Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 36C / b / iii / Solution

Since $⟨z⟩=(\beta mg{)}^{-1}$,Using $\log N!\simeq N\log N-N$ and $S=k_B(\log Z_N+\beta ⟨E⟩)$ givesThis has the Sackur-Tetrode form with the effective volume $A⟨z⟩$, but its additive constant is $7/2$ rather than $5/2$. The extra unit arises because the atmosphere's vertical extent grows with temperature and its mean gravitational energy per atom is $k_{B}T$.