ie800763x

Thermodynamic Consistency Test of Vapor-Liquid Equilibrium Data forMixtures Containing Ionic Liquids

Jose O. Valderrama,*,† Alfonso Reategui,‡ and Wilson W. Sanga§

UniVersity of La Serena, Faculty of Engineering, Department of Mechanical Engineering, Casilla 554,La Serena, Chile, Centro de Informacion Tecnologica, Casilla 724, La Serena - Chile, and UniVersidadNacional del Altiplano, Ciudad UniVersitaria, AV. Floral 1153, Puno-Peru

Vapor-liquid equilibrium of eight binary mixtures containing an ionic liquid and carbon dioxide have beentested for thermodynamic consistency. A method proposed by one of the authors has been modified to beapplied to this special situation in which the gas phase contains practically one component (carbon dioxide)while the liquid phase contains both components in a wide concentration range. The method is based on theGibbs-Duhem equation and on appropriate combination of equations of state, mixing rules, and combiningrules. The Peng-Robinson equation of state with the Wong-Sandler mixing rules including the Van Laarmodel for the excess Gibbs free energy required in the mixing rules, are used. The experimental data wereobtained from literature sources, and the adjustable parameters were found by minimizing the errors betweenpredicted and experimental bubble pressure. It is shown that the proposed modified consistency test is accurateenough to decide about the thermodynamic consistency of this type of data.

Introduction

Ionic liquids are organic salts with some special characteristicsthat make them highly interesting for many applications, suchas media for clean liquid-liquid extraction processes1 and assolvents for several types of reactions.2-8 Some ionic liquidscan be used as biocatalysts with great advantages compared toconventional organic solvents9,10 and some others have liquidcrystal and lubricant properties.11,12 Research work has also beendone on the development of separation methods for soluterecovery from ionic liquids,13 on extractions from azeotropicmixtures,14 and on hydrogen purification using room-tempera-ture ionic liquids.15

Phase equilibrium data of mixtures containing ionic liquidsare necessary for further development of some separationprocesses involving ionic liquids. On this line of research, Akiet al.,16 presented high-pressure phase equilibrium data of CO2

+ ionic liquid mixtures. For all of the ionic liquids studied bythese authors, large quantities of CO2 dissolve in the ionic liquidphase, but no appreciable amount of ionic liquid is found inthe CO2 phase. In addition, the liquid phase volume expansionwith the introduction of even large amounts of CO2 is negligible,in dramatic contrast to the large volume expansion observedfor neutral organic liquids. Other studies on the phase equilib-rium of mixtures containing a high pressure gas and an ionicliquid have been reported in the literature by Shariati andPeters,17 Lee and Outcalt,18 Kumelan et al.,19 and Shiflett andYokozeki,20,21 among others.

As in other areas of phase equilibrium thermodynamics, thedata reported by several authors on mixtures containing an ionicliquid are rarely analyzed from the point of view of theirthermodynamic consistency, that is, a method to discriminateamong published data to use the correct ones in applicationssuch as modeling, simulation, and design. Valderrama and co-

workers have previously presented and applied a sound methodto test the thermodynamic consistency of high pressure gas-liquidand gas-solid equilibrium data.22-24 The method is based onthe Gibbs-Duhem equation, on the fundamental equation ofphase equilibrium and on an appropriate combination betweenequations of state (EoS), mixing rules, and combining rules.

Although the problem is different in some aspects, thetreatment of mixtures containing an ionic liquid and highpressure carbon dioxide (including the near-critical region),present some similarities with those cases treated by the authorsin previous publications: (i) the vapor-phase non-idealities canbe important and a good model to evaluate the fugacitycoefficients, i, is needed; (ii) for isothermal data, the terminvolving the residual enthalpy (HR) vanishes; and (iii) theconcentration of one of the components in one of the phases,the ionic liquid in the gas phase, is very low.

