Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 35C / a

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Solution

The latent heat for converting liquid to gas at coexistence is

L = T (S_{gas} - S_{liq}) = H_{gas} - H_{liq},

with all quantities taken per mole or per particle consistently. Along the phase coexistence curve, the two Gibbs free energy values agree:

G_{gas} (T, p) = G_{liq} (T, p) .

Since

d G = - S d T + V d p,

differentiating the equality along the curve gives

- S_{gas} d T + V_{gas} d p = - S_{liq} d T + V_{liq} d p .

Therefore the Clausius-Clapeyron relation is

\frac{d p}{d T} = \frac{S _{gas} - S _{liq}}{V _{gas} - V _{liq}} = \frac{L}{T ( V _{gas} - V _{liq} )} .

At the critical point the liquid and gas become the same phase, so their entropy discontinuity disappears and

L ⟶ 0 .

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