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

J. L. Lebowitz
Phys. Rev. 133, A895 – Published 17 February 1964
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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 exp[βϕij(r)]Cij(r) equal to gij(r)[eβϕij(r)1], where Cij, gij, and ϕij 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 Cij and gij is solved exactly for a mixture of hard spheres of diameters Ri. The equation of state obtained from Cij(r) via a generalized Ornstein-Zernike compressibility relation has the form pkT={[Σρi][1+ξ+ξ2]18πΣi<jηiηj(RiRj)2×[Ri+Rj+RiRj(ΣηlRl2)]}(1ξ)3, where ηi=π6 times the density of the ith component and ξ=ΣηlRl3. This equation yields correctly the virial expansion of the pressure up to and including the third power in the densities and is in very good agreement with the available machine computations for a binary mixture. For a one-component system our solution for C(r) and g(r) reduces to that found previously by Wertheim and Thiele and the equation of state becomes identical with that found on the basis of different approximations by Reiss, Frisch, and Lebowitz.

  • Received 15 August 1963

DOI:https://doi.org/10.1103/PhysRev.133.A895

©1964 American Physical Society

Authors & Affiliations

J. L. Lebowitz

  • Belfer Graduate School of Science, Yeshiva University, New York, New York

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Issue

Vol. 133, Iss. 4A — February 1964

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