Because of this last problem, the concentration of carbondioxide is close to 1.0, but cannot be considered as a pure gas.For these reasons, the classical differential or integral methodsdescribed in standard books25-27 are not applicable. Therefore,a method specially designed for determining the thermodynamicconsistency of high pressure mixtures containing an ionic liquidis proposed in this work. It should also be mentioned that inthe case of gas + ionic liquid mixtures, only PTx equilibriumdata are commonly available and, therefore, experimental datafor only the liquid phase are used in the proposed test. This isfundamentally different from previous works by the authors whoproposed a consistency test for PTy data.22,28

The thermodynamic relationship that is frequently used toanalyze thermodynamic consistency of experimental phaseequilibrium data is the fundamental Gibbs-Duhem equation.26

This equation, as usually presented in the literature, relates theactivity coefficients of all components in a given mixture. Ifthe equation is not obeyed, the data is declared to be thermo-dynamically inconsistent. The ways in which the Gibbs-Duhemequation is arranged and applied to the experimental data havegiven origin to several Consistency Test Methods, most of themdesigned for low-pressure data. Among these are the Slope Test,the Integral Test, the Differential Test and the Tangent-Intercept

* To whom correspondence should be addressed. E-mail: [email protected].

† University of La Serena, Faculty of Engineering, Department ofMechanical Engineering.

‡ Centro de Informacion Tecnologica.§ Universidad Nacional del Altiplano, Ciudad Universitaria.

Ind. Eng. Chem. Res. 2008, 47, 8416–84228416

10.1021/ie800763x CCC: $40.75 2008 American Chemical SocietyPublished on Web 10/04/2008

Test. Good reviews of these methods are found in theliterature.25,27

Similar to the Van Ness-Byer-Gibbs test,29 the consistencymethod proposed in this work can be considered as a modelingprocedure because a thermodynamic model that can accuratelyfit the experimental data must be used to apply the consistencytest. It should be pointed out that, although the consistencymethod is model-dependent, good fitting does not guarantee thatthe data are consistent. The proposed area test, derived fromthe Gibbs-Duhem equation must also be fulfilled. The fittingof the experimental data requires the calculation of some modelparameters using a defined objective function that must beoptimized.30

The applications done in this work consider isothermal vaporliquid equilibrium data for eight CO2 + ionic liquid mixtures,including 16 isotherms and a total of 120 PTx data points. Thepressure ranges from 1 to 13.2 MPa, the temperature from 313to 333 K, and the mole fractions of carbon dioxide in the liquidphase from 0.06 to 0.8. All of the data were taken from thesame literature source.16

Development of Equations

The Gibbs-Duhem equation for a binary homogeneous liquidmixture at constant temperature can be written as follows:26

[ VE

RT]dP) x1d(Ln γ1)+ x2d(Lnγ2) (1)

being x1 and x2 the mole fraction of components 1 and 2 in theliquid phase, respectively.

Equation 1 can be written in terms of the fugacity coefficients,as follows:31

[(Z- 1)P ]dP)∑ xid(Ln i) (2)

For the situation of interest in this work, a mixture of anionic liquid with a dissolved high pressure gas, this equationcan be more conveniently written in terms of the properties ofthe dissolved gas in the liquid mixture. If the dissolved gas iscomponent 2 in the binary mixture, then the above equationbecomes:

[Z- 1P ]dP ) x2dLn 2+(1- x2)dLn 1 (3)

This equation can be conveniently expressed in integral form,as follows:

∫ 1Px2

dP)∫ 1(Z- 1)2

d 2+∫ (1- x2)

x2(Z- 1)1d1 (4)

In this equation, P is the system pressure, x2 is mole fractionof the dissolved gas in the liquid mixture, 1 and 2 are thefugacity coefficients of components 1 and 2 in the liquid phasemixture, respectively, and Z is the compressibility factor of theliquid mixture. The properties 1, 2, and Z can be calculatedusing an appropriate equation of state and suitable mixing rules.The limits of the integrals are defined by the range of theexperimental data.

In eq 4, the left-hand side is designated by AP and the right-hand side by A, as follows:

AP )∫ 1Px2

dP (5)

A )∫ 1(Z- 1)2

d 2+∫ (1- x2)

x2(Z- 1)1d1 (6)

Thus, if a set of data is considered to be consistent, thenAP should be equal to A within acceptable defined deviations.To set the margins of errors, an overall percent area deviation|%∆A| between experimental and calculated areas is definedas follows:

|%∆A|) 100Σ[ |Aφ-AP|

AP]

i(7)

The maximum value accepted for this deviation has beendiscussed by Valderrama and Alvarez,22 although furtheranalysis and discussion on the particular application beinganalyzed is presented in a separate section.

