The Free Volume Theory of Multicomponent Fluid Mixtures

Energy landscape view of nonideality in binary mixtures

Positive and negative deviations from the prediction of Raoult's Law on the composition dependence of a property of binary mixtures are often explained in terms of structure formation and structure breakage, respectively, upon mixing. However, a detailed theoretical description of these ideas seems to be lacking in the literature. Here we present the energy landscape view of nonideality of the viscosity of the binary mixture using two different models, one for structure former and the other for structure breaker. For both the models, the average inherent structure energy shows an inverse correlation with the viscosity. The inherent structures of the structure former indicate that there is a considerable enhancement of short range order due to stronger attractive interaction between the two constituent species. On the other hand, for the structure breaker, there is no such enhancement of short range order due to weaker interaction between the two constituent species. We find the inherent structures of the structure breaker to be phase separated in many cases where the parent phase is homogeneous. When the configurational entropy of the parent liquid is computed for the two model systems, we find that the configurational entropy also shows an inverse correlation with the viscosity in both the cases.

The statistical thermodynamics of mixtures of Lennard-Jones molecules II. Deviations from random mixing

This paper describes a new statistical treatment of ordering effects in mixtures of spherical molecules, based on a general expansion of the Helmholtz free-energy function about the condition of random mixing. This expansion is similar to that introduced by Kirkwood (1938) for lattice theories, but unlike the latter is based only on the canonical distribution of petit ensemble theory. It is shown that for mixtures of Lennard-Jones molecules the moments in the expansion can be reduced to non-thermodynamic statistical functions of a single reference substance; and by using the assumptions of the cell model, these functions are related to thermodynamic properties. The first approximations for the thermodynamic functions of ordering are discussed for a certain class of mixtures, and it is shown that although ordering effects are small in mixtures of molecules of equal size, they can be large when the molecules differ in size. The equations are used to calculate the ordering effects in liquid mixtures of carbon monoxide and methane.

Cluster integrals and statistical mechanics of solutions

A formally convergent method for calculating the thermodynamic properties of solutions containing molecules not widely different in size and interaction energies is developed using the cluster-integral approach of the theory of imperfect gases. The potential energy of the actual solution is referred to the potential energy of a suitably chosen ‘reference liquid’; the correction terms which are required to give the correct energy for the solution give rise to cluster integrals which contain distribution functions of the reference liquid as factors in the integrand. Even if all the cluster integrals are neglected a useful result may be obtained. The first cluster integral, corresponding to the second virial coefficient in the theory of imperfect gases, is comparatively easy to evaluate. This approach provides a unified and formally exact basis for existing approximate theories of solutions, as well as a method for obtaining more accurate results.

The second-order theory of conformal solutions

This paper describes a theoretical contribution to the statistical thermodynamics of mixtures of spherical molecules. The second-order perturbation free energy of a conformal solution is obtained by a rigorous Taylor-series expansion of the configuration integral in powers of the differences between intermolecular energy and size parameters, about an ideal unperturbed reference solution. Unlike the first-order terms, those of the second order contain statistical functions of the reference solution which cannot, in general, be related to its thermodynamic properties. All but one of these functions are concerned with departures from a random molecular distribution, and have been called molecular fluctuation integrals ; the remaining function can be related exactly to thermodynamic properties for the Lennard-Jones form of the intermolecular potential. The expressions for the molecular fluctuation integrals implied by the full random mixing approximation and by the semi-random mixing approximation of the cell theory, are derived and compared with the correct expressions given by the cell theory. The role of the Taylor series expansion as a critique of solution theories is discussed.