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KINETIC MODELLING OF CHEMICAL PROCESSES IN ACID SOLUTION 401

STEYL, J.D.T. Kinetic modelling of chemical processes in acid solution at t ≤ 200°C. (i) thermodynamics andspeciation in H2SO4-Metal (II) SO4-H2O system. Hydrometallurgy Conference 2009, The Southern AfricanInstitute of Mining and Metallurgy, 2009.

Kinetic modelling of chemical processes in acidsolution at t ≤ 200°C. (i) thermodynamics and

speciation in H2SO4-Metal (II) SO4-H2O system

J.D.T. STEYLAnglo Research, South Africa

Although thermodynamic theory has been widely applied to high-temperature hydrometallurgical processes to describe properties suchas solubility and acidity, few of those studies have attempted todevelop an accurate platform to interpret process kinetics. This studyutilized existing theoretical frameworks to model the ternary H2SO4-MeSO4-H2O system up to 200ºC (where Me = divalent metal ions ofCu, Zn and Fe) and less than 1 mol/kg sulphate, using MgSO4 as aprototype salt. A qualitative interpretation of the chemistry atquantum level was used as a complementary tool to add moreconfidence to the ambiguous nature of the thermodynamic datareported in the literature and to fill the gaps where no such data wasavailable at all. The Pitzer ion-interaction approach was used to builda phenomenological model of this system but with the explicitrecognition of four contact ion pairs, i.e. HSO4

−, H2SO4°, MgSO4°and Mg(SO4)22−. This required an iterative approach around the ion-interaction framework. The interaction parameters simulated the long-range electrostatic interactions and the formation of outer-spherecomplexes, while the explicit inclusion of the above complexesrecognized the important covalent interactions. Once parameterizationwas optimized for the two binary systems (H2SO4-H2O and MeSO4-H2O), three mixing parameters were derived for the ternary system.Very limited experimental information has been reported for themixed system, especially at higher temperature. By selecting aminimum number of adjustable interaction parameters, constrainingthe system to the available thermodynamic data and incorporating theobserved species distributions at lower temperature, predictions athigher temperature were made possible. Consistent and interestingtrends have been generated, some of which are presented in this paper.The overall solution chemistry model was ultimately used as a tool tointerpret the kinetics of ferrous oxidation in acidic sulphate medium,which is discussed in more detail in the second paper of the currentseries.

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HYDROMETALLURGY CONFERENCE 2009402

Introduction

Despite the fact that sulphuric acid is a major chemical commodity, the effort expended todescribe the chemistry of sulphuric acid/base metal sulphate aqueous mixtures does not reflectits industrial importance. This is especially prevalent at higher temperature, under conditionsfrequently encountered in oxidative pressure leach applications. Although thermodynamictheory has been widely applied to describe high-temperature hydrometallurgical processes,focus has usually been directed, either towards calculating specific thermodynamic properties(e.g. solubility in solution) or to calculate solution chemical properties such as acidity, derivedfrom potentiometric or conductivity measurement (examples in Liu and Papangelakis, 2005a,2005b, Seneviratne et al., 2003 and, Baghalha and Papangelakis, 2000). This study attemptsto establish a simple thermodynamic model, with emphasis on the explicit recognition of theminimum, but most important complex species in sulphuric acid/base metal sulphate solution,temperatures below 200ºC and concentrations less than 1 mol/kg total sulphate. The chemicalmodel and peripheral arguments ultimately aims at developing a platform for interpretinghydrometallurgical process kinetics.

Methodology

The methodology of modelling thermodynamic properties has traditionally followed an ion-interaction approach, i.e. based on complete electrolyte dissociation. The virial type ofequations of Pitzer (Pitzer, 1991), which is based on statistical mechanics, have been mostwidely applied. However, certain electrolyte systems, such as H2SO4-H2O, require the explicitrecognition of the strong complexes (bisulphate inner-sphere complex, HSO4

− in this case) inorder to accurately describe its thermodynamic properties (Pitzer et al., 1977). Although such‘modifications’ introduce more interaction parameters, it is compatible with the fullydissociation framework. At the other extreme, many researchers studying hydrometallurgicalsystems frequently opt for a speciation-only type approach, for example to predict solubility(Papangelakis et al., 1994) or to support proposed reaction mechanisms (Crundwell, 1987). Incases where such speciation models also confirm independent measurement, for examplespectroscopic or pH, it becomes an attractive alternative to the interaction-type approach (e.g.Casas et al., 2005a). Casas et al. (2005b) compared the performance of three differentchemical models, i.e. Debye-Hückel (DH) B-dot, Pitzer and Bromley-Zemaitis at hightemperature (>200ºC). Although the three models differed in complexity and recognizedspecies (equilibria), they demonstrated almost the same ability to fit the solubility of Mg andAl in ternary systems. This illustrated that ion interaction parameters can make up fordifferences in the number of explicitly recognized species and vice versa. Measuredinformation on species abundance is severely restricted in the open literature, especially athigher temperature. It is therefore natural to limit the model to explicitly contain only thecomplex species that are most likely to kinetic processes, the driving force behind this study.These complexes are likely to be entropy driven and would therefore become even moresignificant at higher temperature. An added advantage is that the interaction framework, basedon thermodynamic and speciation information at room temperature, can be used to infer thespeciation behaviour at higher temperature where only thermodynamic data are available.

Phenomological model

In view of the above discussions, the following interaction-type (some, also includingspeciation) models have been considered for this study (references indicate recently appliedexamples); Bromley-Zemaitis (Liu and Papangelakis, 2005a), Mixed Solvent Electrolyte



(MSE) (Liu and Papangelakis, 2005b), Pitzer (Casas et al., 2005b) and Electrolyte NRTL(Haghtalab et al., 2004). Although these models have successfully been applied to numerouschemical systems, each having their own subtleties, the relatively low ionic strengths coveredin this study makes the choice somewhat ambiguous. Although the ion-interaction model ofPitzer is more complicated, it has been widely applied to thermodynamic studies over the past35 years and its theoretical framework was also adopted for this study.

Thermodynamic framework

The conventional unsymmetrical reference states of infinite dilution for the solute and purewater for the solvent have been chosen for this low to medium ionic strength application. Forthe same reason, the molality (temperature independent) scale was selected, treating theproton as an unhydrated species. The standard Pitzer ion-interaction model is based on anexpression for the excess Gibbs energy of the solution, which consists of an extended Debye-Hückel (DH) term and virial expansion terms. The relatively dilute electrolyte concentrationalso makes ionic strength dependences of the third virial coefficients (Clegg et al., 1994)redundant; these generalized equations can be found in numerous publications (e.g. Pitzer,1991) and the important functions are reproduced here. The single-ion activity coefficient of acation (M), anion (X) and neutral ion (N) may respectively be represented as follows:

[1]

[2]

[3]

where the primed indices refers to summation over all distinguishable pairs, and:

[4]

[5]

and zi is the valency of ion, i. The water activity (aw), more appropriately expressed as theosmotic coefficient, φ may then be written as:

[6]

The ionic strength (I) dependence of the DH terms may be represented as follows:



[7]

[8]

The universal parameter, b has a value of 1.2 (kg/mol)1/2. In this study, the DH parameter forthe osmotic coefficient (Aφ) was directly calculated, following from the original DHderivation (natural logarithm scale):

[9]

The dielectric properties reported by Bradley and Pitzer (1979) and the density reported byCooper and Le Fevre (1982) was used for pure water as the solvent. These properties forwater and the fundamental constants yielded an Aφ value of 0.3914 (kg/mol)1/2 at 25ºC. Thebinary (Bca) (and its derivative) and ternary (Cca) virial coefficients are expressed in terms oftheir respective interaction parameters (βca and Cφ

ca) and their ionic strength functionalities:

[10]

[11]

[12]

[13]

The values of parameters α1 and α2 are usually set to 1.4 and 12 (kg/mol)1/2, respectively,for 2–2 electrolytes but 2 and 0 (kg/mol)1/2, respectively, for other types of interactions (Pitzer,1991). The functionalities may be represented as follows:

[14]

[15]

In mixed electrolyte systems, binary short-range interaction between ions of like charge (Φ)and various ternary interactions (ψ) may be accounted for. In this study, the long-rangeelectrostatic forces (denoted by E) were also included for unsymmetrical mixing. The binarymixing term, Φ, its derivative and the corresponding mixing term for the osmotic coefficientare then expressed by Equation [16], [17] and [18], respectively. The values of theelectrostatic terms were calculated using Chebyshev approximations (see Pitzer, 1991).

[16]

[17]

[18]

As mention before, the Pitzer framework becomes more complicated when strong ionassociations are explicitly treated as a separate species. Thermodynamically, a complexspecies is considered to be a new entity in solution when the mutual electronic attraction ofthe individual ions is considerably greater that their thermal energy (Robinson and Stokes,1959). In aqueous solution, water molecules are chemically bonded to the ion and this formthe inner coordination sphere. An inner-sphere complex is formed when water molecules arereplaced from the inner coordination sphere of the metal ion by a ligand ion (such as SO42−)

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and then, for example, forming a direct chemical bond with the metal ion. When ligand ionsdo not replace any water molecules from the inner-sphere, and is attached to the positive ionsonly by virtue of electrostatic considerations, it forms an outer-sphere complex (Hancock,1976). Activity coefficients for outer-sphere complexes are strikingly similar up to highconcentrations, and this is reflected in very similar values of their association constants. Inner-sphere complexes on the other hand, always show large differences in stability (Prue, 1966).Inner- and outer-sphere complexes represent two extremes and although most ion complexeswould show predominance of one type, both are likely to be present. Aqueous species aretherefore present as simple and complex ions and the modelling objectives would dictatewhich species need to be explicitly expressed. The ionic strength defines the charge in theelectrolyte solution and would obviously vary between these frameworks:

[19]

The ionic strength value corresponding to complete dissociation is referred to as the nominalor formal ionic strength (independent of temperature). Whatever framework is used todescribe non-ideality in an electrolyte solution, the condition of electrical neutrality has to befulfilled:

[20]

If equilibrium is approached from an association perspective, either a step-wise orcumulative description may be used, i.e., respectively:

[21]

[22]

By definition, the activity of an ion is related to its molality in the following manner (thefinal term on the right incorporates the standard state, which is conveniently chosen at unitmolality, so that fi° = mi° = 1):

[23]

The thermodynamic association constant (denoted by °) at the thermodynamic transcendentcondition of infinite dilution for the step-wise (K°) and cumulative (β°) reaction may then berepresented as follows, respectively:

[24]

[25]

Thermodynamic models are used to set up a framework in which thermodynamic propertiescan be regressed with experimental data. Colligative properties such as freezing pointdepression, boiling point elevation, heat capacity, molar volume, the electromotive force(e.m.f.) of cells, and isopiestic measurement (osmotic coefficient) are most often used incorrelation methods. Only the mean activity coefficient of a salt can be obtained from thesemeasurements (individual ions cannot be isolated during measurement). For a salt, Mν1Xν2 themean activity coefficient is defined as follows (Robinson and Stokes, 1959):



[26]

The ion number (ν) of the salt is equal to the sum of the individual stoichiometriccoefficients (ν1, ν2). Regression to solubility data is popular because its availability for bothpure and mixed electrolyte systems (e.g. Linke and Seidell, 1958 and 1965). The generalizedsolubilization reaction for the hydrated salt may be written as follows:

[27]

The thermodynamic solubility product is derived by taking the activity of the solid phase asunity (pure substance). The right-hand terms were respectively obtained after mathematicalsubstitution of Equations [23] and [26]:

[28]

The molalities and activity coefficients in Equation [28] obviously refer to the individualions, whereas the water activity is directly related to the osmotic coefficient as calculated inEquation [6]. In order to maintain internal consistency, it may be necessary to convert thecalculated osmotic and mean activity coefficients to their formal (also called stoichiometric orobserved) values:

[29]

[30]

Finally, the molality scale is preferred scale in solution thermodynamics because it has theadvantage that it is independent of temperature and pressure. However, the concentrationscale has more practical significance and it may be necessary to convert between the scales.The following relation is easily derived (where C refers to the molarity scale and ρ is thesolution density in g/l):

[31]

The standard thermodynamic properties of an equilibrium reaction are constituted from thesum of the primary components taking part in the reaction. Often, the partial molar propertiesof a species are known only at the reference temperature, Tr (usually 25ºC) and sometimes,not at all. The equilibrium constant can be calculated if the partial molar enthalpy offormation of the constituent species, or the standard molar enthalpy of the reaction is knownas a function of temperature (or at the required temperature). This relationship is known as theVan’t Hoff isochore and is derived from classical thermodynamics (the derivation is beyondthe scope of this study—see Atkins, 1986):

[32]

Often heat capacity data is available and integration of Kirchhoff’s Law (see Atkins, 1986)may be used to calculate the enthalpy at the correct temperature:



[33]

The temperature dependence of the heat capacity cannot be predicted and mathematicalcorrelation is therefore used to represent experimental data. The partial molar heat capacity ofa species is often reported by the following mathematical expression (also known as theKelley equation—original reference in Horvath, 1985):

[34]

The coefficients a, b, c, and d are temperature independent. The standard entropy of areaction is related to the standard heat capacity by the following relation (see Atkins, 1986):

[35]

The definition of the standard Gibbs free energy stipulates:

[36]

When no functional expression for the heat capacity is available, it is often assumed that theheat capacity is constant, i.e. the enthalpy of reaction varies linearly with temperature (afterintegration of Equation [33]):

[37]

The entropy value may then be estimated after integration of Equation [35]:

[38]

When no heat capacity data is available, it is often assumed to be zero, i.e. the enthalpy ofthe reaction is constant. Integration of Equation [32] then yields:

[39]

This equation would give only an estimate over a very short temperature range. Predictivemethods have been developed to estimate the average heat capacity between T and Tr, such asthe method of Criss and Cobble (see Zemaitis et al., 1986). This method, is however,questionable when applied the non-simple cations (like metal sulphate ions) and metal-oxy-anions (like the bisulphate ion) (see Blakey and Papangelakis, 1996). Alternatively, a closeapproximation of the value of Kº at higher temperature may be obtained using Helgesonextrapolation (see description and original references in Liu and Papangelakis, 2005a).However, this method may give an overestimation at higher temperatures (>100ºC)(Papangelakis, 2004). The density function (Anderson et al., 1991) is remarkably accurate inestimating equilibrium constants of reactions at higher temperatures when the molal heatcapacity value is known at the reference temperature. The model is based on the observationof almost linear behaviour of lnK° with lnρ and adopts the form as presented in Equation [40].

[40]

The density (ρ) and coefficient of thermal expansion (α) of water are well documented, e.g.study of Anderson et al. (1991). In another approach, the Balanced Like Charge Method(BLCM) utilizes equilibrium reactions with balanced like charges to extrapolate over



temperature (see e.g. Papangelakis, 2004 and Oscarson et al., 1988). Lindsay (1980), referringto these reactions as isocoulombic equilibria, found that because charge is conserved, theirheat capacity is usually constant with temperature and relatively small. Substitution ofEquations [37] and [38] into Equation [36] and simplifying yields the following relationshipand may be used to test the validity of the BLCM approach:

[41]

The following general equation (after substitution of Equations [33], [35] and [36] intoEquation [32] and rearranging) would still yield the superior result, i.e. if the heat capacity isavailable as a function of temperature:

[42]

No ΔV° terms were included in any of the above equations since temperatures were limitedto 200ºC and, hence, to relatively insignificant pressure effects (at saturation).

Chemistry

This section briefly considers a quantum level interpretation of the chemistry and tries to addmore confidence to the ambiguous nature of the data reported in the literature. A basicunderstanding of ion solvation is an essential part of this analysis. The concept of thehydration shell has been verified for cations and anions by spectroscopic measurement and isalso supported by ab initio calculations (see e.g. Pye and Rudolph, 2001). An ion in watermay be considered to be surrounded by concentric shells of water molecules, where thesuccessive shells become more weakly bound to the preceding shells until a bulk waterstructure is reached (Pye and Rudolph, 1998).

Calculation methodology

All calculations were carried out using the DMol3 (Delley, 1990) density function theory(DFT) code imbedded in the Materials Studio (v.4.2) quantum chemistry package fromAccelrys Inc, i.e. the calculations were conducted within the Kohn-Sham formulism (Kohnand Sham, 1965). In order to increase the chemical accuracy when treating the transitionmetal complexes, an all electron scalar relativistic basis set was employed. The gradientcorrection method (GGA = gradient generalized approximation) was employed throughoututilizing two DFT Hamiltonians, i.e. the local spin density (LSD) method utilizing the Voskoet al. (1980) (VWN) parameterization of the Ceperley-Alder Monte Carlo result for ahomogeneous gas; and the non-local spin density (NLSD) method, utilizing a combination ofthe Becke (B) gradient corrected exchange energy (Becke, 1988) and the Pedrew (P) gradientcorrected correlation energy (Pedrew and Wang, 1992). Fully self-consistent field (SCF)calculations were conducted using the VWN-BP functional throughout. High quality doublenumerical atomic basis sets including polarization functions (DNP) were used throughout, i.e.to yield best accuracy, especially since hydrogen bonding was an important aspect of thisstudy. Because these functions were treated numerically (rather than analytically) in DMol3,basis set superposition errors (BSSE) were minimized (Delley, 1990—after Materials Studiohelp file). The aim of this chemical modelling exercise was more qualitative in nature and noadditional effort was therefore expended to quantify the effect of BSSE errors, or to testalternative basis sets or exchange-correlation functionals.



In addition to the use of explicit water molecules as described above, the Conductor-LikeScreening Model (COSMO) (Klamt and Schüürman, 1993) was used to account for longrange electrostatic solvent effects, using the dielectric constant of pure water. COSMO is anexample of a continuum solvent model where the solvent molecules are not treated explicitlybut are expressed as a homogeneous medium characterized by a bulk dielectric constant. Theeffect of the solvent is modelled by imposing charges on the continuum surface, which leadsto a polarization of the wave function within the solute cavity (Ziegler and Autschbach, 2005).Therefore, the electronic structure and geometry of the solute is still described by the DFTmethod, but the solute is placed inside a cavity which has the same shape of the solutemolecule. Outside of the cavity, the solvent is represented by a homogeneous dielectricmedium (Andzelm et al., 1995). The approach followed in this study is based on the defaultDMol3/COSMO recommendations imbedded within Materials Studio. The DMol3/COSMOcan predict solvation energies for neutral solutes with an accuracy of about 2 kcal/mol(Andzelm et al., 1995) and has been tested extensively (Klamt and Schüürman, 1993, andAndzelm et al., 1995). Although continuum models are effective when the impact of thesolvent is predominantly electrostatic in nature, the lack of an explicit solvent treatment is anoversimplification and neglects specific information on the intermolecular interactions, suchas hydrogen-bonding in the first hydration shell. Therefore, in this study, the totalcoordination of the cation and the composite, polyatomic sulphate anion was restricted byexplicitly using water as a coordination filling species in the first hydration shell, whiletreating the effect of the solvent beyond this level via the COSMO methodology. The mostimportant simpler structures were confirmed to be minima via analytic second derivativecalculation. Since neither the vibrational energy, nor the entropy of the solvated species couldbe obtained using the above methodology, the enthalpy rather than the total free energyformed the basis of comparison with reported literature values. It was assumed that theenthalpy could be represented by the potential energy; hence, it was assumed that the kineticenergy (translational, rotational, etc.) cancels approximately for reactions between solvatedions at constant temperature.

First principles perspective

The objective of this simulation exercise was to create a thermodynamic framework from afirst principles perspective and consistent with reported standard state properties of ions andneutral species in aqueous solution (e.g., NBS tables, Wagman et al., 1982). The modellingwas conducted at the thermodynamic condition of infinite dilution in water. Within thecontext of the conventionally defined zero Gibbs energy of formation of the proton at anytemperature, the hydrogen electrode was the appropriate starting point. The absolute electrodepotential in electrochemistry is the difference in electronic energy between the Fermi levelenergy of a metal electrode and a universal reference system (without any additional metal-solution interface) (see Trasatti, 1986). The standard hydrogen electrode (SHE) is a redox halfcell and forms the basis of the thermodynamic scale of oxidation-reduction potentials:

[43]

This reaction can be achieved at a practical level with the use of a platinum electrodebecause it possesses the ability to catalyse the reduction of the proton to form hydrogen gas ata high exchange current density, and vice versa. The recommended value of the absoluteelectrode potential (Eabs, often referred to as the vacuum scale potential) of this cell in water is4.44±0.02 V at 25ºC (Trasatti, 1986). This number is the required link between theelectrochemical potential scale and the physical energy scale. Reported literature values vary



considerably for such an important quantity, with an average around 4.6 V (see details inBockris and Khan, 1993). The absolute value of the solution energy of a single ion is just oneof the uncertainties that have to be dealt with. The exact nature of the hydrated proton atinfinite dilute is clouded by the fact that the proton is geometrically very small and formsfluctuating hydrogen bonds with the surrounding water molecules. Senanayake and Muir(1988) used e.m.f. measurement (with some extra-thermodynamic assumptions) in chloridemedia to estimate ionic activities and hydration numbers. The hydration number of 7 obtainedfor the proton in dilute solution indicated further hydration beyond the first hydration cell.Based upon the acidity function and other physical evidence (Bell, 1959, and Bockris andReddy, 1977), the existence of this well-defined species is usually assumed to predominate instrongly acid solutions. However, the more recent and insightful paper of Marx et al. (1999)suggests that both the H5O2+ and H9O4+ species are important only in the sense of idealstructures and that numerous unclassifiable situations exist inbetween. Ab initio path integralsimulations were used to show that the hydrated proton actually forms a fluxional defect inthe highly structured, hydrogen-bonded network of water. This ‘protonic defect can assumemany different structures, so that an unambiguous distinction between H5O2+ and H9O4+ canno longer be achieved’ (Marx et al., 1999). In view of these discussions, a thermodynamicbasis needed to be established using the molar enthalpy of hydration as reference. Thethermochemical (Born-Haber) cycle consisted of atomizing the gas, ionizing the gas, followedby solvation in water medium and finally, reduction on a platinum electrode to again formhydrogen gas (see Equation [44]). Since neither the vibrational energy, nor the entropy of thesolvated species could be explicitly calculated within the context of the DMol3/COSMOframework, the enthalpy (rather than the total free energy) formed the basis of comparison.

[44]

Comparison between the calculated and reported enthalpy of hydration (Table I) suggeststhat either the H3O+ or the H5O2+ species could be used as basis. A similar conclusion can bedrawn by comparing ΔH°red, obtained by difference of the reported total ionization enthalpyand the hydration enthalpy. Therefore, for the purpose of this study, the following reactionwas used as thermodynamic basis:

[45]

This approach yields the free enthalpy of hydration, ΔHºhyd of the proton as -1114 kJ/moland the reduction enthalpy, ΔHºred as -429 kJ/mol. If an entropy change of -131 J/mol.K (NBSTables) is used for the hydration reaction, a value of 4.65 V for the absolute potential of thehydrogen reaction is obtained. This is in good agreement with the average value reported inBockris and Khan (1993), i.e. 4.6 V.