In eq 5, AP is determined using the experimental P - x dataat a fixed temperature, while a thermodynamic model, such asan equation of state, is employed to evaluate A in eq 6. If thedata are adequately correlated, meaning that the deviations inthe calculated pressure are within acceptable margins ofdeviations and the individual area deviation %∆Ai are withindefined margins of errors, then the data set is considered to beconsistent. The deviation in the calculated bubble pressure foreach point “i” is defined as follows:

%∆Pi ) 100 ∑[Pcal -Pexp

Pexp ]i

(8)

To evaluate the integrals in eq 6, the following must bedefined: (i) an equation of state; (ii) a set of mixing rules; and(iii) a set of combining rules. In principle, any appropriateequation of state and any mixing and combining rules can beused to evaluate the pressure. It must be mentioned that for aset of N experimental data points, (N - 1) AP areas and (N -1) A areas are calculated for a given mixture at a fixedtemperature.

Vapor-Liquid Equilibrium Calculations

Few reports are available on correlating high pressure VLEof mixtures containing ionic liquids. Simple equation of statemodels such as Redlich-Kwong or Peng-Robinson with vander Waals-type mixing rules give variable deviations.17,15

Deviations as high as 60% are found in some cases, errorsthat are not acceptable for the consistency purposes of thiswork. Our research group has been successfully applying thePeng-Robinson EoS with the Wong-Sandler mixing rulesto model complex high pressure mixtures and to performconsistency tests.22-24 Although, some limitations of theWong-Sandler mixing rule have been pointed out in theliterature,32-34 the results of our previous work have clearlydemonstrated that the Wong-Sandler mixing rule combinedwith the van Laar model (to express the excess Gibbs freeenergy included in the mixing rule), has the accuracy andnecessary flexibility to correlate phase equilibrium variablesin high-pressure systems. Therefore, we have chosen thiscombination, designated as PR/WS/VL, to perform theproposed consistency test.

The fugacity coefficient is calculated from standard thermo-dynamic relations as follows:35

RT ln(i))∫V

∞ [(∂P∂nj

)T,V,n

- RTV ]dV-RT ln Z (9)

The Peng-Robinson equation of state (PR) can be expressedas follows:36

P) RTV- b

- aV(V+ b)+ b(V- b)

(10)

Ind. Eng. Chem. Res., Vol. 47, No. 21, 2008 8417

a) 0.457235(R2TC2 ⁄PC)R(Tr)

b) 0.077796(RTC ⁄PC)

R(Tr)) [1+F(1- T r0.5)]2

F) 0.7464+ 1.54226ω- 0.26992ω2 (11)

For mixtures:

P) RTV- bm

-am

V(V+ bm)+ bm(V- bm)(12)

In this equation, am and bm are the equation of state constantsto be calculated using defined mixing rules.

The Wong-Sandler mixing rules (WS) for the Peng-Robin-son EoS that are used in this work, can be summarized asfollows:37

bm )∑

i

N

∑j

N

xixj(b- aRT)ij

1-∑i

N xiai

biRT-

A∞E(x)

ΩRT

(13)

(b- aRT)ij

) 12

[bi + bj]-√aiaj

RT(1- kij) (14)

am ) bm(∑i

N xiai

bi+

A∞E(x)

Ω ) (15)

In these equations, kij is an interaction parameter, Ω )0.34657 for the Peng-Robinson EoS, and A∞

E

(x) is calculatedassuming that A∞

E

(x) ≈ AoE

(x) ≈ gOE. In this work, gO

E hasbeen calculated using the Van Laar model.

