Contribution to Statistical Mechanics

The method of the grand partition function may be used to calculate distribution functions Fn (z, n) proportional to the probability that n molecules in a system of fugacity z, and fixed temperature, occupy the position of their coordinates symbolized by n. The method makes use of the distribution functions Fn (0, n) at zero fugacity. The distribution functions may be written Fn (z, n) =exp−Wn (z, n) /kT, in which Wn (z, n) is the potential of average force of n molecules at the fugacity z, which becomes equal to the ordinary potential energy at zero fugacity. The equations may be generalized to permit the calculation of the distribution functions at any fugacity assuming a knowledge of them at any other fugacity. Using methods previously employed for imperfect gases, the pressure, and also the density in molecules per unit volume, may be developed in a power series of difference of fugacity around any arbitrary fugacity. The coefficients of these developments are calculable at all fugacities by the same equations always employing the potentials of average force at the fugacity around which the development is made. The power series obtained represent functions which are regular on the real positive axis of fugacity except at the points characteristics of the phase transitions in the system.