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

With wall effects neglected, $a_1=a$. A second hard disc cannot have its centre within distance $2r$ of the first centre. Its excluded volume in this two-dimensional setting is an excluded area $\pi (2r{)}^2=4a_d$, givingThis is exclusion of the second centre, rather than simply subtraction of the first disc's own area.

At sufficiently low packing fraction, neglect overlaps of the excluded regions around different disc centres. Sequential insertion gives the configurational integralTaking logarithms and expanding to first order in density,For large $N$, replace $N-1$ by $N$. The expression $N\ln (a-Nb)$ has the same leading-density expansion when $b=2a_d$. Thus the hard-disc excluded-area approximation yieldsThe factor $N!$ accounts for indistinguishable particles and ${\Lambda }^{2N}$ for two-dimensional momentum integration. The factor one-half between pair-excluded area and $b$ avoids counting every pair twice. Equality of leading expansions does not make this free-volume resummation exact at high density.

The surface pressure follows by differentiating the logarithm of the canonical partition function with respect to container area:Its formal divergence is therefore $\varphi =1/2$. This lies outside the low-density regime used to derive the expression. It is a model pole, not an actual geometrical close-packing fraction.

In the circular hard-disc cage approximation, moving the central disc towards a neighbour brings contact after a centre displacement $R-2r$. ThusAt close packing, $R=2r$ and $\mathcal{A}=0$. 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.

Insert the stipulated near-close-packing relation for $R$ into the hard-disc cage approximation:Writing $y=\sqrt{{\varphi }_{\mathrm{cp}}a/(N{a}_d)}$, we have ${\partial }_{a}y=y/(2a)$ and therefore ${\partial }_a\ln \mathcal{A}=y/[a(y-1)]$. From the supplied crystalline canonical partition function,Its close-packing scale is set by the triangular lattice packing of discs, with ${\varphi }_{\mathrm{cp}}=\pi /(2\sqrt{3})$ in the ideal geometry.

The crystalline expression diverges at $\varphi ={\varphi }_{\mathrm{cp}}\simeq 0.907$, where the cage free area vanishes. This is physically different from the spurious $\varphi =0.5$ pole produced by extrapolating the low-density fluid approximation. A real compression path cannot be inferred by following that fluid pole and then abruptly switching curves. At low density it follows a fluid equation of state, while at sufficiently high density an ordered arrangement can leave more accessible rattling configurations than a disordered crowded arrangement. This is entropy-driven hard-sphere crystallization, here in two dimensions: allowed hard-core configurations have the same interaction energy, so configurational entropy drives the phase competition. An equilibrium first-order coexistence interval would have common pressure and chemical potential, giving a constant-pressure segment as mean density changes. The two asymptotic expressions alone neither locate that interval nor determine the detailed two-dimensional melting sequence. They require a consistent free-energy comparison and trustworthy approximations in the transition region.