= Solution
Since $\langle z\rangle=(\beta mg)^{-1}$,
$$
Z_N=\frac1{N!}\left(\frac{A\langle z\rangle}{\lambda^3}\right)^N.
$$
Using $\log N!\simeq N\log N-N$ and $S=k_B(\log Z_N+\beta\langle E\rangle)$ gives
$$
\boxed{
S=Nk_B\left[\log\left(\frac{A\langle z\rangle}{N\lambda^3}\right)+\frac72\right].
}
$$
This has the Sackur-Tetrode form with the effective volume $A\langle z\rangle$, 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 $k_BT$.
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