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Virial coefficients B_n(T) encode information about n-body interactions in a gas. The second virial coefficient B₂ = -2π N_A ∫₀^∞ [e-u(r)/kT - 1]r²dr depends directly on the pair potential u(r) and can be computed from quantum or classical mechanics.
From your study of the virial expansion, you know that real gas equations of state can be written as a power series in density: P/kT = n + B₂(T)n² + B₃(T)n³ + ..., where n is the number density. The ideal gas law is the first term; each subsequent term adds a correction for interactions among 2, 3, 4, ... molecules simultaneously. The virial coefficients B₂, B₃, ... are functions of temperature alone, and they encode how molecular interactions modify the ideal gas behavior.
The second virial coefficient B₂ has a clean physical interpretation. The integrand [e−u(r)/kT − 1] is called the Mayer f-function. At large separations where u(r) → 0, the f-function vanishes — distant molecules don't interact and don't correct the ideal gas law. Near the hard core where u(r) → +∞, the Boltzmann factor e−u/kT → 0 and the f-function → −1: the two molecules cannot overlap, and this excluded volume reduces the effective space available to each molecule. In the attractive well region where u(r) < 0, the Boltzmann factor exceeds 1 and the f-function is positive: attraction pulls molecules together, increasing the effective density and, at low T, reducing the pressure below the ideal gas value.
Integrating the Mayer f-function over all separations gives B₂. Its sign tells you the dominant effect at that temperature. At high temperature, the attractive well is thermally irrelevant (kT ≫ |u_min|) and the hard-core exclusion dominates: B₂ > 0, and pressure exceeds ideal. At the Boyle temperature, attractive and repulsive contributions cancel exactly: B₂ = 0 and the gas behaves nearly ideally despite having interactions. Below the Boyle temperature, attractions win: B₂ < 0, and the gas is easier to compress than ideal. This temperature dependence connects directly to the van der Waals equation of state — the constants a and b in (P + an²/V²)(V − nb) = nRT can be expressed in terms of the pair potential through the virial coefficient framework.
Third and higher virial coefficients involve three-body clusters and require integrating over all triangular configurations of three molecules. They are computed from the pair-distribution function you have already studied — specifically, the triplet distribution function for B₃. These higher-order terms become important near phase transitions, where density fluctuations are large. The entire virial expansion can be derived systematically using cluster diagrams in statistical mechanics, giving a diagrammatic perturbation theory for gas-phase thermodynamics whose structure anticipates the Feynman diagrams used in quantum field theory.