homogeneousregions in a continuous way. Baiker et al.18,25,26 have stressed the importance ofa proper knowledge of phase equilibria in gas-liquid catalyzed reactions.Advances in this area have been reviewed by Pereda et al.27

There are a few examples in the open literature where equations of statehave been used to seek homogenous operating regions. Camy et al.28 measuredand modelled the phase equilibrium conditions for the synthesis of dimethylcarbonate. The reacting mixture in this case consists of (CO2þmethanolþdimethylcarbonateþwater). For modelling purposes these authors used theSoave-Redlich-Kwong equation of state with modified Huron-Vidal mixingrules (discussed in Chapter 5).29,30 They concluded CO2 has to be used in a largeexcess to ensure the reaction runs under a homogeneous fluid medium. Stradiet al.31,32 studied experimentally the multicomponent phase equilibria of anallylic epoxidation reaction in supercritical carbon dioxide. The Peng-Robinsonequation of state with quadratic mixing rules were used to model the phasebehaviour and then recommend conditions to operate within the homogeneousregion. Ke et al.33 used the same model for the hydroformilation of propene tobutanal to select the proper conditions for single-phase operation. Ke et al.33

measured the variation of the critical point of the reaction with the degree ofconversion. The multicomponent data were successfully predicted by the Peng-Robinson equation of state using binary interaction coefficients fitted to thecritical points of the binary mixtures between CO2 and each of the reactioncomponents. Other binary interaction parameters were obtained from litera-ture data on vapour-liquid equilibria. Chrisochoou et al.34–36 studied experi-mentally the phase equilibrium of binary and multicomponent mixtures foundin the enzymatic production of an enantiopure pyrethroid compound and3-methylbutyl ethanoate carried out under supercritical CO2. The experimentalresults were correlated using the Soave-Redlich-Kwong equation of state withHuron-Vidal mixing rule.The phase equilibria of the hydroformylation of hex-1-ene in supercritical

CO2 was studied experimentally by Jiang et al.37 for different degrees of con-version, involving mixtures of CO, H2, CO2, hex-1-ene and heptanal. Marteelet al.38 conducted the reaction at T¼ 373K and p¼ 18.6MPa and had to use aCO2þ reactant mass ratio equal to 3 to operate within the homogenous phase.Pereda et al.39 modelled the measurements reported by Jiang et al.37 withGroup Contribution Association equation of state40,41 and showed the use of


supercritical propane greatly reduced the amount of solvent required to operatewithin a single phase.Hydrogenations are the most studied gas-liquid reactions under supercritical

medium. The low solubility of hydrogen in liquid substrates is a great drivingforce to use supercritical solvents as reaction media. Baiker et al.42,43, Chouchiet al.44, Wandeler et al.45 and van der Hark et al.46,47, to cite but a few, havestudied the hydrogenation of numerous substrates using supercritical solvents.All verified the phase conditions under which the reaction was taking placeexperimentally but no thermodynamic models were applied to select the opti-mum operating conditions or to understand the experimental outcomes. Peredaet al.48 applied the Group Contribution Association equation of state to modelthe phase behaviour of reacting mixtures typical of the hydrogenation ofvegetable oils and derivatives at supercritical or high-pressure conditions.Yermanova and Anikeev49 have applied the Soave-Redlich-Kwong equation

of state to study the phase behaviour in a Fischer–Tropsch reactor near-criticalphase transition conditions. Typical phase behaviour of multicomponentreactive mixtures and the problems associated with calculations in reactorsoperating under phase-transition conditions were discussed in ref. 49. Theobjective of the work reported in ref. 49 was to localize the critical point of thereactive mixtures and to know the phase compositions in the near-criticalregion. This information allows the correct phase compositions to be providedfor a rigorous kinetic model. Hegel et al.50 and Andreatta et al.51 have studiedthe production of biodiesel using supercritical alcohols. In this case the GroupContribution Association equation of state model was applied to study phasetransitions during reactor warm up and to determine the critical point of thereactive mixture at different operating conditions. The method correlated theexperimental phase equilibrium data at high pressures for the reactive mixturecontaining very asymmetric components (1,2,3-propantriol-alkyl esters,methanol, propane-1,2,3-triol and water).The study of multiphase catalytic reactions in supercritical fluids gives rise to

very challenging problems in the measurement and prediction not only of fluidphase equilibria but also volumetric properties. Both types of information arerequired for the proper design of chemical reactors. Recently Arai et al.21 andNunes da Ponte52 reviewed a wide range of multiphase reactions carried outunder supercritical CO2 and these systems included the following: (i), organicliquids; (ii), polymers; (iii), ionic liquids; (iv), aqueous systems; (v), (solidþfluid) systems. However, few experimental studies covering phase equilibria andvolumetric properties have been reported. Kordikowski et al.53 studied thedensity of several associating, polar and non polar organic liquids, whensupercritical fluids like CO2, ethane or ethene are dissolved in the liquid phase;they also reported experimental (vapourþ liquid) equilibria of these mixtures.The volumetric and solubility data of the aprotic systems were correlated withthe Peng-Robinson equation of state using binary interaction parameters in thecalculation of the mixture co-volume and attractive energy parameter. Asimilar study for aromatic compounds under high pressure CO2 was carried outby Phiong and Lucien54 with the Peng-Robinson equation of state to correlate

438 Chapter 13

data on vapour-liquid equilibria and volumetric properties using binaryinteraction coefficients only for the energy parameter.Eckert et al.22 presented a thorough review on the use of tuneable solvents as

a sustainable way of carrying out chemical reactions and separations, combiningsupercritical fluids and organic solvents. A number of measurements were alsomotivated by the need of reliable information on the behaviour of multiphasereacting systems. For example, Xie et al.55 studied the phase boundaries of(carbon dioxideþmethanolþ hydrogen). Levitin et al.56 measured and mod-elled the phase behaviour of (CO2þmethanolþ tetramethylammoiumþwater)at temperatures of (298 and 353) K and pressures up to 30MPa. In ref. 56 thePeng-Robinson equation of state with Wong and Sandler57 mixing rules wasused to model the experimental results and the predictions were found to bevery sensitive to the adjustable parameters. In the context of using tuneablesolvent methodology, Lazzaroni et al.58 have studied (organicþwater) mix-tures in which the catalyst solubility can be changed by the effect of carbondioxide dissolution. Experimental information in systems containing oxolane,acetonitrile or 1,4-Dioxacyclohexane and water were correlated using Stryjekand Vera59 modification to the Peng-Robinson equation of state with Huronand Vidal60 mixing rules. Lazzaroni et al.61 also measured (vapourþ liquid)equilibria and liquid molar volumes of mixtures of carbon dioxide with severalorganic solvents. To model these results the Patel–Teja62 equation of state wascombined with Mathias-Klotz-Prausnitz63 mixing rules.

13.4 Modelling Reacting Systems with Group

Contribution Equations of State

Prediction of fluid phase behaviour at high pressures must be performedwith an equation of state using the so called f�f approach to attain a con-sistent result near the critical region of the mixture. In this sense, classiccubic equations of state are the more attractive to use due to their simplicity;however, they present some limitations when applied to asymmetricsystems. An alternative approach is provided by group contribution which isparticularly attractive for reacting systems because it is often necessaryto deal with mixtures for which no experimental phase equilibrium data areavailable.Group contribution models allow the prediction of the phase behaviour of

compounds not included in the parameterization procedure. Moreover, thecurrent interest in using renewable products as feedstock to the chemicalindustry has also made the use of group contribution models appealing,because natural products generally contain a large number of similar speciesthat can be represented by a limited small number of functional groups.The cubic Soave-Redlich-Kwong equation of state with the modified Huron-

Vidal mixing rules developed by Michelsen29,30 (herein after assigned theacronym MHV2) is a model that fulfils these requirements and it is veryattractive due to its mathematical simplicity; details of Huron-Vidal mixing


rules are given in Chapter 5. The use of UNIFAC64 to calculate the excessGibbs energy makes this model well suited to predict phase behaviour due tothe large group parameter table available in the literature. The extensionproposed by Dahl et al.65 to include permanent gases like H2, O2, N2, CO, CO2,H2S, CH4, among others, makes UNIFAC adequate to work with(gasþ liquid) reacting systems.One drawback of the MHV2 model is the inability of UNIFAC to

predict (vapourþ liquid) equilibria (VLE) and (liquidþ liquid) equilibria(LLE) conditions using the same set of group-interaction parameters. In gen-eral, cubic equations of state do not provide precise predictions of thephase equilibria when the mixture is asymmetric in size that is attributedto the large differences in the pure-component co-volumes.66 The Carnahan –Starling67 equation for hard spheres is a more realistic model for therepulsive contribution than that proposed by van der Waals. Mansoori et al.68

proposed an equation for mixtures of hard spheres that has been found tocorrelate the phase behaviour of non-polar mixtures with large molecularsize differences.

13.4.1 Group Contribution with Association Equation of State

(GCA-EoS)

The Group Contribution Association equation of state is a model whichcombines a Carnahan Starling repulsive term with a group contributionapproach to describe both the attractive and association contributions to theresidual Helmholtz energy. This model extends the group contribution equa-tion of state Group Contribution Association equation of state proposed bySkjold –Jorgensen69,70 to associating systems, following a group-contributionversion40,71 of the association term in the SAFT equation;72,73 the SAFTequation is discussed in Chapter 8.There are three terms in the expression of the residual Helmholtz function in

the Group Contribution Association equation of state, each one representingthe contributions of different intermolecular forces: (i), repulsive or free volumeAfv; (ii), attractive or dispersive Aatt; and (iii), specific association forces Aassoc.The residual Helmholtz function in the Group Contribution Associationequation of state is given by:

Ar ¼ Afv þ Aatt þ Aassoc ð13:8Þ

The free volume contribution follows the expression developed by Mansooriand Leland74 for mixtures of hard spheres:

ðA=RTÞfv ¼ 3ðl1l2=l3ÞðY � 1Þ þ ðl32=l33ÞðY2 � Y � ln YÞ þ n lnY ; ð13:9Þ

440 Chapter 13

with

Y ¼ 1� pl36V

� ��1; ð13:10Þ

and

lk ¼XNC

j

njdKj ; ð13:11Þ

where each substance is characterized by the hard sphere diameter di. In eqs 9through 11 ni represents the number of moles of component i,NC the number ofcomponents, V the total volume, R the universal gas constant and T thetemperature.The following generalized expression is assumed for the temperature

dependence of the hard sphere diameter:

di=dc;i ¼ 1:065655 1� 0:12 � exp 2 � Tc;i= 3T=Kð Þ½ �f g ð13:12Þ

where dc is the value of the hard sphere diameter of pure component i at thecritical temperature, Tc.Skjold–Jorgensen69,70 calculated values of the critical hard sphere diameter dc

for pure compounds from the corresponding critical properties and vapourpressure. For thermo-liable and high–molecular weight compounds thisinformation is either not available or not reliable because of their very lowvolatility. Bottini et al.75 proposed a method to estimate the dc for high molarmass compounds by fitting experimental data on infinite dilution activitycoefficients obtained by inverse gas chromatography. The critical diametersof alkanes and saturated and unsaturated triacylglicerides are given by thefollowing correlation as a function of the van der Waals volumes76:

lgðdc=1mÞ ¼ 0:4128 lgðrvdWÞ � 1:5848: ð13:13Þ

The attractive term in the evaluation of the Helmholtz energy is a groupcontribution version of a density-dependent NRTL77 expression, where inter-actions are considered to take place through the surfaces of characteristicfunctional groups rather than through the surfaces of the parent molecules sothat:

A

RT

� �att¼ � z

2

XNC

i¼1niXNG

j¼1vijq �

PNG

k¼1ykgkj~qtkj

�RTVð Þ

PNG

l¼1yltlj

8>>><>>>:

9>>>=>>>;; ð13:14Þ


where

yj ¼qj~q

� �XNC

i

nivij ; ð13:15Þ

~q ¼XNC

i¼1niXNG

j¼1vijqj ; ð13:16Þ

tij ¼ expðaDgij~qðRTVÞÞ; and ð13:17Þ

Dgij ¼ gij � gjj : ð13:18Þ

In eq 13.14, z is the number of nearest neighbours to any segment (set equalto 10), nij is the number of groups of type j in molecule i, qj stands for thenumber of surface segments assigned to group j, yk represents the surfacefraction of group k, q is the total number of surface segments, gij stands for theattractive energy between groups i and j and aij is the non-randomnessparameter.The attractive energy between unlike groups is calculated from the corre-

sponding interactions between like groups:

gij ¼ kijðgiigjjÞ1=2 ðkij ¼ kjiÞ; ð13:19Þ

with the following temperature dependence for the energy and interactionparameters:

gjj ¼ g�jj 1þ g0jjðT=T�j � 1Þ þ g

00jj lnðT=T�j Þ

j k; ð13:20Þ

and

kij ¼ k�ij 1þ k0ij ln

2T

T�i þ T�j

!" #; ð13:21Þ

where g*jj is the attractive energy and k*ij the interaction parameter at thereference temperature T*

i .The association term Aassoc is based on Wertheim’s first order perturbation

theory and follows a group-contribution approach:40

Aassoc

RT

� �¼XNGA

i¼1n�i

XMi

k¼1lnX ðk;iÞ � X ðk;iÞ

2

� �þ 1

2Mi

( ): ð13:22Þ

In equation 13.22, NGA represents the number of associating functional groups,n*i the total number of moles of associating group i, X(k,i) the fraction of group i

442 Chapter 13

non-bonded through site k and Mi the number of associating sites in group i.The total number of moles of associating group i is calculated from the numbern(i,m)assoc of associating groups i present in molecule m and the total amount of

moles of specie m (nm):

n�i ¼XNC

m¼1vði;mÞassocnm: ð13:23Þ

The fraction of groups i non-bonded through site k is determined by theexpression:

X ðk;tÞ ¼ 1þXNGA

j¼1

XMj

l¼1r�j X

ðl;jÞDðk;i;l;jÞ" #�1

; ð13:24Þ

where the summation includes all NGA associating groups and Mj sites. TheX(k,i) depends on the molar density of associating group r*j and on the asso-ciation strength D(k,i,l,j):

r�j ¼n�jV

ð13:25Þ

Dðk;i;l;jÞ ¼ kðk;i;l;jÞ exp �ðk;i;l;jÞ.kT

� �� 1

h i: ð13:26Þ

The association strength between site k of group i and site l of group j is afunction of the temperature T and the association parameters k and e, whichrepresent the volume and energy of association, respectively.The cross-association parameters are usually estimated by establishing

appropriate combining rules between the self-association parameters and/or bytreating them as additional adjustable parameters fitted to thermodynamic dataon cross-associating mixtures. Several authors78–80 have concluded that theoptimal combining rules for the cross-association parameters are the arithmeticmean of the self-association energies and the geometric mean of the self-asso-ciation volumes, which is equivalent to the geometric mean of the self-associationstrengths. Therefore, the following combining rules were adopted in the GroupContribution Association equation of state:

�k;i;l;j ¼ �k;i;l;i þ �k;j;l;j

=2 and kk;i;l;j ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikkili � kkjljp

: ð13:27Þ

For the cross-association between a self-associating group (for examplehydroxyl or acid groups) and a group that can solely cross-associate (forexample the ketone or ester groups), it is necessary to fit the cross-associationparameters to experimental data, simultaneously with the attractive


parameters. Ferreira et al.81 discussed in detail the application of the GroupContribution Association equation of state model to cross-associating systemsincluding organo-oxigenated compounds.The thermodynamic properties required to calculate phase equilibria are

obtained by differentiating the residual Helmholtz energy. The associationcontributions to the compressibility factor Z and the fugacity coefficient fi ofcomponent i in the mixture are given by:

Zassoc ¼ �V

n

@ AR�RT

assoc@V

!T ;n

¼ �V

n

XNGA

i¼1n�iXMi

k¼1

1

X k;ið Þ �1

2

� �@X ði;jÞ

@V

� �T ;n

" # ð13:28Þ

and

lnfassocq ¼

XNGA

i¼1v i;qð Þassoc

XMi

k¼1lnX ðk;iÞ � X ðk;iÞ

2

� �þMi

2

" #þ n�i

XMi

k¼1

1

V k;ið Þ �1

2

� �@X ðk;iÞ

@nq

� �T ;V ;nr6¼q

" #( )

