Free energy of classical binary mixtures

The Helmholtz free energy and grand potential of classical binary fluid mixtures are calculated. Only the direct correlation functions of the mixture are required. We follow Ebner's functional expansion method to obtain the free energy and grand potential of a simple fluid and fluid mixtures. Later an exact expression for the Helmholtz free energy is obtained. As an example, the excess free energy per particle of simple and binary hard sphere fluid mixtures are calculated where the generalized mean spherical approximation (GMSA) for simple hard spheres and the Percus Yevick (PY) direct correlation function for both simple and binary hard sphere fluid mixtures are used. The results are compared with those obtained by Carnahan-Starling and Mansoori et al. respectively. We also apply the functional expansion to find the excess free energy per particle of binary hard disk mixtures and later compare the results with some other theories. In both cases the results agree well with the previously mentioned theories.