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

Table of contents
- i d
  - Solution i
- ii d
  - Solution ii
- iii d
  - Solution iii

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}} .

ii

Solution

Differentiating

lo g Z

with respect to

β μ

gives

⟨ N ⟩ = \frac{V e ^{β μ}}{λ _{T}^{3}}, (Δ N)^{2} = \frac{\partial ^{2} l o g Z}{\partial ( β μ ) ^{2}} = ⟨ N ⟩ .

Thus

\frac{Δ N}{⟨ N ⟩} = \frac{1}{⟨ N ⟩},

which vanishes in the thermodynamic limit.

iii

Solution

Combining

Φ = - p V

with part (i) and the expression for

⟨ N ⟩

gives the ideal-gas equation of state

p V = ⟨ N ⟩ k_{B} T .

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