The Guggenheim equation for a system of hard spheres and its generalization

In the present work, for the first time, the generalized Guggenheim equation for a system of hard spheres is obtained based on the use of the Euler accelerated convergence method. This equation allows taking into account an arbitrary number of known virial coefficients. For a metastable region, this equation is generalized to the case of taking into account the asymptotic behavior of the free energy at high densities. The resulting expression for the compressibility of the homogeneous phase of a system of hard spheres describes the data of a numerical experiment within their accuracy.