Chapter 4 Calculation of Standard Thermodynamic Properties of Aqueous Electrolytes and Non-Electrolytes

Vladimir Majer Laboratoire de Thermodynamique des Solutions et des Polymères Université Blaise Pascal Clermont II / CNRS 63177 Aubière, France

Josef Sedlbauer Department of Chemistry Technical University Liberec 46117 Liberec, Czech Republic

Robert H. Wood Department of Chemistry and Biochemistry University of Delaware Newark, DE 19716, USA

4.1 Introduction Thermodynamic modeling is important for understanding and predicting phase and chemical equilibria in industrial and natural aqueous systems at elevated temperatures and pressures. Such systems contain a variety of organic and inorganic solutes ranging from apolar nonelectrolytes to strong electrolytes; temperature and pressure strongly affect speciation of solutes that are encountered in molecular or ionic forms, or as ion pairs or complexes. Properties related to the Gibbs energy, such as thermodynamic equilibrium constants of hydrothermal reactions and activity coefficients of aqueous species, are required for practical use by geologists, power-cycle chemists and process engineers. Derivative properties (enthalpy, heat capacity and volume), which can be obtained from calorimetric and volumetric experiments, are useful in extrapolations when calculating the Gibbs energy at conditions remote from ambient. They also sensitively indicate evolution in molecular interactions with changing temperature and pressure. In this context, models with a sound theoretical basis are indispensable, describing with a limited number of adjustable parameters all thermodynamic functions of an aqueous system over a wide range of temperature and pressure. In thermodynamics of hydrothermal solutions, the unsymmetric standard-state convention is generally used; in this case, the standard thermodynamic properties (STP) of a solute reflect its interaction with the solvent (water), and the excess properties, related to activity coefficients, correspond to solute-solute interactions. For dilute and moderately concentrated solutions, the standard-state functions have a dominant role and can be used as a reasonable approximation for semiquantitative modeling. The solute- solvent interactions particularly prevail at near-critical conditions, where all the STP of solutes undergo rapid variations. As shown in chapter 2, the standard derivative properties of a solute scale with the thermal expansivity and isothermal compressibility of the solvent and diverge at the solvent critical point. The direction of this divergence cannot be unambiguously predicted for certain classes of solutes without experimental evidence. In addition, this extreme behavior strongly affects the properties of solutions in a relatively wide range of conditions below and above the critical point. Thus, the modeling approaches used and the experimental data available at near-ambient conditions, while important in calculations, are

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Calculation of Standard Thermodynamic Properties of Aqueous Electrolytes and Non-Electrolytes

not sufficient for extrapolation of STP towards high temperatures and pressures, particularly up to the supercritical region. This chapter reviews the status of approaches to the determination, correlation and prediction of STP of aqueous solutes at superambient conditions. Its organization is in two parts. The thermodynamic background is presented in the first part, which introduces the essential terms regarding STP of aqueous species, shows their relationship to quantities accessible from experiments, and outlines the strategy for building up prediction schemes at superambient conditions. The second part discusses the physico- chemical concepts that have been used for description of STP over the last thirty years. It focuses particularly on practical models that allow calculation of STP for a variety of solutes and that are in some cases available as a software package. Main sources tabulating STP of aqueous electrolytes and nonelectrolytes are listed, and examples of calculations for selected systems are given. The focus of this chapter is on the transfer of a solute from its pure or ideal-gas state to the standard state in aqueous solution. The determination of standard thermodynamic properties of pure substances is beyond the scope of this chapter and is not addressed. Approaches to the thermodynamic description of aqueous systems at different conditions ranging from ambient to supercritical were developed by authors belonging to different communities: physical chemists, chemical engineers, geologists and environmental chemists. Each discipline addresses the issue of aqueous systems from its own perspective, using specific terminology and concepts regarding model formulation, standard-state conventions and concentration scales. The underlying objective of this chapter is to present a synthetic view of the topic with an effort to identify common denominators of various approaches and to unify the description of STP of aqueous solutes.

4.2 Thermodynamic Background

4.2.1 Basic definitions Any thermodynamic function characterizing an aqueous system can be expressed as a linear combination

of the partial molar property of water (solvent) X1 and the partial molar properties of dissolved ionic or molecular species (solutes) Xi (i > 1). In physical chemistry of solutions, it is usual to consider the solvent separately from solutes present in the system:

n = + X n1 X 1 ∑ ni X i . (4.1) i=2

Each partial molar property can be divided for convenience into a standard-state term and an activity (a) term expressing its variation from the standard state due to changes in concentration and nonideality of the system. Since water is generally present in much greater quantity than other species, it is common practice to treat it differently from solutes. The unsymmetric standard state convention is adopted where the solvent is referenced to its pure state, complying with Raoult’s law, while the solutes are referenced to the state of “infinite dilution,” complying with Henry’s law.1 We will focus first on the partial molar Gibbs energy (chemical potential); all other partial molar thermodynamic properties can be obtained by derivations with respect to temperature and pressure. For water:

1 IUPAC recommends defining standard thermodynamic properties at the temperature of the system and standard pressure of 0.1 MPa. However, when describing aqueous systems over a wide range of conditions, it is more convenient to introduce standard thermodynamic properties that are both temperature and pressure dependent; this approach is adopted in this chapter.

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Calculation of Standard Thermodynamic Properties of Aqueous Electrolytes and Non-Electrolytes

= • + = • + γ R → γ R → G1 G1 RT ln a1 G1 RT ln x1 1 x1 1, 1 1, (4.2)

• γ R where G1 is the molar Gibbs energy of pure water and x1 and 1 are its mole fraction and Raoult’s activity coefficient, respectively. In the literature, thermodynamic functions of solutes are expressed on three concentration scales. While mole fraction xi is mainly used by physical chemists and process engineers for nonelectrolytes, molality mi (moles of solute per kilogram of solvent) is popular with geochemists, and molarity ci (moles of solute per liter of solution) dominates as a concentration unit in environmental and analytical chemistry. The latter two scales are almost always used for electrolyte concentrations. Since often xi<

µ = = o + = o + γ H → γ H → i Gi Gix RT ln aix Gix RT ln(xi ix ) xi 0, ix 1

= o + = o + γ H → γ H → Gim RT ln aim Gim RT ln(mi / m0 im ) mi 0, im 1 (4.3)

= o + = o + γ H → γ H → Gic RT ln aic Gic RT ln(ci / c0 ic ) ci 0, ic 1.

o γ H The standard Gibbs energy (standard chemical potential) Gi and the Henry’s activity coefficient i of a solute are specific for a selected concentration scale. Since both activity and activity coefficient are by ⋅ −1 ⋅ −3 definition dimensionless, the introduction of constants m0 = 1 mol kg and c0 = 1 mol dm is necessary (although often neglected) for obtaining dimensionless concentration variables. It is apparent that ideal behavior of the system is attained at infinite dilution of solutes where activity coefficients of both solvent and solutes are unity. The relationship between the standard chemical potential of solute for different concentration scales can be derived by writing Eq. (4.3) in the limit of infinite dilution. After introducing for mi and ci the limiting conversion relations x x ρ = i = i 1 lim mi lim ci , (4.4) x →0 x →0 i M1 i M 1

ρ ⋅ −1 ⋅ −3 where molar mass M1 and density 1 of water are in kg mol and kg dm , respectively, it follows:

o = o − = o − ρ Gix Gim RT ln(M 1m0 ) Gic RT ln(M 1c0 / 1 ) (4.5)

o o Thus, the difference between Gix and Gim is an additive constant linearly dependent on temperature, o while for conversions where Gic is involved this constant also changes with pressure at conditions of high water compressibility. It should be stressed that the standard chemical potential of a solute must have a finite value and therefore it is not equivalent to the chemical potential of solute at infinite dilution (which is minus infinity as suggested by Eq. (4.3)). For any concentration scale, the standard chemical potential can be expressed generally as

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Calculation of Standard Thermodynamic Properties of Aqueous Electrolytes and Non-Electrolytes

G o = lim(G - RT ln (dcv ) ) , (4.6) i → i i x1 1

where dcv stands for a dimensionless concentration variable of a solute (xi, mi/m0 or ci/c0). It means that o Gi corresponds to the behavior of an infinitely dilute solution whose chemical potential is extrapolated to

unit concentration (xi=1, mi=m0 or ci=c0). In other words, the standard chemical potential relates to a γ hypothetical solution of unit activity referenced to infinite dilution where i=1. It means that there is no solute-solute interaction and the STP of a solute reflect solely the interactions between the dissolved species and water. After this short presentation of properties related to the Gibbs energy, let us turn our attention to the standard derivative properties (standard enthalpy, standard heat capacity and standard volume), which can be obtained from the standard chemical potential as temperature and pressure derivatives using basic thermodynamic relationships. Since the infinite-dilution limit of Eq. (4.3) is valid at any temperature and pressure, it follows that the standard derivative properties (unlike the standard chemical potential and entropy to be discussed below) are directly equal to the partial molar derivative properties at infinite ∞ dilution, X i :

∞ H o = −T 2 (∂(G o /T ) / ∂T ) = −T 2 lim (∂(G /T ) / ∂T ) = lim H = H (4.7) i i p → i p → i i cvi 0 cvi 0

∞ C o = (∂H o / ∂T ) = −T (∂ 2G o / ∂T 2 ) = −T lim (∂ 2G / ∂T 2 ) = lim C = C (4.8) p,i i p i p → i p → p,i p,i cvi 0 cvi 0

∞ V o = (∂G o / ∂p) = lim (∂G / ∂p) = lim V = V , (4.9) i i T → i T → i i cvi 0 cvi 0 where cv denotes a concentration variable. From Eq. (4.5), the standard derivative properties are identical for the mole fraction and molality concentration scales. Corrections involving derivatives of water density with respect to temperature or pressure must be applied for conversion to the molarity scale. Since molarity changes with temperature and pressure, this concentration scale is not useful for calculations at conditions remote from ambient, and no special attention will be given to standard derivative properties on the molarity concentration scale. In the context of this chapter, we do not discuss the standard entropy among the derivative properties, although it is directly related to the temperature derivative of the chemical potential:

o = −(∂ o ∂ ) = o − o Si Gi / T p (H i Gi ) /T . (4.10)

o Unlike the above three derivative properties, the value of Si is always concentration-scale dependent:

o = o + = o + ρ − ∂ ρ ∂ Six Sim Rln(M 1m0 ) Sic Rln(M 1c0 / 1 ) RT( ln 1 / T ) p (4.11) and is not equal to the infinite-dilution limit of the partial molar entropy of a solute, which makes it closer in character to the standard chemical potential. In different theoretical developments, the standard Helmholtz energy is of major importance; its relationship to other STP is straightforward:

o = o − o Ai Gi pVi . (4.12)

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Calculation of Standard Thermodynamic Properties of Aqueous Electrolytes and Non-Electrolytes

o For determining STP Xi , it is necessary to characterize the process of transfer of a solute into its standard state in an aqueous solution. This can be done with help of the standard thermodynamic property ∆ o of dissolution sol X i :

o = • + ∆ o X i X i sol X i , (4.13)

• where X i is the molar thermodynamic property of the pure solute at the same temperature and pressure ∆ o as the solution. An alternative way is to use a thermodynamic property of hydration hyd X i :

o = ig o + ∆ o X i X i [ p ] hyd X i , (4.14)

ig o where X i [ p ] is the property of the ideal gas i at the solution temperature and at the standard pressure of pº=0.1 MPa. It should be noted that, since the molar volume of hydration is given by the pressure ∆ o derivative of hydGi (where the ideal-gas standard Gibbs energy is at constant pressure), it follows that ∆ o = o hydVi Vi . Comparison of Eqs. (4.13) and (4.14) leads to the relation

• ∆ o = ∆ o + ∆ hyd X i sol X i res X i , (4.15)

• • ∆ = − ig o where the residual thermodynamic function res X i X i X i [ p ] corresponds to the difference between the state of pure solute (fluid or a solid) at a given temperature and pressure and that of an ideal gas at the same temperature and pressure pº.

o o o While absolute values of certain properties ( Si , Vi , C p,i ) can be obtained from an experiment and/or a o o o calculation, others ( Gi , Ai , H i ) are, by the virtue of their definition, only related to a selected original ∆ o state. They are often related to the properties of formation f X i of aqueous solutes expressing the difference between the standard property of a solute in a solution and the sum of thermodynamic ν •el properties of pure elements ∑ j X j comprising the solute: j

∆ o = o − ν el •el = ∆ o + ∆ • = ∆ o + ∆ ig f X i X i ∑ j X j sol X i f X i hyd X i f X i X=G,A,H (4.16) j

ν ∆ • ∆ ig where j are the stoichiometric coefficients of elements in i and f X i and f X i are the standard thermodynamic properties of formation of solute in the pure state or ideal-gas state, respectively. Relations among various properties discussed above are depicted in Fig. 4.1.

