Past exam of the chemistry A course of the University of Cambridge / 2023 / Paper 2 / A7 / d / Solution

At low temperature, $x=\epsilon /(k_{B}T)\gg 1$, so the denominator in the exact expression approaches one:The exponential suppression dominates the $T^{-2}$ factor. Almost all particles remain in the ground state, so very little volume change is thermally accessible.

At high temperature, $e^{-x}\to 1$, and thereforeThe excited fraction has already approached $3/4$, so further heating barely changes it. Both limits are consistent with a bounded two-level system; the high-temperature vanishing does not imply that ordinary materials cease expanding when additional levels or other mechanisms become accessible.