= Second virial coefficient of a bounded cubic repulsion
{title2=$B_2=\frac{2\pi a^3}{3}[1-(1-e^{-x})/x]$}
For a classical dilute <gas> with <pair potential> $u(r)=\epsilon[1-(r/a)^3]$ for $r<a$ and zero otherwise, put $x=\epsilon/(k_BT)>0$. Substitution $t=(r/a)^3$ in the <Mayer f-function> formula gives
$$
B_2=\frac{2\pi a^3}{3}\int_0^1[1-e^{-x(1-t)}]dt
=\frac{2\pi a^3}{3}\left[1-\frac{1-e^{-x}}x\right].
$$
It is positive for finite positive <temperature>, tends to $2\pi a^3/3$ as $T\to0$ in the classical model, and behaves as $\pi a^3\epsilon/(3k_BT)$ at high <temperature>. The low-temperature mathematical limit need not remain in the dilute classical regime for a real molecular <gas>.
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