4.2.2 Special Considerations for Ionic Species In the case of ionic species dissociating in aqueous solution:

=ν + z+ + ν − z− ν + + =ν − − Cν + Aν - C (aq) A (aq) z z (4.17)

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Calculation of Standard Thermodynamic Properties of Aqueous Electrolytes and Non-Electrolytes

any STP of a solute may be replaced by that of the corresponding ions

o =ν + o + ν − o X X z + X z − , (4.18) Cν + Aν - C A

where z+ and z− are absolute values of ion charges. Since each ionic solute is composed of at least one cation and one anion, it is not in principle possible to determine the contribution of each ion separately, and some extrathermodynamic assumption must be made regarding the ionic values in order to break the logjam. One way is to operate with so-called conventional properties relating the absolute STP of ions to those of the hydrogen ion, for which

o o o o 1 • o 1 • ∆ X (conv) = ∆ X - z ∆ X + = (X − X ) − z (X + − X ) X=G,A,H (4.19) f ion f ion ion f H ion ν ion H 2 H2

o o o = + X=S,V,C (4.20) X ion (conv) X ion - zion X H p

• where zion is the charge of the ion (positive for cations and negative for anions) and X is the molar property of an uncharged element (or a sum of elements) with a stoichiometry ν towards the ion (for ν − example =2 for Cl since it relates to Cl2). While the absolute ionic properties remain inaccessible, the conventional properties can be calculated and are consistent with an arbitrary assumption o o ∆ + = + = . It is apparent that, when calculating a change of STP in a chemical reaction or a f X H X H 0 physical process, the equation characterizing this change is mass balanced. All terms in Eqs. (4.19) and ∆ o o (4.20) must cancel each other except f X ion and X ion , respectively. More information on this concept, largely used by geochemists, can be found for instance in the monograph by Anderson and Crerar (1993). A different approach was adopted in the physical chemistry of solutions. An effort was made to combine different experimental techniques and to use theoretical considerations in order to estimate STP of individual ions (Marcus, 1985). The extrathermodynamic assumption most frequently used introduces a + − reference electrolyte tetraphenylarsonium tetraphenylborate Ph4As Ph4B (TATB). Due to the similarity of the anion and cation, it is expected that hydration of the two ions is identical and therefore:

∆ o = ∆ o = ∆ o hyd X + hyd X − (1/ 2) hyd X . (4.21) Ph 4As Ph 4B Ph 4AsPh4B

After having determined the hydration properties for TATB and for at least one additional salt containing a cation or anion of the reference electrolyte, absolute hydration properties of all ions can be calculated within this convention. For any salt consisting of a cation Cz+ and a monovalent anion A− (i.e., ν−=z+):

− − ∆ o = ∆ − ∆ o + ∆ X z + X v X (v / 2) X , (4.22) hyd C hyd CAν - hyd Ph 4As A hyd Ph 4AsPh4B and an analogous equation can be written for anions. Once the hydration properties of ions are determined, STP of an ion can be expressed from Eq. (4.14) for S, V, Cp and from Eq. (4.16) for G, A, H:

o = ig o + ∆ o ∆ o = ∆ ig o + ∆ o X ion X ion [ p ] hyd X ion f X ion f X ion [ p ] hyd X ion . (4.23)

Properties of ions in an ideal-gas state can be calculated from spectroscopic data. More detailed discussion of STP for ionic species, including extensive tabulations of recommended values, can be found in the monograph by Marcus (1997).

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Calculation of Standard Thermodynamic Properties of Aqueous Electrolytes and Non-Electrolytes

Each of the two approaches outlined above has advantages and drawbacks. Use of conventional properties is possible over a wide range of temperature and pressure without introducing any speculative assumption. It is satisfactory for calculating a change in STP accompanying a reaction or physical process, but does not produce absolute values for individual ions. However, it is useful to obtain these values for testing and development of ionic solution theories. This is possible in the latter approach, at the cost of introducing an extrathermodynamic assumption whose general validity cannot be proved unambiguously. In addition, this approach is limited to near ambient conditions; its adoption at high temperatures is hampered by limited temperature stability of the reference ions.

4.2.3 Relation of the standard thermodynamic properties to quantities accessible from experiments It is important to show how the STP of an aqueous solute can be obtained from experimentally o o measurable quantities. Both Gi and H i require the combination of the thermodynamic property of a pure o solute with that characterizing dissolution or hydration, using Eq. (4.16). On the other hand, C p,i and o Vi can be obtained directly from experiments on a solution without needing to know the thermodynamic o o property of the pure solute. The remaining STP ( Si and Ai ) are calculated with the help of Eqs. (4.10) and (4.12). STP for aqueous solutes are available as tables of recommended data at 298.15 K and 0.1 MPa; properties of ionic species are particularly up to date thanks to the meticulous work of Marcus (1997). Many fewer data are available at superambient conditions, as discussed in more detail below. We focus here exclusively on the issue of determining STP in an aqueous solution, assuming that thermodynamic functions are available for the pure solutes. Methods of obtaining the standard chemical potential are outlined first, and then the determination of the standard derivative properties. Since data on binary solutions are used as the main source of information, the general subscript i for a solute is replaced in this section by 2.

4.2.3.1 Standard Chemical Potential In order to obtain the standard chemical potential of an aqueous solute, it is necessary to evaluate either ∆ o ∆ o the standard Gibbs energy of dissolution solG2 or the Gibbs energy of hydration hydG2 . The path selected depends largely on the state of the pure solute at the given temperature and pressure, and on the availability of the data. While near ambient conditions the input data are known for a variety of solid or liquid solutes, as well as for the corresponding dissolution process, the combination of ideal-gas and hydration properties can be a better option at elevated temperatures and for gases and volatile solutes in general. The dissolution and hydration properties are similar for gases at low and medium pressures • ≈ ig where G2 G2 , but they differ strongly for liquids and solids. Typically, three types of data from phase-equilibrium measurements can be used in calculations: Raoult’s limiting activity coefficients, solubility and Henry’s constant. For electrolytic solutes, electrochemical measurements with galvanic cells are an additional source of information. The most useful relationships are presented here; more detailed discussion can be found in textbooks of applied thermodynamics (Prausnitz et al., 1999; Sandler, 1999; Pitzer, 1995). When not otherwise indicated, all equations are written for the mole fraction concentration convention, the conversion to other concentration scales being straightforward (see Eq. 4.5).

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Calculation of Standard Thermodynamic Properties of Aqueous Electrolytes and Non-Electrolytes

An exact thermodynamic relationship links the standard Gibbs energy of dissolution with the Raoult’s γ R∞ 2 limiting activity coefficient 2 of a liquid solute (symmetric standard-state convention):

• ∞ ∆ o = o − l = γ R solG2 G2 G2 RT ln 2 . (4.24)

Different experimental approaches are described in the literature for determination of limiting activity coefficients at temperatures below 373 K and near ambient pressure; these are mainly based on vapor- liquid equilibrium and chromatographic experiments (Gmehling et al., 1994a; Eckert and Sherman, 1996; Kojima et al., 1997). Data are available for several hundred organic liquids miscible with water; some of the data for aqueous systems were compiled and discussed in the literature (Gmehling et al., 1994b; Kojima et al., 1997). Values from different sources differ, however, and no exhaustive critical review of ∆ o limiting activity coefficients is currently available. Equation (4.24) is useful for determining solG2 at near-ambient conditions for hydrophilic and moderately hydrophobic organic nonelectrolytes in the liquid state. sol < For hydrophobic solutes exhibiting low solubility in water (say x2 0.001 ), it is appropriate to use the condition of phase equilibrium between the liquid or solid solute and the aqueous solution. The standard Gibbs energy of dissolution can then be calculated directly from the solubility data using the relationship

• ∆ o = o − = − solγ H ≅ − sol solG G2 G2 RT ln x2 2 RT ln x2 , (4.25) which, unlike Eq. (4.24), is only approximate. First, it assumes that the Henry’s activity coefficient of the solute (the unsymmetric standard-state convention) is unity, which is strictly true only at infinite dilution. Thus, the degree of validity of Eq. (4.25) increases with decreasing solubility. Second, it inherently assumes no dissolution of water in the solute phase, which is valid for solids. This is also an acceptable simplification for highly hydrophobic liquids, although the solubility of water in organics such as hydrocarbons is generally one order of magnitude higher than the solubility of the hydrophobic substance in water (Tsonopoulos, 1999; 2001). A huge amount of aqueous solubility data is available at near- ambient conditions, and solubility values have also been published at superambient conditions. The most thorough sources are the multi-volume Solubility Data Series published under the auspices of IUPAC starting in 1979 and continued since 1998 as a NIST-IUPAC series of review articles in the Journal of Physical and Chemical Reference Data. The high-temperature data should be used, however, with caution, since the miscibility of organic liquids and water increases with temperature and the simplifying assumptions of Eq. (4.25) may not be valid. It is also obvious from Eqs. (4.24) and (4.25) that, for highly hydrophobic liquid substances, the inverse γ R∞ ≅ sol solubility approximates the limiting activity coefficient: 2 1/ x2 . For solids, a correction for the change of the Gibbs energy between the solid and hypothetical liquid state requires the use of additional •l − •s thermal data (heat of fusion and the heat capacity difference [ C p C p ]) that are not always available. Most limiting activity coefficients published in the literature for highly hydrophobic solutes were in fact

2 In the symmetric standard-state convention, all components in a liquid mixture are treated in the same way with the standard state being the pure substance in the liquid state (real or hypothetical) at the temperature and pressure of the system. An excess property expresses the deviation from Raoult’s law and the symmetric activity coefficients γ R reflect interactions between different species of a mixture. The difference between the standard thermodynamic property of a solute in the unsymmetric convention and that of the pure liquid solute is then equal to the partial o − •l = E∞ molar excess property at infinite dilution in the symmetric convention X 2 X 2 X 2 (sym. conv.) and γ R γ R∞ = γ H 2 / 2 2

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Calculation of Standard Thermodynamic Properties of Aqueous Electrolytes and Non-Electrolytes

∆ o obtained from solubility measurements; for the purpose of solG2 calculation, it is, however, preferable sol to use directly the measured values of x2 and Eq. (4.25).

∆ o The Gibbs energy of hydration hydG2 is related to the Henry’s constant kH by an exact thermodynamic relationship:

∆ o = o − ig = o hydG2 G2 G2 RT ln(kH / p ) , (4.26)

valid at any temperature and pressure. This equation can be derived by combining Eq. (4.3) with the = ig + o expression G2 G2 RT ln( f 2 / p ) , where f2 is the fugacity of the solute in the aqueous phase. The Henry’s constant is mainly used to express the solubility of gases in water; this is discussed in chapter 3. A thermodynamically rigorous treatment of the vapor-liquid equilibrium between water and a highly sat volatile solute leads to the Henry’s constant at p1 , the saturation pressure of the solvent. The correlations presented in Table 3.1 (for detailed analysis, see Fernandez-Prini et al., 2003) can thus be ∆ o used for calculation of hydG2 as a function of temperature along the saturation curve of water. The o sat standard volume V2 of the dissolved solute is necessary for conversion to a pressure far from p1 :

p ∆ G o = RT ln(k [ p sat ]/ p o ) + V odp (4.27) hyd 2 H 1 ∫ 2 sat p1

o The determination of V2 is not easy for aqueous gases, and only a very limited amount of data from volumetric experiments is available in the literature. Predictive correlation methods are mentioned below. The Henry’s constant concept has frequently been used for characterizing aqueous organic solutes that are liquid or solid in the pure state. For a solute sparingly soluble in water, the data resulting from liquid- org sol liquid or solid-liquid equilibria are used to approximate kH by the ratio f 2 / x2 . The fugacity of a solute org • in the organic phase, f2 , is often replaced by the fugacity of the pure solute, f2 ; this assumption is fully justified for solids but can be an oversimplification for liquids, particularly at elevated temperatures. org It has been shown that cubic equations of state can be used for calculating f2 as a function of temperature and pressure in hydrocarbon-water systems, provided at least one data point for solubility of water in the organic phase is available (Plyasunov and Shock, 2000b; Sedlbauer et al., 2002). This ∆ o approach allowed calculation of hydG2 from aqueous solubility over a wide range of temperature and ≅ sat sol pressure. At near-ambient conditions, the simple relation kH p2 / x2 is an acceptable approximation for highly hydrophobic solutes of low volatility. The data on Henry’s constants in the literature should be used with much caution, since it is often not clear how they were derived from experiments. In addition, alternative definitions of kH are used as well as different concentration scales, causing particular confusion regarding this parameter (Majer et al. 2003). The general equation relating the standard Gibbs energy of dissolution and the Gibbs energy of hydration for a solid or liquid solute is obtained from Eq. (4.15) as

p • • • ∆ G o = ∆ G o + RT ln( f / p o ) = ∆ G o + RT ln(φ p sat / p o ) + V dp , (4.28) hyd 2 sol 2 2 sol 2 2 2 ∫ 2 sat p2

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φ • sat → φ • → where 2 is the fugacity coefficient of the pure solute on its saturation curve (for p2 0, 2 1). The last term on the right side (corresponding to the Poynting correction) is often approximated as • − sat V2 ( p p2 ) , since the volume of a condensed phase changes little with pressure at conditions remote from the critical point. For supercritical solutes, it is not very useful to make the distinction between a ∆ o dissolution and hydration property, since experiments permit obtaining hydG2 directly via the Henry’s constant. The above discussion was focused mainly on nonelectrolytes. For electrolyte solutes, the standard chemical potentials of ions can be derived mainly from solubility and electrochemical measurements. In the first approach, the dissolution equilibrium between a solid salt and aqueous ionic species, corresponding to Eq. (4.17), is described by the expression:

+ − ν + ν − ∆ o = − + +ν − sol γ (ν +v ) =ν + ν − solGm (v )RT ln(q(m / m0 ) ± ) q , (4.29) Cν + Aν - where the dissolution property relates to the molality concentration scale. The salt solubility msol leads directly to the Gibbs energy of dissolution, provided the concentration is low and the mean activity coefficient γ ± approaches unity. Otherwise, the value of γ ± must be determined by an independent method. Measurement with galvanic cells is an approach specific for aqueous electrolytes. The ° relationship is exploited between the standard potential E of the cell and the standard Gibbs energy of an ° ∆ o = − electrochemical reaction, r G zFE . Finally it should be mentioned that the STP of an aqueous solute, electrolyte or nonelectrolyte, can be also obtained from thermodynamic equilibrium constants Kr in combination with the standard thermodynamic properties available for reactants and products, see Eq. (4.46) below. Dissociation or ion pairing reaction constants are examples. In addition, this approach can also be useful for calculation of STP for nonvolatile hydrophilic nonelectrolytes for which the methods described above are not easily applicable.

