Excluded volume

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.