 Solution

ID: past-exam-of-the-chemistry-a-course-of-the-university-of-cambridge/2021/paper-2/a6/b/solution

Write

β = (k_{B} T)^{- 1}

and

x = β δ

. The one-site partition function includes both degenerate lower states,

q = e^{βε} (2 + e^{- x}) .

The occupied adsorption sites are distinguishable positions, so the canonical partition function is a product, without an additional $N!$ divisor:

Z_{N} = q^{N} = e^{βNε} (2 + e^{- β δ})^{N} .

Differentiating

ln Z_{N}

U = - (\frac{\partial ln Z _{N}}{\partial β})_{N} = - Nε + \frac{N δ e ^{- β δ}}{2 + e ^{- β δ}} = - Nε + \frac{N δ}{1 + 2 e ^{β δ}} .

The final factor is the population of the perpendicular orientation. The shared binding offset

- ε

shifts all energy levels equally and does not determine their relative populations.

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