Past exam of the thermal and statistical physics course of the University of Cambridge / 2019 / Section A / 3 / Solution

Use the classical dilute-gas conventionso the second virial coefficient has units of volume. The Mayer f-function formula for an isotropic pair potential isOnly separations smaller than $a$ contribute. Define $x=\beta \epsilon >0$ and substitute $t=r^3/a^3$, so $r^{2}dr=(a^3/3)dt$:This is the second virial coefficient of a bounded cubic repulsion. For every finite positive $T$, $1-e^{-x}<x$, so $B_2(T)>0$. There is no finite positive Boyle temperature. At high temperature, expanding the exponential givesThus the intended limiting statement isAt the other extreme the classical expression tends to $2\pi a^3/3$, the hard-exclusion value for separation $a$. This is a check on the integral, not a claim that a real molecular gas stays classical at arbitrarily low temperature. A molar virial expansion would instead use $B_{2,\mathrm{m}}=N_A{B}_2$.