The statistical thermodynamics of mixtures of Lennard-Jones molecules I. Random mixtures

Abstract

This paper describes a development in the statistical theory of mixtures of spherical molecules. It is shown that for a mixture of molecules interacting according to the Lennard-Jones inverse-power potential, the assumption of random mixing is sufficient to relate the thermodynamic properties of the mixture exactly to those of a reference substance, after the manner of the law of corresponding states; and it is proved in an appendix that only the Lennard-Jones form of the potential energy function leads to this simple result. If the molar configurational Gibbs function of the reference substance is G ₀ (T,P) ,then that of the random mixture is G(T, P, x) = f _x G ₀ (T/f _x , Ph/f _x ) - RT In h _x + RT zeta x _x where x _a is the mole fraction of component c and where f and h _x are dimensionless functions of the composition involving the characteristic molecular energy and size constants for the interactions of the various species. This equation is used to discuss the phenomena peculiar to mixtures of substances, under the headings: mixing effects, phase equilibria, and critical phases. A necessary condition on the intermolecular forces for azeotropy to occur in binary mixtures is derived in a simple form which can be appreciated intuitively; the possibility of a lower critical solution point in these mixtures is examined and shown to be unlikely; and the difficulties in the way of deriving the critical or plait-point curve are outlined. The liquid mixing properties of the system carbon monoxide + methane are calculated from the theory, and shown to be in fair agreement with experiment. The Gibbs function of the mixture is analyzed by a Taylor-series expansion, and it is shown that the first-order terms of the present theory are identical with those of the theory of conformal solutions, due to Longuet-Higgins, but that the second-order terms involve approximations, resulting from the assumption of random mixing. Expanded forms of the mixing functions are derived for the special class of binary mixtures whose characteristic energy and size constants obey geometric and arithmetic mean rules respectively, and the signs of these functions are discussed.