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

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.

Equilibrium compositions and compressibility factors for air in the temperature range 1500 to 13000 Kelvin and for densities from 10¯⁶ to 10³ times standard density

“…We shall consider (12) as the only reasonable model, particularly since it is the only one that leads to results which are satisfactory.…”

“…Thus, if this were a binary system, it would appear to be made up of two pure fluids of different diameters, namely a 3 = (Tjj + X as "experimental" data, it is then reasonable to compare them with (12) and (13) to see which is the better approximation to a mixture of hard spheres. This…”

“…For Curve 1, the reference potential is the Lennard-Jones (18,6); for Curve 2, the Lennard-Jones (12,6). The numerical results are from calculations furnished by Toxvaerd23 using potential parameters e/k = 119.8, a = 3.405 for the latter and equation (12) for the former…”

“…The estimates of the intermolecular potentials used are based on an old analysis by Wbolley in terms of the (12,6) potential function. As a result of our earlier work under AEDC support, better potentials are now known and should be used for the major species and these same potentials should also be used in place of the (12,6) for the new estimates of those associated with the minor species. As already mentioned, we used these older estimates so as to ensure consistency between these tables and the earlier tables.…”

“…They argue, that the (12,6) pair-potential with the above parameters happens to be an "effective potential" that, to first order for argon, accounts for high density non-additive effects.…”