The next step was to define a thermodynamic basis for the sulphate dianion. The most directapproach was to use the enthalpy of dilution of concentrated sulphuric acid to ‘calibrate’ thesystem:

[46]

The ions refer to the formal species in excess water. The structure of a H2SO4 molecule inconcentrated sulphuric acid consists of extensive hydrogen-bonded networks. This is evidentfrom its exceptionally high dielectric constant (ε) of 110 (Lide, 1991). Kazansky and Solkan



(2003) used ab initio quantum chemical calculations, whereas Walrafen et al. (2000) usedexperimental methods, to show that these hydrogen bonds form very definite ionic clusters.These localized structures may be cationic, anionic or neutral in nature, e.g., in the case of thedimer, [H3SO4 H2SO4]+, [HSO4 H2SO4]

− or [2H2SO4], respectively. The modelling of periodicstructures, using a number of interacting molecules (e.g., Arrouvel et al., 2005), wouldobviously yield the best results. In the context of this study, only the most stable electro-neutral complexes, as reported by Kazansky and Solkan (2003), were used. The average O−Hdistance of 1.74 Å is close to the average calculated values of Kazansky and Solkan (2003) of1.8 Å. The total electronic energies (Table II) suggests that the neutral trimer is a sufficientlystable (23% change per mole sulphur) to be used as basis to define the sulphate dianionstructure in water solvent.

A description of the structure of the sulphate ion in water is important because it wouldaffect its interaction with cations, such as the proton and divalent metal ions. The formation ofa contact ion pair between magnesium and sulphate ion was postulated as a temperaturedependent equilibrium with the hydrated sulphate and the hexaaquo magnesium ion (see Pyeand Rudolph, 1998). In a later publication (Pye and Rudolph, 2001), water was proposed tocoordinate with the sulphate ion in a bidentate fashion to form two hydrogen bonds per pair of

Energy Step (Born-Haber Cycle) Calculated Literature References

(kJ/mol) (kJ/mol)

ΔHºat+ΔHºion H(g) → H+(g) 1543 1536 Wagman et al. 1982 (NBS tables)

ΔHºhyd, 25ºC H+(g) → H+

(aq) -524 1103 Marcus, 1987

H+(g) + H2O(aq) → H3O

+(aq) -1081

H+(g) + 2H2O(aq) → H5O2

+(aq) -1147

H+(g) + 3H2O(aq) → H7O3

+(aq) -1185

H+(g) + 4H2O(aq) → H9O4

+(aq) -1216

ΔHºred, 25ºC H+(aq) → H(g) -1019 a-424 Bockris and Khan, 1993: 4.6 Vred

H3O+

(aq) → H(g) + H2O(aq) -462

H5O2+

(aq) → H(g) + 2H2O(aq) -396

H7O3+

(aq) → H(g) + 3H2O(aq) -358

H9O4+

(aq) → H(g) + 4H2O(aq) -327

Molecule Etot (hartrees) Ediff (hartree) (per sulphur) Distance (Å) H-Bond(O−O) Distance (Å) H-Bond(O−H)

[H2SO4] -702.0775021 − − −

[H2SO4]2 -1404.1582559 -0.0016259 2.74 (avg.) 1.72, 1.74

[H2SO4]3 -2106.2385225 -0.0003795 2.74 (avg.) 1.70, 1.78

Table I

Calculated and literature values of the various energy contributions to the Born-Haber cycle for the hydrogen system

aCalculated (Reaction [45]) from average reported absolute potentials and the entropy of hydration, ΔShyd = -131 J/mol.K

(Wagman et al., 1982)

Table II

Calculated molecular energies and average hydrogen bond lengths of the symmetrical complexes of H2SO4 in 100%

sulphuric acid (bulk permittivity of 110)



sulphate oxygen atoms. Structural energetic and vibrational characteristics of isomerscontaining less than six waters in the primary hydration shell were considered. However,comparison between calculated and experimental S-O bond lengths suggested that more thansix waters should be present in the primary hydration shell. Cannon et al. (1994) presentedevidence, using ab initio calculations, that this tetrahedral dianion strongly interacts withwater; roughly three water molecules were coordinated with each of the sulphate oxygenatoms. Therefore, unlike most anions, sulphate can be considered a structure-making ion withrespect to the solvent around it (Cannon et al., 1994). The model developed by Cannon et al.orientated the water with one hydrogen pointing towards sulphate oxygen and the otherhydrogen pointing towards the neighbouring water oxygen. The number of waters in the firsthydration shell of the sulphate ion has been reported to be approximately 6 to 14, based ondiffraction data (see Cannon et al., 1994). A more recent study by Vchirawongkwin et al.(2007), the hydrated sulphate ion was characterized using ab initio quantum mechanical (QM)charge field (QMCF)—molecular dynamics (MD) simulation and large angle X-ray scattering(LAXS) methods. The LAXS data showed an average coordination number (i.e. hydrogenbonded to sulphate ion) of up to 12, while the QMCF-MD simulation displayed a range ofbetween 8 and 14. Experimental investigations in several sulphate solutions (see references inVchirawongkwin et al., 2007) have been conducted and coordination numbers between 6 and8 were assumed. The study of Wang et al. (2000) indicated that isolated SO4

2− is unstable inthe gas phase (strong Coulomb repulsion between the two excess electrons) and a minimumof three water molecules is necessary to stabilize the dianion. Around 12 water moleculeswere claimed to be present in the first hydration shell, with four water molecules, eachforming two direct hydrogen bonds with the sulphate oxygen atoms. The first principles studyof Gao and Liu (2004) considered a large number of isomers in two structural series, using abinitio molecular dynamics simulations. They considered up to 12 water molecules in theprimary hydration shell at 100K. However, when the induced conditions approached 200 K, a‘crowding-out’ of the first hydration shell was observed, with four molecules moving to thesecond shell and eight remaining the primary hydration shell. As pointed out by Cannon et al.(1994), the above wide range of values for the hydration number in the primary hydrationshell exemplifies the difficulties involved in building solvation models for polyatomic ions.The more accurate approach would be use the QM-MD formulation (as suggested by Sterzeland Autschbach, 2006), but since this was outside the scope of this study, the thermodynamicreference framework defined above (for the sulphuric acid system) was expanded to definethe sulphate ion and its complexation with cations.

The sulphate ion is a polyatomic group with the double charge evenly distributed over thefour oxygen atoms in tetrahedral symmetry. The hydrogen bond network in this first shellmust adapt to the geometry imposed by the sulphate group (Gao and Liu, 2004). The Td

structure of Pye and Rudolph (2001) was used as starting point and geometrically optimizedusing the calculation methodology described earlier. The starting structure consisted of eachH2O molecule acting as a bidentate hydrogen bond donor to two sulphate oxygen atoms. Eightoptimizations, each time modifying the starting structure slightly, were conducted and on eachoccasion the structure relaxed to form only one direct hydrogen bond to sulphate oxygen foreach H2O molecule, while the other hydrogen orientated itself towards the dielectriccontinuum. However, at least one and often two H2O molecules donated their ‘free’ hydrogento another H2O oxygen in the same group for hydrogen bonding. This phenomenon suggestedthat water molecules are likely to form clusters in the primary hydration sphere. In fact, as



illustrated by Gao and Liu (2004), the geometrical coincidence of the sulphate group with itstetrahedron structure either form cyclic rings (up to 3 molecules) to accommodate more H2Oin the primary hydration shell, or alternatively a ‘crowding-out’ effect of the first hydrationshell is observed at finite temperature, with some molecules moving to the second shell. Withthis in mind, the H2O molecules was arranged either in 2 or 3 clusters; for example, with 6 molecules in the primary hydration shell, either 2 clusters of 3 H2O molecules or 3 clustersof 2 H2O molecules could be considered. As expected, the highest combination of 3 cyclicrings always resulted in the lower overall energy for the total structure. In most cases, one ofthe H2O molecules in the 3 cyclic clusters orientated itself towards the dielectric continuum.This is in line with the observations of Gao and Liu (2004): ‘the hydrogen bond network inthe first shell must adapt to the geometry imposed by the requirement to solvate the sulphategroup, which introduces strain in the network and even repulsion among the first H2Omolecules’. The calculated energies are listed in Table III. It is important to realise that thesestructures may not be the true minima but that further ‘modifications’ are likely to result onlyin marginal improvements of the calculated reaction enthalpy for the general associationreaction:

[47]

Besides disagreement between calculated and reported reaction enthalpies, the asymmetricalnature of the lower complexes suggests at least 8 H2O molecules in the inner hydration sphereat infinite dilution. This is a simplified approach and if the reality is in fact more molecules inthe primary hydration sphere, the bonding is likely to be less structured than depicted inFigure 1. This may be the case since the calculated value of the absolute standard molarenthalpy of hydration for the sulphate ion (-1213 kJ/mol) is lower than the reported value (-1026 kJ/mol, Marcus, 1987, based on H+

(g)→ H+(aq): ΔH° = -1103 kJ/mol). Figure 1 presents

a static snapshot of the sulphate ion in water solvent; the more accurate approach would be toconsider a dynamic simulation (e.g., Vchirawongkwin et al., 2007), as mentioned above.Nevertheless, the static approach was considered sufficient for the purpose of this study.Typical calculated hydrogen bond lengths (Steyl, 2008) were in reasonable agreement withmeasured (Vchirawongkwin et al., 2007) and calculated (Gao and Liu, 2004) values from theliterature.

Molecule aEtot (hartrees) bΔHºrx,calc (kJ/mol) cNBS tables (kJ/mol)

H2O -76.6240464 − −

SO42−(2H2O)3 -1160.989182 38.8 95.3

SO42−(3H2O)2 -1160.993309 49.6 95.3

SO42−(2H2O)2(3H2O) -1237.624055 67.2 95.3

SO42−(2H2O)4 -1314.254551 84.1 95.3

SO42−(2H2O)(3H2O)2 -1314.260114 98.7 95.3

SO42−(2H2O)3(3H2O) -1390.889044 111.6 95.3

SO42−(3H2O)3 -1390.894635 126.2 95.3

Table III

Calculated molecular energies of SO42−(H2O)n (n = 6-9) and the corresponding reaction enthalpy (Reaction [47])

aCalculated (absolute) energies of molecular species; bCalculated (potential) energies of Reaction 47; cWagman et al. (1982)



The bisulphate, HSO4− ion and H2SO4° molecule were the next species to consider. In view

of the strong interaction between the acidic hydrogen and the oxygens around the sulphur, thesmaller overall charges, and the unsymmetrical geometries, that these species would be lesshydrated than the solvated ‘free’ sulphate ion. The bond distance between the proton and thesulphate oxygen is so short (~1Å) that the proton and one water is not enough to form a cyclicring between two sulphate oxygens. However, the proton and two or three waters may formcyclic rings, as suggested in the study of Arrouvel et al. (2005) for the hydrated H2SO4°molecule. The measured Raman frequencies of Walrafen et al. (2000) in sulphuric acidsolutions pointed to strong hydrogen bonding, resulting from direct H3O+-HSO4

− ion pairinteraction. They suggested 3 or 4 H2O molecules in association with the bisulphate ion inmore dilute acid environment. To be consistent with the way the hydration of the proton wasdealt with in this study, a single H2O molecule was bound to each of the O-H groups of thebisulphate ion and acid molecule, i.e. to form the hydronium ion, H3O

+. The rest of the H2Omolecules were then arranged in either 2 or 3 cyclic rings, similarly to the approach followedfor the hydrated sulphate ion. The acid molecule was modelled by adding one H2O moleculeto each of the protons. It should be noted that these ‘unconstraint’ hydronium ions resulted inseveral structures which were very close in energy. The overall association reaction for thefirst protonation of the sulphate ion may be represented as follows:

[48]

Comparison between the calculated and literature data (Table IV) suggests that the sulphateion sheds half its water upon protonation, i.e. forms HSO4

