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

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 $a_c/{\Lambda }^2$ in two dimensions. Neglecting correlations between different cages,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.

The particle centre can move approximately $\ell -\sigma$ from the cell centre towards each of its four nearest fixed neighbours. Approximating the actual curved boundary by a square givesThis is the area accessible to the centre, not the geometrical area enclosed by the fixed particle centres or an area computed using $\sigma /2$. Two equal particles must keep their centres at least $\sigma$ apart.