The statistical thermodynamics of multicomponent systems

This paper describes a new statistical approach to the theory of multicomponent systems. A 'conformal solution' is defined as one satisfying the following conditions: (i) The mutual potential energy of a molecule of species L$_{r}$ and one of species L$_{s}$ at a distance $\rho $ is given by the expression u$_{rs}$($\rho $) = f$_{rs}$u$_{00}$(g$_{rs}\rho $), where u$_{00}$ is the mutual potential energy of two molecules of some reference species L$_{0}$ at a distance $\rho $, and f$_{rs}$ and g$_{rs}$ are constants depending only on the chemical nature of L$_{r}$ and L$_{s}$. (ii) If L$_{0}$ is taken to be one of the components of the solution, then f$_{rs}$ and g$_{rs}$ are close to unity for every pair of components. (iii) The constant g$_{rs}$ equals ${\textstyle\frac{1}{2}}$(g$_{rr}$ + g$_{ss}$). From these assumptions it is possible to calculate rigorously the thermodynamic properties of a conformal solution in terms of those of the components and their interaction constants. The non-ideal free energy of mixing is given by the equation $\Delta ^{\ast}$G = E$_{0}\underset r<s\to{\Sigma \Sigma}$ x$_{r}$x$_{s}$ d$_{rs}$, where E$_{0}$ equals RT minus the latent heat of vaporization of L$_{0}$, x$_{r}$ is the mole fraction of L$_{r}$ and d$_{rs}$ denotes 2f$_{rs}$-f$_{rr}$-f$_{ss}$. This equation resembles that defining a regular solution, with the important difference that E$_{0}$ is a measurable function of T and p, which makes it possible to relate the free energy, entropy, heat and volume of mixing to the thermodynamic properties of the reference species; and the predicted relationships between these quantities agree well with available data on non-polar solutions. The theory makes no appeal to a lattice model or any other model of the liquid state, and can therefore be applied both to liquids and to imperfect gases, and to two-phase two-component systems near the critical point.