Classical thermodynamic basis of activity coefficients: Predictive and consistency rules for binary and ternary mixtures based on the relation between excess Gibbs free energies of ( c)- and ( c — 1)-component mixtures

Abstract

A working equation in which the molar Gibbs free energy of a ( c)-component mixture is related to that of a ( c — 1)-component mixture, both at the same pressure and temperature, is developed. This working equation is applied to binary and ternary mixtures. For binary mixtures, a reformulated integral version of the Gibbs—Duhem equation for activity coefficients is derived which is readily applicable for consistency tests and calculation of one binary activity coefficient from another. In the case of ternary mixtures a number of rigorous equations are derived for relating ternary- and binary-mixture activity coefficients. It is concluded that, generally, a ternary-mixture activity coefficient can be calculated from the knowledge of another ternary coefficient, plus a binary-mixture activity coefficient. Also, the resulting equations allow us to perform consistency tests of ternary activity coefficients against the binary-mixture data. The limiting forms of the ternary-mixture activity coefficients, when the mole fraction of one of the three components approaches to unity or to its infinite dilution limit, are derived. The form of the ternary-mixture activity coefficients and the conditions under which they may be calculated, solely from the binary activity coefficient data, is presented.