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The nonlinear integral equation of the hypernetted chain (HNC) approximation is solved for a fluid of hard spheres. An inconsistency introduced by the approximation results in a disagreement between two forms of the equation of state. The resulting pair of equations of state are compared with each other, with those obtained for the Born—Green and (approximate) Kirkwood equations by Kirkwood, Maun, and Alder, and with a reference isotherm made up of a combination of the five-term virial series at low and intermediate densities and the gas-like isotherm of molecular dynamics at high densities. The HNC equations of state are found to be a considerable improvement over those of the Born—Green and Kirkwood theories. At intermediate and high densities, the two HNC isotherms bracket the reference isotherm, their mean being a very good approximation to it. The HNC equation exhibits a tendency toward a gas—solid phase transition at extremely high densities. A transition is not actually predicted, however.
Max Klein (Sun,) studied this question.