ð13:29Þ

The final expressions of these contributions depend on the number NGA ofassociating groups and on the number Mi of associating sites assigned to eachassociating group i.Michelsen and Hendriks82 demonstrated that the calculation of the association

contributions to pressure and chemical potential from first order perturbationtheory can be simplified by the minimization of a conveniently definedstate function, which does not require the calculation of first derivatives of thefraction of non-associating molecules XAi

. For the Group Contribution Asso-ciation equation of state model the expressions for Zassoc and ln fassoc

q arereduced to:

Zassoc ¼ � 1

2

XNGA

k¼1

XMi

l¼1

n�kn

1� X ðl;kÞ� �

ð13:30Þ

and

lnfassocq ¼

XNGA

k¼1vðk;iÞassoc

XMi

l¼1lnX ðk;iÞ� �( )

: ð13:31Þ

Phase equilibrium calculations normally involve the computation of firstderivatives of fugacity coefficients with regard to process variables such as

444 Chapter 13

temperature, density and composition. This requires the calculation ofthe first derivatives of the fraction of non-bonded associating groups X(k,i) withrespect to these variables. According to eq 13.24, the fractions of non-bondedassociating groups X(k,i ) are obtained from a set of implicit equations thatdepend on the number of associating groups and on the assignment ofassociating sites to each associating group. The conventional procedureto solve for X(k,i)and its derivatives is to obtain a specific set of equationsfor each type of associating system, which makes the computation ofassociation effects problem-dependent.81,83 Tan et al.84 proposed a gene-ralized numerical procedure to calculate the fraction of non-bonded moleculesand their derivatives, applicable to all associating systems, regardless ofthe number and type of associating sites and the number of components in themixture.Based on Michelsen and Hendriks82 simplifications and the methodology

reported by Tan et al.84, a computational procedure was implemented byAndreatta85 to calculate phase equilibria in associating systems with the GroupContribution Association equation of state model. The computational routineallows the calculation of any type of association (self- and cross-association) insystems having up to a maximum of 12 different associating groups. Asso-ciating groups are considered to have one (electron donor or electron acceptor)or two (an electron donor and an electron acceptor) associating sites. A directsubstitution numerical method is used to solve the system of equations given ineq 13.24 for X(k,i). Rapid convergence is achieved for any arbitrary set ofX(k,i) initial values between 0 and 1.Two group-contribution models were discussed in this section: the MHV2

based on a cubic equation of state and the Group Contribution Associationequation of state. Compared to Group Contribution Association equation ofstate, MHV2 has a larger table of known parameters, which makes this modelready to be used in a wide range of systems. However, predictions obtainedmight become uncertain in systems with partial liquid miscibility or in mixtureshaving diluted associating species. Table 13.1 compares both equations withrespect to model requirements for phase equilibrium engineering of reactivesystems.

Table 13.1 Model requirements for phase equilibrium engineering.

MHV2 GCA-EoS

Predictions at near critical conditions | |Group contribution approach | |Large parameter data base |Prediction of highly asymmetric systems (supercriticalsolventþ high molar mass substratesþ gases)

|

Specific contribution for association effects in mixtures |Prediction of multiphase equilibria using a single set ofparameters

|


13.5 Phase Equilibrium Engineering of Supercritical

Gas-Liquid Reactors

Phase Equilibrium Engineering (PEE) of chemical processes involves thedevelopment of general criteria for phase condition design, allowing an efficientuse of process simulators and experimental data banks. The final goalof phase equilibrium engineering criteria is to set the boundaries of processfeasible operating regions, bringing knowledge about the process potentials andlimitations. Therefore, Phase Equilibrium Engineering is the systematic studyand application of phase equilibrium tools to the development of chemicalprocesses.There are four main steps in the PEE procedure: (1), select an adequate

thermodynamic model to represent process mixtures and conditions; (2), buildan experimental data bank on phase behaviour in order to tune the model tosensitive system properties; (3), select a solvent when it is required; and (4),search for feasible and optimum operating regions.In this section specific criteria for gas-liquid reactions under supercritical

conditions will be presented. Also, several applications of the MHV2 andGroup Contribution Association equation of state models to different stages inthe study of these reactions will be discussed.

13.5.1 Solvent Selection

The selection of an adequate solvent, capable of bringing a gas-liquid or gas-solid mixture to a homogenous phase is a critical step towards the design ofsupercritical reactors. In principle, the solvent should be inert; however, thereare some examples where it is possible to make use of the reactants assupercritical medium, like in transesterification of vegetable oils with super-critical alcohols,86 direct ammination with NH3,

87,88 alkylation using olefins oralcohols.89,90 The solvent must be completely miscible with the reactants andproducts in order to get homogenous operation along the reaction path. It isimportant to highlight that the selection of an inappropriate solvent results inthe need of a high solvent to reactant ratio to get homogenous operation, whichmay decrease the reaction rate due to dilution and it will certainly increaseoperating costs (large solvent recycle).The critical temperature of the solvent has a significant impact on solvent

selection; it is intimately related to the reactor operating temperature Top. Ingeneral, Top is set by the kinetics of the reaction under study (reaction acti-vation temperature). The solvent should have a critical temperature lower thanTop to ensure that it will be supercritical during operation and therefore,completely miscible with all gaseous reactants. On the other hand, the criticaltemperature of the solvent should not be far below Top because it will lose theliquid-like density and its solvent capacity towards the liquid reactants.Another reason to avoid very high reduced temperatures is that they canaccelerate catalyst poisoning due to loss of solvent capacity to extract deacti-vation agents adsorbed in the catalyst.91–94 In summary, the reduced

446 Chapter 13

temperature should be greater than one, but it is not convenient to exceed bymuch the values 1.1 to 1.15.Water and carbon dioxide are the most used solvents due to their low price

and environmental friendliness. The critical temperature of water is 473K andit is used for reactions under extreme conditions.95–97 Carbon dioxide, on theother hand, presents a very low critical temperature and it is adequate forreactions carried out under mild conditions, for example, selective hydro-genations. Unfortunately, it is well known that CO2 is not a good solvent forhigh molar mass organic compounds. Liquid carbon dioxide is miscible withalkanes with up to approximately 10 carbon atoms, while the range of mis-cibility increases for ethane up to 18 carbon atoms, and for propane up to 30carbon atoms.98,99 Thus, the application of CO2 as reaction media is limited tolow molar mass hydrocarbons if a homogenous operation is desired, whileethane and propane are a better option for higher molar mass hydrocarbons.The phase behaviour of the binary mixtures between the potential solvents

and the system components (reactants and reaction products) should be studiedfirst; later the study should be extended to the multicomponent reactive mix-tures for definition of feasible operating regions.

Phase Behaviour and Solvent Selection

In order to select an adequate solvent, it is necessary to understand the phasebehaviour of mixtures showing a different degree of asymmetry in size orintermolecular interactions. Van Konynenburg and Scott100 showed that thefluid phase behaviour observed in binary mixtures can be classified in five maintypes. Later Bolz et al.101, proposed a more detailed classification of each typeand subtype of phase behaviour, describing more than sixteen types of phasediagrams. In the following discussion the new nomenclature is used. In type 1P

phase behaviour, complete liquid miscibility is observed at all temperatures.When there is partial liquid miscibility at low temperatures, the system is oftype 1Pl. Type 1P phase behaviour is usually found in systems with componentsof similar chemical nature and molecular size, like mixtures of hydrocarbons,noble gases or systems that do not deviate greatly from ideal behaviour. Type1Pl is typical of non-ideal mixtures of similar size compounds, in which non-ideality leads to liquid phase split at subcritical conditions. When the liquidimmiscibility persists even at high pressures and temperatures, the systems areof type lPAlnQ. This behaviour is characteristic, for example, of mixturesbetween CO2 and high molar mass alkanes or vegetable oils (triglycerides).When the difference in molecular size becomes significant in almost ideal sys-tems, liquid-liquid immiscibility is observed near the light-component criticaltemperature (solvent Tc in supercritical processes). However, complete mis-cibility is recovered at lower temperatures; this corresponds to type 2P phasebehaviour. Type 2Pl, on the other hand, shows discontinued liquid-liquidimmiscibility; there is liquid immiscibility at low and high temperatures but notat intermediate temperatures. Figure 13.1 is a master chart of the different types


of binary fluid phase diagrams102. The arrows in this figure qualitativelyindicate the type of fluid phase behaviour that can be expected when thesystem components exhibit greater molecular interactions, size differences, orboth.The vapour-liquid-liquid equilibrium lines are limited by the upper critical

(UCEP) and the lower critical (LCEP) end points, where a liquid phasebecomes identical to the vapour phase or two liquids become identical andmerge into one phase. Each type of phase diagram could be recognizedby its UCEP and LCEP. For instance, type 1P has neither UCEP nor LCEP;type 1Pl and 1PAlnQ have only one UCEP (in the case of type 1PAlnQ theUCEP is above the light component critical temperature); type 2Pl is the onlysystem with two UCEP and one LCEP and type 2P has one UCEP and oneLCEP.Molecular size asymmetry is always present in gas-liquid reactions performed

under supercritical media. It is important to have prior knowledge of thepotential type of phase behaviour that a system can present. In general, type1PAlnQ phase behaviour should be avoided because it presents liquid-liquidimmiscibility even at extremely high pressures. In contrast, if a type 2P phasebehaviour is found, the region of partial liquid miscibility can be avoided byincreasing the pressure to reach an homogenous region. Peters et al.98,99 presenta very useful information about the type of phase behaviour that should beexpected for different binary systems between supercritical fluids and homo-logues families (alkanes, alkanols, alkyl benzenes) with different number ofcarbon atoms. For a given family of compounds and a specific supercriticalsolvent, the authors report the UCEP and LCEP versus the carbon numbers inthe homologues series. This information gives the trend of the system to shiftbetween different types of phase behaviour. Moreover, it provides the carbonnumber at which a certain change occurs. The lack of high-pressure experi-mental data makes this type of information more relevant since it allowsinferring the phase behaviour of a given binary from existent experimental dataon a system with similar chemical nature and or size asymmetry. On the otherhand, the knowledge of the basic principles that rule phase behaviour isimportant to detect possible experimental errors found in high-pressureoperation.The phase diagrams in Figure 13.1 only take into account fluid phase

behaviour. In the case of solid reactants103 equations of state allow the eva-luation of solute solubility in the solvent. The conditions of phase equilibriumbetween a supercritical fluid (1) and a solid component (2) are formulated onthe basis of the isofugacity criterion. If the solid phase is assumed to be a purecomponent, the solubility in the gas phase can be directly obtained as:

y2 ¼ Eps2P; ð13:32Þ

where E is the enhancement factor that corrects the ideal solubility and ps2 is thesolute sublimation pressure. For a low-volatile incompressible solid solute, the

448 Chapter 13

CL

CH

CL

CH

T

P

Typ

e 1p

T

P

Typ

e 1P l

T

P

Typ

e 2P

T

P

Typ

e 2P l

T

P

Typ

e 1P A

lnQ

Mol

ecul

arin

tera

ctio

n

Mol

ecul

arin

tera

ctio

n

Mol

ecul

arin

tera

ctio

n

+ Si

ze

Mol

ecul

arin

tera

ctio

n

+ Si

ze

Size

Size

CL

CL

CH

CH

CL

CH

Pure

com

pone

nt v

apou

r

Cri

tical

locu

s

Thr

ee p

hase

reg

ion

(LL

V)

Figure

13.1

Changes

inbinary

phase

behaviourwithsize

andenergyasymmetries

labelled102(phase

type)

classificationofBolz

etal.101CL

andCHare

thecriticalpointsofthelightandheavycompounds,respectively.Thearrowsqualitativelyindicate

thetypeoffluid

phase

behaviourthatcanbeexpectedwhen

thesystem

components

exhibitgreatermolecularinteractions,

size

differences,

or

both.—

—,vapourliquid

equilibria;��,

criticallocus;--------threephase

region(LLV).


enhancement factor can be calculated as follows:

E ¼ expp� ps2

vsol2

RT

� �f2; ð13:33Þ

where f2 is the fugacity coefficient of the solid solute in the gas phase and vsol2 isthe solid molar volume. f2, which can be calculated with an equation of state, ishighly dependent on density. Therefore, the strong variation of fluid densitywith pressure and temperatures in the near critical region greatly affects thereactor operating conditions.

13.5.2 Boundaries of Feasible Operating Regions

Phase diagrams can be used to follow the progress of a given reaction and alsoto select adequate reaction conditions. Ternary Gibbs diagrams, at a giventemperature and pressure, are useful for reaction problems where one of theproducts is very similar to one of the reactants therefore, we deal with ternarymixtures including the solvent. Figure 13.2 shows a scheme of a Gibbs triangle,where the two corners at the base represent a liquid substrate and a gas reactant(having a limited solubility in the substrate) and the top corner corresponds to

Solvent

Gaseousreactant

Liquidreactant/product

xmin

xmin

Heterogeneousregion

Figure 13.2 Pseudo ternary mixture phase diagram at constant temperature andpressure. The two corners on the base represent a liquid substrate and agas reactant (having a limited solubility in the substrate) and the topcorner corresponds to the solvent. The complete miscibility in(gasþ solvent) and (substrateþ solvent) ensures the presence of ahomogenous region around the solvent corner that can only be obtainedfor compositions greater than xmin.

450 Chapter 13

the solvent. The complete miscibility in (gasþ solvent) and (sub-strateþ solvent) ensures the presence of a homogenous region around thesolvent corner. If we represent the binodal curve for both reactants and pro-ducts mixtures, the heterogeneous region will grow with the progress of thereaction when the products are less soluble than the reactants and will shrink inthe opposite case. Moreover, if reactants and products are of the same chemicalnature, their binodal curves will collapse into a single one. This is the case of thehydrogenation of vegetable oils, where reactants (unsaturated triglycerides) andproducts (saturated triglycerides) have almost the same phase behaviour at hightemperatures. The maximum in the binodal curve corresponds to the minimumsolvent composition (xmin) required to work under homogenous conditions,regardless of the reactants molar ratio. The prediction of xmin for differentpressures at a given temperature will set the boundary of the homogeneousoperating region. Pereda et al.39,104 applied this methodology to the phaseequilibrium engineering of the hydrogenation of vegetable oils and (1S,5S)-2,6,6-trimethylbicyclo[3.1.1]hept-2-ene (a–pinene). For vegetable oils it isshown that solvent requirement decreases exponentially with pressure in thelower pressure range and goes to an asymptotic minimum at higher pressures.Therefore, for each reaction temperature there is a pressure range beyondwhich no significant reduction in the solvent requirement is achieved by apressure rise. In the hydrogenation of a-pinene, a reaction pathway with highH2-to-a-pinene molar ratio is proposed based on the phase behaviour of thereactive system.When the number of reaction components is greater than 2, the best way to

describe the single-phase boundaries is by using a phase envelope diagram. Foreach composition (isopleth) this diagram gives the bubble and dew point lines,as well as the critical point of the mixture in pressure vs. temperature coordi-nates. Michelsen105 proposed an algorithm of fast convergence for the calcu-lation of these diagrams. Figure 13.3 shows a scheme of three phase envelopesthat represent isopleths at different degrees of conversion between x0¼ 0 andx2¼ 1. In this example the solubility of the system decreases as the reactionprogresses; that is, products are less soluble than reactants in the supercriticalfluid. However, the solubility will not always increase or decrease mono-tonically; it can show a maximum or minimum at intermediate conversions,according to the balance between the consumption of gaseous components andthe appearance or disappearance of more insoluble components. At each degreeof conversion, the maximum in the phase envelope represents the minimumpressure required to ensure a homogenous phase for any operating tempera-ture. In the case of Figure 13.3, if a pressure higher than p(min, x2) is applied,the reactor will always operate under a single phase, regardless of temperatureand conversion. The operating variables can thus be selected so as to avoidentering the heterogeneous region. Pereda et al.39,104 discuss the use of phaseenvelope diagrams in the hydrogenolysis of fatty acid methyl esters undersupercritical propane and the hydroformylation of 1-hexene under supercriticalCO2 and propane. The Group Contribution Association equation of statemodel was used to construct the diagrams.


