Second virial coefficient of a bounded cubic repulsion ($B_2=\frac{2\pi a^3}{3}[1-(1-e^{-x})/x]$)

For a classical dilute gas with pair potential $u(r)=ϵ[1-(r/a{)}^3]$ for $r<a$ and zero otherwise, put $x=ϵ/(k_{B}T)>0$. Substitution $t=(r/a{)}^3$ in the Mayer f-function formula givesIt is positive for finite positive temperature, tends to $2\pi a^3/3$ as $T\to 0$ in the classical model, and behaves as $\pi a^3ϵ/(3k_{B}T)$ at high temperature. The low-temperature mathematical limit need not remain in the dilute classical regime for a real molecular gas.