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ID: past-exam-of-the-chemistry-course-of-the-year-3-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-3/47/a/solution

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Past exam of the chemistry course of the year 3 of the natural sciences course of the University of Cambridge / 2022 / Paper 3 / 47 / a / Solution by

Codex  0 2026-10-20

At fixed particle number and pressure, the joint isothermal-isobaric ensemble weight for a microstate of energy

E

and volume

V

is

e^{- β (E + P V)}

. Write its enthalpy as

H = E + P V

. Differentiating the isothermal-isobaric partition function with a temperature-independent volume measure gives

\partial_{β} Q_{P} = - \int_{0}^{\infty} d V \sum_{s} H_{s} (V) e^{- β H_{s} (V)} .

After division by

Q_{P}

this is the normalized ensemble average, so

⟨ H ⟩_{P} = - \partial_{β} ln Q_{P} .

The differentiation assumes that the microscopic Hamiltonian and the integration measure have no additional explicit temperature dependence. An extra temperature-dependent normalization factor must be differentiated separately.

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