By working under homogenous conditions it is possible to uncouple processvariables like temperature and composition, which are intimately relatedin heterogeneous systems. Hitzler et al.19 showed for the hydrogenation of3-methylphenol this permits better control of reaction selectivity. Bhanageet al.20 reported for the selective hydrogenation of unsaturated aldehydesto produce unsaturated alcohols under supercritical CO2, big differences inselectivity with slight changes in experimental conditions. Pereda et al.39,106

used the MHV2 model to show that a possible explanation for the differentselectivity achieved by Bhanage et al.20 arose from operation under single-phase(high selectivity) or bi-phasic (low selectivity) conditions. This exampledemonstrates the importance of using phase diagrams and thermodynamicmodels to understand experimental results and to do a proper solvent selection.In addition to phase behaviour, it is also important to evaluate the density of

the reaction mixture. This variable plays a major role in both reaction equili-brium and kinetics. The control of density is more complex in supercriticalreactors, where it can change dramatically with small perturbations in tem-perature, pressure or composition. The more direct application of density, inthe case of continuous reactors, is to calculate the residence time of the reactionmixture for a given operating pressure and temperature. It is important to keepin mind that the volumetric flow measured downstream of the reactor can bevery different from the flow inside the reactor due to the high variability of thedensity at supercritical conditions. In the case of batch reactors there are two

P

T

HeterogeneousRegion

conversion

p (min,x2)

p (min,x1)p (min,x0)

Figure 13.3 Pressure p as a function of temperature T at constant chemical compo-sition. Three compositions are shown x0, x1 and x2 for a reaction inwhich the products are less soluble that the reactants. To maintain ahomogeneous single phase in the reactor requires the application of apressure that increases as the reaction progresses from p(min, x0) top(min, x1) and p(min, x2). If the pressure applied to the reactor is greaterthan p(min, x2,), the reactor will always operate under a single phase,regardless of the stage of the reaction or temperature.

452 Chapter 13

important applications where the knowledge of density plays a significant role.The first is related to the knowledge of the real feed composition loaded into areactor, based on the measurement of the partial pressures of the gaseouscomponents. Pereda et al.39 provided an iterative procedure to calculate feedcompositions, following the experimental feed process. In general, in a batchreactor of total volume (Vt), a known amount of liquid substrate (nsub) ischarged before each gaseous component is fed up to a certain partial pressure.The partial pressure of solvent (Psol) and the total pressure in the reactor afterfeeding the gaseous reactant (Pt) are the control variables to fixthe molar ratio of solvent to reactant. The number of solvent moles (nsol) in thereactor is calculated first by the following iterative procedure: (1) the initialnumber of moles nsol is guessed; (2) the molar volume (V1) of the mixture at thefeeding temperature and intermediate pressure Psol is calculated with anequation of state. The iterative process stops when (nsubþ nsol) �V1¼Vt. Asecond iterative process gives the amount of gaseous reactant (ngas) using thefollowing procedure: (1), the initial number of gas moles is guessed; (2), anequation of state is used to calculate the molar volume (V2) of the mixture(nsubþ nsolþ ngas) at feeding temperature and final pressure Pt. Again, theiterative process stops when (nsubþ nsoltþ ngas) �V2¼Vt. If the feeding com-position corresponds to a two-phase region, the total volume is calculatedfrom the molar volumes of the two coexisting phases liquid (l) and vapour (v):that is nl �Vlþ nv �Vv¼Vt.The second application for which knowledge of density is required for batch

supercritical reactor design is related to the control of the operating pressure,while the reactor is heated to reach near-critical conditions. If a reactor is fed atlow temperatures without taking into account that the liquid phase will expandas the temperature increases, there is a possibility the expanded liquid willcompletely fill the reactor. Figure 13.4 shows a scheme of two possible reactiontrajectories, in a pressure-temperature diagram where the phase envelopes ofhypothetic reactants and products mixtures are also shown. If the initial vapourfraction in the system is low (high global density), the reactor pressure willfollow trajectory (a) as the temperature increases; it is initially very close to thebubble point of the mixture and ends up as a dense liquid showing an isochoricincrement of pressure in the last stage of the reaction. As a result it is impossibleto be near the mixture critical conditions. If the global density is low, the typicalreactor trajectory will follow curve (b). Hegel et al.50 showed the importance ofcontrolling the feed global density to avoid the pressure run away during thecatalyst-free transesterification of vegetable oils to produce biodiesel. Thisstudy allowed the explanation of controversial results previously published inthe literature.Neither the Group Contribution Association equation of state nor Soave-

Redlich-Kwong equations of state with MHV2 mixing rules are recommendedmethods to predict mixture densities. The Peng-Robinson equation of statewith classical mixing rules is the more precise model among the van der Waalsfamily of equation of state to predict molar volumes of mixtures particularlywhen the volume correction has been used as proposed by Peneloux et al.107


that can be introduced to any cubic equation of state to improve the predictionof liquid density without changing the phase-equilibrium conditions.

13.6 Concluding Remarks

In this chapter the application of equations of state to model reactive mixtureshas been discussed. The chemical and phase equilibria reactions operatingunder homogenous or multiphase regimes have been presented and appropriateequations of states required to predict the phase behaviour in a reactordiscussed.Substantial research on near-critical reactions has been published in the lit-

erature in recent years, and important advantages have been found. Althoughthe importance of phase behaviour is generally accepted, its complexity is oftenunderestimated. Ignoring phase equilibrium may lead to false interpretation ofobserved effects, particularly in high-pressure chemistry.For the simultaneous solution of phase and chemical equilibria special

emphasis has to be given to the development of algorithms that solve thecomplex non-linear problem. Frequently, traditional cubic equations of statewith classical mixing rules have been applied to highly non-ideal mixtures; thisinvalidates the precise numerical solutions obtained.

P

T

HeterogeneousRegion

(a)

(b)

Figure 13.4 Pressure p as a function of temperature T illustrating the global effect ofdensity on batch reactor pressures. Path labelled (a) is for high initialdensity and that labelled (b) is for a low initial density. -------, mixtureconditions while system temperature is increased. ------- phase envelopesfor reactants and products mixtures; and K, mixture critical points.

454 Chapter 13

In this Chapter the group contribution equations of states are proposedbecause of their predictive capacity for components for which thermodynamicdata is either limited or non-existent. MHV2 and Group Contribution Asso-ciation equation of state are both suitable models to describe highly non-idealmixtures. The use of MHV2 is more attractive due to its mathematical sim-plicity and wider parameter table; however, Group Contribution Associationequation of state is more appropriate for mixtures that have association effectsand or big size-asymmetry. In this chapter several applications of both modelsto reactive mixtures have been given.Besides phase equilibria, volumetric properties play an important role in the

design of near-critical reactors. This type of information is seldom available inthe literature. Cubic equations of state with the volume correction proposed byPeneloux et al.107 can give predictions accurate enough for design purposes.Finally, it was shown that the use of phase equilibrium engineering tools

provides a good understanding of reaction processes, giving an insight onprocess limitations and potentials.

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458 Chapter 13

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CHAPTER 14

Applied Non-EquilibriumThermodynamics

SIGNE KJELSTRUPa, b AND DICK BEDEAUXa, b

aDepartment of Chemistry, Norwegian University of Science and Technology,NO-7491 Trondheim, Norway; bDepartment of Process and Energy, DelftUniversity of Technology, Leeghwaterstraat 44, 2628 CA Delft,The Netherlands

14.1 Introduction

Non-equilibrium thermodynamics describes all kinds of transport processes.This chapter must focus on a few, namely transport of heat and mass inhomogeneous and heterogeneous systems, in the absence or presence ofchemical reactions. This introduction gives a brief history of the field, a listof good reasons for why the field is important, and a discussion of a basicassumption (See section 14.1.2). We then proceed to examples of applications inthe three sections that follow.The field resulted from the work of many scientists with the objective to find

a more useful formulation of the second law of thermodynamics than thefamiliar inequality DSZ 0. The effort started in 1856 with Thomson’s studies ofthermoelectricity.1 Onsager is, however, considered as the founder of the fieldwith his papers2 from 1931, see also his collected works,3 because he put earlierresearch by Thomson, Boltzmann, Nernst, Duhem, Jauman and Einstein intothe proper perspective. Onsager was given the Nobel prize in chemistry in 1968for this work.




460

14.1.1 A Systematic Thermodynamic Theory for Transport

In non-equilibrium thermodynamics, the second law is reformulated using thelocal entropy production in the system, s, which is given by the product sum ofthe so-called conjugate fluxes, Ji, and forces, Xi, in the system. Using theassumption of local equilibrium, the second law becomes

s ¼Xi

JiXi � 0: ð14:1Þ

In non-equilibrium thermodynamics, we first need to choose a complete set ofindependent extensive variables, ai. The fluxes and forces are then determinedfrom

Ji ¼ dai=dt; Xi ¼ dS=dai: ð14:2Þ

In eq 14.2, S is the entropy of the system and t is the time. Several equivalentvariable choices can be made. The choice is usually made for practical reasons.The fluxes and forces are linearly related:

Ji ¼Xj

LijXj : ð14:3Þ

In eq 14.3, Lij are the Onsager phenomenological coefficients (conductivities).The linear nature of these equations means that the Onsager coefficients do notdepend on the fluxes or the forces. They generally do depend on the variables,ai, however. Examples are shown in Figures 14.3 to 14.5. As a consequence, theresulting descriptions of processes in the system are not linear. For furtherclarification, the reader should refer to the preface of the Dover edition of themonograph by de Groot and Mazur.4

Onsager2 assumed, in the so-called regression hypothesis, that the rate lawswere the same on the macroscopic and the fluctuation level. Making use of theprinciple of microscopic reversibility, he proved that the conductivity matrix ineq 14.3 is symmetric:

Lij ¼ Lji: ð14:4Þ

Without this symmetry, an isolated system will not relax towards equilibrium,but may show oscillatory behaviour.5 Following Onsager, a systematic theoryof non-equilibrium processes was developed in the 1940’s by Meixner6,7 andPrigogine.8 They obtained the entropy production for many physical problems.Prigogine received the Nobel prize in 1977 for his work on the structure ofsystems that are not in equilibrium (dissipative structures), and Mitchell theyear after for his use of the driving-force concept for transport processes inbiology.9

The first books were written by Denbigh10 and Prigogine.11 The most com-plete description of classical non-equilibrium thermodynamics is still the 1962

461Applied Non-Equilibrium Thermodynamics

monograph of de Groot and Mazur.4 Haase’s book12 contains many experi-mental results for systems that experience temperature gradients. Katchalskyand Curran13 developed the theory for biological systems. Their analysis wascarried further by Caplan and Essig,14 and Westerhoff and van Dam.15 Thebook of Førland, Førland and Ratkje,16 which gave various applications inelectrochemistry, biology and geology, presented the theory in a form suitablefor chemists. In 1998, Kondepudi and Prigogine17 presented a first integratedapproach to the teaching of equilibrium and non-equilibrium thermodynamics.An introduction to non-equilibrium thermodynamics for engineers is given byKjelstrup, Bedeaux and Johannessen.18 Demirel19 produced a recent text aboutthe field. Ottinger addressed the non-linear regime in his book.20 A book byKjelstrup and Bedeaux5 extended non-equilibrium thermodynamics to het-erogeneous systems. An excellent overview of the various extensions of non-equilibrium thermodynamics was given by Muschik et al.21

Classical thermodynamics deals with equilibrium states. Entropy changes arethen calculated via reversible processes. By contrast, non-equilibrium thermo-dynamics deals with systems that are not in global equilibrium. The entropyproduction can then be calculated from actual fluxes and forces. Real systems,for instance in biology or in industry, are not in equilibrium and are of coursemore interesting. Transport phenomena are always irreversible, and we shallsee how they are contained in non-equilibrium thermodynamics. The list a to ebelow gives the main reasons for why non-equilibrium thermodynamics isimportant.

a. The theory gives an accurate description of coupled transport pro-cesses. Many processes are adequately described only if couplingamong fluxes is taken into account. Water transport, that accom-panies transport of electric charge in polymer electrolyte fuel cells, is atypical example of coupled transport. Transport of heat and charge insuch cells does not occur without the transport of water. The simpleflux equations of Fick, Fourier, and Ohm do not describe this sce-nario. The coupling of heat and mass transport across phase bound-aries is significant in magnitude and leads to inconsistencies in thethermodynamic description, if neglected, see subsection 14.2.3 and ref.5 for more examples.

b. A framework is obtained for definition of experiments. A transportproperty is well defined only when its measurement is unique. Thethermal conductivity at uniform concentration is, for instance, dif-ferent from the thermal conductivity at zero mass flow.5,12 Non-equilibrium thermodynamics defines each measurement and gives arelation between the two coefficients. The theory is thus indispensablefor definition of membrane transport properties.16 From the theory wecan understand whether a transport property measured under onecondition, is usable under other conditions or not.

c. The theory quantifies the lost work done everywhere in the process.The entropy production defines the dissipated energy or lost work.

462 Chapter 14

According to the Guy-Stodola theorem, recognized by Denbigh,10,22

the lost work (wlost) per volume is given by:

wlost ¼ T0

ZV

sdV : ð14:5Þ

The lost work and the ideal work define the energy efficiency of aprocess; see Box 1 for further explanation. We can calculate theentropy production per volume from knowledge of the fluxes andforces in the system (i.e., the transport properties). We integrate theproduct sums over the volume V of the system and multiply with thetemperature of the surroundings, T0, to find the lost work. The effi-ciency defined from the second law of thermodynamics, is alwaysmeaningful, as it is always less than unity. A completely reversibleprocess has unit efficiency, but this is not a realistic target, as we pointout in section 14.4.3. The efficiency is a measure of the degree ofreversibility of a process, and we can find the maximum efficiency byminimizing eq 14.5. section 14.4.3). First-law efficiencies may be largerthan unity, even infinitely high, and do not instruct us on how toimprove the process.

d. The theory allows us to verify the thermodynamic model is consistent.For stationary states, the local entropy production integrated over thevolume of the system (see Box 1) is equal to the entropy flux out minusthe flux in:

dS=dtð Þirr¼ZV

sdV ¼ _Sout � _Sin: ð14:6Þ

This relation gives two routes to the same quantity. The left-hand sidecalculates the entropy production from actual transport properties,fluxes and forces inside the system, via eqs 14.1 and 14.3. The right-hand side calculates the entropy production from flows of entropyinto and out of the surroundings. When the two calculations give thesame result, the data and the model (eqs 14.1 and 14.3) are consistentwith the second law. Absence of agreement must lead to modelimprovements. An example that illustrates this calculation is given insection 14.4.2.

e. The theory gives the same systematic framework for nano-scale phe-nomena. Recent advances of non-equilibrium thermodynamics intothe mesoscopic domain23�25 allow us to give a thermodynamic basis tophenomena which occur on smaller time and length scales. The basicidea is to introduce internal variable(s), which do not equilibrate onthese shorter scales. The macroscopic description is then found byintegration over the internal variables. As chemical reactions are fre-quently activated, this development can be useful for descriptions of


chemical reactors. The degree of reaction is then the internal variable,and the reaction rate is a linear function of the chemical driving forcealong the reaction coordinate, as discussed in Section 14.3.2.

