Simplification of the Clegg-Pitzer-Brimblecombe Mole-Fraction Composition Based Model Equations for Binary Solutions, Conversion of the Margules Expansion Terms into a Virial Form, and Comparison with an Extended Ion-Interaction (Pitzer) Model

Abstract

Clegg, Pitzer, and Brimblecombe (J. Phys. Chem. 96:9470–9479, 1992) described a thermodynamic model for representing the activities of solutes and a solvent, for a single electrolyte and for mixtures of arbitrary complexity, which is valid to very high concentrations including electrolytes approaching complete mutual solubility. This model contains a Debye-Huckel term along with two ionic-strength-dependent virial terms and a Margules expansion in the mole fractions of the components at the four-suffix level, with ionic strengths expressed on the mole-fraction composition scale. This model is an extension of earlier work by Pitzer and Simonson (J. Phys. Chem. 90:3005–3009, 1986). However, Pitzer’s molality-based ion-interaction model (Activity Coefficients in Electrolyte Solutions, 2nd edn., CRC Press, 1991) is more commonly used for thermodynamic modeling calculations. In this paper we recast the Margules expansion terms of the mole-fraction-based model equations for a single electrolyte in a single solvent into simpler virial expansions in powers of the mole-fraction-based ionic strength. We thereby show that these reformulated equations are functionally analogous to those of Pitzer’s standard ion-interaction model with an additional virial term added that is cubic in the ionic strength. By using a series of algebraic transformations among composition scales, we show that the pairs of terms involving the $B_{\mathrm{M,X}}^{(1)}$ and the $B_{\mathrm{M,X}}^{(2)}$ parameters in the original mole-fraction-based model expression for the natural logarithm of the mean activity coefficient (and consequently for the excess Gibbs energy) differ from each other only by a simple numerical factor of −2 and, therefore, these four terms can be replaced by two terms yielding simpler expressions. Test calculations are presented for several soluble electrolytes to compare the effectiveness of the reformulated mole-fraction- and molality-based models, at the same virial level in powers of ionic strength, for representing activity data over different ionic strength ranges. The molality-based model gives slightly better fits over the ionic strength range 0 mol⋅kg−1≤I≤6 mol⋅kg−1, whereas the mole-fraction-based model is generally better for more extended ranges.