A <hard-sphere gas> is a classical model of identical impenetrable <spheres>, each of diameter $\sigma$ and radius $\sigma/2$. Its <pair potential> is $u(r)=+\infty$ for centre separation $r<\sigma$ and $u(r)=0$ for $r\ge\sigma$; the convention at exact contact has no effect on the <integral>. It isolates the effect of particle size without an attractive <potential energy>.

\Image[/second-virial-core-and-attraction.png]
{title=Hard-sphere centre exclusion, the square-well Mayer f-function, and the exact temperature dependence of the toy second virial coefficient}

The upper-left panel is a cross-section of a three-dimensional forbidden region: the second centre cannot enter a <sphere> of radius $\sigma$ around the first centre. The particles themselves have radius $\sigma/2$. Consequently, with particle <volume> $v_{\mathrm s}$ and pair <excluded volume> $v_{\mathrm{ex}}$,
$$
v_{\mathrm s}=\frac{4\pi}{3}\left(\frac\sigma2\right)^3
=\frac{\pi\sigma^3}{6},\qquad
v_{\mathrm{ex}}=\frac{4\pi\sigma^3}{3}=8v_{\mathrm s},
\qquad B_2=\frac12v_{\mathrm{ex}}=4v_{\mathrm s}=\frac{2\pi\sigma^3}{3}.
$$
The last equality follows directly from the <Mayer f-function>: $f=-1$ throughout the forbidden <volume> and zero outside it. Confusing particle radius with contact separation changes the answer by a factor of eight; omitting the unordered-pair factor changes it by a further factor of two.

Define the <packing fraction> $\eta=nv_{\mathrm s}$, where $n$ is <number density>. Then the <virial expansion> begins
$$
\frac{p}{nk_BT}=1+4\eta+O(\eta^2).
$$
For example, at the toy value $\eta=0.01$, the leading correction predicts $Z\approx1.04$; the omitted terms still matter if accuracy beyond this approximation is needed. The classical <second virial coefficient> is independent of <temperature> because the geometry of forbidden configurations does not change when the <gas> is warmed. The <pressure> itself still scales with <temperature> at fixed <number density>.

This model makes the positive correction particularly clear: eliminating overlapping configurations changes the dependence of the available configuration space on <volume (thermodynamics)>, even though the allowed configurations all have zero interaction <energy>. The figure's other panels add a toy attractive shell and show how its <Boltzmann factor> competes with that geometry. The <hard-sphere gas> is a useful reference model, while a low-order <virial expansion> remains a dilute-gas approximation rather than a dense <sphere packing> formula.


 Hard-sphere gas (source code)