4.2.3.2 Standard Derivative Properties While conditions of phase equilibrium are used for obtaining properties related to the Gibbs energy, calorimetric and densimetric measurements with dilute aqueous solutions are the main source of information for calculating standard derivative properties. This takes advantage of the fact that the partial molar properties at infinite dilution are equal to the standard derivative properties, as shown in Eqs. (4.7) Φ to (4.9). A major tool for their derivation from experimental data are the apparent molar properties X :

• Φ X − n X = 1 1 X X = H, Cp, V, (4.30) n2

comparing a property of an investigated solution X with that of the solvent and expressed per mole of solute. In the infinite-dilution limit, both the denominator and numerator of Eq. (4.30) approach zero and application of l’Hospital’s rule shows that the limiting values of the apparent and partial molar properties are identical:

Φ lim X = lim X = X o . (4.31) → → 2 2 n2 0 n2 0

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Calculation of Standard Thermodynamic Properties of Aqueous Electrolytes and Non-Electrolytes

It is also useful to examine the relationship between the apparent and the excess properties XE in a binary system. Since the derivative properties do not involve any ideal concentration term, an ideal system is characterized by a simple linear combination of the standard terms for a solute and solvent:

• E = − + o X X (n1 X 1 n2 X 2 ) . (4.32)

Combination of Eqs. (4.30) to (4.32) gives

Φ X = X o + (X E / n ) ⇒ lim(X E / n ) = 0 . (4.33) 2 2 → 2 n2 0

E When n2 is replaced by x2 or m2, the excess property X relates to one mole of a solution or a solution containing 1 kg of water, respectively. Equation (4.31) indicates the procedure necessary for characterizing a binary system on the level of derivative properties. The measurements should be carried out over a range of concentrations down to high dilution, leading to the apparent molar properties calculated from experimental data. The corresponding standard derivative property is obtained by Φ extrapolation of X to infinite dilution, and finally the excess property characterizing nonideality can be calculated for finite concentrations from Eq. (4.33). While the excess Gibbs energy can be determined without direct use of the standard terms (see chapter 6), knowledge of the standard derivative property is needed for determining the corresponding excess property. As outlined in chapter 2, the standard derivative properties scale with thermal expansivity and isothermal compressibility of the solvent and diverge at its critical point. This critical behavior is reflected along the o o critical isochore of water by extremes in the first derivative properties ( H 2 , V2 ) and by an “S”-like o behavior in the second derivative properties ( C p,2 ), as illustrated in Figs. 2.16, 2.17 and 2.19. While the o shape of the curves X 2 (T, p) is controlled by the solvent properties in the critical region of water, their sign and size away from the critical region depend on the volatility of the solute and its interaction with the solvent. Figure 4.2 illustrates the evolution of the standard derivative properties with temperature in the subcritical region of water, where they become increasingly positive for volatile nonelectrolytes and negative for nonvolatile solutes, like strong electrolytes. The behavior of polar nonelectrolytes is more difficult to predict, even qualitatively, since it is not usually possible to determine a priori the sign of the critical divergence. Only experimental data can give quantitative information about the values of the standard derivative properties at superambient conditions, particularly in the region of high compressibility of the solvent. Let us show how the apparent and standard derivative properties can be determined from calorimetric and densimetric measurements. Since it is always necessary to perform measurements in the region of high dilution (down to 0.1 to 0.01 molal, depending on the solute, temperature and pressure), flow methods allowing alternation between solution and solvent in a measuring device are appropriate for detecting weak effects with good precision. Mixing calorimetry makes it possible to determine the enthalpy of dissolution, defined as ∆ = − • + • sol H H (n1H1 n2 H 2 ) . (4.34)

Combination with Eqs. (4.30) and (4.33) leads to the relationship

Φ • • ∆ = − = o − + E sol H / n2 H H 2 (H 2 H 2 ) (H / n2 ) . (4.35)

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Calculation of Standard Thermodynamic Properties of Aqueous Electrolytes and Non-Electrolytes

E Since the unsymmetric standard state is applied, the infinite-dilution limit of (H / n2 ) must be zero (see ∆ Eq. 4.33) and the extrapolation of sol H / n2 leads directly to the standard enthalpy of dissolution:

• lim(∆ H / n ) = (H o − H ) = ∆ H o . (4.36) → sol 2 2 2 sol 2 n2 0

Such an experiment can be performed in a flow mixing calorimeter dissolving small amounts of a fluid solute in water; it is, however, more difficult to realize for solids. In that case, it is possible to combine a ∆ calorimetric batch measurement yielding the value of sol H / n2 at a finite concentration with the limiting value of the heat-of-dilution experiments. Water is continuously added to a stream of a solution and the results are extrapolated to zero concentration of a solute:

∆ H / n + lim(∆ H / n ) = ∆ H / n − H E / n = ∆ H o . (4.37) sol 2 → dil 2 sol 2 2 sol 2 n2 0

∆ o The enthalpy of hydration can be obtained according to Eq. (4.15) by adding to sol H 2 the residual ∆ o enthalpy res H 2 , leading to

p • • ∆ H o = ∆ H o + (V −T(∂V / ∂T ) )dp . (4.38) hyd 2 sol 2 ∫ 2 2 p 0

The correction is not significant for gaseous solutes at low and medium pressures, since at ideal-gas ∆ o = ∆ o ∆ o conditions hyd H 2 sol H 2 . For liquid or solid solutes, res H 2 is close to the value of the enthalpy of vaporization or sublimation, respectively. These can be determined by calorimetry or calculated from the temperature derivative of vapor pressure via the Clapeyron equation. Equation (4.38) can be written for liquid solutes as

sat p2 • • • • ∆ H o = ∆ H o − ∆ H + (V −T (∂V / ∂T) )dp ≅ ∆ H o − ∆ H , (4.39) hyd 2 sol 2 vap 2 ∫ 2 2 p sol 2 vap 2 0 and an analogous equation holds for solids. The correction term for the nonideality of vapor can be sat neglected provided the solute vapor pressure p2 is low. An eminent contribution to the determination of enthalpies of hydration for highly hydrophobic fluids was made in the 1970s and 1980s by the groups of I. Wadsö at the University of Lund (Sweden) and S.J. ∆ o Gill at the University of Colorado (see Gill, 1988). Their measurements of sol H at near ambient conditions for inorganic gases and several low and medium molar mass hydrocarbons are a unique source ∆ o of information. Degrange et al. (2003a) calculated hyd H 2 for aqueous hydrocarbons based on calorimetric enthalpies of dissolution measured up to the supercritical region of water. Near ambient conditions, the enthalpy of dissolution is generally negative for gases and much smaller in absolute value than the enthalpy of vaporization/sublimation for organic liquids/solids, so the enthalpy of hydration is generally negative. It has, however, the opposite sign at high temperatures for all volatile solutes, due to ∆ o ∆ o the positive divergence of hyd H 2 at the critical point of water; this change of sign of hyd H 2 implies a ∆ o maximum in hydG2 for this type of solute.

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Calculation of Standard Thermodynamic Properties of Aqueous Electrolytes and Non-Electrolytes

ρ Measurements of the density and specific heat capacity cp lead to the apparent molar volume and the apparent molar heat capacity of a solute from Eq. (4.30): x M (c − c ) (c − c ) Φ = + 1 1 p p,1 = + p p,1 C p c p M 2 c p M 2 (4.40) x2 m2

Φ M x M (ρ − ρ ) M (ρ − ρ ) V = 2 − 1 1 1 = 2 − 1 , (4.41) ρ ρρ ρ ρρ x2 1 m2 1 where concentration is expressed in terms of mole fraction or molality. The main condition for successful o o calculation of C p,2 and V2 is the ability to determine precisely the density and heat capacity difference between the solution and water down to low concentration. Picker-type flow calorimeters and vibrating- ∆ ∆ρ tube flow densimeters directly supplying c p and values are well-suited for this type of experiment, and have yielded an important amount of data at near-ambient conditions for a variety of aqueous solutes. Data above 373 K are sparse and originate from a limited number of laboratories. Sources of high- temperature heat capacity measurements were reviewed by Hnedkovsky et al. (2002), and the evolution of vibrating-tube densimetry at superambient conditions was described by Hynek et al. (1997) and by Majer and Padua (2003). In extrapolating the experimental data to infinite dilution, it is possible to obtain the standard derivative properties for aqueous nonelectrolytes simply from the limits of Eqs. (4.40) and (4.41) as:

 c − c   c − c  o = + p p,1 = + p p,1 C p,2 c p,1M 2 M 1 lim  c p,1M 2 lim   (4.42) x →0  m →0  2  x2  2  m2 

M M  ρ − ρ  M M  ρ − ρ  o = 2 − 1 1 = 2 − 1 1 V2 lim  lim   . (4.43) ρ ρ 2 x →0  ρ ρ 2 m →0  1 1 2  x2  1 1 2  m2 

In the case of aqueous electrolytes, it is better to perform an extrapolation constrained by the Pitzer ion- interaction model to account properly for interionic interactions:

Φ = o + ν + +ν − + − C + 0.5 − 2ν +ν − C C p C p,2 ( ) z z (ADH / 2b)ln(1 bI ) 2RT B m (4.44)

Φ + − + − + − = o + ν +ν V + 0.5 + ν ν V V V2 ( ) z z (ADH / 2b)ln(1 bI ) 2RT B m (4.45)

X where I is the ionic strength, b=1.2 and ADH is the Debye-Hückel slope for property X as defined by Pitzer (1991). The standard property and the second virial coefficient B (generally considered a function of I) are adjustable parameters. While extrapolation to infinite dilution is straightforward at ambient and moderately elevated temperatures, it is much more complicated to obtain data at infinite dilution in the near-critical region where the concentration slopes are steep (see chapter 2). The only heat capacity and volumetric data for dilute aqueous solutions in the near-critical region of water were determined at the University of Delaware (Wood and collaborators), USA and at the Blaise Pascal University (Majer and collaborators) in Clermont-Ferrand, France. Densimetric and heat capacity flow measurements where a solution flows through the instrument are the most important sources of standard derivative properties for both electrolyte and nonelectrolyte aqueous solutions at high temperatures. Their solubility in water at room temperature must be, however, high

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Calculation of Standard Thermodynamic Properties of Aqueous Electrolytes and Non-Electrolytes

enough so that the preparation of several solutions is possible for studying the apparent molar properties as a function of concentration. Heat-of-mixing calorimetry is convenient when investigating gaseous solutes, where preparation of a solution prior to the experiment is complicated. High-temperature mixing measurements are also well-suited for investigation of highly hydrophobic liquids whose miscibility with water increases with increasing temperature; the ability to “prepare” a solution at elevated temperature inside a mixing calorimeter permits experiments with systems that would otherwise be difficult to study. An example is a simultaneous determination of the enthalpies of dissolution and densities for liquid hydrocarbons in water in an original flow instrument combining a heat-compensation calorimeter with a vibrating-tube densimeter in one thermostated environment (Hynek et al., 1999; Degrange et al., 2003b). 4.2.4 Calculation of equilibrium constants from standard thermodynamic properties at superambient conditions The calculation of thermodynamic equilibrium for a physical or chemical process in an aqueous system at superambient conditions requires either reliable experimental data or a sound model for expressing STP as a function of temperature and pressure. The objective is to determine the standard chemical potentials of individual aqueous species at a given T and p, which are then used in calculation of the equilibrium constant Kr[T, p]:

− = ∆ o = ν o RT ln K r [T, p] r G [T, p] ∑ iGi [T, p] . (4.46) i

For processes such as dissolution, precipitation and association where the number of species in a solution changes, i.e., ∆ν (aq) = ν (aq) ≠ 0 , the value of K depends on the concentration scale used, as ∑ i r i suggested by Eq. (4.5). The relationship between the thermodynamic equilibrium constants for mole = ∆ν (aq) = ρ ∆ν (aq) fraction, molality and molarity scales is K rx K rm (M 1m0 ) K rc (M 1c0 / 1 ) . The calculation of Kr from STP is otherwise independent of the choice of concentration scale. Generally, the thermodynamic data on aqueous and pure species are better known at the reference

temperature Tr = 298.15 K and pressure pr = p° = 0.1 MPa than at any other conditions. It is therefore ∆ o,app useful to define an apparent Gibbs energy of formation f Gi of a solute as the difference between the standard chemical potential at the temperature and pressure of the system and that of the constituting elements at Tr, pr. This concept leads to

T , p ∆ G o,app [T, p]= ∆ G o [T , p ]+[]G o , − RT ln K [T, p]= ν ∆ G o,app [T, p] . (4.47) f i f i r r i Tr , pr r ∑ i f i i

∆ o,app o f Gi is rigorously applied in the calculation of the equilibrium constant, because the change in Gi of constituting elements between (T, p) and (Tr, pr) cancels out in any mass-balanced process. The possible ∆ o,app thermodynamic pathways for obtaining f Gi at superambient conditions are illustrated in Fig. 4.3. The Gibbs energy difference can be expressed from basic thermodynamics as

T , p T , p T , p []G o = [H o ]− (T − T )S o [T , p ] −T []S o i Tr , pr i Tr , pr r i r r i Tr , pr

T T p = − − o + ()o − ()o + ()o (T Tr )Si [Tr , pr ] C p,i dT T C p,i dlnT Vi T dp , (4.48) ∫ pr ∫ pr ∫ Tr Tr pr

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Calculation of Standard Thermodynamic Properties of Aqueous Electrolytes and Non-Electrolytes

where the subscripts indicate the integration path. An analogous, but somewhat different, relationship is

obtained when temperature and pressure integration are performed at constant p and Tr, respectively. The standard entropy at the reference state can be determined according to Eqs. (4.10), (4.13) and (4.14) as

o = ∆ o − ∆ o + • Si (Tr , pr ) ( sol H i [Tr , pr ] solGi [Tr , pr ]) /Tr Si [Tr , pr ]

= ∆ o − ∆ o + ig ( hyd H i [Tr , pr ] hydGi [Tr , pr ]) /Tr Si [Tr , pr ] . (4.49)

∆ o,app It is obvious from Eqs. (4.48) and (4.49) that f Gi [T, p] is obtained by combining STP at the reference state, where they are tabulated for many solutes, with the temperature and pressure integrals of the standard heat capacity and volume. As these latter data are not always known at superambient conditions, it should be examined, at least qualitatively, what error is introduced by using instead the values available at Tr and pr. Such approximation is generally acceptable for calculations up to about 423 ∆ o,app o o K. For accurate calculation of f Gi [T, p] at higher temperatures, data on C p,i and Vi determined at superambient conditions become increasingly important and are indispensable above 473 K. Correct description of the standard heat capacity and volumetric data as a function of temperature and pressure is crucial for reliable extrapolations in the region of high compressibility of water. Equation (4.48) also o suggests that a model expressing C p,i as a function of temperature at one pressure and a volumetric o = equation Vi f (T, p) are sufficient for obtaining the whole standard chemical potential surface, ∆ o o 3 provided the integration constants f Gi [Tr , pr ] and Si [Tr , pr ] are known.

