Efficient Implementation of Wertheim’s Theory: 2. Master Equations for Asymmetric Solvation

Abstract

A general method is developed to solve the nonlinear equations derived from Wertheim’s theory of associating molecules [Wertheim, M. S. J. Stat. Phys. 1984, 35 (19), 19–31]. The method assumes the factor characterizing the strength of association can be factored according to a geometric mean. Under this condition, the large system of equations for the extents of bonding for all types of electron donors and acceptors can be minimized. In the case of symmetric solvation, the number of acceptors and donors on each molecule must be equal. In that case, the system of equations reduces to a single master equation as shown in the first paper of this series [Elliott, J. R. Ind. Eng. Chem. Res. 1996, 35 (5) 1624–1629]. The present work extends to asymmetric solvation, in which case the system reduces to two master equations, one covering all acceptors and one covering all donors. The simplification has several ancillary benefits. For example, it suggests characterization of individual acceptor and donor strengths rather than just the strength of the combined energies. These acceptor and donor strengths can be related to acidity and basicity scales that have been developed previously based on spectroscopic measurements and infinite dilution activity coefficients. Expressions are derived here for infinite dilution activity coefficients inferred from Wertheim’s theory under the geometric mean condition. Guidelines are provided for predicting acceptor and donor strengths from previously existing resources. Sample calculations are presented for several systems exhibiting asymmetric solvation with comparisons to predictions based on symmetric solvation. Comparisons are also presented for correlation of vapor–liquid equilibria relative to a standard reference database.