 Solution

ID: past-exam-of-the-thermal-and-statistical-physics-course-of-the-university-of-cambridge/2019/section-a/1/solution

Use the Gibbs-Duhem equation for a homogeneous, extensive, single-component thermodynamic phase:

S d T - V d p + N d μ = 0.

At fixed temperature, this becomes

d p = n d μ

n = N / V

. Separately, at fixed particle number,

(\frac{\partial n}{\partial p})_{T, N} = - \frac{N}{V ^{2}} (\frac{\partial V}{\partial p})_{T, N} = n κ_{T} .

(\frac{\partial n}{\partial μ})_{T} = (\frac{\partial n}{\partial p})_{T} (\frac{\partial p}{\partial μ})_{T} = n κ_{T} n .

Consequently

κ_{T} = \frac{1}{n ^{2}} (\frac{\partial n}{\partial μ})_{T} .

P a^{- 1}

, as required. The derivation assumes a smooth bulk equation of state; changing the fractions of coexisting thermodynamic phases at fixed pressure needs separate treatment.

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