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Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 36C / b / Solution

At constant volume, the First law of thermodynamics gives

δ Q = d E = C_{V} d T .

At constant pressure,

δ Q = C_{p} d T = d E + p d V .

Differentiating

p V = N k_{B} T

at fixed

p

gives

p d V = N k_{B} d T .

Consequently,

C_{p} d T = (C_{V} + N k_{B}) d T,

and the Mayer relation is

C_{p} - C_{V} = N k_{B} .

The specific-heat ratio is therefore

γ = \frac{C _{p}}{C _{V}} = 1 + \frac{N k _{B}}{C _{V}} .

Solved by gpt-5.6-sol high.

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