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 / a

For a hard disc configuration, the potential energy is zero if no particles overlap and infinite otherwise. All allowed configurations therefore have the same configurational Boltzmann factor. At fixed $N,A,T$, their relative free energies differ through the volume of accessible configuration space:The kinetic contribution is common to the phases at the same temperature. “Entropy-driven crystallization” does not mean that particles develop an energetic attraction; excluded volume changes the number of possible many-particle arrangements.

At high density, disorder can frustrate efficient local packing and leave particles with very little room to move. An ordered arrangement can give every particle a larger cage of allowed displacements. The extensive gain in vibrational/configurational freedom within these cages can outweigh the loss of distinct spatial arrangements of cages. Visual order is therefore not a reliable measure of entropy. At low density the fluid's freedom to rearrange dominates, whereas sufficiently dense packing can favour a crystal through accessible free volume.