−(H2O)4. Taking into considerationthe crude assumptions made thus far, the calculated pK2 of 2.2 is in reasonable agreement

Figure 1: Optimized structures for the lowest energy complexes of SO42−(H2O)8

Molecule a½Etotb½ΔHºrx,calc

c½ΔHºrx,lit Log (Kº) Log (Kº)

(hartrees) (kJ/mol) (kJ/mol) (Calculated) (Literature)

HSO4−(H2O)3 (a) -931.546511 51.3 (Rx 52) 73.35 − −

HSO4−(H2O)3 (b) -931.547928 55.1 (Rx 52) 73.35 − −

HSO4−(H2O)4 -1008.180816 78.3 (Rx 52) 73.35 dpK2: 2.2 c1.99; e1.96

HSO4−(H2O)5 (a) -1084.809288 89.9 (Rx 52) 73.35 − −

HSO4−(H2O)5 (b) -1084.811148 94.8 (Rx 52) 73.35 − −

H2SO4(H2O)2 -855.3544882 7.7 (Rx 53) − fpK1: -1.3 g-4.7 to -2.0

Table IV

Calculated molecular energies of HSO4−(H2O)n (n = 3-5) and H2SO4(H2O)2, reaction enthalpies and equilibrium

(protonation) constants

aCalculated (absolute) energies of molecular species; bCalculated (potential) energies of corresponding reactions; cWagmanet al. (1982); dΔSºrx = 111.7 J/mol.K (Wagman et al. 1982); eDickson et al. (1990); fΔSºrx = 0 J/mol.K; gPerrin (1982)



with values listed in the literature (~2) (the pK refers here to the negative logarithm of theionization constant). This calculation included the reaction entropy of 111.7 J/mol.K fromWagman et al. (1982). Assuming that the H2SO4(H2O)2 molecule (as modelled above) is anaccurate enough description, the following reaction may be written for the protonation of thebisulphate ion:

[49]

The calculated pK1 of -1.3 (assuming zero reaction entropy) is slightly higher than typicalvalues listed in the literature, but is considered acceptable in view of the spread of thereported values (see Figure 6).

The next and final step in the quantum level description of the chemistry of the system wasto build a basis for the metal ion and its interaction with the sulphate ion to form contact ionpairs. Since Mg2+ fulfilled the role of surrogate metal for the all the other divalent metal ionsin solution, a description of its solvation was the logical starting point. This alkaline earthelement has a hexagonal crystal structure. The unit cell, consisting of two atoms, wasmodelled using space group P63/mmc and the experimental cell dimensions of a = 3.2088 Åand c = 5.2099 Å (ICSD Database, 2008). The optimized energy, using periodic DFTcalculation, was calculated as -400.837182 hartrees, while keeping the number of k pointswithin reasonable limits. The optimized cell dimensions were calculated as a = 3.1811 Å andc = 5.1493 Å. The optimized energy of the Mg atom in vacuo was calculated as -200.360445hartrees, allowing the cohesive energy, Ecoh to be obtained as follows:

[50]

where n in this case is the number of atoms in the unit cell. The calculated value of -1.58 eV isin good agreement with the literature value (Kittel, 1996) of -1.51 eV. The work required todecompose the metal into a single atom (atomization) is thus 153 kJ/mol, which compareswell with the NBS Tables value of 148 kJ/mol. Divalent transition metal cations, like Mg2+

and the other metals encountered in this study (Cu2+, Fe2+ and Zn2+), are strongly hydrated atroom temperature in dilute aqueous environment and consist of 6 waters (see Horvath, 1985)in an octahedral arrangement in the primary hydration shell. The studies of Pye and Rudolph(1998) and Rudolph et al. (2003) confirmed the hexaaquo magnesium complex by RamanSpectroscopy (RS) and ab initio calculation. The calculated energy of the Mg2+ ion in the gasphase of -199.533566 hartrees may be compared to the gas phase value of the atom (listedabove) to yield the calculated enthalpy of ionization of 2171 kJ/mol. This compares well withthe literature value of the ionization enthalpy of 2201 kJ/mol (Wagman et al., 1982). The freeenthalpy of hydration may be calculated according to the following reaction and is seen to bein close agreement with the literature (Table V):

[51]

In order to complete the thermodynamic cycle, the standard reduction potential may bewritten as follows:

[52]

Since the difference in the energy of the electron between the gas (vacuum) and the metal(Fermi level) of the hydrogen electrode has been calculated as 4.65 V (previous discussions),the standard reduction potential (Reaction [52]) is easily calculated as -2.57 V. This isreasonably close to the literature value of -2.36 V (Atkins and De Paula, 2006). Theconventional molar entropy of hydration of Mg2+(aq) (-138.1 J/mol.K) and the entropy change



for the hydration reaction, H2(g) → 2H+(aq) (-131 J/mol.K) are very similar (values from

Wagman et al., 1982). The fact that the entropy change for the overall reaction is small (~7J/mol.K), explains why the calculated reduction potential agrees reasonable well with theliterature value. As mentioned before, this study also neglects any kinetic energy changes; it isassumed that these energies cancel approximately, at least for reactions between solvated ionsand at constant temperature. The reaction between the solvated Mg2+ and SO4

2− ions to formthe first metal contact ion pair, may now be evaluated:

[53]

If the Mg2+ ion bonds to a sulphate oxygen in a monodentate fashion, the removal ofsulphate water may be analogous to the way the bisulphate molecule was formed, i.e. half ofthe H2O molecules from the SO4

2−(H2O)8 ion were shed. The primary hydration shell aroundthe Mg2+(H2O)6 ion is stronger and more structured compared to the SO4

2−(H2O)8 ion.Assuming only one H2O molecule will be lost from the Mg2+(H2O)6 ion due to thismonodentate bond, three cases may be considered for the contact pair: (H2O)5MgSO4(H2O)3,(H2O)5MgSO4(H2O)4 and (H2O)5MgSO4(H2O)5. The calculated energies are summarized inTable V. Comparison between the estimated reaction enthalpy from Akilan et al. (2006a) andthe calculated values of the complexes, suggests 8 or 9 waters of hydration for the contact ionpair. The optimized structure of the MgSO4(H2O)9 complex, is illustrated in Figure 2.

Molecule aEtotbΔHºrx,calc ΔHºrx,lit Log (Kº) Log (Kº)


Mg(H2O)62+ -660.024857 -1961 (Rx 55) c-1949 − −

MgSO4(H2O)8 (a) -1514.529050 − − − −

MgSO4(H2O)8 (b) -1514.523722 44.55 (Rx 57) d32 e-2.06 −g-0.96 f1.21

MgSO4(H2O)9 -1591.1545255 26.74 (Rx 57) d32 e1.05 −g2.15 f1.21

MgSO4(H2O)10 -1667.787060 -10.08 (Rx 57) d32 − −

Table V

Calculated molecular energies of Mg(H2O)62+ and monodentate MgSO4º contact ion pairs, reaction enthalpies and

equilibrium constants

Figure 2. Optimized structure for the lowest energy complex of MgSO4(H2O)9

aCalculated (absolute) energies of molecular species; bCalculated (potential) energies of corresponding reactions; cMarcus (1987) (expt.), based on H+

(g) → H+(aq): ΔHº = -1103 kJ/mol; dAkilan et al. (2006a); eΔSºrx = 110 J/mol.K (Akilan

et al., 2006a); fAtkinson and Petrucci (1966); gΔSºrx = 131 J/mol.K (see discussion below)



Dielectric spectroscopy (DRS) (Akilan et al., 2006a) and ultrasonic relaxation values(Atkinson and Petrucci, 1966) have proven the existence of a stepwise mechanism, commonlyreferred to as the Eigen mechanism:

[54]

[55]

[56]

The above mechanism presents a general description of the association of free hydrated ionsto first form a double solvated ion pair, then solvent-shared ion pair, and ultimately, thecontact ion pair. The enthalpy and entropy terms for the overall inner-sphere complexreaction, as reported in Table V, was obtained by summing the thermodynamic contributionsof each of the mechanistic steps. The alternative value of 131 J/mol.K for the reaction entropywas obtained from the ultrasonic relaxation data reported by Atkinson and Perucci (1966): theenthalpy for each of the mechanistic steps were fixed at the values reported by Akilan et al.(2006a), while the entropy was calculated to give the average thermodynamic equilibriumconstants in Atkinson and Perucci (1966). This value of the reaction entropy gives log(Kº) =2.15 and illustrates the sensitivity of the equilibrium constant to the reaction entropy.

Various studies (e.g. Zhang et al., 2002 and Rudolph et al., 2003) speculate on thepossibility of a bidentate contact ion pair. The optimized structure (9 waters of hydration) isillustrated in Figure 3, while the calculated energy values are listed in Table VI. By comparingthese energies with the literature enthalpies (Akilan et al., 2006a), it becomes clear that thebidentate complex may explain the formation of a second ‘type’ of contact ion pair. Althoughthe reported thermodynamic values of Akilan et al. (2006a) are qualitative in nature, they doprovide estimates of the enthalpy and entropy contributions of the various equilibria steps.

The DRS study of Akilan et al. (2006) identified the formation of a possible triple ion athigher ionic strengths, assumed to be Mg2SO42+. Although the existence of a triple ion issupported by RS (Rudolph et al., 2003), the presence of the corresponding anion, Mg(SO4)2

2−

cannot be ruled out. This is because the geometry of this anion would be close to linear andtherefore, have a zero dipole moment (cannot be detected by DRS—see Akilan et al., 2006a and Buchner et al., 2004). The study of Zhang et al. (2002) even consideredlarger Mg2+-SO4

2− clusters. Since modelling of the primary contact ion pair has revealed thatthe SO4

2−(H2O)8 ion may lose half its waters to form the Mg(H2O)5SO4(H2O)4 monodentateor the Mg(H2O)4SO4(H2O)5 bidentate complex, no waters are likely to be directly associatedwith the anionic component if the Mg2SO42+ triple ion is formed, i.e.:

Figure 3. Optimized structure for the lowest energy complex of MgSO4(H2O)9 (bidentate)



[57]

[58]

Various structures were considered for the Mg2SO42+(H2O)10 (monodentate) andMg2SO42+(H2O)8 (bidentate) ion pairs, but the minor electronic energy differences betweenthe structures had almost negligible effects on the calculated equilibrium constants. Thevalues reported in Table VI clearly suggest that the Mg2SO42+ complex is an unlikelycandidate for the second contact ion pair, even when using a larger reaction entropy change of130 J/mol.K. The more likely triple ion contact ion pair is the Mg(SO4)2

2− species, as can beseen from the values reported in Table VI. The corresponding optimized symmetricalstructure is illustrated in Figure 4 and the overall reaction is represented by Equation [59].

