Past exam of the chemistry A course of the University of Cambridge / 2021 / Paper 2 / A8 / b / i

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Solution

For a fixed number of non-interacting molecules,

Z_{N} = q^{N} / N!

in the dilute ideal gas limit. The temperature-independent factorial drops out of the internal energy derivative:

U_{m} = R T^{2} (\frac{\partial l n q}{\partial T})_{V} .

(1)

Differentiate once more at fixed volume to obtain

\frac{C _{V, m}}{R} = 2 T \frac{q ^{'}}{q} + T^{2} [\frac{q ^{''}}{q} - (\frac{q ^{'}}{q})^{2}] = \frac{d}{d T} (T^{2} \frac{d ln q}{d T}) .

(2)

Primes denote fixed-volume derivatives. If

q

factorizes into translational, rotational and other independent contributions, their heat capacities add. In particular, the three-dimensional translational factor contributes

3 R /2

, while a constant spin degeneracy factor contributes zero. The formula also follows from the positive canonical energy variance,

C_{V} = Var (E) / (k_{B} T^{2})

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Past exam of the chemistry A course of the University of Cambridge / 2021 / Paper 2 / A8 / b / i

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