Rigorous Basis of the Frenkel-Band Theory of Association Equilibrium

By means of the general theory of the classical grand ensemble, the complete formal relations for pressure and number of molecules in an open system, as a function of activity, are derived without approximation within the scheme of the Frenkel-Band equilibrium cluster theory of association. The physical clusters, defined geometrically by overlap of spheres drawn about each molecule, are not regarded as statistically independent, but their mutual exclusion property gives rise to free energies of formation work for cavities necessary to contain those clusters. A connection is established between these free energies and a suitable generalization of the Mayer irreducible cluster sum. The appropriate modification of the theory to include external forces (such as the gravitational field) is displayed. A rigorous expansion of molecular distribution functions in terms of physical (rather than Ursell-Mayer) cluster integrals is also presented.