[59]

The structure presented in Figure 4 was obtained by adding another partially hydratedSO4

2−(H2O)4 ion to the optimized Mg(H2O)5SO4(H2O)4 monodentate structure in asymmetrical fashion. The presence of this species is in line with the potentiometric study ofFedorov et al. (1973), who assumed the formation of a series of anionic sulphate complexes

Molecule aEtotbΔHºrx,calc

cΔHºrx,lit Log (Kº) cLog (Kº)


MgSO4(H2O)8 (bidentate) -1514.511298 77.17 (Rx 57) 43 d -6.21 -0.22

MgSO4(H2O)9 (bidentate) -1591.1467755 47.14 (Rx 57) 43 d -0.95 -0.22

Mg2SO42+(H2O)10 (monodent.) -1868.033131 68.32 (Rx 61) 43 e-10.40 -0.36

43 f -5.18

Mg2SO42+(H2O)8 (bidentate) -1791.381005 142.04 (Rx 62) 43 e-23.32 -0.36

43 f-18.10

Mg(SO4)22– (H2O)12 (monodent.) -2522.275826 48.77 (Rx 63) 43 e -6.98 -0.36

g-3.5

Table VI

Calculated molecular energies of possible second Mg-SO4x contact ion pairs, reaction enthalpies and equilibrium

constants

aCalculated (absolute) energies of molecular species; bCalculated (potential) energies of corresponding reactions; cAkilan et al. (2006a); dΔSºrx = 140 J/mol.K for the (overall) second contact ion pair (Akilan et al., 2006a). eΔSºrx = 30 J/mol.K (Akilan et al., 2006a) for ‘triple’} ion formation: Reactions [57], [58] and [59]; fΔSºrx = 130 J/mol.K;gΔSºrx = 100 J/mol.K

Figure 4. Optimized structure for the lowest energy monodentate complex of Mg(SO4)22−(H2O)12



for Zn(SO4)n{2-2n} and Cd(SO4)n{2-2n}, up to n=5 at high ionic strengths. It should, however, bementioned that this claim of a high degree of anionic pairing is controversial and has beenquestioned on the basis anion exchange data (see Rudolph, 1998). On the other hand, theformation of inner ferric, Fe(SO4)n{3-2n} complexes are known to form (e.g. Magini, 1979) andalthough it can be partially explained by virtue of the high positive charge of the hydratedferric ion, it does suggest that this kind of anionic pairing may, although to a lesser extent, bepresent in the divalent case. This may specifically be true at higher temperature, especially ifthe reaction entropy is higher than the estimated value of 30 J/mol.K of Akilan et al. (2006a).A reaction entropy of about 100 J/mol.K yields log(Kº) of -3.5. It is concluded that thepresence of this species could be significant at higher temperature, especially in the case of asofter transition metal cation (cf. the harder Mg alkali-earth cation), e.g. Cu2+. That theentropy for the formation of the anionic triple ion may be closer to 100 J/mol.K (cf. 30 J/mol.K), stems from a comparison between the change in the solvent cavity volume of thevarious reactions, as computed within the continuum solvent methodology. The calculatedvolume increase for the first contact ion pair formation (Reaction [53]) was approximately20Å3, while the entropy change for this reaction was estimated to be about 130 J/mol.K (Table V). The calculated volume change for the formation of the anionic triple ion(Mg(SO4)2

2−(H2O)12), via Reaction [59], was very similar, i.e. around 23Å3. The triple anioncarries a net negative charge that would result in some structuring effect in the secondarysolvation shell and beyond. An estimated entropy change of 100 J/mol.K is justified on thegrounds that the ‘disruptive’ effect caused by the accommodation of this large molecule in thesurrounding solvent is only partially compensated for by its structuring effects.

The above results should be viewed as qualitative in nature due to the static modellingapproach followed. It should also have been extended to the divalent transition metals (Cu2+,Fe2+ and Zn2+). However, available thermodynamic data were largely limited to the Mg case(next section), especially for the ternary system, and such a study was not justified within thecurrent context (a static model of the Fe2+/sulphate system is presented in Steyl, 2008).

Model regression

This section requires some quantitative thermodynamic values, while the chemical modelling(previous section) has generated only qualitative information.

Thermodynamic data

The first association of the proton (second protonation of acid, H2SO4) may be written asfollows:

[60]

The thermodynamic values of this association constant (K1º) was obtained from the study ofDickson et al. (1990). They measured (potentiometrically) the dissociation constant of HSO4

−

in NaCl background solution and fitted the data to a series of equations. The followingexpression (molality scale) was directly incorporated into the solution chemistry model andgives the thermodynamic association constant up to 250°C:

[61]

where T (kelvin) and the vector, p = [562.7097,-13273.75,-102.5154,0.2477538,-1.117033E-4]. Various researchers have in the past used the thermodynamic values of Marshall and Jones(1966). They determined the equilibrium constant from solubility measurements of CaSO4 insulphuric acid solution (0–1 mol/kg) from 25–350°C. However, their results have beencriticized by various authors (e.g. Shock and Helgeson, 1988 and Dickson et al., 1990)



because they ignored the presence of the neutral aqueous species, CaSO4° and H2SO4°, whichmay exist in significant amounts, especially at high temperatures. The study of Dickson et al.(1990) also acknowledged the most important results from the literature between 1952 and1990 and incorporated some of it into their expression (Equation [61]). One of the mostimportant studies during that period was the work of Pitzer et al. (1977), but their equationsare valid only to 55°C.

Figure 5 compares the performance of the Equation [61] against other thermodynamic datareported in the literature. The broken lines represent the calculated equilibrium constant afterintegration of the enthalpy and entropy terms (Equation [42]), using reported heat capacityparameters (Equation [34]). The data points were obtained from a random scan of publishedvalues in the open literature (Steyl, 2008). The equation of Dickson et al. (1990) performedwell up to high temperatures (>200ºC) and was therefore adopted for this study. The HSC(2006) heat capacity parameters, imbedded within the software, were obtained from Helgesonextrapolation (Shock and Helgeson, 1988). The second association of the proton (firstprotonation of acid, H2SO4) may be represented as follows:

[62]

Reported thermodynamic values for this association constant (K2º) is very rare because theconcentration of the neutral species, H2SO4º, can be accurately determined only at hightemperature and concentration. In most studies of the dilute sulphuric acid system, even athigh temperature (e.g. Marshall and Jones, 1966, Dickson et al., 1990 and Rudolph, 1996),the formation of the neutral species was ignored. Some of the older Raman studies (e.g.Young and Blatz, 1948 and Young et al., 1959) and the Nuclear Magnetic Resonance (NMR)study of Hood and Reilly (1957) quantitatively took the existence of the neutral ion pair intoaccount, even at low temperatures (<50ºC), albeit at high acid concentrations. The RS studyof Rao (1940) suggested (qualitatively) around 95% dissociation of the neutral species in 1 mol/litre acid. More recent work include the RS (structural) investigation of Walrafen et al.(2002), the spectroscopic measurements of Xiang et al. (1996) and the flow calorimetric workof Oscarson et al. (1988), all at high temperatures (>150ºC). Agreement between the studiesof Xiang et al. and Oscarson et al. is good, and the published equations of latter were used asthe primary source of data for this study. The isocoulombic reaction was obtained byconsidering the water dissociation reaction in addition to Reaction [62], i.e.:

[63]

Both the assumption of zero reaction heat capacity and the BLCM model for theisocoulombic reaction fit their respective linearity assumptions very well over the 150–200ºCrange (Steyl, 2008). The thermodynamic data reported by Sweeton et al. (1974) was used for

Figure 5. Effect of temperature on the thermodynamic association constant (K1°)



the water dissociation reaction. The Density and BLCM models predicted very similar valuesof log(K2º) at 25ºC, i.e. -1.7 and -1.8, respectively. The considerable spread in reported valuesexemplifies the difficulties in measuring this equilibrium constant in dilute sulphuric acidsolution, i.e. the neutral molecule is not present in measurable quantities at room temperature.The thermodynamic values reported by Oscarson et al. (1988) at the reference temperature of150ºC (see Table VIII) were therefore used in the phenomenological model, along with theDensity model to extrapolate to other temperatures.

Earlier discussions highlighted the lack of information on the structure and thermodynamicproperties of contact ion pair formation of the divalent metal sulphates. Spectroscopic studiesof MgSO4 solutions (Akilan et al., 2006a, Buchner et al., 2004, Rudolph et al., 2003), CuSO4

solutions (Akilan et al., 2006b), ZnSO4 solutions (Rudolph et al., 1999a and 1999b), CdSO4

solutions (Rudolph, 1998), FeSO4 solutions (Rudolph et al., 1997) and, NiSO4 and CoSO4

solutions (Chen et al., 2005) have provided a major contribution to the understanding ofspecies complexation in the metal sulphate self-medium. However, the reportedthermodynamic values undoubtedly contain uncertainties (see Akilan et al., 2006a). The RSstudy of Rull et al. (1994) has even gone as far as dismissing the presence of contact ion pairsaltogether in solutions, up to 2.9 mol/kg MgSO4 and 80ºC. Figure 7 compares the overall(SIPs+CIPs) equilibrium data with various extrapolated lines. The data (open symbols) at hightemperature, used in the models of Baghalha and Papangelakis (1998) and Casas et al. (2005),were calculated using Helgeson extrapolation, while the model of Liu and Papangelakis(2005) used the Density model. The lines in Figure 7 were calculated using the Density modeland thermodynamic reference data from various spectroscopic studies. This figure exemplifies

Figure 6. Extrapolation of the thermodynamic association constant (K2°) to 25ºC

Figure 7. Experimental data vs. extrapolated values of the overall thermodynamic constant, K3°(tot)



the disagreement in the literature, especially when extrapolating to higher temperature. From aphenomenological modelling perspective, Casas et al. (2005) illustrated that explicitrecognition of species are not actually required. In fact, numerous studies (e.g. Holmes andMesmer, 1983, Phutela and Pitzer, 1986 and Reardon and Beckie, 1987) used an ioninteraction approach without any explicit species recognition to describe the thermodynamicsof metal sulphate systems up to high temperatures. However, as pointed out by Akilan et al.(2006), models that take no note of the actual species present ‘cannot be much more thanexercises in numerology, with little physical significance’.

Since this study was concerned with building a framework to assist with the interpretation ofkinetic processes, it was important to recognize the existence of the contact ion pairs, despitethe lack of thermodynamic data. In view of the rationale followed thus far, the total ‘free’metal ion molality refers to the sum of the actual free hydrated ions and all the solventseparated ion pairs (SSIPs):

[64]

where y refers to the mole fraction of apparent unassociated metal ions (m) and mf refers to theformal metal salt molality (CIP refers here to the first contact ion pair, MgSO4º and αCIP is thefraction contact ion pairs):

[65]

and z is the true mole fraction of unassociated ions. This equation may be combined withsome of the thermodynamic relationships presented earlier, the yield the following equation ofequilibrium constant (Steyl, 2008):

[66]

The activity coefficient of the neutral contact ion pair is not likely to be 1, especially sincethe previous section suggested that the ion pair may still be hydrated. Errors in speciesdistributions would ultimately be absorbed in the rate constants of the kinetic processes. Themore important issue was an accurate description of the changes in species predominancewith solution concentration and temperature changes. The value of log(β3º) of 1.5 wastherefore retained for the MgSO4º contact ion pair (Table VIII). No distinction betweendifferent bonding mechanisms, i.e. monodentate or bidentate was made in thephenomenological model. This is because monodendate bonding was assumed to be onlypresent in the case of the alkali earth element, Mg and not in the case of the divalent metalions of Cu, Fe and Zn (Steyl, 2008).

System Method alog(β3º) Reference

MgSO4-H2O RS ~1.5 Rudolph et al. (2003)

DRS ~1.6 Akilan et al. (2006a)

FeSO4-H2O RS b~l.5 Rudolph et al. (1997)

CuSO4-H2O RS c1.0 Akilan et al. (2006b)

ZnSO4-H2O RS ~1.5 Rudolph et al. (1999a)

aAverage calculated value (Equation [66]), using reported data above 0.1 mol/kg (from the corresponding reference) and assuming unit activity coefficient for the contact ion pair; bStoichiometric mean activity coefficient fromReardon and Beckie (1987); cValue taken directly from the literature

Table VII

Reported thermodynamic values from the literature



Ab initio calculation suggested the possibility of triple-ion pair formation. This wasconfirmed for the Mg system (Akilan et al., 2006a, Buchner et al., 2004 and Akilan et al.,2006a) and the Ni and Co systems (Chen et al., 2005), but not in the Cu system (Akilan et al.,2006b). However, calculations in the previous section suggested that this triple ion may be theanionic complex, Mg(SO4)2

2−, which would not be detectable by DRS. The thermodynamicdata for this stepwise reaction, as estimated by crude ab initio modelling (previous section),was therefore adopted for the phenomenological model. The positive enthalpy values of allthe above reactions may be ascribed to the energy required to break the coordinated H2Omolecules of the constituent species prior to bonding. However, the positive entropy valuesmay be ascribed to the release of H2O molecules (reflecting less structure around the ionpairs), which compensates for the positive enthalpies, especially at higher temperature, i.e.these reactions can be viewed as being entropically driven. No explicit interactions betweenHSO4

− and the metal cations were taken into account because the HSO4− anion has a noble

gas electronic structure (similar electronic structure to perchlorate, ClO4− ion) and is not

expected to form contact ion pairs (Tremaine et al., 2004 and Rudolph et al., 1997). The following subsections discuss the binary and ternary systems up to 200ºC, i.e. the

H2SO4-H2O, MgSO4-H2O and H2SO4-MgSO4-H2O systems, respectively.