14.1.2 On the Validity of the Assumption of Local Equilibrium

A central assumption in non-equilibrium thermodynamics is the assumption oflocal equilibrium. This assumption states that the normal thermodynamicrelations are valid in a volume element of the system. So, when is it possible tofind ensemble averages, or proper time averages of the thermodynamic vari-ables, in such volume elements? Generally speaking, the volume element shouldbe large compared to molecular sizes and small compared to the macroscopicdistances involved, and the time averaging should be over a sufficiently longperiod. In order to find a more precise meaning of the words large and small, wehave studied systems that were exposed to severe temperature gradients.26 Non-equilibrium molecular dynamics simulations, which are not biased by thethermodynamic theory, were used in the tests. Some results are summarized inTable 14.1. We found that for stationary states an accurate determination of athermodynamic property P is possible in rather small volume elements, with asurprisingly small number of particles, N. The particle fluctuation need not besmall (see the last column of Table 14.1). For stationary states, we found that a

Box 1. The Energy Efficiency According to the Second

Law

The energy available to do work on a process per unit of time Dt is ideallyequal to

wideal ¼E

Dt¼ DU

Dtþ p0

DVDt� T0

DSDt

where E is the exergy of the system, and the other symbols are thoseadopted by IUPAC, for example, see below eq 14.7. In reality, morework must be done during the time Dt to overcome the entropy production,giving

w ¼T0ðdS=dtÞirr þDUDtþ p0

DVDt� T0

DSDt

¼wlost þ wideal

The lost work is the difference between these expressions, leading to thedefinition of the second law efficiency: wideal/w. Maximum efficiency meansminimum entropy production.

464 Chapter 14

volume element need only be a fraction of a nanometre, or include around 10particles, to satisfy the condition of local equilibrium.26 The presence of afast chemical reaction (row four of Table 14.1) does also not affect the validityof the assumption in spite of large fluctuations in temperature and particlenumber, see Section 14.3 for further evidence. The potential energy forinteraction of argon with zeolite varies considerably over short distances.Nevertheless, it was possible; as shown in Figure 14.1, to establish that thethermodynamic properties in the system at a given temperature were the same,in the absence and presence of a temperature gradient (last line of Table 14.1);additional evidence can be found elsewhere.27–32 Local equilibrium does notnecessarily imply chemical equilibrium. Chemical equilibrium imposes anadditional restriction on the system, as discussed in section 14.3. A differencebetween local equilibrium and global equilibrium can be found in the corre-lation functions.33 These are no longer represented by their equilibriumexpressions in a system that is not in global equilibrium.

14.1.3 Concluding Remarks

Non-equilibrium thermodynamics is a systematic theory of transport, whichsatisfies the second law, because the transport equations are always derivedfrom the entropy production. When forces and fluxes are constructed accordingto eq 14.2, the theory gives relations which are linear. The theory can alsohandle non-linear processes, discussed in 14.3.2. When the prescription to findthe entropy production is followed, the advantages listed in a to e above follow.

Figure 14.1 The potential energy Up as a function of distance from the crystal surfacel for the interaction of argon with silicalite-1 inside silicalite pores.


To start from the dissipation function, may lead to erroneous results, see Ref 5pages 55 and 56. The assumption of local equilibrium is sound, as tested byapplications of temperature gradients much larger than realizable in industrialcontexts (cf. Table 14.1).

14.2 Fluxes and Forces from the Second Law of

Thermodynamics

Fluxes and forces are derived from the entropy production in a volume elementas follows:

a) Formulate the Gibbs relation and find the time rate of change of theentropy density.

b) Substitute the laws of mass, momentum and energy conservation intothe Gibbs relation.

c) Identify the entropy flux and the entropy production by comparing theresult with the entropy balance equation.

The derivation of the equations for homogeneous systems can be found inmost of the literature cited. Kjelstrup and Bedeaux5 have given expressions forheterogeneous systems. We shall apply both sets of equations to both systems,considering transport of heat and mass at mechanical equilibrium.As an example, we take the heterogeneous system studied by Inzoli et al.34

and illustrated in Figure 14.2. The system consisted of a microporous zeolitephase (silicalite-1) in contact with butane gas. The crystal and the gas phase areshown in the lower part of Figure 14.2, while the upper part shows the con-centration profile of butane through the heterogeneous system at equilibrium.We see the constant average concentration of the gas adsorbed inside, and the

Table 14.1 Characteristic properties of five selected systems in a temperaturegradient, as studied by molecular dynamics simulations.The systems’ particles obey equipartition of kinetic energy for alldirections and have linear flux-force relations. The quantity P is athermodynamic property of a volume element with thickness l.The temperature is T, and N is the particle number. The ratio DP/P gives the uncertainty in the determination of an equilibriumproperty in the element, while d denotes the fluctuation in theproperty (adapted from Ref. 26).

System (dT/dx)/K �m�1 l/nm N DP/P dT/T dN/N

Binary mixture 108 0.18 16 0.03 0.03 0.28Vapour-liquid, Ar 3 � 108 0.34 10 0.05 – –Vapour-liquid, octane 109 0.13 8 0.05 – –Fast chemical reaction 6.6 � 1011 0.12 8 – 0.31 0.37Ar adsorbed in zeolite 8.7 � 109 0.34 18 0.03 0.20 0.18

466 Chapter 14

low gas concentration outside. The surface is determined by excess variablesaccording to Gibbs5 that are introduced in Section 14.2.2. Ref 5 gives defini-tions and further explanation.

14.2.1 Continuous Phases

In order to obtain a proper set of fluxes and forces for the continuous phases,(e.g. the silicalite or gas phase) we start with Gibbs’ equation for the system:

dUm ¼ TdSm � pdVm þXCj¼1

mjdnj ; ð14:7Þ

where Um is the molar internal energy, T the temperature, p the pressure andVm the molar volume, as recommended by IUPAC. Furthermore nj the amount

Figure 14.2 The number of butane molecules, n, as a function of the distance fromthe crystal surface l. The average concentration is plotted inside thecrystal. –––– (solid black line), results for a flat surface; and solid greyline, results for a zig-zag textured surface. Reprinted with permission ofJ. Phys. Chem. B.34


of substance and mj the molar chemical potential of component j. There are Ccomponents. By integration with constant intensive variables the internalenergy becomes

Um ¼ TSm � pVm þXCj¼1

mjnj : ð14:8Þ

The summation is carried out over all C independent components. We need alocal description, so we introduce the densities per unit of volume; s¼Sm/Vm,u¼Um/Vm, cj¼ nj/Vm, which gives, using also eq 8:

du ¼ TdsþXCj¼1

mjdcj : ð14:9Þ

The time rate of change of the entropy density is thus:

@s

@t¼ 1

T

@u

@t� 1

T

XCj¼1

mj@cj@t: ð14:10Þ

Energy conservation gives for a system in mechanical equilibrium:

@u

@t¼ �div Jq; ð14:11Þ

where Jq is the internal energy (or total heat) flux. Mass conservation in thepresence of one chemical reaction gives:

@cj@t¼ �divJ j þ njr: ð14:12Þ

Here Jj is the mass flux of j, r is the rate of the chemical reaction and nj is thestochiometric coefficient. We introduce the last equations into eq 14.10, andcompare with the entropy balance

@s

@t¼ �divJ s þ s: ð14:13Þ

The entropy flux, Js, and the entropy production, s, are then found to be:

Js ¼1

TJq �

XCj¼1

mjJ j

!; ð14:14Þ

and

468 Chapter 14

s ¼ Jq�grad1

T

� ��XCj¼1

J j�gradmjT

� �� r

DrG

T: ð14:15Þ

The dot indicates a scalar product between two vectors and DrG ¼PCj¼1njmj is the

reaction Gibbs energy. Alternatively; we can introduce the measurable heat flux

via, Jq ¼ J 0q þPCj¼1

J jHj, and write eq 14.15 in the form:

s ¼ J 0q�grad1

T

� �� 1

T

XCj¼1

Jj�gradmj;T � rDrG

T: ð14:16Þ

In eq 14.16 the subscript T of the chemical potentials indicates that the deri-vative is taken keeping the temperature constant. The last form is convenientbecause it can be related to experimental results. All variables are measurableor can be calculated from experimental data. The force-flux relations were

grad1

T¼rqqJ 0q þ

XCm¼1

rqmJm;

� 1

Tgradmj;T ¼rjqJ 0q þ

XCm¼1

rjmJm and

�DrG

T¼ rrrr:

ð14:17Þ

The first two forces are vectors, while the last is a scalar, so they do not couple(The Curie principle). Possible choices for the frame of reference4 are the centreof mass, the average volume, the average molar and the solvent velocity. Inheterogeneous systems, the natural frame of reference is the surface.5

Inzoli et al.34 determined the resistivities, rij, in eq 14.17 for transport ofbutane inside silicalite-1 at T¼ 400K. The resistivities to heat transfer, masstransfer and the coupling coefficient are shown in Figures 14.3a to c. Theresistivity to mass transfer changed linearly with the concentration, the numberof molecules in the unit cell (m.u.c), as shown in Figure 14.3a. This coefficientwas almost independent of temperature. The resistivity to heat transport(shown in Figure 14.3b) and the coupling coefficient (shown in Figure 14.3c)depended on temperature, but not significantly on the butane concentrationinside silicalite-1. It was verified34 that the sum of the measurable heat transferand the partial molar enthalpy of the butane in silicalite-1 was independent ofthe temperature, and that the sum of the heat transfer and partial molarenthalpy of butane was constant,


q�mðcmÞ þHmðcmÞ ¼ C; ð14:18Þ

at a given temperature. This relation may be useful for estimates of the heattransfer. It is not possible from eq 14.17, to conclude that heat flux is smallfrom a small temperature gradient only. A large mass flux can contribute bycarrying a large heat transfer.

Figure 14.3a The resistivity rbb to transfer of butane as a function of the number ofmolecules in a unit cell Cm in a chemical potential gradient inside sili-calite at T¼ 400K. Reprinted with permission of J. Phys. Chem. B.34

Figure 14.3b The resistivity rqq to heat transfer of butane inside silicalite-1 as afunction of temperature Tm at zero mass flux and Cm¼ 3.89 moleculesper unit cell. ––––, linear representation of the data. Reprinted withpermission of J.Phys.Chem. B.34

470 Chapter 14

A test was made on the statement below eq 14.3, that the coefficients areindependent of the driving forces. The results in Figure 14.4 show that this istrue for the coefficient in Figure 14.3b. The resistivity to heat transfer is not afunction of the temperature difference across the system.

Figure 14.3c The coefficient for coupled heat transport rqb for butane inside silicalite-1 as a function of temperature Tm for Cm¼ 3.89 molecules per unit cell;––––, linear representation of the data. Reprinted with permission of J.Phys. Chem. B.34

Figure 14.4 The thermal resistivity rqq of silicalite with adsorbed butane takenfrom Figure 14.3b, as a function of the driving forces the temperaturegradient of (TA�TB). The resistivity at T¼ 400K does not depend onthe temperature gradient applied in the investigation and confirmseq 14.3; ––––, linear representation of the data. Reprinted with permis-sion of J. Phys. Chem. B.34


A chemical reaction is normally not described by the linear relation in thethird line of eq 14.17. The rate is a non-linear function of DrG/T on the mac-roscopic level, and the law of mass action is used. We explain in subsection14.3.3 how chemical reactions far from equilibrium also can be included intothe scheme of non-equilibrium thermodynamics, cf. also point e of section14.1.1.

14.2.2 Maxwell-Stefan Equations

An immediate consequence of non-equilibrium thermodynamics is that trans-port in multi-component systems can be described in a homogeneous phasewith the so-called Maxwell-Stefan equations. These equations can be cast in away that obeys Onsager relations, and such that the transport coefficients canbe estimated from coefficients for binary systems. We proceed to give theseequations because of their potential usefulness.We first note that mechanical equilibrium imposes a restriction on the driving

forces through the Gibbs-Duhem equationPj

cj gradmj;T ¼ 0. We also note that

the Onsager relations can be used for the resistivity matrix in eq 14.17, when thethermodynamic forces depend on each other, as discussed in Ref 4, Chapter VI,y3 , meaning that:

rmq ¼ rqm and rmj ¼ rjm ð14:19Þ

By using Gibbs-Duhem’s equation, it follows that the sum of the terms on theleft hand side of the second line of eq 14.17 times cj is zero. As the second line ofeq 14.17 holds for an arbitrary measurable heat and molar fluxes it follows thatthe resistivities in the matrix are dependent, and satisfy

XCj¼1

cjrjq ¼ 0 andXCj¼1

cjrjm ¼ 0: ð14:20Þ

From the Onsager relations, it also follows that

XCj¼1

cjrqj ¼ 0 andXCj¼1

cjrmj ¼ 0: ð14:21Þ

Once the C(C� 1)/2 independent resistivities have been obtained fromexperiments, the others can be calculated using the above relations. Theresistivity matrix has an eigenvalue equal to zero, and thus a zero determinant.It can therefore not be inverted into a conductivity matrix without first elim-inating all linear dependencies among the coefficients.57 In order to relate theresistivities to better known transport coefficients we write eq 14.17 in the

472 Chapter 14

following form:

gradT ¼� 1

lJ 0q �

XCm¼1

q�mJm

!; and

1

Tgradmj;T ¼�

q�jT2

gradT �XCm¼1

RjmJm;

ð14:22Þ

where the thermal conductivity, the measurable heat transfer and the resistiv-ities for component fluxes at constant temperature are defined by:

l � 1

T2rqq; q�j �

J 0qJ j

� �@T=@x¼0;Jm¼0

¼ � rqj

rqqand

Rjm � rjm �rjqrqm

rqqx:

ð14:23Þ

The measurable heat transfer and the resistivities to mass flow at constanttemperature satisfy, (given by eqs 14.20 and 14.21),

XCj¼1

cjq�j ¼ 0 and

XCj¼1

cjRjm ¼XCj¼1

cjRmj ¼ 0: ð14:24Þ

The measurable heat transfer is related to the thermal diffusion, and the Dufourand the Soret coefficients.5 By using eq 14.24 in eq 14.22, one may verify thatthe right (and therefore the left) hand sides are independent of the choice of theframe of reference for the fluxes.The Maxwell-Stefan diffusion coefficients, Djk, are defined byi

Rjm � �R

cDjmfor j 6¼ m: ð14:25Þ

The diagonal coefficients Rmm are found using eq 14.24. We see that the dif-fusion coefficients are symmetric, Djk¼Dkj. In order to obtain the Maxwell-Stefan equations for multi-component diffusion, we introduce the velocitiesvm¼ Jm/cm. Using eq 14.24, we can write eq 14.22 in the form

gradT ¼� 1

lJ 0q �

XC�1m¼1

q�mcm vm � vnð Þ !

;

1

RTgradmj;T ¼�

q�jRT2

gradT �XCm¼1

xm

Djmvm � vj�

for jon:

ð14:26Þ

iWe follow Krishna and Wesselingh in this definition rather than Kuiken who uses the pressure pinstead of cRT.