∆ o,app Figure 4.3 also shows an alternative way to obtain f Gi [T, p] by combining the thermodynamic data of a solute in an ideal-gas state with the Gibbs energy of hydration at conditions remote from ambient:

T , p ∆ G o,app [T, p]=∆ G ig [T , p ]+[]G ig r + ∆ G o [T, p] . (4.50) f i f i r r i Tr , pr hyd i

The Gibbs energy difference for a solute in an ideal-gas state can be expressed by a relationship analogous to Eq. (4.48). This approach does not require knowledge of dissolution or hydration data at the reference state and allows calculation without using the heat capacity and volumetric data of the aqueous solute. This can be useful for obtaining the standard chemical potentials of gases or solids dissolved in supercritical water from experimental solubility or Henry’s-constant data with the aid of Eqs. (4.26) to ∆ o,app (4.28). This relationship is also a straightforward way to calculate f Gi [T, p] from an equation of o state for the standard volume Vi of a solute consistent with the ideal-gas limit at low solution density. In this case,

3 These two quantities are often calculated by combining the values of hydration and ideal-gas properties at the reference conditions Tr and pr (see Eqs. 4.14 and 4.16). For ions, the apparent Gibbs energies of formation should be conventional while hydration properties tabulated in literature may be absolute, obtained from an extrathermodynamic assumption as in the compilation by Marcus (1997). It is therefore necessary to use Eqs. (4.19) and (4.20) for converting Gibbs energy and entropy to the values appropriate for use in Eq. (4.48): o o ig o ig o ig ∆ G (conv) = ∆ G (abs) + ∆ G − z ()∆ G + + ∆ G + = ∆ G (abs) + ∆ G − z (467.2) f ion hyd ion f ion ion hyd H f H hyd ion f ion ion o = ∆ o + ig − ()∆ o + ig = ∆ o + ig − − S (conv) S (abs) S z S + S + S (abs) S z ( 22.2) ion hyd ion ion ion hyd H H hyd ion ion ion Both equations are used at reference conditions 298.15 K and 0.1 MPa. Gibbs energy in kJ⋅mol−1 and entropy in J⋅K−1⋅mol−1 are converted using the data for hydrogen ion tabulated by Marcus (1997).

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Calculation of Standard Thermodynamic Properties of Aqueous Electrolytes and Non-Electrolytes

0 p p RT ∆ G o [T, p] = RTd lnp + V odp = RT ln( p / p o ) + (V o − )dp . (4.51) hyd i ∫ ∫ i ∫ i po 0 0 p

The two approaches to the determination of the apparent Gibbs energy of formation via Eqs. (4.47) and (4.50) are interrelated, as is apparent from Fig. 4.3:

T,p T,p ∆ G o [T, p] = ∆ G o [T , p ] + []G o − [G ig ]r = ∆ G o [T , p ] − (T − T )∆ S o [T , p ] hyd i hyd i r r i Tr , pr i Tr , pr hyd i r r r hyd i r r T T p + ()∆ C o dT − T ()∆ C o dlnT + ()V o dp . (4.52) ∫ hyd p,i pr ∫ hyd p,i pr ∫ i T Tr Tr pr

The above relationships illustrate the importance of heat capacity and volumetric data in calculation of the standard chemical potential at superambient conditions. High-temperature Gibbs energies of hydration obtained from experimental Henry’s constants and enthalpies of hydration from calorimetric experiments also contribute to the description of the standard chemical potential over a wide range of temperature and pressure. In this context, a good approach is to propose a thermodynamic model of hydration, valid over a wide range of conditions. The parameters of such a model are obtained by fitting simultaneously different STP obtained from phase-equilibrium, calorimetric and volumetric experiments. The minimized objective function for correlation usually has the form

2 2  ∆ G oexp − ∆ G ocor   ∆ H oexp − ∆ H ocor  F =  hyd hyd  +  hyd hyd  ∑ σ∆ o  ∑ σ∆ o  j G k H  hyd  j  hyd k

2 2  ∆ C oexp − ∆ C ocor  V oexp −V ocor  +  hyd p hyd p  +   , (4.53) ∑ σ∆ o  ∑ σ o  l C m V  hyd p l  m where X is the estimated uncertainty of a given data point and the variables denoted by superscripts exp and cor are calculated from the experimental data and from the model, respectively. For the sake of consistency with Eq. (4.47), it is necessary to combine a hydration model with the recommended values ∆ o ∆ o ∆ o of hydGi [Tr , pr ] and hyd H i [Tr , pr ] , and hence also of hyd Si [Tr , pr ] (Eq. 4.49). In this case, tabulated literature data or an independent prediction scheme is used for the two reference terms (lit) in Eq. (4.52), while the sum of three integrals is obtained from the proposed hydration model (mod), leading to the relationship

∆ ocor = ∆ olit − − ∆ olit hydGi [T, p] hydGi [Tr , pr ] (T Tr ) hyd Si [Tr , pr ]

+ ∆ omod − ∆ omod − − ∆ omod hydGi [T, p] ( hydGi [Tr , pr ] (T Tr ) hyd Si [Tr , pr )]) . (4.54)

In an analogous way, the enthalpy of hydration can be constructed as

∆ ocor = ∆ olit + ∆ omod − ∆ omod hyd H [T, p] hyd H [Tr , pr ] [ hyd H (T, p) H (Tr , pr )]. (4.55)

This approach, linking the high-temperature hydration model with the best available data and/or predictions at the reference state, was explored by Sedlbauer et al. (2002).

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4.3 Models For Standard Thermodynamic Properties As A Function Of Temperature And Pressure

4.3.1 Description of Models

4.3.1.1 Background Attempts at physically realistic modeling of STP at high temperatures can be traced back to the pioneering work of Criss and Cobble (see Fernandez-Prini et al., 1992), but major developments in the field occurred during the last two or three decades, starting with the landmark work of Helgeson et al. (1976; 1981). The Helgeson-Kirkham-Flowers (HKF) thermodynamic model has since been revised several times (Tanger and Helgeson, 1988; Shock et al., 1992; Plyasunov and Shock, 2001a) and has been used in many, often geochemical, applications. Despite its success it has some important inherent limitations, raising a need for further development. A few promising thermodynamic models have been proposed in the last fifteen years or so based on various theoretical considerations (e.g., Tanger and Pitzer, 1989; Harvey et al., 1991; O’Connell et al., 1996; Sedlbauer et al., 2000; Plyasunov et al., 2000a,b; Akinfiev and Diamond, 2003). It should be emphasized that all these models, sometimes called equations of state in analogy with the thermodynamics of one-component systems, are to some extent empirical. However, the leading terms of these equations arise from theoretical considerations, and these fundamental assumptions are a prerequisite of their success and determine the possible application range of the model. First, a so-called “standard-state term” (Ben-Naim, 1987; Wood et al., 1994), corresponding to a point mass, should be present in any theoretically founded model for STP of aqueous solutes; SS = 3 Λ Gx RT ln(Λ /V ) , where is the de Broglie wave length and V is the molar volume in a given state. For a hydration process (Eq. 4.14):

• ∆ ss = ig o = ρ o hydGx RT ln(V1 [ p ]/V1 ) RT ln( 1RT / M 1 p ) , (4.56) which corresponds to the Gibbs energy change of compression of an ideal gas from the standard pressure pº = 0.1 MPa to a pressure corresponding to the liquid water density. This equation is valid for the mole fraction concentration scale; the standard-state terms for molality and molarity scales can be obtained easily by combination of Eqs. (4.6) and (4.56). The temperature and pressure derivatives then yield the corresponding terms for standard derivative properties. A thermodynamic model should reduce to the standard-state term at the low-density limit. The magnitude of the standard-state term is small at ambient o conditions, but increases with temperature and diverges at the solvent’s critical point for Vi (and also o o for H i and C p,i ), so it becomes a leading part of the model in the near-critical region. Models that include this term are thus more reliable for extrapolation in the high-temperature and supercritical regions. It should be noted that the standard-state term is not functionally additive, because there is clearly only one translational degree of freedom per independent particle. Therefore, in any group additivity scheme, this contribution should always be subtracted first, before applying decomposition of the solute into functional groups (Criss and Wood, 1996). The same holds true for models of aqueous electrolytes, which should always include as many standard-state terms as there are independent particles (ions) present in the solution. In general, development of a model for STP may begin from two different perspectives. The most desired STP, leading directly to the thermodynamic equilibrium constants characterizing chemical and phase equilibria, is the standard chemical potential, represented by the Gibbs energy of formation of an aqueous

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Calculation of Standard Thermodynamic Properties of Aqueous Electrolytes and Non-Electrolytes

solute. Therefore, the initial interest is often to find a model that accurately describes the properties at the ∆ o Gibbs energy level, such as hydGi . Another approach is to look first for equations to correlate the o o derivative properties, typically Vi and C p,i , which then lead to the Gibbs energy by integration (see Eqs. 4.48 or 4.52). Although the two approaches are ultimately equivalent, often models formulated at the derivative level are more flexible for simultaneous correlation of data for several thermodynamic properties. On the other hand, models at the Gibbs energy level are more straightforward and often simpler when expressing the standard chemical potential and thermodynamic equilibrium constants. Below, we briefly discuss the theories that have been employed for the development of different thermodynamic models for STP. Then we describe in more detail three models that are promising for application and provide an overview of easily available sources of data and software tools for practical calculations. Finally, we give a few examples of calculations with the selected models to illustrate their accuracy and limitations.

4.3.1.2 Scaled Particle Theory One of the oldest theories explicitly involving the standard-state term is the Scaled Particle Theory. It was originally formulated for the standard volume, consisting of three contributions (Pierotti, 1976):

o = + + κ Vi Vca Vin 1RT , (4.57)

where the last term on the right-hand side is the standard-state term, given by the pressure κ derivative of Eq. (4.56) ( 1 is the isothermal compressibility of water). Vca is the contribution due to the finite volume of the solute and Vin is the volume contribution due to solute-solvent interaction. Expressions for Vca and Vin are available from the theory. However, these equations include properties that are generally unknown and must be estimated empirically. Because its use in practical calculations is limited, we mention the Scaled Particle Theory only as a historical source of inspiration in model development.

4.3.1.3 Models Based On A Charging Process For charged species, the Born equation provides the simplest way to calculate STP of hydration of an aqueous ion:

1 ∆ o = ω −  hyd Gi  1 , (4.58)  ε 

where ω = N z 2e 2 /()8πε r , ε is the dielectric constant (relative permittivity) of water and z , and r are A i o i i i the ion charge and radius, respectively. In such an electrostatic model, the hydration process is reduced to the difference in work between charging an ion in a solution and in a vacuum, where water is considered an incompressible continuum and its dielectric constant is not affected by the presence of an ion. This is, of course, a crude simplification. The Born equation does produce divergence of the STP at the water’s critical point, but not in a quantitatively correct manner. Equation (4.58) has nevertheless served as a foundation for several semiempirical equations. Among these are the semicontinuum models, where the ∆ i.s. contribution of the inner shell Gi close to the ion is treated differently from the contribution of the

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Calculation of Standard Thermodynamic Properties of Aqueous Electrolytes and Non-Electrolytes

∆ bulk bulk region Gi , which is generally considered as a dielectric continuum described by the Born equation:

∆ o = ∆ i.s. + ∆ bulk + ∆ ss hydGi Gi Gi G . (4.59)

The simplest approach is to describe the first term by Eq. (4.58), using a modified value of ε (Abraham et al., 1983). A more sophisticated approach was used by Tanger and Pitzer (1989), who treated the solvent in the inner shell as a sum of discrete particles; the successive hydration of the ion by water molecules was accounted for using mass spectroscopic data. The explicit estimation of energetic interactions between the ion and water molecules was attempted by Goldman and Bates (1972) and extended to high temperatures with the use of a thermodynamic cycle by Tremaine and Goldman (1978). The effect of high solvent compressibility on ion hydration was examined by Wood et al. (1994). Among the electrostatic approaches, the HKF model, which combines the modified Born equation with an empirical correction function, is by far the most widely used, despite its simplicity and inherent limitations as discussed below.

4.3.1.4 Density-Based Models Another approach is represented by a broad family of so-called density models, where empirical temperature and density functions are used for correlating Gibbs energies or ionization or association constants of weak electrolytes in aqueous solutions (see chapter 13). A good example is the concept of the total equilibrium constant, proposed by Marshall and Franck and later worked out by Marshall, Mesmer and others (e.g., Marshall, 1970; Anderson et al., 1991). These models often perform well, as demonstrated by the equation of Marshall and Franck, which was applied in the IAPWS formulation for the ionization constant of pure water (Marshall and Franck, 1981):

3 2 ∆ o = − = − j + ρ − j ionG / RT ln Kion ∑a jT ln 1 ∑b jT , (4.60) j=0 j=0

ρ where Kion is the ionization constant, 1 is the density of water and ai, bi are adjustable parameters. The density models are able to describe with more or less flexibility the standard derivative properties for a variety of solutes, as documented by Tremaine and collaborators (e.g., Tremaine et al., 1997; Clarke et al., 2000). However, they lack generality since the number of adjustable parameters is typically high, limiting their use to systems well defined by experimental determinations. For correlation of aqueous nonelectrolytes, Majer (1999) and Schulte et al. (1999) proposed equations that approximately follow the behavior of solute derivative properties near the solvent’s critical point while keeping the number of adjustable parameters limited:

c  ∂α  C o [T, p] =c+ T −(ω+ω T)T 1  − 2ω Tα (4.61) p,i − Θ T ∂ T 1 T  T  p

o = + − ω+ω κ Vi [T, p] a aT T ( T T ) 1 . (4.62) κ α The incorporation of the isothermal compressibility 1 and thermal expansivity 1 of water provides for better scaling at near-critical conditions compared to the use of dielectric properties (see Eqs. (4.67) and (4.68) below). Thus, it makes possible a reasonable description of the standard chemical potential over a ω ω wide range of T and p with a limited number of adjustable parameters (a, aT, c, cT, , and T) after integration using Eqs. (4.47) and (4.48).

