The <Clausius gas> equation can be written
$$
P=\frac{NT}{V-Nb}=\frac{nT}{1-nb}=nT\left(1+nb+O(n^2)\right).
$$
Comparison with the <virial expansion> gives
$$
\boxed{B(T)=b}.
$$
For a <hard-sphere gas> of atomic radius $r_0$, the <pair potential> is infinite when the center separation is below $2r_0$ and zero otherwise. Thus
$$
B=\frac12\int_{r<2r_0}d^3r
=\frac12\frac{4\pi}{3}(2r_0)^3
=\boxed{\frac{16\pi r_0^3}{3}}.
$$
This is four times the volume of one atom and is the per-particle <excluded volume> appearing in the dilute <equation of state>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 4 / 35A / iii / Solution

A hard-sphere gas has $u(r)=+\infty$ below the contact separation and $u(r)=0$ above it, so particles cannot overlap.


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 Incoming links: Hard-sphere gas