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A virial coefficient quantifies an interaction correction to the dilute-gas equation of state. In the number-density convention, $p/(n{k}_{B}T)=1+B_2(T)n+B_3(T)n^2+\cdots$, where $n=N/V$. Thus $B_2$ has units of volume; a molar-density expansion uses differently scaled coefficients.

For classical particles with a pair potential $u(r)$, $B_2=-\frac{1}{2}\int [e^{-u(r)/(k_{B}T)}-1]d^{3}r$. Hard spheres of diameter $d$ exclude separations below $d$, giving $B_2=2\pi d^3/3>0$. Attractive interactions can instead contribute negatively. The expansion is a low-density organization of correlations, not a guarantee of convergence near a phase transition.

For a dilute gas with the truncated virial expansionlet $N$ be fixed particle number and let $C_p$ be the whole sample's heat capacity at constant pressure. The Joule-Thomson coefficient isThis follows by differentiating the equation of state at fixed pressure and substituting into ${\mu }_{\mathrm{JT}}=[T(\partial V/\partial T{)}_p-V]/C_p$. On a mechanically stable dilute branch with $C_p>0$ and $1+2B_{2}n>0$, a pressure drop cools the gas exactly when $T{B}_2^{\prime }-B_2>0$, equivalently $d(B_2/T)/dT>0$. The expression is exact for the displayed truncated model; neglected higher virial coefficients limit its accuracy at greater number density.