Liquid

In a homogeneous isotropic one-component fluid, $g(r)$ is the conditional number density at separation $r$ from a tagged particle divided by the bulk number density $n$. Thus the displayed shell contains the mean number of neighbours. For an uncorrelated ideal gas $g(r)=1$ away from the tagged particle; repulsive cores suppress small separations, while liquid coordination shells can produce damped oscillations approaching one at large distance. This is a pair correlation, not a normalized one-particle radial probability density. The static structure factor measures its Fourier-space counterpart.

For a differentiable central pair potential in a classical homogeneous three-dimensional fluid, the mechanical virial theorem gives the displayed pressure. The radial distribution function accounts for the probability of each pair separation. In the dilute limit, $g(r)=e^{-v(r)/(k_{B}T)}+O(n)$, and integration by parts yields $B_2(T)=-2\pi {\int }_0^{\infty }[e^{-v(r)/(k_{B}T)}-1]r^{2}dr$. Thus the second virial coefficient controls $p/(n{k}_{B}T)=1+B_{2}n+\cdots$. A few virial terms are a low-density approximation, not generally a converged description of a dense liquid. A discontinuous hard core needs its contact form or a limiting argument.

For a classical monatomic homogeneous fluid with pairwise central pair potential $v(r)$, the mean interaction energy is one half the sum over neighbours of a tagged particle. Using the radial distribution function gives the displayed formula; the factor one half prevents counting each pair twice. The kinetic term follows from the equipartition theorem. Its use requires the relevant integral to exist and a thermodynamic state specified by number density and temperature. Quantum kinetic contributions, many-body forces and internal molecular modes require modifications.