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Japas and Levelt Sengers (1989) investigated the asymptotic behavior of the Henry's constant kH near the critical point of water, which led to a relationship for the limiting temperature dependence of the Gibbs energy of hydration (Harvey et al., 1991):

∆ o = o + ∆ o = • + + ρ − ρ + hydGi [T, p] RT ln( f1 / p ) res Ai RT ln f1 a b( 1 c,1 ) ... (4.63)

∆ o Analysis of the residual Helmholtz energy res Ai at the solvent critical point suggests that the leading term is linear in solvent density, with parameters a and b having exact thermodynamic interpretation. By adding an empirical term to represent behavior farther from the critical region, a three-parameter equation was obtained that was successful in correlation of the Henry's constants of volatile solutes down to room temperature (for details see chapter 3). Such a simple relationship is not, however, flexible enough to represent simultaneously the standard derivative properties (Sedlbauer and Majer, 2000). Recently, use was also made of Fluctuation Solution Theory (FST) (Kirkwood and Buff, 1951), which relates the spatial integral of the infinite-dilution solute-solvent direct correlation function o o C1,i with a dimensionless parameter A1,i , often called the modified Krichevskii parameter (O’Connell, 1995):

V o 1− C o = 2 = Ao . (4.64) 1,i κ 1,i 1RT

Both the standard volume of the solute and the solvent compressibility diverge at the solvent’s critical o point, but their ratio remains finite and relatively well-behaved, thus suggesting that A1,i can be correlated. In addition, this property can be expressed in terms of a virial expansion valid for low solvent densities:

 ∂ ()pV / RT  Ao = lim  = 1+ ()2 / M ρ B + ..., (4.65) 1,i →   1 1 1,i ni 0 ∂n  i T ,V

where B1,i is the second cross (solute-water) virial coefficient (O’Connell et al., 1996; Plyasunov et al., 2000a). An analogous procedure can be adopted for the pure solvent, where the water-water direct

correlation function is linked with the A11 parameter and a similar virial series is obtained. By comparing the virial expansions for the aqueous solute and for pure water, one obtains the relationship

o = + − + ρ − + A1,i 1 d(A11 1) 1 (2 / M 1 )(B1,i dB11 ) ... , (4.66) where the d parameter is a scaling factor related to the difference between the “cavity creating” volume of the solute and that of a water molecule. Equation (4.64) established a basis for two promising thermodynamic models describing the difference of the virial terms. The approach of Plyasunov et al.

(2000a), denoted below as POCW, is more explicit and requires knowledge of B1,i and B11 as input, approximating the higher order terms empirically. This way of anchoring the low-density limit of the model allows reduction of the number of adjustable parameters, but limits use of the equation to volatile nonelectrolytes for which data on cross virial coefficients are available or can be estimated and for which the virial expansion converges quickly. The approach of Sedlbauer et al. (2000), denoted below as the SOCW model, does not explicitly include virial coefficients and can be applied for all types of solutes at the cost of a higher number of adjustable parameters (see below). In both cases, the resulting equations for the standard volume are integrated (Eq. 4.51) to obtain the standard Gibbs energy of hydration.

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Calculation of Standard Thermodynamic Properties of Aqueous Electrolytes and Non-Electrolytes

4.3.1.5 Considerations For Practical Use Of Models Despite the growing number of thermodynamic models in the literature, only a few find broad application in the description of all STP and of the standard chemical potential in particular. The reasons are as follows: i) Some models were proposed for one STP and were never extended to cover other properties of interest, or are not flexible enough for such extension. For example, the equation of O’Connell et o al. (1996) is available only for Vi , and the model of Harvey et al. (1991) was proposed for the Gibbs energy of hydration and can be in principle used for all related thermodynamic functions, o o but its simultaneous use with Vi or C p,i data does not give satisfactory results (Sedlbauer and Majer, 2000). ii) Application of some models is limited to a selected group of solutes. For example, the equations of Harvey et al. (1991) or those of Plyasunov et al. (2000a) are recommended for volatile nonelectrolytes and cannot account for the behavior of non-volatile or ionic solutes. Many density models were applied exclusively to the correlation of ionization constants of weak electrolytes, etc. iii) The parameters of many models were published for just a few solutes or sometimes not at all (if only qualitative features were discussed by the authors). Collecting available data and treating them properly requires some proficiency and is not effective for a person who needs to calculate ad hoc some STP for a given solute at a few state points. iv) Implementation of some model equations is prohibitively complex. For example, the semicontinuum model of Tanger and Pitzer (1989) or the model of Plyasunov et al. (2000a,b; 2001) were successful in correlation of experimental data for selected systems, but their broader use by non-experts is hindered by the relative complexity of the equations. The necessity of calculating thermodynamic properties of water is a drawback of most published models. In general, for a model to be widely useful, it should describe as many aqueous systems as possible based on parameters obtained by correlation of extensive experimental data or by some proven predictive scheme, and should be accessible to non-experts. Success can be achieved for models that are sufficiently simple such as that of Harvey (1996) or some density models, or for those that are available as a user-friendly software tool. The latter condition is best exemplified by the HKF model that keeps a prominent position in geochemical applications, although it is increasingly criticized for its weak theoretical foundations and lower correlation accuracy compared to newer models. In addition to the HKF model, we present in the next section the SOCW concept (Sedlbauer et al., 2000) that has found application mainly in the prediction of thermodynamic properties of aqueous organic solutes based on a functional group-contribution scheme. Finally, we discuss the POCW model of Plyasunov et al. (2000a,b), which represents a successful approach for correlation of data on nonelectrolyte solutes, particularly at supercritical conditions.

4.3.1.6 Recent Models For Practical Use The HKF model is the most widely adopted model in geochemistry when treating new experimental data or pursuing calculation of thermodynamic equilibrium constants for

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Calculation of Standard Thermodynamic Properties of Aqueous Electrolytes and Non-Electrolytes

hydrothermal reactions. In this model, all STP are considered as the sum of “solvation” and “non-solvation” contributions, the latter being represented by an empirical function that reconciles the simple solvation model based on the Born equation with reality. The equations for o o Vi and C p,i proposed by Helgeson et al. (1976; 1981) were later revised by Tanger and Helgeson (1988):

a  a  1  ω  ∂ε   1   ∂ω  V o [T, p]= a + 2 +a + 4   −   + −1   (4.67) i 1 ψ +  3 ψ +  −Θ  ε 2  ∂  ε ∂ p  p  T   p T    T  p

c  ∂ 1  ∂ε  1  ∂ε   ∂ω   1   ∂ 2ω  C o [T] = c + 2 +ωT   + 2T     −T  −1   , (4.68) p,i 1 2  ∂ ε 2 ∂  ε 2 ∂ ∂ ε  ∂ 2  ()T − Θ T  T   T  p  T  p   T   p   p

where the heat-capacity equation is at a reference pressure of 0.1 MPa. The aj and cj are six adjustable parameters in the “non-solvation” part of the equation and Θ = 228 K, Ψ = 260 MPa. The solvation terms containing the Born parameter ω are obtained as derivatives of Eq. (4.58). The relationship for the change o o in standard chemical potential used in Eq. (4.47) is obtained by integration of Vi and C p,i (see Eq. 4.48):

T , p  T  ψ + p []o = − ()− o − − + + − + − Gi T Tr Si [Tr ,pr ] c1 T ln T Tr a1 ( p pr ) a2 ln Tr,pr   ψ +  Tr  pr  1 1  Θ − T T T (T −Θ) 1  ψ + p  c  −  − ln r + a ( p − p ) + a ln + (4.69) 2  −Θ −Θ  Θ Θ 2 −Θ  −Θ  3 r 4 ψ +   T Tr  T(Tr ) T  pr   1   1  (T − T )  ∂ε[T ,p ]  ω −1 −ω[T ,p ] −1 + r  r r   , ε r r  ε  ε 2  ∂     [Tr ,pr ] [Tr ,pr ]  p     Tr 

o ε ω where Si [Tr , pr ], [Tr , pr ] , and [Tr , pr ] represent standard entropy, dielectric constant of water and

the Born coefficient, respectively, at reference conditions Tr = 298.15 K and pr = 0.1 MPa. In the framework of the HKF model, the parameter in Eqs. (4.67) to (4.69) is calculated from the so-called effective electrostatic radius, which is the ionic radius ri modified by a complex temperature and pressure function, based on correlation of thermodynamic data on NaCl(aq) at elevated conditions (Tanger and Helgeson, 1988; Shock et al., 1992). For nonelectrolytes, ω is considered another adjustable parameter ω = ω independent of T and p ( [Tr , pr ] at all temperatures and pressures, Shock et al., 1989). Since the concept of solvation in the HKF model is tied to the Born equation describing ion solvation in terms of dielectric properties of the solvent, extension to nonelectrolytes was done by analogy with no theoretical justification. The model is not reliable at low densities and at near-critical conditions where the solvent compressibility is high. This has been criticized in recent literature, and the HKF approach was reported to have particular difficulties in the description of the standard derivative properties of aqueous nonelectrolyte solutes when extrapolating toward the critical point of water (O’Connell et al., 1996; Schulte et al., 1999; Plyasunov et al., 2000a,b; Sedlbauer et al., 2000; Clarke et al., 2000). The parameter vectors a and c in the HKF model are most often obtained from empirical correlations using standard thermodynamic data at Tr and pr. The original algorithm for adjusting the parameters of the revised model was described by Shock and Helgeson (1988). A major revision of the approach to parameter adjustment,

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Calculation of Standard Thermodynamic Properties of Aqueous Electrolytes and Non-Electrolytes

valid for aqueous nonelectrolytes, was published by Plyasunov and Shock (2001a). They used recent high-temperature experimental data for generating new parameters for the HKF model and have also

proposed a novel correlation algorithm for parameter estimation using the Gibbs energy of hydration at Tr and pr. Empirical observation of linear dependence between the modified Krichevskii parameter, defined in Eq. (4.64), and an exponential term in solvent density led O’Connell et al. (1996) to propose a volumetric equation of state for aqueous nonelectrolytes:

o = • + κ ρ ()+ []ϑρ − Vi V1 1RT 1 a b exp( 1 ) 1 , (4.70) where the constant ϑ = 0.005 m3⋅kg−1 was obtained by simultaneous correlation of standard volumes for aqueous gases CH4, H2S, NH3, CO2 and aqueous H3BO3 produced by experiments up to the critical region of water. It was shown later that this simple equation provided good description of standard volumes for various aqueous solutions, but only in a limited range of conditions. A new equation of state was proposed (Sedlbauer et al., 2000) by introducing some additional constraints into Eq. (4.70). First, the standard-state volume was separated from the molar volume of water, and this latter property was adjusted to better mimic the volume of a cavity created by insertion of a solute molecule in bulk water. Second, the effect of solute-solvent interaction was modeled by a linear combination of exponential terms in temperature and density:

V o = (1− z)κ RT + d(V • − κ RT) + i 1 1 1 , (4.71) κ ρ ()+ []ϑρ − + θ + δ []λρ − 1RT 1 a b exp( 1 ) 1 cexp( /T ) exp( 1 ) 1

κ λ − 3⋅ −1 θ where 1 is the isothermal compressibility of water, = 0.01 m kg and = 1500 K, a, b, c, and d are adjustable parameters of the model, specific for each solute, and z is the charge of a particle (z = 0 for neutral molecules, z ≥ 1 for cations and −z ≥ 1 for anions). The (1 − z) factor is needed in order to comply + with the hydrogen convention for aqueous ions, which requires V o (H ) = 0 . Using a = b = c = d = δ = 0 for H+(aq) would leave the standard-state term in the equation; it is removed by formal assignment of two standard-state terms in a 1-1 electrolyte to the anion. Parameter δ depends on the charge of the solute, for nonelectrolytes with zero charge δ = 0.35 a, for anions δ = −0.645 m3⋅kg−1, for cations δ = 0. Again, the unsymmetric choice of δ for cations and anions is dictated by the hydrogen convention. When embedded into Eq. (4.51), the volumetric Eq. (4.71) leads to the Gibbs energy of hydration:

p RT ∆ G o =RT ln( p / p o ) + (V o − )dp + G corr , (4.72) hyd i ∫ i i 0 p

∆ o and the other STP are obtained by appropriate temperature and pressure derivatives of hyd Gi analogous to Eqs. (4.7) to (4.9).

corr The correction term Gi is needed at conditions where the two-phase boundary is crossed during integration from p° = 0.1 MPa to p, i.e., at temperatures lower than solvent critical temperature. It is needed because the relatively simple volumetric Eq. (4.71) is unable to describe solution properties accurately on both sides of the phase-transition boundary. The correction becomes smaller with

increasing temperature, and is by definition zero at the solvent critical temperature Tc and at supercritical conditions. Its functional form, which differs for aqueous ions and for nonelectrolyte solutes, is given here for the standard heat capacity:

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Calculation of Standard Thermodynamic Properties of Aqueous Electrolytes and Non-Electrolytes

corr = − 2 − Θ = < C p,i e(T Tc ) /(T ) z 0, T Tc , (4.73) corr = ()− Θ + − ≠ < C p,i e /(T ) g) (T Tc ) z 0, T Tc where e and g are additional adjustable parameters of the model and Θ = 228 K is a universal constant. corr corr The correction function G2 is calculated from C p,2 using basic relations along with appropriate corr = corr = integration limits ( H i [Tc ] Si [Tc ] 0 ):

T T G corr = H corr −T S corr H corr = C corr dT S corr = C corr dlnT . (4.74) i i i i ∫ p,i i ∫ p,i Tc Tc

In applications, the calculation of the standard chemical potential is usually constrained by data at the reference conditions Tr=298.15 K and pr = 0.1 MPa and use is made of Eq. (4.54) where, in addition to ∆ omod ∆ omod hydGi , hydSi is also needed. The explicit form of the final equation for Gibbs energy and entropy of hydration varies for electrolytes and for nonelectrolytes due to different correction functions, so here we give only key parts of the final equation:

∆ G o mod = (1− z)RT ln[ρ RT /( p o M )]+ d()G • − G ig − RT ln[ρ RT /( p o M )] + hyd i 1 1 1 1 1 1 (4.75) {}()+ θ − − δ ρ + ϑ[][]ϑρ − + δ λ λρ − + corr RT a c exp( /T ) b 1 b / exp( 1 ) 1 / exp( 1 ) 1 Gi

∆ o mod = − − ()[ρ o ]+ − α + hyd Si (1 z)R ln 1 RT /( p M 1 ) 1 T 1 ()• − ig + ()[]ρ o + − α + θ θ ρ − d S1 S1 R ln 1RT /( p M 1 ) 1 T 1 R c exp( /T ) 1 /T {}()+ θ − − δ ρ + ϑ[][]ϑρ − + δ λ λρ − − R a c exp( /T ) b 1 b / exp( 1 ) 1 / exp( 1 ) 1 , (4.76)  ∂ρ  RT 1  {}a + b[]exp(ϑρ ) −1 + c exp(θ /T ) + δ []exp(λρ ) −1 + S corr ∂ 1 1 i  T  p

