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 4-1 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. 4-2 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<<x1, and at ambient ≈ ≈ conditions mi ci 55xi , the molality and molarity scales are able to express concentration in magnitudes not far from 1. However, molarity changes with temperature and pressure, which makes this concentration scale impractical at elevated conditions. The partial molar Gibbs energy (chemical potential) of a solute is expressed as 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 4-3 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.