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The second virial coefficient B₂(T) represents the leading correction to ideal behavior and reflects two-body interactions. It changes sign at the Boyle temperature where it vanishes, and its temperature dependence reveals the balance between repulsive and attractive forces in intermolecular interactions.
The virial expansion writes the pressure of a real gas as a power series in density: PV/NkT = 1 + B₂(T)/V + B₃(T)/V² + …, where each virial coefficient captures the effect of increasingly complex multi-particle encounters. The ideal gas result (1) corresponds to particles that never interact. The second virial coefficient B₂(T) is the first correction and accounts for two-body interactions — collisions between pairs of molecules. At low enough densities, three-body encounters (B₃) are so rare that they can be ignored, making B₂ the dominant correction in most practical situations.
B₂(T) has a statistical mechanical expression: B₂(T) = −½ ∫ [exp(−u(r)/kT) − 1] 4πr² dr, where u(r) is the pair potential — the interaction energy between two molecules separated by distance r. The integrand, known as the Mayer f-function f(r) = exp(−u(r)/kT) − 1, vanishes when molecules don't interact (u = 0) and is nonzero only where they do. At short distances, repulsive interactions (u >> kT) make f(r) ≈ −1, contributing a positive term to B₂. At intermediate distances, attractive interactions (u < 0) make f(r) > 0, contributing a negative term. The sign and magnitude of B₂ reflect which effect dominates.
At high temperatures, kT >> |u(r)|, so the attractive well has negligible effect. The hard-core repulsion dominates, making B₂ > 0 — the gas behaves as if molecules simply exclude each other's volume, so pressure is higher than ideal (PV > NkT). At low temperatures, the attractive well matters: molecules linger near each other, reducing the effective pressure below ideal, making B₂ < 0. The Boyle temperature T_B is where B₂(T_B) = 0 — repulsive and attractive corrections exactly cancel, and the gas obeys PV = NkT to first order regardless of density. This is not because the gas is ideal; it is a coincidental cancellation. Real gases like nitrogen have T_B ≈ 327 K and are studied near this temperature to isolate higher-order effects.
The practical value of B₂(T) goes beyond corrections to the ideal gas law. Its temperature dependence is a fingerprint of the intermolecular potential u(r): measuring B₂(T) at many temperatures can be used to infer the shape of u(r) without directly measuring molecular forces. The Lennard-Jones potential u(r) = 4ε[(σ/r)¹² − (σ/r)⁶] — with its characteristic hard-core repulsion and shallow attractive well — was refined historically by fitting its parameters ε and σ to experimental B₂(T) data. The second virial coefficient thus bridges macroscopic thermodynamic measurements and microscopic molecular physics.