The equation for grad mn,T follows directly from the Gibbs-Duhem equationand is not given here. Only velocity differences enter in eq 14.26 and thedescription is thus independent of the frame of reference. The diffusion coef-ficients are symmetric and they contain in essence binary interdiffusion coeffi-cients. Such coefficients are available from experiments.35 They are found to besurprisingly independent of the concentrations of the various components.35,36

The Maxwell-Stefan equations give, therefore, a convenient way to describemulti-component diffusion.5,35–38

14.2.3 Discontinuous Systems

When heat and mass are transported across heterogeneous systems, the inter-face may pose a barrier to transport. In Figure 14.2 this happens with lowbutane concentration. Governing equations are needed for the interface, asthese equations give boundary conditions for the transport processes in thehomogeneous phases on each side. The boundary equations are determinedfrom the excess entropy production for the interface.5

Consider transport into and through a flat surface. Excess fluxes along thesurface are two-dimensional vectors. Though very interesting, they will not beconsidered here. The fluxes in the homogeneous phases (described above) arenormal to the surface, and these normal components are scalars. This has animportant consequence: the normal components of heat and mass fluxes coupleto the driving force for the chemical reaction at a surface.For a two-dimensional surface, as described by the excess densities intro-

duced by Gibbs, in a thermodynamic system the Gibbs equation is:

dUs ¼ T sdSs þ gdOþXCj¼1

msjdnsj ; ð14:27Þ

where Us, S s and nsj are the excess internal energy, entropy and amount ofsubstance of component j. Furthermore T s is the temperature, g the surfacetension, O the surface area and msj the molar chemical potential of component j.Superscript s denotes the surface. By integration with constant intensive coef-ficients one obtains for the excess internal energy:

Us ¼ T sSs þ gOþXCj¼1

msj nsj : ð14:28Þ

The local description uses excess densities per unit of surface area, ss¼Ss/O,us¼Us/O, Gj¼ nsj /O, giving:

dus ¼ T sdss þXCj¼1

msjdGj : ð14:29Þ

474 Chapter 14

In the system shown in Figure 14.2, the excess concentration can be negative,when the surface concentration is smaller than the concentration in the silica-lite, as indicated for two surfaces. The time rate of change of the entropydensity is thus:

dss

dt¼ 1

T s

dus

dt� 1

T s

XCj¼1

msjdGj

dt: ð14:30Þ

Straight derivatives are used, since there is no position dependence of thesurface variables. Energy conservation (in the absence of electric fields) for thesurface as a discrete system is:

dus

dt¼ �Jo

q þ J iq: ð14:31Þ

Superscript i denotes the homogeneous phase for xo0, while o denotes thehomogeneous phase for x40. The fluxes are the values of the normal com-ponents of the fluxes in the homogeneous phases. Mass conservation in thepresence of chemical reactions is given by:

dGj

dt¼ �Jo

j þ J ij þ nsj rs: ð14:32Þ

A chemical reaction takes place in the surface with a reaction rate rs where nsj isthe stochiometric coefficient. Substituting eqs 14.31 and 14.32 into eq 14.30,and comparing with the entropy balance:

ds

dt¼ �Jo

s þ J is þ ss; ð14:33Þ

one finds for the excess entropy production of the surface is:

ss ¼J iq

1

T s� 1

T i

� �þ Jo

q

1

To� 1

T s

� �

�Xj

J ij

msjT s�

mijT i

!�Xj

Joj

mojTo�

msjT s

� �� rs

DrGs

T s;

ð14:34Þ

where DrGs ¼

PCj¼1nsjm

sj : The entropy production for the surface contains double

the number of heat and component flux terms as did eq 14.15, since we mustdistinguish between fluxes coming in from xo0 and going out to x40. Thisform of the entropy production takes into account discontinuities in the fluxes


at the interface. The measurable heat fluxes are given by

J iq ¼ J 0

iq þ

Xj

J ijH

ij and Jo

q ¼ J 0oq þ

Xj

Joj H

oj : ð14:35Þ

They will in general differ substantially between the two sides, because of theenthalpy changes at the interface. Replacing the energy fluxes with the mea-surable heat fluxes with eq 14.35, we obtain:

ss ¼J 0iq1

T s� 1

T i

� �þ J 0

oq

1

To� 1

T s

� �

� 1

T s

Xj

J ij msj � mij T

sð Þn o

� 1

T s

Xj

Joj moj T sð Þ � msjn o

� rsDrG

s

T s;ð14:36Þ

where the chemical potentials are now all calculated at the temperature of thesurface.Experiments are often done using stationary-state conditions. In that

case, it follows from eqs 14.32 and 14.35 that Joj ¼ Jijþ nsjrs and

J 0oq ¼ J 0iq �Pj

J ijDHj�

Pj

nsjHoj r

s. These relations make it possible to eliminate

the fluxes on the o side in the entropy production given in eq 14.36. In this wayone obtains the following alternative equation:

ss ¼J 0iq1

To� 1

T i

� �� 1

To

Xj

J ij moj � mij T

oð Þ� �

� 1

TorsXj

nsjmoj ;

ð14:37Þ

where mij(To) means that the chemical potentials are calculated at the local

concentrations of all components in the i phase at the temperature To. Thistemperature is easier to measure. Eliminating the fluxes on the i side one findssimilarly

ss ¼ J 0oq

1

To� 1

T i

� �� 1

T i

Xj

Joj moj T i

� � mij

n o� 1

T irsXj

nsjmij ; ð14:38Þ

where similar definitions are used for moj (Ti). The number of independent force-

flux pairs has been reduced from 2Cþ 3 to Cþ 2. Another simplification is thatneither of the two expressions for the entropy production in eq 14.37 containsthe temperature or the chemical potentials of the surface. In particular this isthe case for the stationary state reaction term, in which the reaction Gibbs

energy DrGs;k �

Pj

nsjmkj for k¼ i or o, contains the chemical potentials in the i

or o phase right outside the surface rather than those of the surface.

476 Chapter 14

The resulting force-flux relations for the film, when we use the measurableheat flux on the i side, are

1

To� 1

T i¼rs;iqqJ 0

iq þ

XCm¼1

rs;iqmJim þ rs;iqrr

s;

�moj � mijðToÞ

To¼rs;ijq J 0

iq þ

XCm¼1

rs;ijmJim þ rs;ijr r

s; and

�DrGs;oðToÞTo

¼rs;irqJ 0iq þ

XCm¼1

rs;irmJim þ rs;irr r

s;

ð14:39Þ

for j¼ 1, . . . ,C. When we use the measurable heat flux on the o side, we obtain

1

To� 1

T i¼rs;oqq J 0

oq þ

XCm¼1

rs;oqmJom þ rs;oqr r

s;

�moj ðT iÞ � mij

T i¼rs;ojq J 0

oq þ

XCm¼1

rs;ojm Jom þ rs;ojr r

s; and

�DrGs;iðT iÞT i

¼rs;orq J 0oq þ

XCm¼1

rs;ormJom þ rs;orr r

s:

ð14:40Þ

The Onsager relations39 make both matrices symmetric, but unfortunatelythere are still many coefficients to be determined and little work has been doneto determine them. Inzoli et al.34,40 reported coefficients for heat and masstransfer (coefficients in the first two lines in eqs 14.39 and 14.40) for butaneadsorption from the gas phase to the zeolite phase. The system, shown inFigure 14.2, was investigated with a flat and a zig-zag textured interface. Theauthors determined the coefficients rs,iqq, r

s,iqb and rs,ibb and coefficients derived from

these, for eqs 14.41 to 14.43.Figure 14.5a shows the resistivity to heat transfer, rs,iqq, this is an overall

resistivity, of the first line of eq 14.41. The large value at low gas pressures p canbe related to negative excess surface concentrations. The surface may be rate-limiting to heat transport at such conditions. Figure 14.5b shows the resistivityto mass transfer on the gas side, Rs,i

bb, for isothermal conditions, that is thesecond line of eq 14.42. The coupling coefficients for the transfer of heat andmass for the two sides of the surface were divided by the thermal resistivity togive the heat transfer, of eq 14.43 below. Both heat transfers, shown in Figure14.5c, are larger than corresponding values for homogeneous phases andconfirm eq 14.44 which states the difference of the two coefficients is theenthalpy of adsorption and in this case results in a value of -52 kJ �mol�1. Thesecoefficients give a new route to determine the enthalpy of adsorption, analternative to a plot of lnp versus 1/T. In modelling of transport across aheterogeneous system like that in Figure 14.2, we need results like those inFigures. 14.5 and 14.3a through 14.3c plus gas-phase coefficients (not shown).


Kuhn et al.41 modelled the transport of heat and water across a silicalitemembrane with estimates of such data.All fluxes in eqs 14.39 and 14.40 are scalar, so all processes couple, unlike in

eq 14.17. The chemical reaction can therefore drive heat and mass fluxesthrough the surface and vice versa. This fact has not been used in a systematicmanner before. A simple application is discussed in section 14.3.3. The reaction

Figure 14.5a The resistivity to energy transfer in the form of heat rqq at zero massflux from the zeolite surface as a function of butane gas pressure p/p

�J

where p�J ¼ 0.1MPa.B, Ts¼ (362� 3)K; &, Ts¼ (382� 5)K; and *,

Ts¼ (402� 4)K. Figure reprinted with permission of Mesoporous andMacroporous Materials.40

Figure 14.5b The resistivity to mass transfer rqq at zero mass flux from the zeolitesurface as a function of butane gas pressure p/p

�J where p�J ¼ 0.1MPa.

B, Ts¼ (367� 3)K;&, Ts¼ (389� 5)K; and *, Ts¼ (408� 4)K. Figurereprinted with permission of Mesoporous andMacroporousMaterials.40

478 Chapter 14

Gibbs energy in eq 14.40 refers to the temperature next to the catalyst. Thistemperature may not be the same as the temperature of the surface.42 Inbiology, active transport means that an ion or a compound is transportedagainst its chemical potential gradient with the help of energy from a sponta-neous chemical reaction.25,43 We see from eqs 14.39 and 14.40, that a positivecoupling coefficient, rs,ijr , can maintain temperature and chemical potentialdifferences, a situation that is not possible in an isotropic homogeneous phase.Conservation equations and Gibbs-Helmholtz equation gives relations

between these coefficients:

rs;oqq ¼rs;iqq � rsqq; rs;oqm ¼ rs;omq ¼ rs;iqm þ DHmr

sqq;

rs;oqr ¼rs;orq ¼ rs;iqr þ DrHs;irs;iqq þ

XCm¼1

rs;iqmnsm;

rs;ojm ¼rs;omj ¼ rs;ijm þ DHmr

s;ijq þ DHjr

s;iqm þ DHjDHmr

sqq;

rs;ojr ¼rs;orj ¼ rs;ijr � DHjr

s;iqr � DrH

s;i rs;ijq þ DHjrs;iqq

� �þXCm¼1

rs;ijm þ DHjrs;iqm

� �nsm; and

rs;orr ¼rs;irr þ 2DrHs;irs;irq � DrH

s;i� 2

rs;iqq � 2XCj¼1

rs;ijr þ DrHs;irs;ijq

� �nsj þ

XCj;m¼1

rs;ijmnsm;

ð14:41Þwhere DHm�Ho

m�Him and DrH

s;i �PCm¼1

nsmHim.

Figure 14.5c The energy of transfer in the form of heat for the whole surface q*s as afunction of butane gas pressure p/p

�J where p�J ¼ 0.1MPa. The energy

flux from the zeolite side is the lower data at q*so10 while the energyflux on the gas side is the upper data at q*s450. The difference betweenthe enthalpies of transfer gives the enthalpy of adsorption. B,Ts¼ (36� 4)K; &, Ts¼ (382� 4)K; and *, Ts¼ (402� 5)K. Figurereprinted with permission of Mesoporous andMacroporous Materials.40


The thermal and chemical driving forces in eqs 14.39 and 14.40 are differ-ences across the surface rather than gradients, like in eqs 14.17 and 14.18. Thedimension of the surface resistivity is therefore the dimension of the corre-sponding coefficients in the homogeneous phase times a length. Knowing thecoefficient in the homogeneous phase, one could multiply it with the surfacethickness and estimate the surface coefficients. This operation gives the surfacesimilar properties to the homogeneous phase, however, when the surface is abarrier to transport, the surface resistivity is large compared to the bulk valuestimes the thickness.5 A convenient form of eq 14.39 is:

DT ¼� 1

lsJ 0

iq �

XCm¼1

q�s;im Jm � q�s;ir rs

!;

moj � mijðToÞTo

¼� q�s;ij

DTT iTo

�XCm¼1

Rs;ijmJm � Rs;i

jmrs and

DrGs;oðToÞTo

¼� q�s;ir

DTT iTo

�XCm¼1

Rs;irmJ

im � Rs;i

rr rs;

ð14:42Þ

where DT�To�Ti. Equation 14.40 can be rewritten in a similar form. Thethermal conductivity, the measurable heat transfer and the resistivities to massflow of the interface are defined by:

ls � 1

T iTorsqq; q�s;km �

J 0kqJm

" #DT¼0;Jk

l¼rs¼0

¼ �rs;kqm

rs;kqq; q�s;kr �

J 0kqrs

" #DT¼0;Jkm¼0

¼ �rs;kqr

rs;kqq;

Rs;kjm � rs;kjm �

rs;kjq rs;kqm

rs;kqq; Rs;k

jr � rs;kjr �rs;kjq r

s;kqr

rs;kqq; Rs;k

rr � rs;krr �rs;krq r

s;kqr

rs;kqqfor k ¼ i or o:

ð14:43Þ

The thermal conductivity ls has the dimension of the bulk thermal conductivitydivided by a length. The heats of transfer have the same dimensionality as theyhave in the bulk regions. It follows from eqs 14.41 and 14.43 that

q�s;om � q�s;im ¼ �DHm and q�s;or � q�s;ir ¼ �DrHs;i �

XCm¼1

q�s;im nsm

¼ �DrHs;o �

XCm¼1

q�s;om nsm: ð14:44Þ

The enthalpy difference between two phases is normally large. This leads tolarge heat effects at surfaces as shown in Figure 14.5c. It is a commonassumption to neglect the heat transfer at the surface. Equation 14.44 showsthat this assumption has large consequences for the heat flux calculations at the

480 Chapter 14

interface and it makes a proper description of the thermodynamic properties ofthe system impossible.


We have seen in sections 14.2.1 to 14.2.3 how fluxes and forces can be derivedfrom the entropy production. In the description of heat and mass transport inheterogeneous systems, we have shown that it is necessary to include couplingcoefficients, as neglect of these can lead to erroneous results. We need non-equilibrium thermodynamics to give equations that include the essential cou-pling of the fluxes. The laws of Fick and Fourier are not enough. A large effortis needed to determine relevant interface coefficients, but the effort may also berewarding in the sense that a better understanding is obtained of the nature ofthe interface. A surface can have a negative excess adsorption and be rate-limiting for heat and/or mass transfer. Information on the exact surface tem-perature is important in catalysis.At a surface, the normal components of vectorial fluxes are scalar. Differ-

ences in temperatures and chemical potentials across or with a surface cantherefore drive a chemical reaction in heterogeneous catalysis. Chemical reac-tions can similarly drive heat and mass transport into and through a surface.This has not been studied before. A chemical reaction will lead to changes in theconcentrations (see eq 14.12) and thereby modify the chemical potentials. Inthis manner a chemical reaction may also modify heat and mass transport inhomogeneous systems. This will be discussed further in subsection 14.3.2.

14.3 Chemical Reactions

Chemical reactions have long been regarded as outside the regime of non-equi-librium thermodynamics. However recent developments may alter this view.23–25,43 Grossly speaking, there are two regimes for chemical reactions. When theGibbs energy of the reaction is small compared to RT, the reaction is described bythe third line of eq 14.17. This is the case when the reaction is close to equilibrium.We discuss such a system in subsection 14.3.1, and show that the presence of achemical reaction changes the transport coefficients in the system. When thereaction is far from chemical equilibrium the third line of eq 14.17 is notappropriate. The progress of a chemical reaction can be measured by the degree ofreaction,A. This can be done in a description on the mesoscopic level discussed insection 14.3.2.This is a prerequisite for energy efficiency optimizations of chemical reactors,

discussed in section 14.4.

