 Solution

ID: past-exam-of-the-thermal-and-statistical-physics-course-of-the-university-of-cambridge/2015/paper-1/1/a/solution

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 gives

p = k_{B} T [\frac{N}{V - N b} - \frac{a N ^{2}}{k _{B} T V ^{2}}] .

(p + \frac{a N ^{2}}{V ^{2}}) (V - N b) = N k_{B} T, V > N b .

The 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.

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