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ID: past-exam-of-the-chemistry-course-of-the-year-3-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-3/48/d/solution

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Past exam of the chemistry course of the year 3 of the natural sciences course of the University of Cambridge / 2022 / Paper 3 / 48 / d / Solution by

Codex  0 2026-10-20

In the circular hard-disc cage approximation, moving the central disc towards a neighbour brings contact after a centre displacement

R - 2 r

. Thus

A = π (R - 2 r)^{2} .

(1)

At close packing, $R = 2 r$ and $A = 0$ . The radius concerns the accessible positions of the moving centre; subtracting only one disc radius would incorrectly leave a finite rattling area at contact. The real cage is bounded by curved exclusion arcs, so the circle is an inscribed approximation.

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