H2SO4-H2O system

Although the thermodynamic properties of this system have been studied extensively, therestill exists discrepancy in the actual species molalities in the dilute system. At highertemperatures, very little data are available at all. The MSE model of Liu and Papangelakis(2005b) suggests almost 40% H2SO4º in a 0.5 mol/kg acid solution at 200ºC. Wang et al.(2006) adjusted this model by explicitly treating of the proton as the hydrated species, H3O

+,and adjusting the interaction parameters to reflect experimental measurements (e.g. Hood andReilly, 1957, Young et al., 1959 and Walrafen et al., 2000). However, as pointed out earlier,current experimental techniques have not yet developed to a level where quantitativeinformation about minor species abundance, such as H2SO4º, can be provided in the dilute (<1 mol/kg) range at high temperatures. Due to the relative dilute range of this study, it wasnot deemed necessary to follow the hydrated cation approach of Wang et al. (2006). With, (i)careful selection of the important equilibrium constants (discussed previously), (ii) selecting aminimum number of adjustable interaction parameters, (iii) constraining the system tomeasured thermodynamic properties (salt and water activity), and (iv) incorporating theexperimentally observed (at lower temperature) distribution of the primary species (HSO4

−

and SO42−), predictions at higher temperature should be realistic.

Reaction No. log(Kº) ΔHºrx (kJ/mol) ΔSºrx (J/mol.K) Reference

H+(aq) + SO2-

4 (aq) = HSO-4 (aq) K1º 1.964 22.8 114 aDickson et al. (1990)

H+(aq) + HSO-

4(aq) = H2SOo4 (aq) K2º

b-1.05 b19.62 b26 Oscarson et al. (1988)

Mg2+(aq) + SO2-

4(aq) = MgSOo4 (aq) β3º 1.5 c10 c62 This study

SO2-4(aq) + MgSOo

4(aq) = Mg(SO4)2-2 (aq) K4º -3.5 d49 d97 This study

Table VIII

Thermodynamic values used in the regression of the phenomenological model

aEquation [61] is used to extrapolate over temperature; bValue at 150ºC (ΔCpº = 113 J/mol.K), using Equation [40] toextrapolate over temperature; cAverage enthalpy from RS measurement ~10 kJ/mol for various divalent metal sulphates(see Rudolph, 1998, Rudolph et al., 1997, 1999a and 1999b) and ΔSº calculated (no ΔCpº estimate); dΔHº estimated from ab initio calculation, ΔSº calculated and assumed ΔCpº = 0 J/mol.K (Akilan et al., 2006a)

401-444_Steyl:text 2/17/09 10:45 AM Page 423


The studies of Pitzer et al. (1977), Clegg et al. (1994) and Clegg and Brimblecombe (1995)present excellent literature reviews of the important thermodynamic studies of the H2SO4-H2O system over the years, all of which were limited to below 60ºC. The only hightemperature study based on own experimental measurement and where a description of theself-medium was attempted, is the isopiestic study of Holmes and Mesmer (1992). The mostimportant experimental data were incorporated into a regression analysis, using the modeldescribed above and minimizing the objective function (the relevant regression information ispresented in Table X). The symbol α1´ refers to the degree of dissociation of the bisulphateion. Various combinations of the interaction parameters (β(0)HB, β(1)HB, CφHB, β(0)HS, β(1)HS,CφHS, θBS, ψHBS) were attempted. The activity coefficient of the neutral complex was assumedto be unity, i.e. no interaction parameters involving the neutral acid molecule was included.The subscripts H, B and S refer to the proton (H+), bisulphate (HSO4

–) and sulphate ions(SO4

2–), respectively. In order to simplify this task, a scan of their relative influences wasconducted, both with and without the inclusion of the long-range electrostatic parameters. Thebest fit with the minimum number of adjustable parameters was achieved with the parameterspresented in Table IX. A power series was used to calculate the temperature dependence of theinteraction parameters (after Holmes and Mesmer, 1992):

[67]

with the maximum value of i = 4, Td = T – 298.15K and To = 1K. Due to the lack ofthermodynamic data and species abundance, especially at higher temperatures, the interactionparameters were allowed to vary only linearly (i = 2). The lower end of the regression waslimited to 0.1 mol/kg H2SO4 at higher temperatures (>50ºC) because isopiestic measurements(Holmes and Mesmer, 1992) are generally not regarded as accurate at the lower molalities. Acomparison between the experimental data and the model output are represented in Figures 8and 9. Figure 10 presents the speciation diagram at 1 mol/kg H2SO4 and suggests that theneutral species (H2SO4º) plays a minor role (less that 5%), even at a temperature of 200ºC.This is in line with the predictions of Wang et al. (2006).

p1(kg/mol) 103· p2 (kg/mol)

β(0)HB 0.2291 -0.4641

β(1)HB 0.3736 0.0498

θBS 0.1151 -0.1788

Temp (ºC) aObjective function bAARD (%) (φ, γ±, α) Temp (ºC) aObjective function bAARD (%) (φ, γ±, α)

25 1/3⟨φ⟩ + 1/3⟨γ±⟩ + 1/3⟨α1´⟩ 0.31, 0.22, 2.49 125 ½⟨φ⟩ + ½⟨γ±⟩ 0.13, 0.42, n/a

50 ½⟨φ⟩ + ½⟨γ±⟩ 0.49, 0.40, n/a 150 ½⟨φ⟩ + ½〈γ±⟩ 0.09, 0.22, n/a

75 ½⟨φ⟩ + ½⟨γ±⟩ 0.12, 0.34, n/a 175 ½⟨φ⟩ + ½⟨γ±⟩ 0.14, 0.05, n/a

100 ½⟨φ⟩ + ½⟨γ±⟩ 0.06, 0.39, n/a 200 ½⟨φ⟩ + ½⟨γ±⟩ 0.67, 0.74, n/a

Table X

Objective functions and their relative deviations for the H2SO4-H2O system

Table IX

Optimized interaction parameters for the H2SO4-H2O system (with inclusion of the long-range electrostatic terms)

aThe error of property, p, defined as: ⟨p⟩ = ∑(|pi calc/pi exp – 1|; bAbsolute average relative deviation of φ, γ± and α,

respectively

401-444_Steyl:text 2/17/09 10:45 AM Page 424


MeSO4-H2O system

Thermodynamic data of the base metal sulphates encountered in this study (CuSO4, ZnSO4,FeSO4 and MgSO4) in water medium is scattered through the literature, with significantvariances. The most important sources of measured and reported osmotic and mean activitycoefficient data can be found in Robinson and Stokes (1959), Pitzer (1972) and Majima et al.(1988). The more recent paper of Quendouzi et al. (2003) also gives a good summary ofreported data and it compares well with their calculated values for various metal salts in water.

Figure 8. Comparison between the experimental values of the osmotic coefficient and the model output (this study)for the H2SO4-H2O system at 25ºC intervals

Figure 9. Comparison between the experimental values of the fraction HSO4– dissociated (α´1) and the model output

(this study) for the H2SO4-H2O system at 25ºC intervals

Figure 10. Sulphate species (fraction of the total sulphate molality) as a function of temperature for the H2SO4-H2Osystem at 1 mol/kg acid



Additional data for the CuSO4-H2O system was found in Harned and Owen (1958), Downesand Pitzer (1976) and Majima and Awakura (1988). The more recent measurements ofAlbright et al. (2000) and Miladinovic et al. (2002) provided useful information on theZnSO4-H2O system. However, only the study of Oykova and Balarew (1974) revealedinformation about the FeSO4-H2O system, which exemplifies the difficulties in obtainingaccurate experimental data for this system. Additional information for the MgSO4-H2Osystem was obtained from the measurements of Snipes et al. (1975), Archer and Wood(1985), Phutela and Pitzer (1986) and, especially, Archer and Rard (1998) for the osmoticcoefficient, and Pitzer and Mayorga (1974), Rard and Miller (1981) and Holmes and Mesmer(1983) for the mean activity coefficient. Agreement between the above studies is reasonable,except for the measurements of Majima et al. (1988). Their data were not included for theZnSO4 and MgSO4 systems. Figure 11 presents the average reported values of the meanactivity coefficients (at 25ºC). The observed mean activity coefficient is relatively insensitiveto the type of divalent metal salt. Although uncertainty exists to the degree of contact ion pairformation, earlier discussions emphasized that it would vary significantly between thedifferent salts. Notwithstanding, Figure 11 suggests that contact ion pair formation is notsignificant at 25ºC (~10% at 1 mol/kg MgSO4; see Rudolph et al., 2003) and also reflects thesimilarities in the solvent-separated ion pairs between these salts. This is also evident from thevery similar values of the total thermodynamic equilibrium constants of these salts at 25ºC, asreported in various publications (e.g. Nair and Nancollas, 1958 and 1959, Helgeson, 1967,Pitzer, 1972 and Högfeldt, 1982). The overall equilibrium constants are therefore dominatedby electrostatic forces (see Pitzer, 1972). Previous discussions and information from theliterature (e.g. Rudolph et al., 1997, 1999a, Rudolph, 1998 and Akilan et al., 2006a) suggeststhat contact ion pair formation increases rapidly with temperature. In an attempt to gauge if anincrease in temperature would transpire into significant differences between salts, theirreported thermodynamic values were compared. For example, the emf study of Nair andNancollas (1959), conducted up to 45ºC, yielded very similar enthalpy (20.3 kJ/mol and 16.8kJ/mol) and entropy (111 J/mol.K and 102 J/mol.K) values for MgSO4 and ZnSO4,respectively. Helgeson extrapolation (Helgeson, 1967) yielded similar equilibrium constantvalues at 100ºC (consistent with other reported conductivity results, see e.g. Högfeldt, 1982),and log(Kº) = 4.8 and 4.6, respectively, at 200ºC.