14.3.1 Thermal Diffusion in a Reacting System

Consider a box with a chemical reaction, and with a temperature gradient in thex-direction. As a consequence, all vectorial fluxes inside the box are in the


x-direction only. The reaction we take as:

2AÐB: ð14:45Þ

We shall apply equations in section 14.2.1 in a study of the transport coeffi-cients. The system was studied by molecular dynamics simulations to providemolecular insight.44–46 The entropy production from eq 14.17 is

s ¼ J 0qd

dx

1

T

� �� 1

TJA

dmA;Tdxþ JB

dmB;Tdx

� �� r

DrG

T: ð14:46Þ

For a stationary state, the reaction and the closed system impose the followingmass balance:

JA ¼ �2JB: ð14:47Þ

This leads in a reduction of the expression for the entropy production:

s ¼ J 0qd

dx

1

T

� �� 1

TJB

dDrGT

dx� r

DrG

T: ð14:48Þ

The resulting force-flux equations for coupled heat and mass transport are thengiven by eq 14.17, which becomes:

d

dx

1

T¼rqqJ 0q þ rqmJB; and

� 1

T

d

dxDrGT ¼rmqJ 0q þ rmmJB;

ð14:49Þ

for the chemical reaction by the third line of eq 14.17. In the stationary state,the energy flux is constant and we have:

Jq ¼ J 0qðxÞ þ JAðxÞHA þ JBðxÞHB ¼ J 0qðxÞ þ JBðxÞDrH: ð14:50Þ

The enthalpy of reaction is DrH¼HB� 2HA.The interesting question is now related to the coefficients. We know that the

coefficients rij do not depend on the driving forces. But are they the same in thepresence and absence of the chemical reactions? To be more precise; are theresistivities in eq 14.49 affected by the presence of the chemical reaction? Inorder to answer this question, consider first the limiting case of chemicalequilibrium in the reaction in eq 14.45. This limiting case serves to illustrate apoint. With the condition DrG¼ 0, it follows that

d

dxDrGT ¼

d

dxmB � 2mAð ÞT¼ �

1

THB � 2HAð Þ dT

dx¼ � 1

TDrH

dT

dx: ð14:51Þ

482 Chapter 14

The entropy production reduces further to

s ¼ J 0q þ JBDrH� d

dx

1

T

� �¼ Jq

d

dx

1

T

� �: ð14:52Þ

The force-flux relation that follows from this expression is

d

dx

1

T¼ RqqJq ¼ RqqJ

0qðxÞ þ RqqDrHJBðxÞ: ð14:53Þ

Comparing with eq 14.49 it follows that rqq¼Rqq and rqm¼RqqDrH. It followsfor the limiting case of chemical equilibrium, that the measurable heat oftransfer is equal to the enthalpy of reaction q*¼DrH.44 A mixture of non-reacting gases has normally relatively small measurable heats of transfer (Soretcoefficients). The enthalpy of reaction is a large quantity. It is clear that thepresence of a chemical reaction can increase the coupling coefficient sig-nificantly. The molecular mechanism for the transport processes is changed bythe chemical reaction. In the absence of a chemical reaction, the thermal con-ductivity of a gas mixture can be explained, say, by kinetic theory, as energytransfer at zero net mass movement. In the presence of a chemical reaction anadditional effect appears. Heat is transferred as enthalpy of the reaction byinterdiffusion of A and B by eq 14.53. This interdiffusion takes place at zero netmass flux and is illustrated in Figure 14.6.

Figure 14.6 The interdiffusion of A and B in a temperature gradient at zero mass fluxas a function of the number of layers n. Reprinted with permission ofPhys. Chem. Chem. Phys.45


This means that transport properties of reacting mixtures (e.g., flames)cannot be modelled as properties of a mixture of gases. The heat of transfermust be included and the thermal conductivity obtains additional terms.

14.3.2 Mesoscopic Description Along the Reaction Coordinate

Many chemical reactions take place far from equilibrium. In that case thereaction rate is given by the law of mass action, which is a very non-linearfunction of the chemical driving force, DrG. In order to understand the origin ofthis law in the context of non-equilibrium thermodynamics4,47 we must describethe chemical reaction process along the mesoscopic reaction coordinate A,which varies between 0 (reactants) and 1 (products). Along the reaction coor-dinate the chemical potential is given by m(A). At the ends of the reactioncoordinate, m(0)¼ 2mA and m(1)¼ mB. The total entropy production along thereaction coordinate is given by

sr ¼ �1

T

Z10

rðAÞ dmðAÞdA

dA: ð14:54Þ

For so-called ideal systems the chemical potential can be written as

mðAÞ ¼ RT ln cðAÞ þ FðAÞ; ð14:55Þ

where c(A) is the molar density of complexes in the state characterised by thecoordinateA (see eq 14.55). Furthermore F(A) is a potential, which can dependon the temperature and the pressure in the system. Because the chemicalpotential is constant along the reaction coordinate in equilibrium it follows thatmeq¼RT ln ceq (A)þF(A).Along the reaction coordinate the progress of the reaction is hindered by a

large energy barrier in F(A). This barrier is illustrated in Figure 14.7 Due tothis energy barrier a quasi-stationary state develops for which the reaction ratebecomes constant, r(A)¼ r, along the reaction coordinate. Using this propertyand the boundary conditions we can integrate eq 14.54 to give

sr ¼ �rDrG

T; ð14:56Þ

which is the contribution given in eq 14.48.On the mesoscopic scale the flux-force relation becomes, following Onsager:

rðAÞ ¼ �lðAÞ dmðAÞdA

: ð14:57Þ

Following Kramers’ analysis48 we now use reaction rate l that is to a goodapproximation proportional to the density and introduce the following constant

484 Chapter 14

diffusion coefficient along the reaction coordinate

Dr �RTlðAÞcðAÞ : ð14:58Þ

Substituting eq 14.58 into eq 14.57, using eq 14.55 and the fact that r is con-stant, we can integrate eq 14.57 and obtain

r ¼ lrr 1� expDrG

RT

� �¼ lrr 1�

c2A;eqcB

c2AcB;eq

!; ð14:59Þ

where

lrr ¼1

rrr¼ Dr exp

2mAkBT

Z10

expFðAÞkBT

dA

24

35�1

: ð14:60Þ

Equation 14.59 is the law of mass action, (discussed by de Groot and Mazur4

[pages 226 to 232]). The expression for the entropy production, eq 14.56, is alsovalid when the rate is given by the law of mass action.We conclude that the law of mass action also has a basis in non-equilibrium

thermodynamics. According to Ross andMazur49 this also follows from kinetictheory. The reason is attributed to the distribution of the particle velocities inthe ensemble. A small deviation in the Maxwell-distribution can be tolerated, as

Figure 14.7 The activation energy F as a function of the degree of reaction A.Introduces the internal coordinate to describe chemical reactions.


long as the deviation is proportional to the chemical driving force. Such adeviation was documented for this case45,46 and does not violate the conditionof local thermodynamic equilibrium. This means that we also can expect theOnsager relations to apply at the mesoscopic level.

14.3.3 Heterogeneous Catalysis

The fact that the normal components of the vectorial fluxes are scalar andcouple to a chemical reaction at an interface (cf. eq 14.40 and 14.42), haveimplications for descriptions of heterogeneous catalysis. We demonstrate herethe application of eqs 14.40 to a simple example.Consider, as a specific example, a flat catalytic surface at x¼ 0, with an

adjacent film layer. A chemical reaction takes place at this surface at a rate r.The surface is a good heat conductor and has the temperature To. A diffusionlayer develops in front of the surface with thickness d. The reactants enter withtemperature Td. The concentrations of the reactants and products as well as thetemperature in x¼ � d are known. We consider a stationary state in which thetotal heat flux Jiq, as well as the mass fluxes Jij, are independent of position anddirected in the x-direction. Other variables depend on x only. On the o-side ofthe catalyst the mass fluxes are zero. It follows from eq 14.31 that Jiq¼ Joq¼ J0oqin a stationary state. The mass fluxes become proportional to the reaction rate,eq 14.32, giving Jij¼ � njr. All chemical potential gradient terms in eq 14.15 canthen be contracted, and we obtain for the entropy production in the i-phase:

si ¼ J iq

d

dx

1

T iþ r

d

dx

DGi

T i; ð14:61Þ

where DGi ¼P

vsjmsj is the reaction Gibbs energy of the gas supplied in x¼ � d.

The gradient in this combination of chemical potentials is the effective drivingforce for diffusion. The resulting linear force-flux relations are:

d

dx

1

T i¼Ri

qqJiq þ Ri

qdr; and

d

dx

DGi

T i¼Ri

dqJiq þ Ri

ddr:

ð14:62Þ

With constant fluxes and constant resistivities, we integrate and obtain:

1

T iðxÞ ¼1

Tdþ 1

Td� 1

Tr

� �x

d; and

DGiðxÞT iðxÞ ¼

DGið0ÞT ið0Þ þ

DGið0ÞT ið0Þ �

DGi

Tr

� �x

d:

ð14:63Þ

In eq 14.63, Tr is the temperature of the right-hand side of the layer.

486 Chapter 14

Substitution of eq 14.63 into eq 14.62 gives:

1

T ið0Þ �1

Tr¼dRi

qqJiq þ dRi

qdr; and

DGið0ÞT ið0Þ �

DGi

Tr¼dRi

dqJiq þ dRi

ddr:

ð14:64Þ

For the surface we obtain the excess entropy production from eq 14.34,

ss ¼ J iq

1

To� 1

T ið0Þ

� �� r

DGið0ÞT ið0Þ : ð14:65Þ

The last term comes from the chemical reaction at the surface. This gives aslinear force-flux relations

1

To� 1

T ið0Þ ¼RsqqJ

iq þ Rs

qrrs; and

�DGið0ÞT ið0Þ ¼R

srqJ

iq þ Rs

rrrs:

ð14:66Þ

Combining eqs 14.64 and 14.66, we obtain a lumped expression, where thechemical driving force takes care of the reaction at the surface as well as dif-fusion in the film layer. The thermal driving force refers to the same distance,between 0 and d:

1

To� 1

Tr¼ Rs

qq þ dRiqq

� �J iq þ Rs

qr þ dRiqd

� �r; and

�DGi

Tr¼ Rs

rq þ dRidq

� �J iq þ Rs

rr þ dRidd

� r:

ð14:67Þ

From eq 14.67 we can calculate the effect of a total heat flux on the reaction ratein terms of the inverse temperature difference between the incoming gas and thesurface, and how much the incoming gas is away from equilibrium, DGi. Thetotal heat flux can be related to the measurable heat flux (and experiments) viathe energy balance, Jiq¼ Joq¼ J0oq. The choice of the film thickness d was dis-cussed by Taylor and Krishna37 who use a value between 0.1 and 1 mm. Theinverse temperature difference can drive the reaction.44–46


When chemical reactions occur in homogeneous solutions, their impact ontransport is indirect, via the conservation laws and the driving forces. Themechanism of transport and thus the transport coefficient can change in the


presence of a chemical reaction when a gradient is applied to the homogeneousphase. In heterogeneous catalysis, non-equilibrium thermodynamics predicts adirect coupling of heat and mass transport into the surface with a chemicalreaction at the surface. For instance, a difference in temperature of the catalystand adjacent films contributes directly to the rate of the reaction. This has notyet been used in analysis of experimental results.We have also seen that chemical reactions can be dealt with in the same

systematic manner as other transport processes. The progress of a reaction asmeasured along the reaction coordinate, gives a form equivalent to the law ofmass action, but with the rate as a non-linear function of the driving force. Asimple case was presented, but the procedure has been extended successfully tothe description of active transport through Ca-ATPase.25,43 The fact that wecan give chemical reactions a thermodynamic basis, is important for entropyproduction minimizations, discussed in section 14.4.

14.4 The Path of Energy-Efficient Operation

We saw above that non-equilibrium thermodynamics is required for a properdescription of transport processes in many common cases, because couplingcoefficients cannot be neglected. Coupling leads to reversible contributions inprocesses and plays, therefore, an important role in problems that addressenergy efficiency.Once the entropy production of a system is well described and calculated at

any position throughout the system, the next step is to ask: Is it possible tominimize its total value, while keeping the performance? We shall see how thiscan be done, using the familiar heat exchange process as a pedagogicalexample, because it can be solved analytically.18 We proceed to show resultsfrom studies of chemical reactors, which are converters of thermal to chemicalenergy. An interesting hypothesis has emerged from these studies, namely thehighway hypothesis, which will be discussed.

14.4.1 An Optimisation Procedure

A robust mathematical tool is needed to perform an optimisation. We havefound that optimal control theory provides such a tool.50,51 The system isconstrained by the requirement to balance energy, momentum and mass. Theseconstraints must be specified for each particular case. In optimal control ter-minology there are two classes of variables. The first class are the state vari-ables, for instance the temperature, T(z,t), the pressure, p(z,t), and theconcentrations, cj(z,t), in a tubular reactor. The second class are the controlvariables, which are determined from the outside. An example is the tem-perature, Tc(z,t), on the outside along the tubular reactor. Optimal controltheory in this case provides a general method to obtain Tc(z,t) such that thetotal entropy production is minimal, given certain constraints.

488 Chapter 14

We clarify this for stationary plug flow along a tubular reactor.52 In optimalcontrol theory one rewrites the balance equations in the form

dT

dz¼fT ðTðzÞ; pðzÞ; cjðzÞ;TcðzÞÞ;

dp

dz¼fpðTðzÞ; pðzÞ; cjðzÞÞ and

dck

dz¼ fkðTðzÞ; pðzÞ; cjðzÞÞ:

ð14:68Þ

One then defines the Hamiltonian50,51 by:

H ¼ sðzÞ þXi

liðzÞfi; ð14:69Þ

where s is the entropy production and li are Lagrange multipliers. The mini-mum of the entropy production is now found by calculating the minimum valueof the Hamiltonian, that is, by solving the set of equations:

Energy balance gives : fT ¼dT

dz¼ @H

@lT;

dlTdz¼ � @H

@T;

Momentum balance gives : fp ¼dp

dz¼ @H@lp

;dlpdz¼ � @H

@p; and

Mass balances for each j: fj ¼dcj

dz¼ @H@lj

;dljdz¼ � @H

@cj:

ð14:70Þ

Also the control variable is chosen such that the Hamiltonian has a minimum.In this case it implies that

@H

@Tc¼ 0: ð14:71Þ

In the minimum, the Hamiltonian is constant along the z-axis. Using theminimum value of the Hamiltonian one can calculate the optimum value of thecontrol variable(s) along the z-axis.

14.4.2 Optimal Heat Exchange

Let us apply eqs 14.70 and 14.71 to the simple heat exchange process18 illu-strated in Figure 14.8. A hot fluid is cooled from the temperature Th,in to thetemperature Th,out by a cold fluid across a metal plate. The subscript h is shortfor hot. We neglect any work connected with the pumping of the fluids (zeroviscosity and constant pressure). Conservation of energy in the hot fluid impliesthat:

FCpdTh ¼ J 0qðzÞDydz) fT ðzÞ ¼dTh

dz¼ J 0qðzÞDy

FCp;m; ð14:72Þ


where F is the constant molar flow,Cp,m is the heat capacity, J0q(z) is the value ofthe heat flux at position z, transferred across the area Dydz from the hot to thecold fluid. For the whole system, we have the overall entropy balance:

dSirr

dt¼F So � Sið Þ þ dSc

dt¼ �Dy

ZL0

J 0qðzÞ=ThðzÞ �

dz

þ DyZL0

J 0qðzÞ=TcðzÞ �

dz:

ð14:73Þ

The first term on the right-hand side of the first equality is the difference of theentropies flowing out and into the hot side per unit of time. The second term is theentropy increase on the cold side per unit of time. In the last equality both theseterms are written in terms of the measurable heat flux from the hot to the coldfluid for stationary state conditions. The entropy production takes place in themetal plate and adjacent films indicated in Figure 14.8. Using the relation betweenthe measurable heat flux and the inverse temperature difference we can also write:

dSirr

dt¼ Dy

ZL0

Zd0

sðx; zÞdxdz ¼ DyZL0

lqq ThðzÞð Þ 1

TcðzÞ� 1

ThðzÞ

� �2

dz: ð14:74Þ

It is now interesting to ask: Can we obtain the same cooling with better effi-ciency? The cooling capacity is unaltered if for the same flow F and initialtemperature Th,in the final temperature Th,out is the same. As the molar entropydepends only on the temperature, this implies that the entropy flows in and out

Figure 14.8 The heat exchange process. A hot fluid is cooled from temperature Th,in

to Th,out by a cold fluid across a metal plate. Reprinted with permissionof Tapir Academic Publishers.18

490 Chapter 14

must be the same in the optimization. The problem is therefore equivalent to

finding the minimum of dScdt¼ Dy

RL0

J 0qðzÞ=TcðzÞ �

dz. The temperature dis-

tribution of the cold fluid is the control variable of this problem. For the heatexchange problem, we now have to solve;

dTh

dz¼ J 0qDy

FCp;m¼ @H

@lT;

dlTdz¼ � @H

@Tand

@H

@Tc¼ 0; ð14:75Þ

where H ¼ sðzÞ þ lT ðzÞfTðzÞ. The exact result found18 is that the entropyproduction is constant as a function of z. This result is approximatelyequivalent to keeping the thermal driving force constant, see Figure 14.9.

