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 $p_c$ and the critical density ${\rho }_c$ are calculated. One can then deduce the correct shape of the critical isotherm $p-p_c\propto \mathrm{sgn}(\rho -{\rho }_c){|\rho -{\rho }_c|}^{\delta }$ and determine the related critical index $\delta$. The ratio $\frac{p_c}{{\rho }_c{k}_B{T}_c}$ is found to be $\frac{\zeta (2+{\delta }^{-1\mathrm{}})}{\zeta (1+{\delta }^{-1\mathrm{}})}$, where $\zeta \mathrm{}\left(x\right)\mathrm{}$ is the Riemann zeta function, and its universality is thus related to that of $\delta$. 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 $b_l$ and the asymptotic behavior of $b_l\left(T_c\right)$.