 Solution

ID: past-exam-of-the-chemistry-course-of-the-year-4-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-2/section-e/26/b/i/solution

Freeze neighbouring particles and approximate the mobile particle's accessible area by

a_{c}

. Its one-particle translational contribution to the canonical partition function is

a_{c} / Λ^{2}

in two dimensions. Neglecting correlations between different cages,

Z_{cell} ≃ (a_{c} / Λ^{2})^{N}, F = - N k_{B} T ln (a_{c} / Λ^{2}) .

There is no additional uncancelled

N!

: assigning particles to cages removes permutations, while restoring all

N!

assignments cancels the indistinguishability factor. The result is a free-volume approximation; collective cage motion is excluded by construction.

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