Past exam of the thermal and statistical physics course of the University of Cambridge / 2015 / Paper 1 / 1 / a / Solution

For a canonical ensemble, $F=-k_{B}T\ln Z$ and $p=-(\partial F/\partial V{)}_{T,N}=k_{B}T({\partial }_V\ln Z{)}_{T,N}$. The kinetic and factorial factors do not depend on $V$, so differentiating only the volume-dependent terms givesThus the equation of state isThe excluded-volume correction raises pressure relative to the same gas without that correction; the attractive mean-field term lowers it. No Stirling formula is required for this derivative.