• • ig ig where G1 and S1 are the molar Gibbs energy and entropy of water, G1 and S1 are the same properties α of water in the ideal-gas state at pº = 0.1 MPa, and 1 is the thermal expansivity. While the SOCW model accounts for the difference of the second virial coefficients in Eq. (4.66) using a semiempirical function, the POCW model of Plyasunov et al. (2000a) is directly linked to the low-density limit by incorporating experimental or estimated values of the pure-solvent second virial coefficient B11 and of the solute-solvent cross virial coefficient B1i

V o = κ RT + N(V • −κ RT) + i 1 1 1 , (4.77) κ ρ ()[]− − ρ + 5 + [ρ − ] 1RT 1 2 B1i NB11 exp( c1 1 ) (a /T b) exp(c2 1 ) 1

3⋅ −1 3⋅ −1 where c1 = 0.0033 m kg and c2 = 0.002 m kg are universal constants and a, b, N are adjustable parameters. The term with the a parameter is needed mainly to compensate for an unrealistically large ∆ o contribution of the second virial coefficients at lower temperatures. Calculating hyd Gi from Eq. (4.51) and using appropriate derivations with respect to temperature, one can obtain all other thermodynamic properties of hydration. Again, this simple procedure can be used with success only at supercritical conditions; as with the SOCW model, the pressure integration of the volumetric Eq. (4.77) fails at subcritical temperatures. Plyasunov et al. (2000b, 2001) have circumvented this problem by introducing

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Calculation of Standard Thermodynamic Properties of Aqueous Electrolytes and Non-Electrolytes

for the heat capacity of hydration an empirical expression, valid below a “switching” temperature Ts of

658 K and at a reference pressure pref = 28 MPa:

− ∆ o = + + φ − − − n 2 hydC p,i d 0 d1T d 2T exp(T / ) d 3Tn(n 1)(T0 T ) T < 658 K, (4.78)

φ where = 40 K , T0 = 669 K, n = 0.6 are constants and d0 to d3 are four additional adjustable parameters. The choice of 28 MPa as a reference pressure is because many experimental data at elevated conditions were obtained along this isobar. Temperature integration of Eq. (4.78) produces the Gibbs energy of hydration as well as other thermodynamic functions of hydration at the given isobaric condition. Their combination with the appropriate pressure integral of the volumetric equation then leads to the thermodynamic function of hydration at the temperature and pressure of interest. The four d parameters ∆ o are constrained by combining the literature values of hydration functions hydGi [Tr , pr ] , ∆ o ∆ o ∆ o hyd H i [Tr , pr ] , hydC p,i [Tr , pr ] , with the calculated values based on Eq. (4.77) of hydGi [Ts , pref ] , ∆ o ∆ o hyd H i [Ts , pref ] , hydC p,i [Ts , pref ] . The final equations for thermodynamic properties are complex and will not be given here. The model was proposed for nonelectrolyte solutes, preferably those for which a good estimate of the second cross virial coefficient as a function of temperature is possible.

4.3.2 Data on and Prediction of Standard Thermodynamic Properties of Aqueous Solutes The amount of experimental data on aqueous solutions is growing rapidly; the bulk of those data are at

reference conditions Tr = 298.15 K and pr = 0.1 MPa. Data at these conditions are often sufficient for realistic estimation of standard chemical potentials of aqueous solutes at moderately elevated temperatures and pressures. In addition, the standard chemical potential and standard entropy at Tr and pr serve as integration constants in most models discussed above. In the last twenty years or so, comprehensive compilations were published by Marcus (1997, ion properties), Wagman et al. (1982, standard thermodynamic properties of inorganic and some organic solutes), Cabani et al. (1981, hydration properties of nonelectrolyte organic solutes, extensively updated by Plyasunov and Shock, 2000a o [hydrocarbons and alcohols]; 2001b [ketones]), Hoiland (1986, V i of organic nonelectrolytes) and o Gianni and Lepori (1996, Vi of organic ions). The sources of limiting activity coefficients and solubility data were discussed above. Henry’s constants for organic compounds of environmental interest were compiled by Mackay and Shiu (1981) and by Shiu and Ma (2000a,b). Important sources of information on reaction constants for inorganic and some organic aqueous solutes are the compilation of Baes and Mesmer (1976, hydrolysis of metal cations) and the series of articles by R.N. Goldberg in the Journal of Physical and Chemical Reference Data on biochemical reactions. Another important data collection is the database accompanying the SUPCRT92 software and based on a series of articles by Helgeson, Shock and collaborators (see below for details). A review of procedures to calculate thermodynamic data for aqueous electrolytes over a wide range of conditions was published by Rafal et al. (1994). Measurements at high temperatures are more difficult and thus relatively scarce. However, at least some high-temperature information is crucial for reliable prediction of STP in a wide range of conditions. Because the data at elevated conditions on standard chemical potential are often insufficient or missing completely, the available experimental results on standard derivative properties are often utilized in correlations. These thermodynamic properties are interconnected by a number of familiar thermodynamic relations, most of which were given above. It is evident that the most reliable representation of different types of experimental data can be achieved by their simultaneous treatment as outlined by Eq. (4.53). This approach requires an equation-of-state model that is flexible enough to capture the evolution of different

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Calculation of Standard Thermodynamic Properties of Aqueous Electrolytes and Non-Electrolytes

thermodynamic properties with respect to T and p. This or a similar procedure has been applied for solutes for which predictions are available within the framework of the HKF, SOCW and POCW models. No systematic compilation of high-temperature STP of aqueous solutes is available; most references to high-temperature determinations, in some cases reaching the critical region of water, can be found in the article by Plyasunov and Shock (2001a). Fernández-Prini et al. (2003) published recommended Henry’s constants for gases along the saturation line of water (see chapter 3). Articles of Helgeson, Shock and their collaborators published mainly in Geochimica et Cosmochimica Acta over the past 20 years contain an important amount of high-temperature reaction constants characterizing dissociation, ion pairing and complexation. The HKF model has been implemented into the software package SUPCRT92 (Johnson et al., 1992), which includes data, parameters and procedures leading to predictions of STP for hundreds of aqueous ∆ o ionic and nonelectrolyte solutes. The freeware program is accompanied by a database of f Gi [Tr , pr ] o and Si [Tr , pr ] values, along with parameters of the HKF model for the solutes. The database is regularly updated and made available by the group of E.L. Shock, while the program itself can be obtained upon request at the Laboratory of Theoretical Geochemistry at the University of California, Berkeley. Data at reference conditions in the database are typically values selected from the literature; in some cases these properties are estimated from empirical regularities valid for specific groups of solutes. The papers ∆ o o o o typically contain recommended values of f Gi [Tr , pr ], Si [Tr , pr ], Vi [Tr , pr ] , C p,i [Tr , pr ] and summaries of available high-temperature data for these properties and for equilibrium constants of reactions (complexation, dissociation, etc.) involving concerned species. Also included are parameters of the HKF model, which were in most cases estimated using empirical correlation algorithms among equation parameters or parameters and reference state properties as mentioned above (Shock and Helgeson, 1988; Plyasunov and Shock, 2001a). A review of the apparent Gibbs energies of formation for a variety of important solutes was published by Oelkers et al. (1995). The major contributions for ionic species and their complexes are: Shock and Helgeson (1988) and Shock et al. (1997a) for inorganic ions; Sverjensky et al. (1997) and Shock et al. (1997a) for metal complexes; Shock and Koretsky (1993; 1995) and Prapaipong et al. (1999) for metal organic complexes; Shock et al. (1997b) for uranium compounds; Haas et al. (1995) for rare earth complexes; Sassani and Shock (1998) for platinum group elements; and Murphy and Shock (1999) for americium ions and complexes. Inorganic nonelectrolytes were treated by Shock et al. (1989) and Schulte et al. (2001). The basic contributions regarding organic solutes are by Shock and Helgeson (1990) and Amend and Helgeson (1997a). In addition, several papers were devoted to specific classes of organic solutes: Shock (1995) for carboxylic acids, Schulte and Shock (1993) for aldehydes, Dale et al. (1999) for alkyl phenols, Amend and Helgeson (1997b) for amino acids, Haas and Shock (1999) for chloroethenes and Amend and Plyasunov (2001) for carbohydrates. The SOCW model has been used for correlation of experimental data on various aqueous solutes from dissolved gases to strong 1-1 electrolytes. The accuracy of correlation was usually satisfactory and sometimes superior to other models. This can be attributed especially to a better description near the solvent critical point and to a flexible functional form. General applicability of the model allowed its use in the development of a group-contribution scheme for aqueous organic solutes:

N o = ss + o , (4.79) Yi Y ∑n jYi, j j=1

where N is the total number of functional groups present in a given compound, nj is the number of o SS occurrences of each functional group, and Yi, j stands for the j-th group of the Y property. Y accounts for the intrinsic contribution to the Y property that is equal to the contribution of a point mass. More than

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Calculation of Standard Thermodynamic Properties of Aqueous Electrolytes and Non-Electrolytes

6000 experimental data points for several STP were collected into databases and part of them were used in simultaneous adjustment of model parameters for selected functional groups. In the first attempt by Yezdimer et al. (2000), the types of compounds included were aliphatic hydrocarbons, alcohols, primary amines and amides, carboxylic acids, amino acids and carboxylates. Substantial additions to databases for ∆ o hyd Gi values of aqueous hydrocarbons allowed refinement of functional group parameters by Sedlbauer et al. (2002). New or more accurate predictions were presented for alkanes, alkenes, cycloalkanes and alkylbenzenes at temperatures to 570 K and pressures to 100 MPa. A software tool is to be introduced (Mayer et al., 2003) that implements the method and enables simple calculation of the Henry’s constant for organic solutes at user-defined conditions. For academic users, the program can be obtained upon request at [email protected] or [email protected]. Other extensions of the new group additivity scheme at elevated temperatures are in progress for different structural elements of organic nonelectrolytes and ions.

4.3.3 Examples of Standard Thermodynamic Properties Calculated from Selected Models This section illustrates predictions by the HKF, SOCW and (for volatile solutes only) the POCW models when used with parameters retrieved from the literature and/or from available freeware codes that apply these equations. The use of models is illustrated by a few typical examples, including nonpolar, strongly polar and ionic solutes. Properties related to the Gibbs energy are given here in the molality standard state, which dominates the field of solution thermodynamics at high temperatures. Conversion from the mole fraction standard state was done with Eq. (4.5).

∆ o,app Figure 4.4 displays the apparent Gibbs energy of formation f Gi , calculated from Eqs. (4.47) or (4.50), of carbon dioxide and benzene along the saturation line of water and at an elevated pressure. The agreement of predicted values with the selected literature data at the vapor pressure of the solvent is very good for all three methods considered. In the critical and supercritical regions, the HKF results begin to ∆ o,app deviate from the predictions by the two FST models. The estimates of f Gi for methane at extreme conditions where experimental data are rare is shown in Fig. 4.5. Comparison with values obtained by molecular simulation suggests the superiority of the SOCW and POCW models compared to the SUPCRT92 results. This trend is clearly confirmed by predictions of the standard volume of carbon dioxide and benzene near the critical point of water (Fig. 4.6). It is a stringent test for a model to describe o an extremum in Vi occurring at the combination of temperature and pressure corresponding to the critical density of water (see chapter 2). It should be noted that parameters for benzene in the SOCW model were obtained from the group-contribution scheme of Sedlbauer et al. (2002) and are not specifically fitted for benzene. Important disagreement between the FST and electrostatic models is observed for boric acid (Fig. 4.7), o for which HKF predicts the same type of Vi divergence at the critical point of water as for volatile nonelectrolytes (Fig. 4.8). Although B(OH)3 is completely associated at high temperatures, its high polarity and the size of the molecule lead to the same type of near-critical behavior as for the strong electrolyte NaCl, whose volumetric behavior is also depicted in Fig 4.8. This confirms the finding that the negative sign of the critical divergence is due to sufficiently attractive solute-solvent interaction, which does not necessarily imply an electrolytic character for a solute (Wheeler, 1972).

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Calculation of Standard Thermodynamic Properties of Aqueous Electrolytes and Non-Electrolytes

As mentioned above, application of the electrostatic Born model of solvation for aqueous nonelectrolytes has no theoretical justification, and this seriously hampers utilization of the HKF equations for these solutes. On the other hand, when calculations are not needed at conditions far from ambient, the results for all models are mainly governed by the standard Gibbs energy and the standard entropy values at the

reference condition of Tr, pr (see Eqs. 4.48, 4.52 and 4.54). In this case, the SUPCRT92 program can be used with reasonable confidence for most nonelectrolytes up to 473 K. At elevated conditions, equations that comply with the correct low-density limit such as SOCW or POCW are preferred for aqueous nonelectrolytes, provided the parameters for the solutes are available from the literature. The published parameters cover most common inorganic gases (Sedlbauer et al., 2000; Plyasunov et al., 2000a,b) and some organic molecules (Plyasunov et al., 2001). Predictions are available for a wide array of organic solutes by the group-contribution method based on the SOCW model (Yezdimer et al., 2000; Sedlbauer et al., 2002). The situation is more favorable for ionic solutes, where both HKF and SOCW provide consistent and fairly accurate predictions to very high temperatures and pressures. The correlations for NaCl shown in Fig. 4.9 indicate similar results for both models at temperatures at least up to 600 K. Even at extreme conditions (Fig. 4.10), the estimates are comparable and in semiquantitative agreement with the values obtained from ab initio calculations. Figure 4.11 provides an estimate of the ionization constant for water at two high-pressure isobars. The properties of the OH− ion were calculated from the models, while the needed thermodynamic functions for pure water were obtained from the equation of state of Hill (1990). To summarize, the HKF model was primarily designed for ionic species and is still the method of choice for practical use for ionic solutes due to incorporation of several hundred charged solutes into the SUPCRT92 software package. A comparable alternative is not available for any other model at this time. The software with a group contribution method based on the SOCW model (Majer et al., 2003) is focused mainly on Henry’s constant of organic nonelectrolyte solutes (essentially hydrocarbons).

4.3.4 Tabulation of Standard Thermodynamic Properties Calculated from the SOCW Model for Selected Solutes In this final section, we give examples of calculations with the SOCW model for several nonelectrolyte and electrolyte solutes. Tabular presentation gives a better perception of numerical values of various functions, and can also be useful for readers who might wish to program the model. While implementing the SOCW equations is not difficult, an obstacle may arise from the need for thermodynamic properties of pure water. For this purpose, the equations based on the IAPWS formulation for water are most appropriate, and the package available from NIST (Harvey et al., 2000) yields all the necessary data for water. Table 4.1 tabulates (for standard states on the molality scale) for five nonelectrolyte solutes and three ions the four adjustable parameters a, b, c, d of Eq. (4.71), as well as parameters e and g of the correction corr corr functions defined by Eq. (4.73) and given below the tables for Gi and Si . These parameters and terms are used in Eqs. (4.75) and (4.76) for calculating the Gibbs energy and entropy of hydration from the SOCW model. These values are then used in Eq. (4.54) for obtaining the Gibbs energy of hydration ∆ o ∆ o consistent with the recommended literature values of hydGi and hyd Si at the reference conditions Tr

=298.15 K and pr=0.1 MPa; these latter values are presented in the last two columns of Table 4.1.

