Critical parameters of a Lennard-Jones gas

The critical parameters of a gas with molecules interacting via the Lennard-Jones (6, 12) potential are calculated by analyzing the irreducible cluster integrals bₗ of the fugacity expansions. The method used here has earlier been successfully applied to the Ising model, and locates the critical temperature Tc as the limit of the zeros of the cluster integrals bₗ. The asymptotic behavior of bₗ at Tc is numerically found, and from the fugacity series the critical pressure pc and the critical density ρc are calculated. One can then deduce the correct shape of the critical isotherm p-pc∝sgn(ρ-ρc)|ρ-ρc|^δ and determine the related critical index δ. The ratio pcρckBTc is found to be ζ(2+δ^-1)ζ(1+δ^-1), where ζ(x) is the Riemann zeta function, and its universality is thus related to that of δ. The numerical values of the critical parameters are then compared with the experimental data on rare gases; the agreement is good and represents an improvement over the methods based on equations of state. This can be taken to be a strong support for the numerical conjectures about the behavior of zeros of bₗ and the asymptotic behavior of bₗ(Tc).