Past exam of the chemistry A course of the University of Cambridge / 2025 / Paper 2 / A6 / a

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Solution

The internal energy is the population-weighted sum of the molecular energy levels:

U = \sum_{i} n_{i} ε_{i} = \frac{N}{q} \sum_{i} ε_{i} e^{- ε_{i} / k T} .

At fixed volume, assume the energy levels themselves are temperature-independent. Differentiating the molecular partition function gives

(\frac{\partial q}{\partial T})_{V} = \frac{1}{k T ^{2}} \sum_{i} ε_{i} e^{- ε_{i} / k T}, U = N k T^{2} \frac{\partial lo g q}{\partial T} .

Choosing the zero of energy changes

U

by the corresponding constant per molecule, so the ground-level convention must be consistent throughout.

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