7.8 For a real gas at low density obtain the internal energy of the gas up to terms proportional to its density, in terms of the intermolecular potential u(r).

The radial distribution function g(r, p, T) is defined as follows.

Consider a gas at temperature Tin a volume F and let p = N/V be its mean particle density. Then is the number of molecules which lie within a spherical shell of radius r and thickness dr, given that there is a molecule at the origin r = 0. For a uniform distribution we would have g= 1.
Thus g allows for deviations from uniform density due to the intermolecular forces. (Experimentally one obtains g from x-ray diffraction experiments.) Show that for a real gas the internal energy is related to the radial distribution function by the equation Obtain a relation between the radial distribution function in the limit of zero density and the intermolecular potential u(r).