Virial Expansion of the Solution of the Kirkwood Integral Equation for the Radial Distribution Function of a Fluid

Abstract

The solution gK(r) of the Kirkwood integral equation for the radial distribution function of a fluid is expressed as a series in the density. This series is compared with the virial expansion of the exact radial distribution function g(r). It is shown that the term g2(r) proportional to the square of the density in the expansion of gK(r) differs from the corresponding term in the expansion of g(r). For the special case of hard-sphere particles g2K(r) is computed and used to obtain two approximations to the fourth virial coefficient of the equation of state. These results are compared with corresponding results for an approximate Kirkwood equation and for the Born-Green-Yvon and Percus-Yevick integral equations. The best agreement with exact theory is obtained from the Percus-Yevick equation; the poorest from the Kirkwood equation and the approximate Kirkwood equation.