Analytic Solution to Integral Equations of Liquid State Theories for Potentials with a Hard Core at Low Densities

Abstract

We present in this paper a general analytical solution to the integral equations of liquid state theories (Born–Green–Yvon, hyper-netted-chain, and Percus–Yevick Equations) at low-density limit for potentials with a hard core. For the specific case of the Lennard-Jones potential with a hard core, we have derived an analytical function for the radial distribution function at high temperature and low density. We have noted that this function has two humps which is the characteristic feature of the radial distribution function at low densities. In addition, this function has been used to calculate the third virial coefficient for such a fluid exactly. We see that for the especial case of Lennard-Jones fluid with a hard core, which its radial distribution function has explicitly been calculated at high temperatures, the correct behavior of the third virial coefficient with temperature is obtained. The magnitude of hard-core diameter has significant effect on the thermodynamic properties of fluid: for instance, when the diameter changes only by a few percent the third virial coefficient may change more than 100%. The hard-core diameter decreases when temperature increases. The reduction is less than 20%. For the supercritical fluid, the calculated compression factor and internal energy are in good agreement with those obtained from the simulation for the Lennard-Jones fluid.