Free Energy and the Equation of State for Two-Dimensional Rigid Sphere Systems for Stable and Metastable Phases

In this work, free energy and state equation are obtained for homogeneous phase of two-dimensional hard spheres systems that are effective for both stable and metastable phases. To do this, the virial series was first transformed into a new series, the coefficients of which differ little from one another for the virial coefficients known at the present time. A method of accelerated Euler convergence is applied to this series. Performing an inverse transformation, we obtain for the compressibility a new equation, which, as calculations show, is much more accurate than the virial equation. The resulting equation is similar to the Carnahan-Starling equation for a three-dimensional system. But unlike the latter, it accurately reproduces all known virial coefficients. Further, this equation is generalized to the case of reproduction of the asymptotic behavior of the free energy at high densities. This makes it possible to describe a metastable region with a high degree of accuracy.