Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 4 / 34D / i / 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 $\beta =1/(k_{B}T)$ and horizontal area $A=L^2$, the partition function isThis follows by independent Gaussian momentum integrals and the elementary height integral. Taking $E=-{\partial }_{\beta }\log Z_N$ givesFor an arbitrarily tall box compared with the gravitational scale height, $L\gg k_{B}T/(mg)$, the upper-wall correction is negligible andThe energy includes gravitational potential relative to the floor; the kinetic part alone is $3N{k}_{B}T/2$. Retaining the finite-height correction also gives the ordinary ideal-gas limit as $g\to 0$.