Read interactions through virial coefficients

Write pressure divided by thermal energy as density plus a second coefficient times density squared and further terms. Equivalently, the compressibility factor is one plus a correction linear in density. The second coefficient has volume units and the third has volume squared units when density counts particles per cubic metre. Molar-density conventions use different units and numerical factors involving Avogadro's constant. State the density convention before substituting a tabulated coefficient.

The dilute classical regime needs both weak quantum exchange and controlled interaction corrections. Small density times thermal wavelength cubed suppresses exchange statistics. Small successive virial contributions control the interaction truncation, provided the relevant cluster integrals exist. Near a Boyle temperature the second coefficient can vanish by cancellation between attraction and repulsion while higher coefficients remain nonzero. A zero leading correction is therefore not proof that the interacting gas is exactly ideal at every density.

For particles interacting through a spherically symmetric pair potential u of separation, factor each Boltzmann pair weight into one plus a Mayer function. The function vanishes when the pair does not interact, making connected contributions identifiable. In the dilute thermodynamic limit the second virial coefficient is minus one half its integral over separation. The half removes double counting of unordered pairs; omitting it gives an excluded-volume coefficient twice too large.

Repulsive energy makes the Mayer function negative and contributes positively to the second coefficient. Attractive energy makes it positive and contributes negatively. The exponential must be retained for a strong repulsive core; expanding it in inverse temperature is not valid where potential energy diverges. An integrable short-range tail permits a finite pair integral. Unscreened long-range interactions may violate that condition and require neutralization, screening, or resummed methods rather than this simple ordinary virial formula.

For hard spheres of diameter d, separations below d are forbidden and all larger separations have zero pair potential. The Mayer function is minus one inside the excluded separation sphere and zero outside. Its integral gives a second coefficient equal to half that excluded sphere volume. This is four times the physical volume of one sphere, not the entire excluded sphere volume and not the particle's own volume. Keeping diameter distinct from radius prevents another factor-of-eight error.

At sufficiently low density the positive correction raises pressure at fixed density and temperature. It describes the first pair contribution, not an exact equation of state for close packing. Attractions can lower the pressure and may make the coefficient negative, but a negative coefficient alone does not establish phase separation. The quadratic lab isolates the second-order pressure term with a chosen coefficient. It cannot derive a pair potential, calculate higher clusters, or determine convergence near a critical point.