On the Theory of the Virial Development of the Equation of State of Monoatomic Gases

Abstract

The problem of the condensation of a gas is intimately related to the asymptotic behavior of the virial coefficients, B_m, as m→∞. The problem of the evaluation of the virial coefficients may be divided into two distinctly different ones. The first of these, which is purely combinatorial in nature and is independent of the intermolecular force law, is that of determining the number of a certain type of connected graphs of l points and k lines which are called ``stars.'' This problem is solved by means of generating functions, with the result that the total number of such stars is asymptotically equal to $$\left(\begin{array}{c}\frac{1}{2}\text{l(l−1)}\\ k\end{array}\right),$$ for almost all k. Arguments are also presented which indicate that the total number of topologically different stars is $$\frac{1}{\mathrm{l!}}\left(\begin{array}{c}\frac{1}{2}\text{l(l−1)}\\ k\end{array}\right).$$ With these results the combinatorial problem is essentially solved. The second problem is that of evaluating certain integrals of functions which depend on the intermolecular potential. This problem is not so near to a solution. For a purely repulsive force, asymptotic expressions are obtained for k=l, and k=l+1. The partial contributions to the virial coefficient in these two cases are: $${\text{(−1)}}^l·\frac{5}{3}{\left(\begin{array}{c}5\\ \text{2π}\end{array}\right)}^{\frac{1}{2}}{\text{(2b)}}^{\text{l−1}}\frac{\text{(l−1)}}{l^{\text{5/2}}},$$ and $${\text{(−1)}}^l\frac{{\text{2·5}}^3}{\text{24π}\sqrt{3}}{\text{(2b)}}^{\text{l−1}},$$ respectively. Results for some simple one‐dimensional rigid lines are also given.