 Solution

ID: past-exam-of-the-chemistry-a-course-of-the-university-of-cambridge/2023/paper-2/a7/b/solution

Use the additive-volume two-level system implicit in the model: each excited particle increases the volume by the same increment. The difference between the two endpoint volumes is

V_{0}

, so that increment is

v = V_{0} / N

N_{1}

particles are excited,

V = V_{0} + (V_{0} / N) N_{1}

. Linearity of the expectation gives

⟨ N_{1} ⟩ = N P_{1}

, hence

V (T) = V_{0} (1 + \frac{3 e ^{- x}}{1 + 3 e ^{- x}}) = V_{0} \frac{1 + 6 e ^{- x}}{1 + 3 e ^{- x}} .

T \to 0

V \to V_{0}

T \to \infty

, only three quarters of the particles are excited on average, giving

V \to 7 V_{0} /4

, rather than the all-excited endpoint

2 V_{0}

. These limits also check the inclusion of state degeneracy.

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