The Van Laar model (VL) for a binary mixture is describedby the following equation:

goE

RT)

(L12 ⁄ RT)x1x2

x1(L12 ⁄ L21)+ x2(16)

The expressions for the fugacity coefficient using the PRequation with the above-described WS mixing and combinationrules can be obtained from eq 12. The problem is then reducedhere to determine the parameters L12 and L21 in the Van Laarmodel and the k12 parameter included in the combining rule for(b - a/RT)12, using available high pressure vapor-liquidequilibrium data for the mixtures of interest. The optimumparameters are those that make minimum the difference betweencalculated pressure Pcal and experimental pressure Pexp. Thisdifference is expressed through an objective function W, definedas follows:

W) 100ND

∑i)1

ND |Pcalc -Pexp

Pexp |i (17)

In this eqn., ND is the number of points in the experimentaldata set.

Consistency Criterion

Although the concept of consistency seems to be differentfrom consistency tests applied to low pressure vapor-liquidequilibrium data, the situation is conceptually the same. At lowpressures, the equilibrium equation is applied and the activitycoefficients are determined and after the modeling the Gibbs-Duhem equation is applied. At high pressures, the equilibriumequation is applied and the fugacity coefficients are determinedto then apply the Gibbs-Duhem equation. Since the gas phasehas low concentration of the ionic liquid, and it is rarelymeasured, it seems more appropriate in this case to use theexperimental data of the liquid phase to do the consistencyanalysis. Therefore, at high pressures, an appropriate model toevaluate the fugacity coefficients of each component in the liquidmixture and the compressibility factor of the mixture are needed.As explained above, the PR/WS/VL model is used to evaluatethese properties.

The model is accepted and the consistency test is then appliedif the average absolute pressure deviations %∆P, defined byeq 8 is below 10%. After the model is found appropriate, it isrequired that the average absolute deviations in the individualareas AP and A, defined by eqs 7 and 8, are below 20% todeclare the data as being thermodynamically consistent. Thedata are considered to be thermodynamically inconsistent (TI)if the deviations in correlating the equilibrium bubble pressureare within the established limits but the individual deviationsin the areas are outside the established limits, for more than25% of the data points in the data set. The test cannot be appliedif the equilibrium pressure is not well correlated, that means ifdeviations in the calculated pressure (eq 17) are greater than10%. If the deviations in correlating the equilibrium pressureare within the established limits ((10%) but the individualdeviations in the areas are outside the established limits for lessthan 25% of the points, then the data are considered to be notfully consistent (NFC).

Data Selection

Data for eight CO2+ ionic liquid mixtures presented by Aki

et al.,16 have been considered in this study. The mixtures arecarbon dioxide with each of the following ionic liquids: [bmim][NO3], [bmim] [BF4], [bmim] [DCA], [bmim] [TfO], [bmim][methide, [bmim] [Tf2N], [hmim] [Tf2N], and [omim] [Tf2N].Table 1 shows the basic properties of the fluid substancesinvolved in the study. In the Table, M is the molecular mass,Tc is the critical temperature, Pc is the critical pressure, Vc isthe critical volume, and ω is the acentric factor. The data forcarbon dioxide were obtained from the DIPPR database,38 whilethe values for the ionic liquids (critical properties and acentricfactors) were taken from the literature.39,40

Table 1. Properties of the Substances Involved in the Vapor-Liquid Calculations

compound IUPAC name formula M TC (K) PC (MPa) ω

CO2 carbon dioxide CO2 44.0 304.2 7.38 0.2240[bmim] [NO3] 1-butyl-3-methylimidazolium nitrate C8H15N3O2 201.1 946.3 2.73 0.6039[bmim] [BF4] 1-butyl-3-methylimidazolium tetrafluoroborate C8H15N2BF4 226.0 632.3 2.04 0.8489[bmim] [DCA] 1-butyl-3-methylimidiazolium dicyanamide. C10H15N5 205.1 1035.8 2.44 0.8419[bmim] [TfO] 1-butyl-3-methylimidiazolium triflate. C12H13N2O3SF3 322.0 1158.0 2.90 0.4118[bmim] [methide] 1-butyl-3-methylimidiazolium tris(trifluoromethylsulfonyl)methide C12H15 F9N2O6S3 550.0 1571.4 2.40 0.1320[bmim] [Tf2N] 1-butyl-3-methylimidiazolium bis(trifluoromethylsulfonyl)imide C10H15N3F6S2O4 419.3 1265.0 2.76 0.2656[hmim] [Tf2N] 1-hexyl-3-methylimidiazolium bis(trifluoromethylsulfonyl)imide C12H19N3F6S2O4 447.3 1287.3 2.39 0.3539[omim] [Tf2N] 1-octyl-3-methylimidiazolium bis(trifluoromethylsulfonyl)imide C14H23N3F6S2O4 475.4 1311.9 2.10 0.4453

