Statistical thermodynamic theory of calculating the inner pressure and inner energy of the liquid mixture

The formulas of inner pressure and excess inner energy of the liquid mixture are derived using distribution function theory. During the derivation three characters of interaction potential energy of the liquid mixture are assumed. The first is that the interaction force between molecules is the short - range force. The second is that the many - body potential is only relying on the distances between the molecules. The third is that the potential energy of the liquid mixture can be written as the sum of a series of many - body potential. The inner pressure and inner energy of the liquid mixture can be expressed in the power series of the volume. The coefficients in the series are expressed in many - body potential and radial distribution functions and are only depend on temperature. The expression of the inner pressure and excess inner energy of pure liquid are discussed. The excess inner energy and inner pressure are agree with the result of Egelstaff's perturbation theory and Frank's experiment respectively in two special situations. Two special cases of the liquid mixture are discussed. One of them is that a method to prove the mixing law of the excess inner energy of the liquid mixture in special situation. The other is that the expressions of excess inner energy and the inner pressure have the same form with the result of Frank's experiment.