14.4.3 The Highway Hypothesis for a Chemical Reactor

A system is autonomous50–52 if the Hamiltonian is no explicit function of z andt. In autonomous systems the Hamiltonian is constant, that is, independent of zand t. For the optimal heat exchange process (z dependent), and for the optimal

Figure 14.9 Temperature of the hot side Th and cold side Tc obtained from the exactsolution at constant entropy production and the approximate solutionfor Th and Tc obtained with constant thermal force. ––––, Th exact valueobtained from the equipartition of entropy production;- - - -, Tc, exactvalue obtained from the equipartition of entropy production; – � – � –,Th, approximate value from the equipartition of forces; and � � � � � � � ,Tc, approximate value from the equipartition of forces. Reprinted withpermission of Tapir Academic Publishers.18


expansion of a gas against a piston (t dependent) with a certain equation ofmotion and the external pressure as the control variable,18 this is equivalent tohaving a path with a constant entropy production.Chemical reactor engineering deals with batch or tubular reactors. In

batch reactors, there is a development in time, while a tubular reactoroperated at steady state, has changes along the length of the reactor tube. Forboth reactors systems we can formulate an autonomous Hamiltonian. Forchemical reactors, which normally have less degrees of freedom (fewer controlvariables) the local entropy production becomes constant along the path ofoperation only if there are sufficient degrees of freedom in the system.52

This was proven numerically, and led the authors to formulate a hypothesis forthe state of minimum entropy production. The hypothesis was formulated onthe basis of a large set of solutions of plug flow reactors with minimum totalentropy production:

Equipartition of entropy production, EoEP, but also equipartition of forces,EoF, are good approximations to the state of minimum entropy productionin parts of an optimally controlled system that has sufficient freedom.

The hypothesis was based on the observation; that the optimal path of manythousands of possible reactors, were crowding in on a band (called ‘‘thehighway’’). The band embedding the solutions was near, but not at theequilibrium line. This is a line in state space that defines the equilibriumcomposition of a chemical reaction at a given pressure, temperature. In otherwords, the most energy-efficient operations are carried out at a well-defineddistance from equilibrium, along a path with constant entropy production.This situation, illustrated in Figure 14.10 is similar to driving a car on thereal highway at uniform speed, not in the lane with maximum speed, but withsome speed below that. This is well known to be energy-efficient car driving.The band has a large distance from the line obtained for maximum reactionrate(s).The hypothesis was also used to predict the optimal state of a hypothetical

packed distillation column.53 The optimal path of operation as determinedfrom eqs 14.70 and 14.71, confirmed the hypothesis, giving constant entropyproduction in the rectifying and stripping sections of the column. Doing dis-tillation along these lines, one can save up to 50 % of the lost work.54 Con-structional changes will, however, be needed in the column.To find the optimal profiles of the intensive variables in a chemical reactor,

one solves the balance equations for mass, energy and momentum. Transportlaws are also needed. The coupling of heat and mass is zero for fluxes that havedifferent directions (that are perpendicular to and along the tube). Hetero-geneous catalysis may require that coupling terms are taken along, as describedin section 14.3.3. The local entropy production is found once the fluxes andforces are determined. An example of a set of such profiles is given in the nextsection.

492 Chapter 14

14.4.4 Energy-Efficient Production of Hydrogen Gas

One promising flow sheet for production of hydrogen with the help of thermalenergy is given in Figure 14.11. The last step in this process, the regeneration ofsulphuric acid, was studied by van der Ham and coworkers.55 The authors fol-lowed the procedure outlined above, and calculated first the profiles in intensivevariables, as obtained from the state-of-the-art reactor design. The results for the

Figure 14.10 The highway in state-space for a chemical reactor. Reprinted withpermission of Chem. Eng. Sci.52

Figure 14.11 Flow sheet for production of hydrogen from thermal energy. The threeconsecutive steps are shown, where the first step produces hydrogen atT¼ 573K from HI, while HI is produced at a lower temperature. Theregeneration of sulphur dioxide is the high energy demanding step thatis subject to optimisation.


temperature and composition are given in Figure 14.12a. The system was nextsubject to an optimization of the total entropy production, following eqs 14.70and 14.71. The outcome of these calculations is plotted in Figures. 14.12b and14.13. We see that the original profiles in 14.12a change to be more parallel thanin 14.12b. This led to a drastic change in the variation of the entropy production(compared with Figure 14.13); it became more uniform along the reactor in theoptimal case, than in the outset. The entropy production was not perfectlyconstant, but this was explained by the constraints set on the optimization, andthe fact that there was only one degree of freedom (one control variable). Thereduction in the lost work was between (10 and 20) %. This example shows thatthis tool provided by non-equilibrium thermodynamics may be helpful in aworld that struggles to use its energy resources without excessive waste.

14.4 Conclusions

Non-equilibrium thermodynamics defines the transport equations of an irre-versible process in a systematic way. Some flexibility exists in the choice ofvariables, but equivalent choices must all give the same entropy production. Wehave seen that the basic assumption in the theory; the assumption of localequilibrium is sound, and does not restrict normal use of the theory. The theory

Figure 14.12a Temperature of the reference reactor T and of the heating utility tem-perature THe along the dimensionless reactor length coordinate l/l0

where lo¼ 1m with conversion of the sulfuric acid dissociation x1 and ofthe sulfur trioxide decomposition x2 with the equilibrium conversionx2,eq shown for comparison. (gray line), reactor temperature T; (blackline), THe; (light grey dashed line), sulfuric acid dissociation x1;- - - - -,the sulfur trioxide decomposition x2; and - - - - -, the equilibriumconversion x2,eq. Reprinted with permission of Ind. Eng. Chem. Res.55

494 Chapter 14

can help us to use a consistent set of assumptions, which can be checked againstthe system’s entropy balance and the second law. Applying the theory, we havefound that common approximations used to describe multi-component phasetransition and chemical reactions are incorrect, and that new possibilities arisefor the understanding of experimental results. Only a few examples have beenstudied, heat and mass transport and chemical reactions in homogeneous andheterogeneous systems. By including transport of charge, numerous otherapplication possibilities arise, and many experiences from this study canprobably be transferred.All energy conversion is associated with entropy production. The limit given

by the reversible processes (zero entropy production) is unrealistic. A state ofminimum total entropy production can be realized, so it follows that theunrealistic reversible limit of the second law efficiency definition could bereplaced by this more realistic target for optimal performance.56 This targetmight be developed further to serve as a benchmark for energy-efficientoperation. Altogether, we are looking optimistically at the possibility to pro-vide many problems with a firmer thermodynamic basis, a systematic way todeal with assumptions and experiments, new ways to understand observations,new experiments to do, and better use of our valuable resources.

Figure 14.12b Temperature of the optimized reactor T and of the heating utilitytemperature THe along the dimensionless reactor length coordinate l/l0

where lo¼ 1 m with conversion of the sulfuric acid dissociation x1 andof the sulfur trioxide decomposition x2 with the equilibrium conver-sion x2,eq shown for comparison. (gray solid line), optimized reactortemperature T; (black solid line) THe; (light gray dashed line), sulfuricacid dissociation x1; (black dashed line), the sulfur trioxide decom-position x2; and (gray dashed line), the equilibrium conversion x2,eqshown for comparison. Reprinted with permission of Ind. Eng. Chem.Res.55


Acknowledgements

The Norwegian Research Council is thanked for many years of support forresearch that this chapter builds on, in particular for the Storforsk Grantno 167336/V30.

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498 Chapter 14

Subject Index

absorption, 173total, 195

acentic factor, 48, 50, 88, 114, 118,137, 143, 154

acoustic virial coefficients, 45, 47activity, 27, 433

coefficient, 22, 23, 99, 100, 101,106, 110, 241, 243, 287, 381infinite dilution, 99, 373, 441model, 97, 106

non random two liquid(NRTL), 63, 64, 101, 107,241, 311, 380density dependent, 441electrolyte, 380

universal functional activitycoefficient (UNIFAC), 63,105, 106, 110, 254, 260,308, 381, 440

universal quasi chemical(UNIQUAC), 63, 241, 311,380

segment molar, 290aggregation number, 186, 187amount of substance, 6, 33, 295amphiphilic molecules, 184, 185, 186,187

asphaltene, 281, 283, 308, 310, 311,313flocculation, 309precipitation, 284

associated perturbed anisotropicchain theory, 71

associationinteraction, 221potential, 221

asymmetric coefficients, 344asymptotic

behaviour, 337power laws, 333, 335scaling, 338

aqueous solutions, 184Avagadro’s constant, 39, 86azeotrope, 24, 377, 378, 379azeotropic

composition, 25

Baker-Williams fractionation, 293Benedict, Webb and Rubin equationof state, 111–114, 116, 404, 405,406, 413acentric factor, 114AGA 8, 114Bender, 114, 405, 406Jacobsen and Stuart, 114, 145, 406Lee and Kesler, 114, 117, 118modified, 113–114, 145, 406Morsy, 114Nishiumi and Sato, 114Schmidt and Wagner, 114Starling, 114, 405Starling and Han, 114, 115

Berthelot rule, 47, 92binary interaction parameter, 62, 77,91, 104, 105, 113, 163, 381, 435,437, 438, 439

binodal, 451biodiesel, 438biofuels, 394Boltzmann’s constant, 39, 86, 136,158, 174, 217

Born term, 244

Born-Oppenheimer model, 241, 243,244

Boyleinversion curve, 422temperature, 34volume, 48

Brownian particle, 206bubble point, 283, 370, 381, 451

Cahn equation, 209canonical partition function, 190Carnahan–Starling de Santis equationof state, 154, 155, 156

chemicalequilibria, 27, 217, 436equilibrium, 433, 435, 436, 481, 482potential, 7, 25, 45, 174, 185, 186,208, 209, 229, 286, 287, 290, 328,335, 339, 347, 434, 444, 468, 476,479, 481, 484, 486

reactions, 433, 440, 472, 474, 475,478, 479, 482, 483, 486, 488, 492,495diffusion controlled, 433systems, 434

Clausius-Clapeyron equation, 295Clausius virial theorem, 226cloud point, 281, 282, 298, 300, 302,304, 305, 306

cluster integrals, 38coexistence

boundary, 195curve, 208, 337

diameters, 337, 344liquid+liquid, 199

combining rules, 47, 85, 88, 92–97,443Chueh and Prausnitz, 97Fender and Halsey, 95Halgren, 96hard sphere, 122Hiza, 95Hudson and McCoubrey, 94, 124,253

Kohler, 95Kong, 96

Lorentz-Berthelot, 92, 120, 124,252

non-quadratic, 97SAFT, 123Sikora, 95Waldman and Hagler, 96

compressibility factor, 158, 159, 399,425, 444

compression factor, 22, 33, 36, 164,288, 398, 400

consulate point, 339continuous,

distribution, 280thermodynamics, 280, 281, 285,286, 287, 288, 291, 293, 295, 296,299, 300, 304, 306, 307, 309, 310,311, 313associating systems, 311

correlation length, 178, 192, 194, 196,200

corresponding states principle, 47,135–166, 381, 418applications, 162–166extended, 138, 154, 163, 164, 166,241, 425

four parameter, 137mixtures, 138, 156–161phase equilibrium calculations, 161pure fluid, 138shape factors, 138, 142, 143, 144,145, 146, 153, 155, 163, 164, 166,425apparent, 143, 145, 147exact, 146–154mixtures, 160

simplethree-parameter, 50two parameter, 136, 137, 141, 147

COSMO-RS, 383critical, 26

amplitude, 192, 194, 323, 328, 335behaviour, 321, 353

mean field, 355composition, 281compression factor, 49, 50condition, 31

500 Subject Index

density, 399, 411exponent, 24, 228, 229, 323, 325,359universal, 192, 193, 194, 228,321, 323, 357

fluctuations, 180, 189, 196, 200,228, 325, 347, 354, 356, 357, 358

isochore, 192line, 24locus, 345, 347, 348, 359micelle concentration, 185, 235parameters, 143, 198, 346, 348, 381,382

phenomena, 229, 321, 322, 349point, 24, 36, 136, 178, 182, 189,190, 191, 192, 194, 195, 196, 200,202, 203, 207, 209, 228, 229, 231,232, 233, 293, 294, 295, 299, 305,322, 323, 326, 329, 334, 338, 340,347, 371, 396, 433, 437, 438, 451azeotropic, 340double, 340, 345, 346gas+gas, 339gas+liquid, 30, 136, 178, 202,322, 339, 347

liquid+liquid, 207, 339re-entrant, 340universality, 321

power laws, 202, 323pressure, 88, 118, 228region, 401, 403, 412, 439, 450scaling, 337

correction to, 338, 344densities, 345law, 228, 321, 328, 333fields, 322, 339, 341

slowing down, 207–208state, 28, 29, 31temperature, 88, 94, 118, 178, 187,188, 195, 202, 228, 231, 232, 399,411, 446, 447

volume, 232crossover, 196, 228, 229, 231, 232,233, 234, 322, 339, 349, 350, 351,354parametric, 356, 357

tricriticality in polymer solutions,196–200

cubic equation of state, 53–83,87–88, 97, 106, 284, 357, 381,439a-parameter, 58Elliott, Suresh and Donohue(ESD), 60, 64, 126

Generalized crossover, 165Gibbons and Laughton, 87higher-order, 61mixtures, 62–64Patel-Teja, 58, 64, 65, 87, 299Peng-Robinson, 57–58, 59, 60, 62,64, 65, 77, 80, 87, 103, 105, 108,127, 154, 156, 283, 287, 307, 309,381, 404, 435, 437, 438, 439,453

predictive Soave-Redlich-Kwong,63, 105, 111

Redlich-Kwong, 54, 56, 58, 59, 87,381

Redlich-Kwong-Joffe-Zudkevitch,65, 66, 67, 71

Sako-Wu-Prausnitz, 74, 76, 281,303, 310

Soave-Redlich-Kwong, 56–57, 58,60, 62, 64, 65, 66, 72, 75, 80, 87,108, 154, 156, 283, 287, 307, 309,404, 435, 436, 437, 438, 439, 453predictive, 63

van der Waals, 53–55, 58, 59, 60,77, 80, 84, 88, 102, 106, 158, 166,178

volume correction (translation),59–60, 453, 455

cubic plus association (CPA) equationof state, 61, 71, 125–127combining rules, 125mixing rules, 125

de Broglie wavelength, 223Debye-Huckel 246

limiting law, 240theory, 241, 2442

DECHEMA, 101, 105, 106

501Subject Index

densityamount-of-substance, 33, 41, 45conjugate, 28gradient theory, 247functional theory, 247fluctuations, 230saturated liquid, 250

departure functions, 13, 14, 16deviation functions, 11dew point, 283, 370, 451dielectric constant, 241, 245, 433diffusion coefficient, 81,205, 207, 208diffusivity, 207, 436dipole

interaction, 234, 235, 239moment, 49, 50, 97, 138, 154, 235,237, 372

DIPPR, 111Dirac delta, 218dispersion interaction, 222, 234distribution function, 288, 289, 295,296, 297extensive, 285, 289gamma, 297Gaussian, 296, 297, 298Hosemann-Schramek, 284, 298molar, 286Schulz-Flory, 291, 294, 297, 298,300, 306, 309

Schulz-Zimm, 297Stokmayer, 291, 298Wesslau, 299, 306

Dufour coefficients, 473

electromagnetic scattering, 204electrostatic interactions, 215equation of state

see Benedict, Webb and RubinCarnahan-Starling, 440see Carnahan-Starling de Santissee cubic equation of stategroup contribution association,437, 438, 440, 443, 444, 445, 451,453, 455

hard spheresmean field, 195

Haar Gallagher, 409Keenan, Keyes, Hill and Moore,409

see multi-parameter equations ofstate

Sanchez-Lacombe, 305, 306, 307See SAFTSchmidt Wagner, 411See virial equation of state

equilibriumconditions, 25, 26constant, 434fluctuations, 174liquid+liquid, 280, 281liquid+liquid+vapour, 374solid+liquid, 374, 395

Euler-Lagrange equation, 179Euler’s theorem, 8, 9eutectic, 374excess

functions, 17, 18, 19Gibbs function, 22, 23, 63, 78, 79,91, 100, 102, 103, 105, 106, 107,110, 120

