A closure for the Ornstein–Zernike relation that gives rise to the thermodynamic consistency

Abstract

A closure is proposed for the direct correlation function in the grand canonical ensemble theory that gives rise to the thermodynamic consistency, by which it is meant that identical results are obtained when the equations of state are calculated via the virial and compressibility routes, respectively, and when the excess chemical potentials are calculated by means of the thermodynamic derivative and the statistical mechanical formula, respectively. The integral equation for the pair correlation function under the closure is analytically solved in the case of hard spheres. The equation of state for hard spheres turns out to have the same form as that in the scaled particle theory or the compressibility equation of state in the Percus–Yevick theory although the present closure is quite different from that of the Percus–Yevick theory. The excess chemical potential is also found in the same form as that in the scaled particle theory. It seems to suggest that the present closure produces an integral equation theory equivalent to the scaled particle theory.