Approximations to the Pair Correlation Function for a Hard-Sphere Fluid

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Abstract
The equation of state for hard spheres as predicted by the chain, watermelon, convolution-hypernetted chain (CHNC), Percus-Yevick, and two new approximations to the pair correlation function are compared with each other and with a reference isotherm. The equations of state of the chain and watermelon approximations have been found to be only slight improvements over the linear correction to the ideal gas law. The Percus-Yevick theory has been found to be an improvement over the CHNC approximation for hard spheres in agreement with the results of Hoover and Poirier for hard cubes and of Broyles for several densities of the Lennard-Jones (12,6) gas. The new approximations are designated n = 1 and n = 2. They make use of the solutions to the CHNC integral equation but do not include all the diagrams for the pressure. The approximation n = 2 was found to yield an equation of state in excellent quantitative agreement with the reference isotherm for densities up to 0.707 times the close packed density. Both the nearest and next nearest neighbors were computed as a function of density in each approximation. Only the CHNC and Percus-Yevick results extrapolate to the proper value at the density of close packing. Both of these approximations, furthermore, indicate the existence of a singularity at densities less than or equal to that of close packing. The approximations considered seem to form part of a uniform approach to the reference isotherm. At the same time, the representations of the known virial coefficients associated with each do not approach the correct virial coefficients in a uniform fashion. Therefore, an understanding of this sequence of approximations cannot be had from the usual examination of virial coefficients.