A discussion of Margules-type formulations for multicomponent solutions with a generalized approach

Abstract

The generalized mathematical method for obtaining Margules-type formulations for the excess free energy of solution, Gxs or any other excess molar thermochemical property, and RT ln (γi), or the corresponding partial molar quantity in multicomponent solutions, is presented in this paper. For the ncomponent system, the Gxs function is approximated by a pth-order Taylor series involving (n − 1) independent compositional variables. The expression for Gxs is differentiated with respect to each compositional variable, and the resultant partial derivatives are evaluated at each of the compositional extremes to obtain the Margules parameters (activity coefficients at infinite dilution) replacing the constants in the Taylor series which have no thermodynamic meaning. When p = 2, the solutions are symmetric (strictly regular) and no ternary and higher order interaction parameters exist in ternary and higher order systems. For p = 3, the solutions are asymmetric (subregular) and no quaternary and higher order parameters exist in quarternary and higher order systems. As a special case of the latter system (p = 3) the constituent binaries can exhibit symmetric behavior, but the expressions for Gxsand RT ln (γi) can contain ternary interaction parameters. Thus, the existence of nonbinary interaction parameters is a result of the complexity of the Taylor series approximation to the Gxs function. In general, ternary interaction parameters cannot be completely defined by the binary interaction parameters. Simple expressions for the excess function and the activity coefficients in symmetric and asymmetric multicomponent solutions have been derived. Previously published binary, ternary, and quarternary symmetric and asymmetric solution models are discussed and compared to the solution models derived herein.