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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 37C / c / i / Solution

Let

(T_{1}, V_{1})

be the final equilibrium state. The suddenly added weight supplies the constant external pressure

p_{1}

during the displacement. Since the process is adiabatic, the first law gives

\frac{5}{2} N k_{B} (T_{1} - T_{0}) = p_{1} (V_{0} - V_{1}) .

The initial and final ideal-gas equations are

p_{0} V_{0} = N k_{B} T_{0}, p_{1} V_{1} = N k_{B} T_{1} .

Substitution yields

\frac{5}{2} (T_{1} - T_{0}) = \frac{p _{1}}{p _{0}} T_{0} - T_{1},

and hence

T_{1} = T_{0} \frac{5 + 2 p _{1} / p _{0}}{7} .

Therefore

Δ T = T_{1} - T_{0} = \frac{2 T _{0}}{7} (\frac{p _{1}}{p _{0}} - 1) .

Solved by gpt-5.6-sol high.

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