Exact Solution of Generalized Percus-Yevick Equation for a Mixture of Hard Spheres

Abstract

The Percus-Yevick approximate equation for the radial distribution function of a fluid is generalized to an $m$-component mixture. This approximation which can be formulated by the method of functional Taylor expansion, consists in setting $\mathrm{exp}[\ensuremath{-}\ensuremath{\beta}{\ensuremath{\phi}}_{\mathrm{ij}}(r)]{C}_{\mathrm{ij}}(r)$ equal to ${g}_{\mathrm{ij}}(r)[{e}^{\ensuremath{-}\ensuremath{\beta}{\ensuremath{\phi}}_{\mathrm{ij}}(r)}\ensuremath{-}1]$, where ${C}_{\mathrm{ij}}$, ${g}_{\mathrm{ij}}$, and ${\ensuremath{\phi}}_{\mathrm{ij}}$ are the direct correlation function, the radial distribution function and the binary potential between a molecule of species $i$ and $a$ molecule of species $j$. The resulting equation for ${C}_{\mathrm{ij}}$ and ${g}_{\mathrm{ij}}$ is solved exactly for a mixture of hard spheres of diameters ${R}_{i}$. The equation of state obtained from ${C}_{\mathrm{ij}}(r)$ via a generalized Ornstein-Zernike compressibility relation has the form $\frac{p}{\mathrm{kT}}={[\ensuremath{\Sigma}{\ensuremath{\rho}}_{i}][1+\ensuremath{\xi}+{\ensuremath{\xi}}^{2}]\ensuremath{-}\frac{18}{\ensuremath{\pi}}\ensuremath{\Sigma}{i