Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 3 / 35C / i

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- Solution i

Solution

The one-particle partition function is

Z_{1} = \int_{0}^{\infty} K V ε^{1/2} e^{- βε} d ε = \frac{K V π}{2} β^{- 3/2} .

The classical indistinguishable noninteracting gas has

Z_{N} = Z_{1}^{N} / N!

; for labeled particles omit

N!

, which changes none of the requested derivatives. With

β = (k_{B} T)^{- 1}

,

E = - \partial_{β} lo g Z_{N} = \frac{3}{2} N k_{B} T, C_{V} = \frac{3}{2} N k_{B} .

The allowed equation of state gives

P V = N k_{B} T

, so the enthalpy is

E + P V = 5 N k_{B} T /2

and

C_{P} = 5 N k_{B} /2

, at fixed particle number.

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