Fifth and Sixth Virial Coefficients for Hard Spheres and Hard Disks

Abstract

New expressions for the fourth, fifth, and sixth virial coefficients are obtained as sums of modified star integrals. The modified stars contain both Mayer f functions and f̃ functions (f̃≡f+1). It is shown that the number of topologically distinguishable graphs occurring in the new expressions is about half the number required in previous expressions. This reduction in the number of integrals makes numerical calculation of virial coefficients simpler and more nearly accurate. For particles interacting with a hard‐core potential, values of the modified star integrals are shown to depend strongly on dimension; for example, several modified star integrals are identically zero for hard disks (two dimensions), but give nonzero values for hard spheres (three dimensions). Of all the modified star integrals contributing to the fourth, fifth, and sixth virial coefficients, the complete star integrals are shown to be the largest. Mayer's expressions for these coefficients made the complete star integrals the smallest contributing integrals. The fifth (B₅) and sixth (B₆) virial coefficients of hard‐sphere and hard‐disk systems are obtained by Monte Carlo integration of the modified star integrals. The resulting values are $$\begin{array}{ccc}\mathrm{spheres}:& B_5{\mathrm{/b}}^4\text{=0.1103±0.0003;}& B_6{\mathrm{/b}}^5\text{=0.0386±0.0004}\\ \mathrm{disks}:& B_5{\mathrm{/b}}^4\text{=0.3338±0.0005;}& B_6{\mathrm{/b}}^5\text{=0.1992±0.0008}\end{array}$$ where b is the second virial coefficient. Estimated values of B₇ obtained from a Padé approximation to PV²/(N²kT) — V/N are B₇/b⁶=0.0127 for hard spheres and 0.115 for hard disks. For hard spheres virial series calculations including terms through the sixth virial coefficient give values of PV/(NkT) which agree closely, for densities less than half of closest‐packing, with the molecular dynamics data of Alder and Wainwright. Furthermore the approximate Padé expression agrees within 2% with the molecular dynamics data for all densities on the fluid side of the solid‐fluid transition. This agreement indicates convergence of the virial series along the entire fluid branch of the hard‐sphere equation of state.