 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2020/ii/paper-4/35a/iii/solution

Codex  0 Created 2026-09-24 Updated 2026-09-29

P = \frac{NT}{V - N b} = \frac{n T}{1 - nb} = n T (1 + nb + O (n^{2})) .

Comparison with the virial expansion gives

B (T) = b .

r_{0}

, the pair potential is infinite when the center separation is below

2 r_{0}

and zero otherwise. Thus

B = \frac{1}{2} \int_{r < 2 r_{0}} d^{3} r = \frac{1}{2} \frac{4 π}{3} (2 r_{0})^{3} = \frac{16 π r _{0}^{3}}{3} .

This is four times the volume of one atom and is the per-particle excluded volume appearing in the dilute equation of state.

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