Excluded volume

The excluded volume is the region of relative positions unavailable to a second particle because it would overlap the first. For equal hard spheres of radius $r$, their centres must remain at least $2r$ apart. The forbidden region for the second centre therefore has volume $4\pi (2r{)}^3/3$, eight times the physical volume of one sphere.

The second second virial coefficient is half that pair-excluded volume for hard spheres, because an unordered pair must not be counted twice. Thus $B_2=16\pi r^3/3$, four single-particle volumes. This distinction explains why the excluded-volume parameter entering a dilute equation of state need not equal either the physical occupied volume or the entire forbidden region around one fixed centre. At high density, overlaps among exclusion regions and collective packing make independent-pair counting insufficient.

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.