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Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 36A / d / i / Solution

The one-particle translational partition function is

q_{1} = \frac{V}{h ^{3}} \int_{R^{3}} e^{- β p^{2} / (2 m)} d^{3} p = \frac{V}{λ _{T}^{3}}, λ_{T} = \frac{h}{2 πm k _{B} T} .

For a classical ideal gas,

Z = N = 0 \sum \infty \frac{( e ^{β μ} q _{1} ) ^{N}}{N !} = exp (\frac{V e ^{β μ}}{λ _{T}^{3}}),

and hence

Φ = - \frac{k _{B} T V e ^{β μ}}{λ _{T}^{3}} .

Solved by gpt-5.6-sol high.

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