Critical parameters of a Lennard-Jones gas

Abstract

The critical parameters of a gas with molecules interacting via the Lennard-Jones (6, 12) potential are calculated by analyzing the irreducible cluster integrals b_l of the fugacity expansions. The method used here has earlier been successfully applied to the Ising model, and locates the critical temperature T_c as the limit of the zeros of the cluster integrals b_l. The asymptotic behavior of b_l at T_c is numerically found, and from the fugacity series the critical pressure ρ_cand the critical density ρ_c are calculated. One can then deduce the correct shape of the critical isotherm ρ-ρ_c∝sgn(ρ-ρ_c)|ρ-ρ_c|^δ and determine the related critical index δ. The ratio ρ_c/ρ_ck_BT_c is found to be ζ(2+δ-1)/ς(1+δ¹), where ς(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 b_land the asymptotic behavior of b_l(T_c).