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Past exam of the mathematics course of the University of Cambridge

2025

Paper 2

For a simple compressible system with fixed particle number, the first law is
$$
dE=T\,dS-p\,dV.
$$
Define the <enthalpy> by $H=E+pV$. Then
$$
dH=dE+p\,dV+V\,dp
=T\,dS+V\,dp.
$$
Thus $H$ has natural variables $(S,p)$ and
$$
T=\left(\frac{\partial H}{\partial S}\right)_p,
\qquad
V=\left(\frac{\partial H}{\partial p}\right)_S.
$$
Equality of mixed partial derivatives gives the Maxwell relation
$$
\boxed{
\left(\frac{\partial T}{\partial p}\right)_S
=\left(\frac{\partial V}{\partial S}\right)_p.}
$$

Solution

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

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

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

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