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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/b/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 / b / Solution by

Codex  0 2026-10-20

Differentiate the normalized average, including the derivative of its denominator. The enthalpy fluctuation formula follows from

\partial_{β} ⟨ H ⟩_{P} = - ⟨ H^{2} ⟩_{P} + ⟨ H ⟩_{P}^{2} = - Var (H) .

(1)

Since

d β / d T = - 1/ (k_{B} T^{2})

C_{P} = \frac{Var ( H )}{k _{B} T ^{2}} .

(2)

Thus a larger equilibrium spread of enthalpy corresponds to a larger response to heating at fixed pressure. The variance is nonnegative, so this finite-system equilibrium description has

C_{P} \geq 0

. The relation connects a response coefficient to fluctuations of the appropriate exchanged quantity; at fixed volume the corresponding quantity is internal energy.

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