8418 Ind. Eng. Chem. Res., Vol. 47, No. 21, 2008

Table 2 gives details on the selected experimental vapor-liquidequilibrium data for the eight mixtures studied. As seen in thetable, data for 16 isotherms were considered. The temperaturesfor which data are available are 313 and 333 K, while thepressure ranges from 1 to 13.2 MPa. Bubble pressure calcula-tions for binary mixtures were performed using the PR/WS/VL model. The adjustable parameters of the model (L12, L21,and k12), were determined by optimization of the objectivefunction given by eq 17.

To evaluate the integral in eqs 5 and 6 for a set of ND

experimental points, two consecutive data points are used,obtaining ND - 1 values of the integrals. To evaluate AP (eq 5)only the experimental data are used, while the PR/WS/VL modelis used to evaluate the fugacity coefficients and the compress-ibility factor included in A (eq 6).

Areas and the Limits of Deviations. The margins ofacceptable error for the bubble pressure and for the calculated areas((10% for the calculated pressure and ( 20% for the areas) needto be explained in more detail, as follows. The percentages definedfor the consistency criterion are based on information presented inthe literature related to the accuracy of experimental data in thistype of mixture (for AP) and on the errors in the calculated pressure(for A). Deviations found with the same model used here (PR/WS/VL) for complex mixtures at high pressure have shown thatthis model has the required flexibility to correlate VLE data withlow deviations.23,24 This proved to be equally applicable to thecases of mixtures containing high pressure CO2 and an ionic liquid,as the results show.

To define the margins of errors for the calculated areas, errorpropagation on the measured experimental data and on thecalculated data has been performed as elsewhere explained bythe authors.24 The authors did bubble pressure calculations andtherefore they considered pressure, temperature, and gas phaseconcentration as the independent variables. In the cases con-sidered in this study, the gas phase concentration is unknownand therefore error propagation is done considering the pressure,temperature, and liquid phase concentration as the independentvariables. This is done as follows:

EA ) [∂A∂P]∆P+ [∂A

∂x ]∆x+ [∂A∂T ]∆T (18)

For the errors in AP, which is determined using only experimentaldata, maximum uncertainties of ∆P ) 0.1 bar for the experimentalpressure, ∆T ) 0.1 K for the experimental temperature, and ∆x

) 0.001 for the experimental mole fraction were considered.41 Theerror EA for the experimental area AP is small, considering thesesmall uncertainties (EP below 1%). For the errors in A, the onlycalculated variable is the pressure and therefore maximum uncer-tainties are determined by the model accuracy (accepted up 10%for ∆P in eq 18). The results are similar to those previously foundby the authors for other systems.42 An example of these calculationsis presented in Figure 1. The error in the area A (eq 18) for anerror of 10% in calculated pressure (0.1Pcal) for two mixtures arepresented. As seen in the figure, maximum deviations in thecalculated areas up to 24% are found for errors in the pressure upto 10% (∆P ) 0.1Pcal). Results are similar for all other systems.Considering all of these results, it is established in the consistencytest method that an error of 10% for the absolute pressure in themodeling step and 20% in the areas (eq 7), is acceptable forconsistency purposes.

Results and Discussion

Table 3 presents the results of the consistency test for all themixtures considered in this study. As seen in the table, 13 ofthe 18 data sets were found to be thermodynamically consistent(TC), four sets were found to be not fully consistent (NFC),and one single set was found to be thermodynamically incon-sistent (TI).