No reported data could be found for CuSO4 and FeSO4 at higher temperatures. Since therewere no data available at high temperatures to suggest otherwise, this study assumed thatthese salts behave similarly up to 200ºC, which justifies the Mg surrogate approach, albeitartificially (e.g. Pitzer, 1972, suggested that possible triple ion pair formation is expected to bemore prominent in the case of Cu2+ as compared to Mg2+). Experimental data for MgSO4

were incorporated into a regression analysis. Numerous combinations, from a minimum of

Figure 11. Average mean activity coefficients of various metal sulphates in water (data points represent averages)



3 to a maximum of 6 parameters were tested, involving the main interactions between ions(β(0)MS, β(1)MS, β(2)MS, CφMS, β(0)MT, β(1)MT, CφMT, θST, ψMST). The subscripts M, C and T refer tothe metal, first contact ion pair (MgSO4º) and the triple ion pair (Mg(SO4)2

2–), respectively.Reactions [68] and [69] represent the first and triple contact ion pair reactions, respectively:

[68]

[69]

The above interaction parameters were later extended to include binary interactionsinvolving the neutral contact ion pairs (λMC, λSC, λCC, λCT) and then also to ternaryinteractions (μMCC, μSCC, μCCC, ζMSC). The best regressions at 25ºC with a minimum numberof 4 parameters was obtained with the parameters β(0)MS, β(1)MS, β(2)MS, with explicitrecognition of MgSO4º and Mg(SO4)2

2–, in addition to one of the following CφMS, λSC, or λMC.Although previous studies of MgSO4 at high temperature (e.g. Holmes and Mesmer, 1983,and Phutela and Pitzer, 1986) only utilized the parameters, β(0)MS, β(1)MS, β(2)MS and CφMS, theexplicit inclusion of the two contact ion pairs demanded a modified approach. Excellentresults at high temperatures were obtained with the inclusion of one of the neutral interactionparameters, λSC or λMC. The inclusion of these parameters may be perceived to capture theelectrostatic interactions between the dipole of the first contact ion pair and the chargedsulphate anion or metal cation, respectively. The inclusion of the β(2)MS term in theconventional treatment of 2–2 electrolytes (see Pitzer and Mayorga, 1974) is related to therapid decrease in the ion activity coefficients in the dilute range (~0.03 to 0.1 mol/kg), whichin turn, is related to a maximum degree of association found for typical 2-2 electrolytes in thisdilute range. With optimized β(2)MS values for MgSO4 in the range -37.23 (Pitzer andMayorga, 1974) to -32.7 (Rard and Miller, 1981), its value needed to be adjusted because of explicit contact ionpair formation. In order to prevent the regression from getting stuck in local minima, theinitial β(2)MS values were systematically varied before each optimization. Figure 12 illustratesthe results of this scan, suggesting an optimum β(2)MS value in the region of -29 (kg/mol) andmaintaining the 2–2 electrolyte parameters, α1 and α2, at 1.4 and 12 (kg/mol)1/2, respectively.Although the trough is relatively shallow around the minimum, it is (as expected) differentthan what the conventional treatment requires, i.e. the explicit recognition of ion pairs lowersthe absolute value of β(2)MS.

The optimized interaction parameters at 25ºC are presented in the first column of Table XI.In order to determine the temperature dependence of the various interaction parameters,Equation [67] was again utilized. However, as may be recalled from Table VIII, no heatcapacity information was available for the formation of two contact ion pairs (MgSO4º and

Figure 12. Residual sum of squares versus the second virial coefficient at 25ºC



Mg(SO4)22–), and hence, no accurate description of their enthalpy and entropy changes with

temperature could be made. The heat capacities of these two reactions were thereforeincorporated into the regression via Equations [34] and [42]. Experimental data at highertemperatures for the MgSO4-H2O system were limited to the heats of dilution measurementsof Snipes et al. (1975) (up to 80ºC), the isopiestic vapour pressure measurements of Holmesand Mesmer (1983) (at 110ºC) and the heat capacity measurements of Phutela and Pitzer(1986) (up to 150ºC). The studies of Archer and Wood (1985) and Archer and Rard (1998)also included other data sources (most of them limited to 150ºC) and their model outputsprovided excellent generalizations of all available data up to 150ºC. Although the study ofPhutela and Pitzer (1986) extended up to 200ºC, this information was limited to very lowmolality (~0.1 mol/kg), with questionable accuracy (discrepancies with the Archer and Woodmodel above 140ºC). It was important for this study to include additional information in theregression at high temperatures (>150ºC). Incorporation of the solubility of the monohydrate(kieserite) was ideally suited for this purpose:

[70]

A major unknown was the thermodynamic equilibrium constant, Kºsp at higher temperaturesand more specifically, the partial molal heat capacity of the crystal phase. This study used aCpº25ºC value of 145 J/mol.K, while the study of Archer and Rard proposed a value of 126 J/mol.K and the study of Pabalan and Pitzer (1987) suggested 134 J/mol.K (value adoptedin the study of Liu and Papangelakis, 2005a). The temperature dependence of Kºsp wouldobviously also depend on the specific choice of thermodynamic parameters for the otherspecies taking part in Reaction [70]. Baghalha and Papangelakis (1998) and Casas et (2005b)used Helgeson extrapolation to calculate Kºsp at high temperatures, whereas Liu andPapangelakis (2005a) calculated slightly larger values, using the Density model. The datachosen for this study originated from the HSC (2006) database and resulted in the followingthermodynamic values for Reaction [70] at 25ºC: log(Kºsp) = 0.0662, ΔCpº = -355.92 J/mol.Kand ΔHº = -52.43 kJ/mol. Equation [6] was used to obtain the water activity. This value andthe mean activity coefficient were then plugged into Equation [28] during the regressionanalysis to obtain the calculated value of the solubility at different temperatures. Theexperimental solubility values for the kieserite phase were obtained from Marshall andSlusher (1965), and Linke and Seidell (1965).

Because the exact heat capacity values of the two metal contact ion pair reactions had asignificant influence on the optimized interaction parameters, a systematic ‘manual’ scan wasfirst conducted. Figure 13 presents a surface plot of the area close to the global minimum andsupports the assumption of Akilan et al. (2006a) of zero heat capacity for triple ion formationreaction (Reaction [69]). The overall regression was subsequently conducted, initializing ΔCpº(Reaction [68]) at 270 J/mol.K. In order to prevent overparameterization, no temperature

Table XI

Optimized interaction parameters for the MgSO4-H2O system

p1 (kg/mol) 102· p2 (kg/mol) 105· p3 (J/mol.K) ΔCpº (kg/mol)

β(0)MS 0.2575 0.1532 0.5653 −

β(1)MS 3.1890 0.4970 -0.6343 −

β(2)MS -29.2839 -7.1213 -3.1555 −

λSC -0.0990 -0.5775 1.3769 −

Reaction [68] − − − 267.5036

Reaction [69] − − − −



dependence of the reaction heat capacities was deemed necessary, i.e. the enthalpy andentropy expressions reduced to Equations [37] and [38], respectively. A power series(Equation [67]) was once again used to calculate the temperature dependence of theinteraction parameters. However, higher order (i = 3) terms needed to be included to representthe solubility of kieserite at the lower temperature end, i.e. 170ºC. All the optimizedparameters are listed in Table XI. The objective functions and their relative deviations (at theircorresponding temperatures) are summarized in Table XII. Figure 14 illustrates the goodnessof the regression by comparing the model prediction with the measured solubility of MgSO4

in the kieserite region. The performance of the phenomenological model from this study is compared to the

published data in Figures 15 and 16. Except for the data of Phutela and Pitzer (1986), themodel gives an excellent representation of the experimental osmotic coefficient and observedmean activity coefficient of MgSO4 up to high temperatures. Figure 17 compares thecalculated fraction contact ion pairs with the available experimental data (25ºC). The modelpredictions at higher temperatures are also illustrated, emphasizing the preferential ionassociation at low molalities (discussed previously). These trends also suggest that theformation of the triple ion (Mg(SO4)2

2–) becomes increasingly important at high temperaturesand at the expense of the first contact ion pair (MgSO4º).

Figure 13. Residual sum of squares versus ΔCpº for the first (CIP1) and triple (CIP2) contact ion pairs

Temp (ºC) aObjective function bAARD (%) Temp (ºC) aObjective function bAARD (%)

(φ, γ±, α) (φ, γ±, log Kºsp)

25 1/3⟨φ⟩ + 1/3⟨γ±⟩ + 1/3⟨α⟩ 0.85, 1.65, 1.14 130 ½⟨φ⟩ + ½⟨γ±⟩ 1.34, 3.42, n/a

45 1/3⟨φ⟩ + 1/3⟨γ±⟩ + 1/3⟨α⟩ 1.04, 2 .14, n/a 140 ½⟨φ⟩ + ½⟨γ±⟩ 1.26, 4.32, n/a

65 ½⟨φ⟩ + ½⟨γ±⟩ 1.61, 1.55, n/a 150 ½⟨φ⟩ + ½⟨γ±⟩ 0.35, 6.58, n/a

80 ½⟨φ⟩ + ½⟨γ±⟩ 1.50, 1.63, n/a 170 102· ⟨log(Kºsp)⟩ n/a, n/a, 0.55




Table XII

Objective functions and their relative deviations for the MgSO4-H2O system

aThe error of property, p, defined as: ⟨p⟩ = ∑(|pi calc/pi exp – 1|; bAbsolute average relative deviation of φ, γ± and α (or log

Kºsp), respectively



H2SO4-MeSO4-H2O system

Reliable experimental information on this ternary system is rare, which is surprising in viewof its industrial importance. The most comprehensive set of experimental data was generatedby Majima et al. (1988) at 25ºC, but its accuracy is questionable (elaborated upon later in thissection). Valuable information was obtained from the isopiestic measurements of Rard andClegg (1999) for the H2SO4-MgSO4-H2O system at 25ºC. The isopiestic measurements of

Figure 14. Comparison between the model output (this study) and the measured solubility of the kieserite crystalphase in the MgSO4-H2O system

Figure 15. Comparison between the experimental values of the osmotic coefficient and the model output (this study)for the MgSO4-H2O system at selected temperatures

Figure 16. Comparison between the experimental values of the mean stoichiometric activity coefficient of MgSO4 andthe model output (this study) for the MgSO4-H2O system at selected temperatures (the broken line artificially exceedsthe solubility limit)



Majima et al. (various acid-salt mixtures) and the vapour pressure and emf measurements ofTartar et al. (1941) (H2SO4-ZnSO4-H2O system) is also compared (see the data of Rard andClegg). References to other data sources for these ternary systems can be found in Rard andClegg, but were not considered for this study since they were all limited to 25ºC. Only onepaper was found that reported thermodynamic data at higher temperature, i.e. the study ofJaskula and Hotlos′ (1992) for the H2SO4-CuSO4-H2O system at 60ºC. It was unfortunately oflittle use to this study because no comparative values were reported for the same system at25ºC. Various researchers have applied the Pitzer ion interaction model to these types ofternary systems, e.g. Guerra and Bestetti (2006) (H2SO4-ZnSO4-H2O system, ≤ 45ºC), Rardand Clegg (1999) (H2SO4-MgSO4-H2O system, 25ºC) and Reardon and Beckie (1987)(H2SO4-FeSO4-H2O system, ≤ 90ºC). Due to the relatively dilute nature of the solutions andthe explicit recognition of selected contact ion pairs, this study had to derive its own uniquemixing parameters. The poor agreement between the experimental data of Majima et al. andthe other studies mentioned above prevented an accurate derivation of the mixing parameters.In order to compare the results of the various studies, the relative mean activity coefficient, γ′±was defined as follows:

[71]

This equation allowed the mean activity coefficient in the ternary system to be normalizedto its value in the pure binary system and thus provided a more suitable basis of comparison.Figure 18 compares the relative mean activity coefficient of sulphuric acid in ternary mixturesand suggests reasonably close agreement, even for different divalent metal salt systems.However, Figure 19 illustrates a considerable spread of the relative mean activity coefficientof metal sulphate between the different salts in ternary mixtures with sulphuric acid and water.

