The fifth virial coefficient of a fluid of hard spheres

Abstract

The fifth virial coefficient of a fluid of hard spheres is a sum of 238 irreducible cluster integrals of 10 different types. The values of 5 of these types (152 integrals) are obtained analytically, the contributions of a further 4 types (85 integrals) are obtained by a combination of analytical and numerical integration, and 1 integral is calculated by an approximation. The result is $E = (0\cdot1093 \pm 0\cdot0007) b^4, b = \frac{2}{3} \pi N_A\sigma^3,$ where $\sigma$ is the diameter of a sphere. A combination of the values of 237 of the cluster integrals obtained in this paper with the value of one integral obtained independently by Katsura & Abe from a Monte Carlo calculation yields $E = (0\cdot1101 \pm 0\cdot0003) b^4.$