 Solution

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/e/solution

Insert the stipulated near-close-packing relation for

R

A = 4 a_{d} (\frac{ϕ _{cp}}{ϕ} - 1)^{2} = 4 a_{d} (\frac{ϕ _{cp} a}{N a _{d}} - 1)^{2} .

y = ϕ_{cp} a / (N a_{d})

, we have

\partial_{a} y = y / (2 a)

and therefore

\partial_{a} ln A = y / [a (y - 1)]

. From the supplied crystalline canonical partition function,

Π_{xtl} = N k_{B} T \partial_{a} ln A = \frac{k _{B} T}{a _{d}} \frac{ϕ}{1 - ϕ / ϕ _{cp}} .

Its close-packing scale is set by the triangular lattice packing of discs, with

ϕ_{cp} = π / (23)

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