The relative mean activity coefficient of MgSO4 from Rard and Clegg (1999) is alsosignificantly different from the values reported by Majima et al. (1988). Only the data of Rardand Clegg provided consistency between the mean activity coefficient of MgSO4 and theosmotic coefficient (Figure 20), and was therefore the only source of data used in theregression analysis. Due to the lack of data, only three mixing parameters were selected.Because of the relative obscurity of the contact ion pairs, especially in acidic media (seeRudolph et al., 1997), their mixing parameters were not considered. Only combinations of thefollowing mixing parameters were therefore considered: β(0)MB, β(1)MB, CφMB, θHM, ψHMB,ψHMS and ψMBS. The subscripts H, M, B and S refer to the proton (H+), metal (Mg2+),

Figure 17. Model predictions of the fraction (of total Mg) MgSO4º (α3) and Mg(SO4)22– (α4) and the experimentalvalues for the MgSO4-H2O system. The dotted line refers to α3, while the solid line represents the fraction totalcontact ion pair formation (α3+4) (the broken line artificially exceeds the solubility limit)



Figure 18. Relative mean activity coefficient of sulphuric acid in ternary mixtures from different studies (the surfacerepresents the average fit through all the data points)

Figure 19. Relative mean activity coefficient of the metal sulphate in ternary systems from different studies (thesurface represents the best fit through the data points of Rard and Clegg, 1999, for the H2SO4-MgSO4-H2O system)

Figure 20. The osmotic coefficient of the H2SO4-MgSO4-H2O system. The surface represents the best fit through thedata points of Rard and Clegg (1999), while the solid lines represent the model output from this study for the twobinary systems



bisulphate (HSO4–) and sulphate (SO42–), respectively. Various combinations were tested,

ultimately suggesting that at least two of the binary interaction terms β(0)MB, β(1)MB or θHM

needed to be included. The study of Rudolph et al. (1997) also suggested that mixing of Me2+

and HSO4− ions, although not forming contact ion pairs, should be approached from an ion-

interaction perspective. The limited experimental data available for the mixed system madethe choice of the third (ternary) interaction term somewhat arbitrary. It was therefore decidedto use the same ternary interaction term as used in the work of Reardon and Beckie (1987),i.e. ψHMB, but with the inclusion of the long-range electrostatic terms, EθHM and Eθ′HM. Figure 21 illustrates the goodness of the regression by comparing the model prediction withthe measured data of Rard and Clegg and its comparative offset with the data of Majima et al.(1988).

The optimized parameters (at 25ºC) are summarized in Table XIII, corresponding to an α1

value of 2 (kg/mol)1/2. The objective functions and their relative deviations are summarized inTable XIV. The only thermodynamic data found for the H2SO4-MgSO4-H2O system at highertemperatures was the solubility data of kieserite at 200ºC from Marshall and Slusher (1965).This lack of data could only justify a linear dependency on temperature (i = 2), using Equation[67]. Figure 22 illustrates the excellent agreement between the measured solubility data in theternary system and the model output, even though some areas (dotted lines) lie outside therange used in the regression at 25ºC.

Summary and conclusions

The theoretical framework provided by the Pitzer ion-interaction approach was used to modelthe ternary H2SO4-MeSO4-H2O system, using MgSO4 as the surrogate salt to represent other

p1 (kg/mol) 102· p2 (kg/mol)

β(0)MS 0.2606 0.0121

β(1)MS 2.0949 1.3823

ψHMB 0.2930 -0.1662

Figure 21. Comparison between the experimental values of the mean stoichiometric activity coefficient of H2SO4 internary mixtures with MgSO4 and H2O, and the model output from this study

Table XIII

Optimized interaction parameters for the H2SO4-MgSO4-H2O ternary system (with inclusion of the long-range

electrostatic interaction terms between H+ and Mg2+)



divalent metal sulphate electrolytes. Since this study aimed at developing a platform forinterpreting hydrometallurgical process kinetics in acidic metal sulphate solutions, the mostimportant inner-sphere complexes were explicitly recognized. Thermodynamic data was usedto constrain the phenomenological model, thereby inferring the speciation behaviour at highertemperatures were no qualitative experimental information was available.

Due to the relatively dilute nature of the electrolyte modelled in this study, allthermodynamic expressions referred back to the transcendent position of infinite dilution. Aqualitative interpretation of the quantum level chemistry was used as a complementary tool toadd more confidence to the ambiguous nature of the thermodynamic data reported in theliterature. Molecular geometries were optimized within a DMol3/COSMO framework. Thetotal coordination of aqueous species were restricted by explicitly using water as acoordination filling species in the first hydration shell, while treating the effect of the solventbeyond this level via the continuum solvent methodology. Since neither the vibrationalenergy, nor the entropy of the solvated species could be explicitly calculated within thecontext of the DMol3/COSMO framework, the enthalpy rather than the total free energyformed the basis of comparison with reported literature values. It was assumed that theenthalpy was represented by the potential energy; hence, it was assumed that the kineticenergy (translational, rotational, etc.) cancels approximately for reactions between solvatedions at constant temperature. The thermodynamic basis for the proton yielded an averagehydration of between H3O+ and H5O2+, while a structured and static approach to the solvationof sulphate suggests the dianion to have 8 H2O molecules in the primary hydration shell. Itwas estimated to lose half its H2O molecules when reacting with the proton to form thebisulphate ion. With zero reaction entropy, the pK1 was estimated to be around -1.3, assuming

Figure 22. Comparison between the experimental values of kieserite solubility in the ternary system, H2SO4-MgSO4-H2O at 200ºC and the model output from this study (the dotted line refers to the model prediction outside theregression range)

Temp (ºC) aObjective function bAARD (%) (φ, γ±, log Kºsp)

25 ½⟨φ⟩ + ½⟨γ±, MgSO4⟩ 3.09, 8.03, n/a

200 ⟨log(Kºsp)⟩ n/a, n/a, 0.03

Table XIV

Objective functions and their relative deviations for the H2SO4-MgSO4-H2O ternary system

aThe error of property, p, defined as: 〈p〉 = ∑(|pi calc/pi exp – 1|; bAbsolute average relative deviation of φ, γ± and log Kºsp

respectively



the hydrated sulphuric acid molecule consists of two H2O molecules in its primary shell. Thefirst contact ion pair of MgSO4° was likely a monodendate complex, but the presence of asecond contact ion pair could also be explained by the MgSO4° bidentate complex. Thepresence of a monodentate Mg2SO42+ triple ion (reported in the literature) could not beverified by calculation. Instead, its anionic analogue, Mg(SO4)2

2− seemed more probable,especially at higher temperatures (significant entropy contribution). The log(Kº) wascalculated as -3.5, using an estimated reaction entropy of 100 J/mol.K.

The equation of Dickson et al. (1990) gave a good description of the experimental secondionization constant of acid up to high temperatures, and was adopted for this study. The firstionization constant, estimated from a quantum level treatment, compared very well with thethermodynamic data of Oscarson et al. (1988) (the Density model was used to extrapolate thereported high-temperature data to 25ºC). The association constant for the first contact ion pairfor different metals was estimated from spectroscopic data (reported in the literature) to beabout 1.5, despite having different bonding mechanisms. The Density model was used toextrapolate to higher temperatures, using an average reported reaction enthalpy of 10 kJ/mol.Since no heat capacity value was available, it was considered a variable in the regressionanalysis. The reaction enthalpy of 49 kJ/mol for the formation of the triple ion was adoptedfrom ab initio calculations, with a linear dependence of log(Kº) on the inverse temperature,i.e. zero reaction heat capacity.

The explicit recognition of the four contact ion pairs, i.e. HSO4−, H2SO4º, MgSO4º and

Mg(SO4)22− required an iterative approach around the ion-interaction framework. Using the

thermodynamic values and extrapolation techniques described above, selecting a minimumnumber of adjustable interaction parameters, constraining the system to the availablethermodynamic data and incorporating the experimentally observed lower temperature speciesdistributions in the regression routine, speciation predictions at higher temperature were madepossible. The binary H2SO4-H2O and MgSO4-H2O systems were first regressed, using theinteraction parameters {β(0)HB, β(1)HB, θBS} and {β(0)MS, β(1)MS, β(2)MS, λSC}, respectively. Theneutral acid species, H2SO4°, was found to play an insignificant role, even at 1 mol/kg H2SO4

and 200ºC, which is in line with recent predictions from the literature. Although the pureMgSO4-H2O system introduces an oversimplified approach to complexities of a mixed systemof various divalent metal sulphates, their thermodynamic values agree to such an extent thatthe Mg-surrogate approach could be justified. These values and also very similar overallequilibrium constant values, reflect the fact that these salts consist predominantly of outer-sphere complexes at room temperature, i.e. they are dominated by electrostatic forces. Theinteraction parameters may be regarded as capturing the long-range electrostatic effects andthe formation of outer-sphere complexes, while the explicit inclusion of equilibrium constantsrecognizes the important covalent interactions. The formation of the first contact ion pair,MgSO4º, was found to become more important at higher temperatures (using a regressed andconstant reaction heat capacity of 268 J/mol.K), while the second contact ion pair,Mg(SO4)22−, started to become more prevalent at even higher temperatures (>150ºC) and atthe expense of the first contact ion pair. The disadvantage of the Mg-prototype approach isthat softer transition metal cations, such as Cu2+, may be expected to form more prominentfirst and second contact ion pairs with the sulphate dianion. Since this study was concernedwith trends, rather than exact values, these discrepancies may be absorbed in the kinetic rateconstants when modelling process reactions.

Reliable experimental data was found to be rare for the mixed H2SO4-MgSO4-H2O ternarysystem, especially at higher temperatures. Once parameterization have been optimized for thebinary systems, three adjustable mixing parameters were required for the ternary system(ignoring the role of the two higher-order contact ion pairs), i.e. β(0)MB, β(1)MB and ψHMB.



These parameters were varied linearly with temperature because only one set of data wasavailable in the dilute range (≤ 1 mol/kg total sulphate) at higher temperature, i.e. thesolubility of kieserite at 200ºC. The model was found to behave in a consistent manner, evenbeyond the range used in the regression. In conclusion, the following two figures illustratetypical trends that may be expected. Figure 23 presents the variation in the unassociatedproton fraction (0.2 mol/kg H2SO4) with changing metal sulphate molality. The fractionunassociated H+ is clearly more sensitive to the total metal sulphate molality at higherH2SO4:MeSO4 ratios. Put differently, the fraction unassociated H+ is more sensitive to thetotal acid at lower total MeSO4 molality.

Figure 24 represents the total fraction metal sulphate contact ion pairs(MeSO4º+Me(SO4)22−) with varying acid molality and temperature. Contact ion pair formationis clearly favoured at higher temperature and lower acid conditions. At low metal sulphatemolalities, the temperature and acid concentration has a relatively smaller effect on the totalfraction contact ion pairs present.

The above model provides a powerful tool that may be used to interpret the kinetics ofreactions occurring in hydrometallurgical processes.

Figure 23. Predicted trends of the fraction unassociated H+ (0.2 mol/kg H2SO4) with varying metal sulphate molalityin the ternary system, H2SO4-MgSO4-H2O, at various temperatures: (a) Surface plot; (b) Contour plots (the dottedline represents the 150ºC contour at 0.1 mol/kg H2SO4)

Figure 24. Predicted trends of the total fraction metal sulphate contact ion pairs with varying acid molality in theternary system, H2SO4-MgSO4-H2O, at various temperatures: (a) Surface plot at 0.5 mol/kg MgSO4; (b) Contourplots at varying MgSO4



Acknowledgements

The author is grateful to Dr. Maggie Burger for helpful discussions on the quantum levelchemistry aspects of this study, whilst Prof. Vladimiros Papangelakis is thanked for hisobjective commentary regarding the phenomenological model. Anglo American plc isacknowledged for financially supporting this work. The author would also like to thank LukeNgubane for sourcing the numerous references so efficiently.

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Johann SteylTechnical Specialist, Anglo Research, South Africa

Johann started his employment at Mintek in 1994, after graduatingfrom the University of Pretoria with a degree in ChemicalEngineering. He joined Anglo Research in 2001 and is currentlyworking as technical specialist in the Technology Division. Hisresearch focuses mainly on the development of new processes for thetreatment of ores and concentrates.

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