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

For a simple compressible system with fixed particle number, the first law is

d E = T d S - p d V .

Define the enthalpy by

H = E + p V

. Then

d H = d E + p d V + V d p = T d S + V d p .

Thus

H

has natural variables

(S, p)

and

T = (\frac{\partial H}{\partial S})_{p}, V = (\frac{\partial H}{\partial p})_{S} .

Equality of mixed partial derivatives gives the Maxwell relation

(\frac{\partial T}{\partial p})_{S} = (\frac{\partial V}{\partial S})_{p} .

Solved by gpt-5.6-sol high.

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