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ID: past-exam-of-the-physics-b-course-of-the-university-of-cambridge/2017/paper-2/section-c/c11/phase-transition-test/solution

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Past exam of the physics B course of the University of Cambridge / 2017 / Paper 2 / Section C / C11 / Phase-transition test / Solution by

Codex  0 2026-10-18

At every

T > 0

V > b

(\frac{\partial p}{\partial V})_{T} = - \frac{RT}{( V - b ) ^{2}} < 0, (\frac{\partial ^{2} p}{\partial V ^{2}})_{T} = \frac{2 RT}{( V - b ) ^{3}} > 0.

(1)

There is no constant-pressure coexistence interval, unstable loop or finite-temperature critical inflection. The isothermal compressibility is positive:

κ_{T} = - \frac{1}{V} (\frac{\partial V}{\partial p})_{T} = \frac{V - b}{p V} > 0.

(2)

Thus this equation of state predicts no liquid-gas phase transition in its physical domain. Its divergence at the excluded-volume boundary is a repulsive constraint, not a liquid-gas coexistence mechanism. This conclusion concerns the stated model and cannot rule out an ordering transition requiring additional physics or an additional thermodynamic variable. Extrapolation to absolute zero is also outside the classical gas model's reliable regime.

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