= Solution
Take the floor at $z=0$, gas-particle mass $m$, $N$ atoms and box side $L$. The one-particle <Hamiltonian> is $p^2/(2m)+mgz$. With $\beta=1/(k_BT)$ and horizontal area $A=L^2$, the <partition function> is
$$
Z_N=\frac1{N!h^{3N}}\left[A\left(\frac{2\pi m}{\beta}\right)^{3/2}\frac{1-e^{-\beta mgL}}{\beta mg}\right]^N.
$$
This follows by independent Gaussian momentum integrals and the elementary height integral. Taking $E=-\partial_\beta\log Z_N$ gives
$$
\boxed{E=N\left(\frac5{2\beta}-\frac{mgL}{e^{\beta mgL}-1}\right).}
$$
For an arbitrarily tall box compared with the gravitational scale height, $L\gg k_BT/(mg)$, the upper-wall correction is negligible and
$$
\boxed{E=\frac52Nk_BT.}
$$
The energy includes gravitational potential relative to the floor; the kinetic part alone is $3Nk_BT/2$. Retaining the finite-height correction also gives the ordinary ideal-gas limit as $g\to0$.
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