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Calculation of Standard Thermodynamic Properties of Aqueous Electrolytes and Non-Electrolytes

Tables 4.2 and 4.3 list the standard volume (Eq. 4.71) and the Gibbs energy of hydration (Eq. 4.54) for the selected solutes as a function of temperature and pressure. Table 4.4 presents ideal-gas properties necessary for calculating the apparent Gibbs energy of formation ∆ o,app f Gi (given in Table 4.5) from the relationship

∆ o,app = ∆ olit − − olit + ∆ omod − ∆ omod f Gi [T, p] f Gi [Tr , pr ] (T Tr )Si [Tr , pr ] hydGi [T, p] hydGi [Tr , pr ]

T T − T C ig d lnT + C ig dT, (4.80) ∫ p,i ∫ p,i Tr Tr

ig which was obtained by inserting Eq. (4.54) into Eq. (4.50) where C p,i is the ideal-gas heat capacity of a ∆ olit olit solute. f Gi [Tr , pr ] and Si [Tr , pr ] are the literature values of standard Gibbs energy of formation and standard entropy in aqueous solution, respectively, independent of the SOCW model. These quantities were calculated by combining the values of hydration and ideal-gas properties from Tables 4.1 and 4.4 at the reference conditions Tr and pr (see Eqs. 4.14 and 4.16). In the case of ions, the properties of formation as well as hydration are conventional and were obtained from the tabulation by Marcus (1997).

4.4 Acknowledgments This work was supported by an IAPWS Fellowship and by the Grant Agency of the Czech Republic under contract No. 203/02/0080. The authors are grateful to E.L. Shock for his comments on the geochemical applications of the models.

4.5 References Abraham, M.H., Matteoli, E. and Liszi, J. (1983). J. Chem. Soc., Faraday Trans. I, 79, 2781-2800 Akinfiev, N.N. and Diamond, L.W. (2003). Geochim. Cosmochim. Acta 67, 613-627 Amend, J. P. and Helgeson, H. C. (1997a). Geochim. Cosmochim. Acta 61, 11-46 Amend, J. P. and Helgeson, H. C. (1997b). J. Chem. Soc. Faraday Trans. 93, 1927-1941 Amend, J.P. and Plyasunov, A.V. (2001). Geochim. Cosmochim. Acta 65, 3901-3917 Anderson, G.M. and Crerar, D.A. (1993). Thermodynamics in Geochemistry, Oxford University Press, New York, Oxford Anderson, G.M., Castet, S., Schott, J. and Mesmer, R.E. (1991). Geochim. Cosmochim. Acta 55, 1769- 1779 Baes, C.F., Jr. and Mesmer, R.E. (1976). The Hydrolysis of Cations, Wiley, New York Ben-Naim, A. (1987). Solvation Thermodynamics, Plenum Press, New York Cabani, S., Gianni, P., Mollica, V., Lepori, L. (1981). J. Solution Chem. 10, 563-595 Clarke, R.G., Hnedkovsky, L., Tremaine, P.R. and Majer, V. (2000). J. Phys Chem. B. 104, 11781-11793

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Criss, C.M. and Wood, R.H. (1996). J. Chem. Thermodyn. 28, 723-741 Dale, J.D., Shock, E.L., MacLeod, G., Aplin, A.C. and Larter, S.R. Geochim. Cosmochim. Acta 61, 4017- 4024 Degrange, S., Majer, V., Sedlbauer, J. and Hynek, V. (2003a). J. Phys Chem., submitted Degrange, S., Majer, V., Coxam, J.-Y. and Hynek, V. (2003b). J. Solution Chem., submitted Eckert, C.A. and Sherman, S.R. (1996). Fluid Phase Equil. 116, 333-342 Fernández-Prini, R., Corti, H.R. and Japas, M.L. (1992). High Temperature Aqueous Solutions: Thermodynamic Properties, CRC Press, Boca Raton, FL Fernández-Prini, R., Alvarez, J.L. and Harvey, A.H. (2003). J. Phys. Chem. Ref. Data 32, 903-916 Gianni, P. and Lepori, L. (1996). J. Solution Chem. 25, 1-41 Gill, S.J. (1988). J. Chem. Thermodyn. 20, 1361-1382 Gmehling, J., Menke, J. and Schiller, M. (1994a). Activity Coefficients at Infinite Dilution, Chemistry Data Series, Volume IX, Part 3, Dechema, Frankfurt am Main Gmehling, J., Menke, J. and Schiller, M. (1994b). Activity Coefficients at Infinite Dilution, Chemistry Data Series, Volume IX, Part 4, Dechema, Frankfurt am Main Goldman, S. and Bates, R.G. (1972). J. Am. Chem. Soc. 94, 1476- 1484 Haas, J.R. and Shock, E.L. (1999). Geochim. Cosmochim. Acta 63, 3429-3441 Haas, J.R., Shock, E.L. and Sassani, D.C. (1995). Geochim. Cosmochim. Acta 59, 4329-4350 Harvey, A.H. (1996). AIChE J. 42, 1491-1494 Harvey, A.H., Levelt Sengers, J.H.M. and Tanger, J.C. (1991). J. Phys. Chem. 95, 932-937 Harvey, A.H., Peskin, A.P. and Klein, S.A. (2000). NIST/ASME Steam Properties (NIST Standard Reference Database 10, Version 2.2), Nat. Inst. Stand. Tech., Gaithersburg, MD Helgeson, H.C. and Kirkham, D.H. (1976). Am. J. Sci. 276, 97-240 Helgeson, H.C., Kirkham, D.H. and Flowers, G.C. (1981). Am. J. Sci. 281, 1249-1516 Hill, P.G. (1990). J. Phys. Chem. Ref. Data 19, 1233-1274 Hnedkovsky, L. and Wood, R.H. (1997). J. Chem. Thermodyn. 29, 731-747 Hnedkovsky, L., Majer, V. and Wood, R.H. (1995). J. Chem. Thermodyn. 27, 801-814 Hnedkovsky, L., Wood, R.H. and Majer, V. (1996). J. Chem. Thermodyn. 28, 125-142 Hnedkovsky, L., Hynek, V., Majer, V. and Wood, R.H. (2002). J. Chem. Thermodyn. 34, 755-783 Hoiland, H. (1986). In Thermodynamic Data for Biochemistry and Biotechnology, H.-J. Hinz, ed. Springer-Verlag, Berlin, 17-44 Hynek, V., Obšil, M., Majer, V., Quint, J.R., Grolier and J.-P.E. (1997). Int. J. Thermophys. 18, 719-731 Hynek,V., Degrange, S., Polední ek, M., Majer, V., Quint, J.R. and Grolier, J.-P.E (1999). J. Solution Chem. 28, 631-666 Japas, M.L. and Levelt Sengers, J.M.H. (1989). AIChE J. 35, 705-713 Johnson, J.W., Oelkers, E.H. and Helgeson, H.C. (1992). Computers & Geosciences 18, 899-947

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Kirkwood, J.G. and Buff, F.P. (1951). J. Chem. Phys. 19, 774-782 Kojima, K., Zhang, S. and Hiaki, T. (1997). Fluid Phase Equil. 131, 145-179 Lin, C.-L. and Wood, R.H. (1996). J. Phys. Chem. 100, 16399-16409 Liu, W., Sakane, S., Wood, R.H. and Doren, D.J. (2002). J. Phys. Chem. A 106, 1409-1418 Mackay, D. and Shiu, W.-Y. (1981). J. Phys. Chem. Ref. Data 10, 1175-1199 Majer, V. (1999). In High Pressure Chemical Engineering, Proceedings of the International Meeting of the GVC-Fachausschuss Hochdruckverfahren Technik, Karlsruhe, 145-150 Majer, V. and Padua, A. (2003). In Experimental Thermodynamics: Measurement of the Thermodynamic Properties of Single Phases, Volume VI Editors: Marsh, K., Wakeham, W.A., Goodwin, A.R.H., Elsevier, Amsterdam Majer, V., Hui, L., Crovetto, R. and Wood, R.H. (1991). J. Chem. Thermodyn. 23, 213-229 Majer, V., Bergin, G. and Sedlbauer, J. (2003). Fluid Phase Equil., submitted Marcus, Y. (1985). Ion Solvation, Wiley, Chichester Marcus, Y. (1997). Ion Properties, Marcel Dekker, New York Marshall, W.L. (1970). J. Phys. Chem. 74, 346-355 Marshall, W.L. and Franck, E.U. (1981). J. Phys. Chem. Ref. Data 10, 295-304 Murphy, W. M. and Shock, E.L. (1999). Rev. Mineral. 38, 221-253 O’Connell, J.P. (1995). Fluid Phase Equil. 104, 21-39 O'Connell, J. P., Sharygin, A.V. and Wood, R.H. (1996). Ind. Eng. Chem. Res. 35, 2808-2812 Oelkers, E.H., Helgeson, H.C., Shock, E.L., Sverjensky, D.A., Johnson, J.W., Pokrovskii, V.A. (1995). J. Phys. Chem. Ref. Data 24, 1401-1560 Pierotti, R.A. (1976). Chem. Rev. 76, 717-726 Pitzer, K.S., Ed. (1991). Activity Coefficients in Electrolyte Solutions, CRC Press, Boca Raton, FL Pitzer, K.S. (1995). Thermodynamics, 3rd edition, McGraw Hill, New York Plyasunov, A.V. and Shock, E.L. (2000a). Geochim. Cosmochim. Acta 64, 439-468 Plyasunov, A.V. and Shock, E.L. (2000b). Geochim. Cosmochim. Acta 64, 2811-2833 Plyasunov, A.V. and Shock, E.L. (2001a). Geochim. Cosmochim. Acta 65, 3879-3900 Plyasunov, A.V. and Shock, E.L. (2001b). J. Chem. Eng. Data 46, 1016-1019 Plyasunov, A.V., O’Connell, J.P. and Wood, R.H. (2000a). Geochim. Cosmochim. Acta 64, 495-512 Plyasunov, A.V., O’Connell, J.P., Wood, R.H. and Shock, E.L. (2000b). Geochim. Cosmochim. Acta 64, 2779-2795 Plyasunov, A.V., O’Connell, J.P., Wood, R.H. and Shock, E.L. (2001). Fluid Phase Equilib. 183-184, 133-142 Prapaipong, P., Shock, E.L. and Koretsky, C. M. (1999). Geochim. Cosmochim. Acta 63, 2547-2577 Prausnitz, J.M., Lichtenthaler, R.N. and de Azevedo, E.G. (1999). Molecular Thermodynamics of Fluid- Phase Equilibria, 3rd Edition, Prentice Hall, Upper Saddle River, NJ

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Rafal, M., Berthold, J.W., Scrivner, N.C. and Grise, S.L. (1994). in Models for Thermodynamic and Phase Equilibria Calculations, ed. Sandler, S.I., Marcel Dekker, New York, 601-669 Rard, J.A. and Archer, D.G. (1995). J. Chem. Eng. Data 40, 170-185 Sandler, S.I. (1999). Chemical Engineering Thermodynamics, 3rd Edition, Wiley, New York Sassani, D.C. and Shock, E.L. (1998). Geochim. Cosmochim. Acta 62, 2643-2671 Schulte, M.D. and Shock, E.L. (1993). Geochim. Cosmochim. Acta 57, 3835-3846 Schulte, M.D., Shock, E.L., Obsil, M. and Majer, V. (1999). J. Chem. Thermodyn. 31, 1195-1229 Schulte, M.D., Shock, E.L. and Wood, R.H. (2001). Geochim. Cosmochim. Acta 65, 3919-3930 Sedlbauer, J. and Majer, V. (2000). Europ. J. Mineral. 12, 1109-1122 Sedlbauer, J., Yezdimer, E.M. and Wood, R.H. (1998). J. Chem. Thermodyn. 30, 3-12 Sedlbauer, J., O’Connell, J.P. and Wood, R.H. (2000). Chem. Geology 163, 43-63 Sedlbauer, J., Bergin, G. and Majer, V. (2002). AIChE J. 48, 2936-2959 Shiu, W-Y., Ma, K-C. (2000a). J. Phys. Chem. Ref. Data 29, 41-102 Shiu, W-Y., Ma, K-C. (2000b). J. Phys. Chem. Ref. Data 29, 387-462 Shock, E.L. (1995). Am. J. Sci. 295, 496-580 Shock, E.L. and Helgeson, H.C. (1988). Geochim. Cosmochim. Acta 52, 2009-2036 Shock, E.L. and Helgeson, H.C. (1990). Geochim. Cosmochim. Acta 54, 915-945 Shock, E.L. and Koretsky, C.M. (1993). Geochim. Cosmochim. Acta 57, 4899-4922 Shock, E.L. and Koretsky, C.M. (1995). Geochim. Cosmochim. Acta 59, 1497-1532 Shock, E.L., Helgeson, H.C. and Sverjensky, D.A. (1989). Geochim. Cosmochim. Acta 53, 2157-2183 Shock, E.L., Oelkers, E.H., Johnson, J.W., Sverjensky, D.A. and Helgeson H.C. (1992). J. Chem. Soc. Faraday Trans. 88, 803-826 Shock, E.L., Sassani, D.C., Willis, M. and Sverjensky D.A. (1997a). Geochim. Cosmochim. Acta 61, 907-950 Shock, E.L., Sassani, D.C. and Betz, H. (1997b). Geochim. Cosmochim. Acta 61, 4245-4266 Sverjensky, D.A. Shock, E.L. and Helgeson H.C. (1997). Geochim. Cosmochim. Acta 61, 1359-1412 Tanger, J.C. and Helgeson, H.C. (1988). Am. J. Sci. 288, 19-98 Tanger, J.C. and Pitzer, K.S. (1989). J. Phys. Chem. 93, 4941-4951 Tremaine, P.R. and Goldman, S. (1978). J. Phys. Chem. 82, 2317-2321 Tremaine, P.R., Shvedov, D. and Xiao, C. (1997). J. Phys. Chem. B. 101, 409-419 Tsonopoulos, C. (1999). Fluid Phase Equil. 156, 21-33 Tsonopoulos, C. (2001). Fluid Phase Equil. 186, 185-206 Wagman, D.D., Evans, W.H., Parker, V.B., Schumm, R.H., Halow, I., Bailey, S.M., Churney, K.L. and Nuttall, R.L. (1982). J. Phys. Chem. Ref. Data 11, Supplement No. 2 Wheeler, J.C. (1972). Ber. Bunsenges. Phys. Chem. 76, 308-318

