Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 4 / 35A / iii

Table of contents
- Solution iii

Solution

The Clausius gas equation can be written

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 .

For a hard-sphere gas of atomic radius

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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