Statistical thermodynamics of mixtures with non-zero energies of mixing

Abstract
A combinatory formula is obtained for g(N$_{i}$, X$_{ij}$), the number of ways of arranging a mixture of any number of kinds of molecules on a lattice, the values of N$_{i}$ and X$_{ij}$ being specified, where N$_{i}$ denotes the number of molecules of type i, z denotes the number of sites which are neighbours of one site, and zX$_{ij}$ denotes the number of pairs of neighbouring sites occupied one by a molecule of type i, the other by a molecule of type j. Each molecule of type i is assumed to occupy r$_{i}$ sites, where r$_{i}$ is any integer with different values for different types of molecules. This formula is used to derive the thermodynamic properties of mixtures of molecules occupying various numbers of sites, assuming that the intermolecular energy can be regarded as a sum of terms, each pair of neighbours contributing one term. For binary mixtures the formulae obtained are very similar to those previously obtained for 'regular' solutions where each molecule occupies one site. A rather simple formula is obtained for the critical temperature and the composition of the critical mixture. The degree of accuracy of the treatment is the same as Chang's use of Bethe's first approximation and as the 'quasi-chemical' method of approach. A brief investigation of a higher approximation for a binary regular mixture on a close-packed lattice indicates that the errors due to the approximation used are unlikely ever to be serious.

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