For the isotherms declared to be NFC, [bmim][NO3] + CO2,[bmim][BF4] + CO2 and [bmim][Tf2N] + CO2, all at 313 K, and

Table 2. Details on the Phase Equilibrium Data of Aki et al.,16 forthe Eight Systems Considered in This Studya

range of data

no. CO2+ T (K) P(MPa) xCO2

1 [bmim] [NO3] 313 1.2-9.4 0.10-0.48333 1.3-8.9 0.06-0.42

2 [bmim] [BF4] 313 1.2-8.4 0.13-0.53333 1.4-8.5 0.09-0.45

3 [bmim] [DCA] 313 1.4-11.6 0.15-0.57333 1.8-11.5 0.15-0.53

4 [bmim] [TfO] 313 1.0-8.9 0.10-0.59333 1.5-9.8 0.10-0.54

5 [bmim] [methide] 313 1.8-11.5 0.31-0.78333 1.70-11.2 0.27-0.75

6 [bmim] [Tf2N] 313 1.3-13.2 0.23-0.76333 1.3-13.1 0.17-0.72

7 [hmim] [Tf2N] 313 1.3-1.2 0.25-0.76333 1.70-11.2 0.19-0.72

8 [omim] [Tf2N] 313 1.5-11.5 0.27-0.80333 1.60-11.5 0.24-0.76

a The temperature values have been rounded to the closest integer.

Figure 1. Error in the area A (eq 18) for a deviation of 10% in calculatedpressure for two mixtures s [bmim][NO3] + CO2 and --- [omim][ Tf2N]+ CO2.

Table 3. Calculated Parameters in the Wong-sandler Mixing Rules,Pressure Deviations for the Mixtures, And Consistency Tests for Allthe Systems Studied

mixture, CO2+ T (K) ND k12 A21 A12 |%∆P| |%∆A| result

[bmim][NO3] 313 6 0.4500 2.3300 0.5600 6.5 22.5 NFC333 6 0.0130 1.6466 0.8440 2.7 14.7 TC

[bmim][BF4] 313 8 0.7097 1.5552 0.4364 4.1 22.0 NFC333 7 0.4428 2.8820 0.9144 3.8 19.5 TC

[bmim][DCA] 313 8 0.5600 0.2600 0.0800 6.8 32.3 TI333 8 0.5970 0.0362 -0.9678 2.9 14.1 TC

[bmim][TfO] 313 8 0.2590 1.6988 0.2545 3.5 19.6 TC333 7 0.1088 1.6990 0.2686 4.1 22.0 NFC

[bmim][methide] 313 8 0.4506 -0.1330 -3.6926 2.8 18.2 TC333 8 0.3359 -0.5001 -2.8230 1.1 12.1 TC

[bmim][Tf2N] 313 8 0.3273 -0.2908 -0.2014 2.1 22.9 NFC333 6 0.1939 -0.2706 -0.3778 3.6 17.6 TC333 8 0.2195 -0.4558 -1.0200 3.6 18.9 TC

[hmim][ Tf2N] 313 6 0.2852 -0.3102 -0.5609 1.3 9.1 TC313 8 0.3049 -0.5142 -1.6027 1.4 13.7 TC333 8 0.1749 0.0014 -0.0028 1.5 12.4 TC

[omim][ Tf2N] 313 8 0.2856 -0.4206 -0.2604 1.8 15.2 TC333 8 0.2235 -0.7641 -0.4761 1.1 10.5 TC


[bmim][TfO] + CO2 at 333 K, the modeling gave high pressuredeviations for a single point, resulting average pressure deviationshigher than the established limit of 10%. If that point is notconsidered in the analysis, then the remaining set gave pressuredeviations and area deviations within the established limits.Therefore, the original set of data is declared to be NFC. It shouldbe mentioned, however, that care must be taken with this, sincethe defined percentages have a frontier character only and theexplanatory character of the eliminated data depends on the system,on the pressure and on the temperature of the data. For the caseconsidered to be thermodynamically inconsistent, [bmim][DCA]+ CO2, the average pressure deviation is not very high (6.8%) butthere are three points with pressure deviations higher than 10%and area deviations much higher than the established limits (20%),with an average area deviation of 32.3%.

A graphical description of the results is shown in Figures 2 and3. Results for three representative systems at 313 and 333 K areshown: [bmim][NO3], [bmim][DCA] and [omim][Tf2N]. Figure 2shows the absolute pressure deviation that varies between themargins established to accept the modeling for further analysis.For the three systems shown in the figure, and for all other casesstudied, absolute deviations are below 10%, indicating the goodaccuracy of the PR/WS/VL model used in correlating the data.This fact, however, does not guarantee consistency of the data inthe way defined in this work, as shown in Table 3.

