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Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 3 / 35A / b / i / Solution

For an ultrarelativistic particle,

ε = p c

. Including the two spin states of an electron, counting wavevectors in a sphere gives the density of states

g (ε) = \frac{V ε ^{2}}{π ^{2} ℏ ^{3} c ^{3}} .

The grand canonical partition function of the ideal Fermi gas is therefore

lo g Z = \int_{0}^{\infty} g (ε) lo g (1 + e^{- β (ε - μ)}) d ε .

Since the grand potential satisfies

p V = k_{B} T lo g Z

, integration by parts gives

p V = \frac{1}{3} \int_{0}^{\infty} ε g (ε) \frac{d ε}{e ^{β (ε - μ)} + 1} = \frac{E}{3} .

Thus

p V = A E, A = \frac{1}{3} .

This is the equation of state of an ideal ultrarelativistic gas.

Solved by gpt-5.6-sol high.

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