Hard-sphere gas

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.

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 $v_{\mathrm{ex}}$,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 $\eta =n{v}_{\mathrm{s}}$, where $n$ is number density. Then the virial expansion beginsFor example, at the toy value $\eta =0.01$, the leading correction predicts $Z\approx 1.04$; 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, 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.

The excluded volume is the volume of relative positions forbidden to another particle. For hard spheres of radius $r_0$, that forbidden region is a sphere of radius $2r_0$.

For one mole, a simple repulsive-volume model uses $p(V-b)=RT$ with $b>0$ and domain $V>b$. The Maxwell relation and first law of thermodynamics give $(\partial U/\partial V{)}_T=T(\partial p/\partial T{)}_V-p=0$, so $U=U(T)$. If its molar heat capacity at constant volume $C_V$ is temperature independent, it is constant, $U=C_{V}T+U_{\ast }$ and $C_p=C_V+R$. Reversible adiabatic processes obey $T(V-b{)}^{R/C_V}=\mathrm{constant}$ or $p(V-b{)}^{1+R/C_V}=\mathrm{constant}$. All positive-temperature isotherms are strictly decreasing and convex; this single-phase model contains no attractive liquid-gas coexistence mechanism. The parameter $b$ is an effective unavailable molar volume, not automatically the literal geometric volume of the molecules.