An approximation for the radial distribution function

Abstract

The present paper is concerned with a new approximation for the distribution function of third order. This approximation may be regarded as an improved form of the well-known superposition assumption of Kirkwood. The idea is to add to Kirkwood’s expression a linear combination of distribution functions of the same type, the coefficients of which can be easily determined. The new approximation is introduced in the Bom—Green integral equation for the radial distribution function, for which an expansion into powers of the density is used. From this the terms proportional to the first and second power of the density are calculated. The first four virial coefficients can be expressed as functions of these terms, whether the pressure equation or the compressibility equation is used. Numerical evaluation is performed for the ideal case of a gas of rigid spheres. The value obtained for the fourth virial coefficient is compared with the exact one and those given by naing Kirkwood’s assumption by Rushbrooke & Scoins, and Nijboer & van Hove. It is seen to be more nearly exact and internally consistent. The term proportional to the square of the density, in the expansion of the radial distribution function, appears to be very similar to the exact one as calculated by Nijboer & van Hove. It can be seen to be better than the corresponding term when Kirkwood’s assumption or one proposed by Nijboer & van Hove is used. Finally, an alternative assumption is suggested, and applied to the case of hard spheres.