Suppose that there are long-range interactions between atoms in a gas in the form of centr...

Suppose that there are long-range interactions between atoms in a gas in the form of central forces derivable from a potential

where r is the distance between any pair of atoms and m is a positive integer. Assume further that relative to any given atom the other atoms are distributed in space such that their volume density is given by the Boltzmann factor:

where N is the total number of atoms in a volume V. Find the addition to the virial of Clausius resulting from these forces between pairs of atoms, and compute the resulting correction to Boyle’s law. Take N so large that sums may be replaced by integrals. While closed results can be found for any positive m, if desired, the mathematics can be simplified by taking m = +1.