Fluid Superposition Approximation and the Fourth Virial Coefficient

Abstract

A new superposition approximation is given for relating the triplet distribution for a classical system of identical particles in interaction (fluid) to the pair distribution. The new approximation is based upon the well-known expression due to Kirkwood, but explicitly accounts for the particle density through a power series expansion. It is proposed that the coefficients of this expansion should be chosen in order to ensure internal consistency in the calculation of the virial coefficients of the pressure (when expressed in powers of the density) from either the pressure equation or the relative compressibility equation, and the necessary formulas are given. The equations are checked numerically for the ideal case of a gas of rigid spheres, as far as the fourth virial coefficient. The modified form of the original Kirkwood superposition approximation, which is the new approximation, is found for this case to be g(3)(r, s, t) = g(2)(r)g(2)(s)g(2)(t) [1+0.1014bn] where g(3) is the triplet distribution, g(2) the pair distribution, n the particle number density, and b is equal to four times the volume of each particle. The theory then leads to a fourth virial coefficient of value 0.2885b3 which is to be compared with the exact value 0.2869b3. This value is found to be a marked improvement on the values calculated by previous authors. The higher powers of n in the new superposition approximation necessary for the correct calculation of higher virial coefficients than the fourth are not considered in this paper.