Conclusions

A reasonable and flexible method to test the thermodynamicconsistency of high-pressure phase equilibrium data of binary

mixtures containing an ionic liquid and a high pressure gas hasbeen presented. On the basis of these results, the following threemain conclusions can be drawn: (i) the proposed consistency testmethod allows to globally analyze vapor-liquid equilibrium dataof mixtures containing an ionic liquid; (ii) the numerical techniqueused to find the optimum model parameters for the PR/WS/VLmodel shows to be efficient and accurate for modeling thesolubility; and (iii) the method gives an answer about consistencyor inconsistency of a set of experimental PTx data for all casesthat are well correlated by a thermodynamic model.

Acknowledgment

The authors thank the Direction of Research of the Universityof La Serena for permanent support, the Center for TechnologicalInformation of La Serena-Chile for use of its library and computerfacilities, and the National Council for Scientific and TechnologicalResearch (CONICYT), for its research grant FONDECYT1070025.

Nomenclature

Symbolsa ) Force constant in the PR equation of stateam ) Force constant for a mixtureA∞

E(x) ) Helmholtz free energy at infinite pressureAo

E(x) ) Helmholtz free energy at low pressureAP ) Integral for point x2i to x2i+1 using P - y experimental dataA ) Integral for point x2i to x2i+1 using a thermodynamic model%∆Ai ) Individual percent area deviation

Figure 2. Absolute pressure deviations between calculated and experimentalvalues for three systems: s [bmim] [NO3]; --- [bmim] [DCA]; and - · - · - · -[omim] [Tf2N]. (a) at 313 K and (b) at 333 K.

Figure 3. Absolute area deviations for three systems: s [bmim] [NO3]; ---[bmim] [DCA], - · - · - · - [omim] [Tf2N]. (a) at 313 K and (b) at 333 K.


%∆Aave ) Average relative percent area deviation for the (N-1)areas

|%∆Aave| ) Average absolute percent area deviation for the (N-1)areas

b ) Volume constant in the PR equation of statebm ) Volume constant for a mixtured ) Derivative operatorEA ) Error in the calculated areas (eq 18)F ) Acentric factor parameter for the PR equation of statego

E ) Gibbs free energy at low pressureHR ) Residual enthalpykij ) Binary interaction parameters for the force constant in an EoSk12 ) Interaction parameters for the force constant in an EoS for a

binary mixtureL12, L21 ) Parameters in the van Laar modelLn ) Natural logarithmni ) Number of moles of component “i”nj ) Number of moles of component “j”N ) Number of components in the mixtureND ) Number of data points in a data setP ) Pressure%∆Pi ) Percent deviation in the system pressure for a point “i”Pc ) Critical pressure of componentR ) Ideal gas constantT ) TemperatureTc ) Critical temperatureTr ) Reduce temperature (Tr ) T/Tc)V ) VolumeVE ) Excess volumeW ) Objective functionx ) Mole fraction in the liquid phasexi, xj ) Mole fraction of components i and j in the liquid phasex1, x2 ) Mole fraction of components 1 and 2 in the liquid phaseZ ) Compressibility factor (Z ) PV/RT)

AbbreViations

VLE ) Liquid-Vapor equlibriumEoS ) Equation of stateeq ) EquationNFC ) Not Fully ConsistentPR ) Peng-Robinson EoSTC ) Thermodynamically ConsistentTI ) Thermodynamically InconsistentVL ) Van LaarWS ) Wong-Sandler

Greek lettersR ) Temperature function for the PR equation of stateγ ) Activity coefficient∆ ) Deviation|%∆A| ) Absolute deviation between AP and A (eqn. 7)∂ ) Partial derivative ) Fugacity coefficientω ) Acentric factorΩ ) Constant in the WS mixing rule. Ω)0.34657 for the PR EoS

Super/subscriptscal ) Calculatedexp ) Experimentalgas ) Gasi, j ) Component i or j

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ReceiVed for reView May 12, 2008ReVised manuscript receiVed June 29, 2008

Accepted August 4, 2008

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