Fick’s law, 462, 481Fisher-Tropsch reactor, 438flash calculations, 283Flory-Huggins theory, 107, 183, 196,198, 290, 303, 306, 308, 311

flux,heat, 468, 469, 476, 477mass, 468, 469, 478

Fourier’s law, 462, 481friction theory, 79fugacity, 20, 21, 26, 99, 100, 435

coefficient, 20, 21, 26, 91, 99, 161,287, 289, 293, 434, 450mixtures, 22

gas condensates, 283, 299Gaussian

distribution, 175fluctuation, 182, 186probability distribution function,183

502 Subject Index

gel permeation chromatography, 284,296

Gibbs function (energy), 6, 9, 11, 22,25, 26, 30, 31, 41, 44, 63, 101, 147,173, 183, 184, 187, 188, 198, 209,241, 246, 281, 286, 289, 290, 293,300, 311, 395, 397, 408, 420, 434,435, 436, 440

Gibbs-Duhem equation, 7, 20, 22, 31,287, 289, 348, 472, 474

Gibbs phase rule, 24Gibbs triangle, 450Ginzburg

criterion, 179, 200number, 179, 351, 357

grand partition function, 38group contribution associationequation of state, 437, 438, 440,443, 444, 445, 451, 453, 455

Guy-Stodola theorem, 462

Hamiltonian, 190, 356, 489, 491,492

hard sphere equation of state, 121perturbed, 121

Hartree-Fock, 251heat capacity, 418, 422, 490

constant chemical potential, 176constant pressure, 9, 328, 396, 400,408

ideal gas, 396, 399, 403, 407isochoric, 175, 191, 336, 358, 396,413divergence, 191specific, 335, 336

isomorphic, 341volume, 408

Helmholtz energyequation of state, 419for mixtures, 426

Helmholtz function (energy), 6, 31,41, 44, 101, 102, 140, 173, 178, 179,187, 189, 190, 205, 220, 222, 223,224, 229, 231, 235, 236, 241, 242,243, 247, 249, 258, 294, 328, 329,358, 395, 399, 400, 404, 408, 409,

410, 420, 425, 426, 427, 440, 441,444equation of state, 118

GERG-2004, 118heterogeneous

catalysis, 486, 488, 492systems, 437, 451, 460, 462,495

hierarchical reference theory, 358homogeneous

systems, 437, 446, 451, 452, 460,474, 477, 480, 488, 495

liquid phase, 375, 376homonuclear diatomic, 219Hugoniot curve, 403hydrocarbon

aromatic, 288mixtures, 285

hydrogen bonding, 215, 216, 217hydrophobic, 184, 187, 372hydrophilic, 184, 313, 376, 377hydroscopic, 373

idealgas, 12, 13, 14, 21, 218, 231, 407,411, 435mixture, 12, 18

inversion curve, 422solution, 19

immiscibility, 372, 447gap, 371, 376liquid+liquid, 447

ionic liquid, 368, 370–383catalystselectrolytes, 368entrainers, 378gas absorption, 369heat-transfer fluids, 368liquid-liquid extraction, 369lubricants, 368membrane separation, 369phase behaviour, 368, 369, 379

binary mixtures, 369–374liquid+aliphatic+aromatic,376

liquid+gas, 369–372

503Subject Index

ionic liquid (continued)liquid+organic, 373liquid+water+alcohol, 377

separation, 368solubility with,

CO2, 369, 370synthesis, 368vapour pressure, 368

incompressible liquid, 341integral equations, 135, 358interface

vapour+liquid, 193interfacial tension, 186, 187, 189intermolecular forces, 39

three-body, 39intermolecular potentials, 39, 136,218energy functions, 47, 142hard-core-square-well, 39, 47hard sphere, 219Keesom, 234Lennard-Jones, 37, 92, 94, 97, 124,141, 156, 158, 224, 225, 235, 238,358, 379, 380

Maitland-Smith, 47Mie, 47, 96, 226non-spherical, 39pair additivity, 38polar molecules, 39square well, 223, 226, 358three-body, 47well, 39Yukawa. 240, 244

ionic fluids, 2Ising, 349, 351, 353, 354, 355

amplitudes, 335model, 181, 321, 322, 341, 352critical exponents, 195–196

isopleths, 451isothermal compressibility, 9, 19, 175,204

Joule inversion, 404Joule-Thomson

coefficients, 396inversion curve, 422

kinetic theory, 483Krichevskii parameter, 348, 349

Landau expansion, 178, 180, 198, 231,337, 349, 350, 355

Landau model, 350, 351, 356, 357Landau-Ginzburg functional, 178,179, 187, 209

Landau-Ginzburg local Helmholtzenergy, 190, 209

Laplace equation, 181Legendre transformation, 6, 198,330

Legrange multipliers, 489Lennard-Jones potential, 37, 92, 94,97, 124, 141, 156, 158, 224, 225,235, 238, 358, 379, 380

Lifshitz-point, 187, 188, 189light scattering, 352London theory of dispersion, 75, 92,94

Lorentz rule, 47, 92lower critical

end point, 375, 448point, 301, 302

lower critical solutiontemperature, 77, 301, 303, 345, 373,374

point, 340, 344

mass fraction, 281, 370Mayer cluster, 220Mayer function, 39, 217, 219, 221, 224Maxwell criterion, 395, 397Maxwell relations, 8, 41Maxwell-Stefan equations, 472–474McMillan-Mayer model, 241, 242mean field

theory, 181, 228approximation, 198, 337

melting line, 395metastable, 208micelle, 1, 185, 187Michelsen’s phase stability analysis,436

microemulsion, 189

504 Subject Index

microscopic reversibility principle,461

miscible, 374, 446, 448, 451mixing

functions, 13, 17properties, 18

mixing rules, 54, 62, 85, 88–92, 164,435, 436, 453hard sphere, 122Huron-Vidal, 71, 77, 79, 100, 437,439

linear combination of Huron-VidalMichelson, 109–110

Luedecke and Prausnitz, 71, 90linear combination of Vidal andLuedecke and Prausnitz, 71, 90Mathias and Copeman, 90–91Michelson, 63Michelson-Vidal-Huron 1, 104,109, 111

Michelson-Vidal-Huron 2, 104,109, 111, 439, 440, 445, 446, 452,453, 455

Mullerup, 90non-quadratic, 97quadratic, 437universal, 63, 110–111van der Waals, 49, 62, 71, 88, 89,90, 91, 100, 106, 112, 138, 164

Vidal, 100Whiting and Prausnitz, 90Wong Sandler, 63–64, 105, 106,107, 108, 109, 111, 439

Zhong and Masuoka, 77molar quantity, 6molar

mass, 286volume, 18, 19

mole fraction, 6, 31, 86, 229, 340, 341,345, 346, 370, 373

monodisperse, 281Monti Carlo simulation, 379, 380multi-component mixture, 31multi-parameter equations of state,111–127, 394, 396bank of terms, 402, 404

Helmholtz energyequation of state, 409–411, 419for mixtures, 426

linear fitting, 403non-linear fitting, 400, 402–404,426

optimization algorithm, 402, 404,411

natural gas, 394, 436near critical state, 173neutron scattering, 204non-equilibrium, 25

meso-thermodynamics, 205thermodynamics, 460, 461, 462,464, 465, 494

non random two liquid (NRTL), 63,64, 101, 107, 241, 311, 380density dependent, 441electrolyte (eNRTL), 380

normal boiling temperature, 285, 295nucleation,

homogeneous, 205

Ohm’s law, 462Onsager’s

coefficients, 461kinetic coefficient, 206, 207phenomenological coefficients, 461principle, 206relations, 472, 477

Ornstein-Zernikeexponential decay, 177integral equation, 242wave number-dependentself-consistent approximation, 359susceptibility, 177

Osmoticpressure, 183compressibility, 198susceptibility, 204

Pade approximate, 235, 236, 239pair interaction

energy, 86,parameter, 93, 99

505Subject Index

parametric equations, 328partial molar

properties, 19volume, 22

partition function, 139, 223Patel-Teja equation of state, 58, 64,65, 87, 299

Pauling radii, 244Peng-Robinson equation of state,57–58, 59, 60, 62, 64, 65, 77, 80, 87,103, 105, 108, 127, 154, 156, 283,287, 307, 309, 381, 404, 435, 437,438, 439, 453

Perkus-Yevick approximation, 219perturbed hard chain theory, 75

simplified, 68petroleum, 394phase

equilibrium, 25, 433, 437ionic liquid, 368rule, 25,59, 60, 62, 64, 65transition

second order, 173, 185, 189, 196lambda line, 196

separation, 196perfect gas, 33, 40, 41

heat capacityat constant volume, 41at constant pressure, 44

petroleum fluids, 281Poisson equation, 240polydisperse, 1, 280, 281, 288

fluid, 299mixture, 289polymer, 290, 292, 306systems, 290, 313thermodynamics, 300

polydispersity, 187, 282, 283, 284,304, 305, 313, 314

polymer, 1melts, 173segment fraction, 290segment number, 290solution, 173, 200, 280systems, 281, 284

Poynting factor, 26

Prigogine’s parameter, 74pseudo critical constants, 49pseudo component, 280, 283, 284,285, 303, 304, 305, 307

quadrupolar interaction, 234quadrupole moment, 140quantum mechanics, 251

COSMO-RS, 383

radial distribution function, 158, 159,225

radius of gyration, 183, 196, 200Raoult’s law, 26, 295, 373reactants, 434, 451

gas, 450, 453liquid, 446solid, 448

reactions, 436, 472activation energy, 446diffusion controlled, 436gas+liquid, 437, 438, 446, 448

supercritical, 446multi-phase, 437

catalytic, 438ionic liquid, 438polymer, 438

near-critical, 436supercritical fluid, 436, 437, 438

reactor, 446, 453chemical, 491, 492Fisher-Tropsch, 438supercritical, 452, 453

rectilinear diameter, 181, 195reduced

density, 119pressure, 116temperature, 116, 119

Redlich-Kwong equation of state, 54,56, 58, 59, 65, 87, 381

Redlich-Kwong-Joffe-Zudkevitchequation of state, 65, 66, 67, 71

REFPROP, 428regression hypothesis, 461regular solution theory, 100renormalization group theory, 189

506 Subject Index

residualcanonical-ensemble partitionfunction, 139

compressibility factor, 155functions, 12, 13, 17Gibbs energy, 20Helmholtz energy, 155properties, 40, 44

resistivityheat transfer, 471, 477mas transfer, 469

reversible processes, 25

SAFT see statistical associating fluidtheory

Sako-Wu-Prausnitz equation of state,74, 76, 281, 303, 310

Sanchez-Lacombe, 305, 306, 307saturated

liquid, 402vapour, 37,

density, 402pressure, 335, 402

second virial coefficient, 34, 38, 48, 63,86, 86, 105, 183, 184

scalingcomplete, 333correction to, 338, 344densities, 345fields, 322, 339, 341laws, 228, 321, 328, 333, 338

asymptotic, 229, 338relations, 192–193revised, 332

Scatchard-Hildebrand theory, 309shock-wave measurements, 403

Hugoniot curve, 403Soave-Redlich-Kwong equation ofstate, 56–57, 58, 60, 62, 64, 65, 66,72, 75, 80, 87, 108, 154, 156, 283,287, 307, 309, 404, 435, 436, 437,438, 439, 453Predictive, 63

solubility, 373closed loops, 345mutual, 372

solvent,selection, 446–450

Soret coefficients, 473speed of sound, 45, 396, 411, 413, 421,422

spinodal, 208, 293, 294, 295, 299, 300,302, 303, 305decomposition, 208, 209

stability, 28conditions, 29

statistical associating fluid theory(SAFT), 61, 71, 123–124, 215, 221,222, 224, 228, 231, 243, 247, 249,256, 305, 306, 314, 358, 382, 440combining rules, 123critical region, 228–234crossover-SAFT, 229, 231, 233DFT-SAFT, 247–248DGT-SAFT, 247electrolyte solutions, 240ePC-SAFT, 246GC-SAFT, 253, 255, 257GC-SAFT-VR, 257, 258iSAFT, 248LJ-SAFT, 235mixing rules, 123PC-PSAFT, 237, 309PC-SAFT, 226, 227, 228, 230, 231,236, 237, 238, 239, 240, 247, 256,283, 306

PC-SAFT-GV, 237PC-SAFT-JC, 236, 237PC-SAFT-KEPC-SAFT-SF, 236PCP-SAFT, 236, 237, 238, 239,252

sPC-SAFT, 227SAFT1, 226SAFT2, 246SAFT-HR, 224, 225, 228, 233, 235,236, 239, 249

SAFT-HS, 224, 248SAFT-g, 257, 258SAFT-LJ, 239, 247SAFT-Mie, 250SAFT-QM, 251

507Subject Index

statistical associating fluid theory(SAFT) (continued)

SAFT-VR, 124, 226, 228, 233, 234,238, 239, 240, 244, 245, 247, 248,250, 251, 255, 256, 258

SAFT-VR+D, 239, 240SAFT-VR+DE, 245, 246SAFT-VRE, 244, 246SAFT-VRX, 233soft-SAFT, 225, 230, 231, 382tPC-PSAFT, 237, 382

Stokes-Einstein formula, 206stoichiometric number, 27, 434successive precipitation fraction,292

successive solution fractionation,293

supercritical,fluids, 439solvents, 438

surface tension, 176, 181, 193, 194,195, 474

susceptibility, 176, 177, 326mesoscopic, 177critical behaviour, 326, 351

systematic error, 395, 397

Taylor series, 227, 331thermal conductivity, 207, 473, 480,483, 484

thermal diffusion, 473, 481thermal diffusivity, 207, 473thermodynamics

continuous, 280, 281, 285, 286,287, 288, 291, 293, 295, 296, 299,300, 304, 306, 307, 309, 310, 311,313

first law, 25meso scale, 2, 172, 174, 209nano scale, 2, 172non-equilibrium, 1, 2, 460, 461,462, 464, 465, 494

polymer, 290property formulations, 395second law, 5, 6, 460, 461, 466, 495extensive properties, 19

thetacondition, 184point, 182, 184, 196, 198, 200, 208temperature, 183, 196

third virial coefficient, 34, 86, 183Tolman’s length, 181, 182, 195, 196,200, 201, 209polymer solutions, 200–202divergence, 195–196, 200

transportcoefficients, 78–81, 206, 472heat, 460, 466, 481mass, 460, 466, 481phenomena, 462processes, 460, 474, 488properties, 484theory, 465

Trouton’s rule, 295tricritical point, 24, 189, 196, 200triple point, 233, 396

uncertainty, 397experimental, 397

universal constant, 194coupling, 350

universal, 192, 193, 194, 228,321, 323, 357

universal functional activitycoefficient (UNIFAC), 63, 105,106, 110, 254, 260, 308, 381, 440

universal gas constant, 33, 440universal quasi chemical(UNIQUAC), 63, 241, 311, 380

universal scaling function, 194, 207,228, 321

universality, 192upper critical point, 301, 302, 448upper critical solution

temperature, 77, 301, 303, 345, 372,373, 374

point, 340, 344Ursell-Mayer virial theory, 410

van der Waalsequation of state, 53–55, 58, 59, 60,77, 80, 84, 88, 102, 178

508 Subject Index

one-fluid mixing rule, 49, 62, 71,88, 89, 90, 91, 100, 106, 112, 138,158, 164, 166

volume, 75virial

coefficients, 33, 45, 100acoustic, 45, 47composition dependence, 35temperature dependence, 34third, 34, 86, 183, 404second, 34, 38, 48, 63, 86, 86,105, 183, 184, 404

equation, 33equation of state, 33–52, 85–87,404mixtures, 35–36, 425

series

convergence, 36pressure, 37

viscosity, 78–80volume translation, 59–60, 453, 455

water-gas shift reaction, 435wax, 281

precipitation, 284Wertheim’s theory, 216, 219, 227, 249,257, 442

wetting transition, 173Wilson equation, 380

Yang-Yanganomaly, 336plots, 336relation, 335

509Subject Index

goodwin a.r.h., sengers j.v., peters c.j. (eds.) applied thermodynamics of fluids (rsc, 2010)(isbn...

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