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Wood, R.H., Carter, R.W., Quint, J.R., Majer, V., Thompson P.T. and Boccio, J.R. (1994). J. Chem. Thermodyn. 26, 225-249 Yezdimer, E.M., Sedlbauer, J. and Wood, R.H. (2000). Chem. Geology 164, 259-280

Table 4.1 Parameters of the SOCW model, literature data on hydration properties at reference

conditions Tr, pr and correction functions for selected solutes. a. Nonelectrolyte solutes. 3 4 5 a ⋅ 10 b ⋅ 10 c ⋅ 10 d e ∆ olit ∆ olit hydGi hyd Si

m3⋅kg−1 m3⋅kg−1 m3⋅kg−1 J⋅K−2⋅mol−1 kJ⋅mol−1 J⋅K−1⋅mol−1 − − − CH4 4.2134 1.7672 0.69122 0.41610 0.13768 16.39 102.6 − − − CO2 3.3921 1.4801 1.3880 0.55582 0.26387 8.38 96.1 − − − − NH3 1.3255 0.49738 0.80535 0.88078 0.18104 10.05 81.2 − − − − C6H6 2.3614 4.9737 2.6470 0.31871 0.66370 4.29 120.7 − − − H3BO3 0.61428 0.38857 0.86324 1.8227 0.12002 38.59 140.3 corr (()()()2 2 ()2 T −Θ ) corr G = e 2T − Θ T − T +1/ 2 T − T + T − Θ ln[ −Θ ] − TS i c c c c Tc i 2 T −Θ 2 corr = ( − − Tc (Tc ) T −Θ ) S e T T Θ ln[ ] + Θ ln[ −Θ ] i c Tc Tc

b. Ionic solutes. 4 5 −2 a b ⋅ 10 c ⋅ 10 d e ⋅ 10 g ∆ olit b ∆ olit b hydGi hyd Si

m3⋅kg−1 m3⋅kg−1 m3⋅kg−1 J⋅K−1⋅mol−1 J⋅K−2⋅mol−1 kJ⋅mol−1 J⋅K−1⋅mol−1 Na+ −0.0071136 2.3126 −3.0289 −1.5203 2.9492 −1.1456 −842.4 −89 Cl− −0.65675 −0.86168 −0.30967 2.5615 0.16432 0.26063 110.2 −97.9 (Ac−)a −0.65974 1.2786 −0.017832 2.0193 0.19540 0.16493 94.8 −192.1 corr 2 2 T −Θ corr G = g(T − T )/ 2 + ()()()T −T e − gT + e Θ −T ln[ −Θ ] − TS i c c c c Tc i

corr T Tc T −Θ S = g()T −T + ln[ ]()e − gΘ Θ + e()Θ − T ln[ −Θ ]/ Θ i c Tc c Tc ______a acetate ion b conventional hydration property

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Calculation of Standard Thermodynamic Properties of Aqueous Electrolytes and Non-Electrolytes

Table 4.2 o 3⋅⋅⋅ −1 Standard volume Vi (cm mol ) calculated from the SOCW model for selected solutes.

T/K 298 373 473 573 623

p psat 30 MPa psat 30 MPa psat 30 MPa psat 30 MPa psat 30 MPa

CH4 38.8 38.6 42.9 42.0 57.8 53.7 113 85.4 338 145

CO2 34.7 34.6 39.0 38.3 51.8 48.4 97.4 74.7 277 123

NH3 23.7 23.6 26.0 25.6 31.9 30.4 50.6 42.0 116 61.4

C6H6 81.1 81.6 89.3 88.3 114 107 182 146 386 200

H3BO3 39.4 39.0 39.9 39.4 43.2 42.4 45.0 45.4 4.2 39.9 NaAca 39.3 41.1 39.8 41.3 27.8 32.0 −78.1 −27.6 −747 −195 NaCl 16.6 18.0 16.0 17.7 −1.9 4.1 −120 −61.4 −807 −232 ______a sodium acetate

Table 4.3 ∆ o ⋅ −1 Gibbs energy of hydration hydGi (kJ⋅⋅ mol ) calculated from the SOCW model for selected solutes.

T/K 373 473 573 623

p psat 30 MPa psat 30 MPa psat 30 MPa psat 30 MPa

CH4 22.4 23.6 26.4 28.0 27.0 29.0 25.3 28.0

CO2 14.2 15.3 18.7 20.1 20.3 22.1 19.5 21.8 − − NH3 4.4 3.6 2.0 2.8 6.8 7.8 8.5 9.5

C6H6 11.0 13.7 14.3 17.4 13.5 16.9 12.3 15.7 − − − − − − − − H3BO3 28.7 27.6 17.7 16.5 8.0 7.1 3.0 2.6 NaAca −726.7 −725.4 −698.5 −697.6 −665.2 −666.2 −640.8 −645.5 NaCl −717.2 −716.7 −694.3 −694.3 −665.6 −667.4 −644.0 −649.3 ______a sodium acetate

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Calculation of Standard Thermodynamic Properties of Aqueous Electrolytes and Non-Electrolytes

Table 4.4 Ideal-gas properties for selected solutes; heat capacity as a function of temperature, standard

Gibbs energy of formation and entropy at the reference state Tr and pr.

− − − − x x ⋅ 10 2 x ⋅ 10 5 x ⋅ 10 8 x ⋅ 10 11 ig ig 1 2 3 4 5 ∆ f Gi Si J⋅K−1⋅mol−1 J⋅K−2⋅mol−1 J⋅K−3⋅mol−1 J⋅K−4⋅mol−1 J⋅K−5⋅mol−1 kJ⋅mol−1 J⋅K−1⋅mol−1 − − − CH4 36.87 6.00 25.0 21.6 6.19 34.33 186.3 − − − CO2 19.28 7.80 6.97 3.36 0.681 394.36 213.7 − − − NH3 32.48 0.975 9.23 8.30 2.43 16.45 192.4 − − C6H6 29.52 5.14 119.4 164.7 68.5 129.9 269.3 − − H3BO3 23.56 16.7 8.78 0.122 0.963 930.2 295.1 Na+ 20.78 0 0 0 0 580.5 148 Cl− 20.78 0 0 0 0 −241.4 154.4 (Ac−)a 14.62 16.39 3.727 −12.16 4.514 −464.1 278.7 ig = + + 2 + 3 + 4 C p x1 x2T x3T x4T x5T

______

a acetate ion

Table 4.5 ∆ o,app ⋅⋅⋅ −1 Apparent Gibbs energy of formation f Gi (kJ mol ) calculated from the SOCW model for selected solutes.

T/K 373 473 573 623

p psat 30 MPa psat 30 MPa psat 30 MPa psat 30 MPa − − − − − − − − CH4 42.7 41.4 58.6 57.0 79.0 76.9 91.4 88.7 − − − − − − − − CO2 396.6 395.4 414.8 413.4 436.8 435.0 449.8 447.6 − − − − − − − − NH3 35.6 34.8 49.8 48.9 66.3 65.3 75.6 74.6

C6H6 120.0 122.6 92.8 95.9 58.8 62.2 39.9 43.4 − − − − − − − − H3BO3 981.6 980.5 1002. 1001. 1026. 1025. 1039. 1038. 6 4 8 8 3 9 NaAca −642.9 −641.6 −659.9 −659.0 −673.7 −674.8 −673.6 −678.3 NaCl −401.1 −400.7 −409.9 −409.9 −413.7 −415.6 −408.7 −414.0 ______a sodium acetate

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Calculation of Standard Thermodynamic Properties of Aqueous Electrolytes and Non-Electrolytes

Pure Solute ∆ o • ∆ • solXi Xi resXi Aq. solute Ideal gas solute ∆ X o X o hyd i ig[] i Xi pr

X= S,V,Cp

Pure solute • ∆ o ∆ solXi • resXi Xi Aq. solute Ideal gas solute ∆ o X o hydXi ig [] i Xi pr ∆ • f Xi ∆ X o ∆ ig f i Elements f Xi ν el el• ∑ j X j

X= G,H,A

Figure 4.1 Relations among dissolution, hydration, residual and formation standard properties.

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Calculation of Standard Thermodynamic Properties of Aqueous Electrolytes and Non-Electrolytes

(a)

(b)

(c) Figure 4.2 Standard thermodynamic properties as a function of temperature calculated from experimental data at p = 28 MPa (lines represent the smoothed values).

a. Standard volume, lines from up to down: CH4 (Hnedkovsky et al., 1996); NH3 (Hnedkovsky et al., 1996); NaBr (Rard and Archer, 1995).

b. Standard heat capacity, lines from up to down: CH4 (Hnedkovsky and Wood, 1997); NH3 (Hnedkovsky and Wood, 1997); NaBr (Rard and Archer, 1995). c. Enthalpy of hydration, lines from up to down: benzene (Degrange et al., 2002a); NaBr (Rard and Archer, 1995).

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Calculation of Standard Thermodynamic Properties of Aqueous Electrolytes and Non-Electrolytes

Aq. solute Ideal gas solute ∆ Go[]T, p ∆ GTpo,app [], hyd i ∆ ig,app f i frGTpi [], System state T,p

T, p []ig T , pr []Go Gi i Tr, pr Tr , pr

Aq. solute Ideal gas solute ∆ Go[]T, p ∆ GTpo [], hyd i r r ∆ ig fri r frrGTpi [],

Reference state T = 298.15 K, p = 0.1 MPa r r

Figure 4.3 Thermodynamic pathways for obtaining the apparent standard Gibbs energy of formation ∆ o,app f Gi at superambient conditions

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Calculation of Standard Thermodynamic Properties of Aqueous Electrolytes and Non-Electrolytes

Figure 4.4. Apparent Gibbs energy of formation of nonpolar solutes along vapor-liquid saturation line and at elevated pressure. Circles are representative data.

a. CO2 at psat (left) and p = 30 MPa (right). Full line – SOCW model with parameters from Sedlbauer and Majer (2000); dashed line – HKF model with parameters from SUPCRT92; dotted line – POCW model with parameters from Plyasunov et al. (2000b).

b. C6H6 at psat (left) and p = 30 MPa (right). Full line – SOCW model with parameters from Sedlbauer et al. (2002); dashed line – HKF model with parameters from SUPCRT92; dotted line – POCW model with parameters from Plyasunov et al. (2001).

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Calculation of Standard Thermodynamic Properties of Aqueous Electrolytes and Non-Electrolytes

Figure 4.5 Apparent Gibbs energy of formation of methane at two isotherms (T = 875 K, T = 1175 K). Circles – molecular simulation results by Lin and Wood (1996); full line – SOCW model with parameters from Sedlbauer and Majer (2000); dashed line – HKF model with parameters from SUPCRT92; dotted line – POCW model with parameters from Plyasunov et al. (2000b).

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Calculation of Standard Thermodynamic Properties of Aqueous Electrolytes and Non-Electrolytes

(a)

(b)

Figure 4.6 Standard volume for nonpolar solutes at p = 28 MPa isobar.

a. CO2: Circles – experimental data by Hnedkovsky et al. (1996); full line – SOCW model with parameters from Sedlbauer and Majer (2000); dashed line – HKF model with parameters from SUPCRT92; dotted line – POCW model with parameters from Plyasunov et al. (2000b).

b. C6H6: Circles – experimental data by Degrange (2002a); full line – SOCW model with parameters from Sedlbauer et al. (2002); dashed line – HKF model with parameters from SUPCRT92; dotted line – POCW model with parameters from Plyasunov et al. (2001).

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Calculation of Standard Thermodynamic Properties of Aqueous Electrolytes and Non-Electrolytes

Figure 4.7 Apparent Gibbs energy of formation of polar solute boric acid along vapor-liquid saturation line (left) and at elevated pressure p = 30 MPa (right). Full line – SOCW model with parameters from Sedlbauer et al. (2000); dashed line – HKF model with parameters from SUPCRT92.

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Calculation of Standard Thermodynamic Properties of Aqueous Electrolytes and Non-Electrolytes

(a)

(b)

Figure 4.8 Standard volume for polar and ionic solutes at p = 28 MPa isobar.

a. H3BO3: Circles – experimental data by Hnedkovsky et al. (1995); full line – SOCW model with parameters from Sedlbauer et al. (2000); dashed line – HKF model with parameters from SUPCRT92. b. NaCl: Circles – experimental data by Majer et al. (1991) (extrapolated by Sedlbauer et al., 1998); full line – SOCW model with parameters from Sedlbauer et al. (2000). HKF model is not recommended for NaCl at this pressure and T > 623 K (Johnson et al., 1992).

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Calculation of Standard Thermodynamic Properties of Aqueous Electrolytes and Non-Electrolytes

Figure 4.9 Apparent Gibbs energy of formation of ionic solute NaCl along vapor-liquid saturation line (left) and at elevated pressure p = 60 MPa (right). Full line – SOCW model with parameters from Sedlbauer et al. (2000); dashed line – HKF model with parameters from SUPCRT92.

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Calculation of Standard Thermodynamic Properties of Aqueous Electrolytes and Non-Electrolytes

Figure 4.10 Apparent Gibbs energy of formation of NaCl at two isotherms (T = 573 K, T = 973 K). Circles – ab initio results by Liu et al. (2002); full line – SOCW model with parameters from Sedlbauer et al. (2000); dashed line – HKF model with parameters from SUPCRT92.

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Calculation of Standard Thermodynamic Properties of Aqueous Electrolytes and Non-Electrolytes

Figure 4.11 Logarithm of the ionization constant for water at two isobars (p = 200 MPa, p = 400 MPa). Circles – experimental data as processed by Marshall and Franck (1981); full line – SOCW model with parameters for OH− ion from Sedlbauer et al. (2000); dashed line – HKF model with parameters for OH− ion from SUPCRT92

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