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The classical one-particle partition function, with phase-space cell $h^3$, isFor the homogeneous trap, rescale $r=V^{1/3}(kT{)}^{1/(2n)}u$. The spatial measure contributes $V(kT{)}^{3/(2n)}$, soHere $I_n=(2m\pi /h^2{)}^{3/2}{\int }_0^{\infty }4\pi u^2{e}^{-u^{2n}}du$, as specified; the last integral is $2\pi \Gamma (3/(2n))/n$.

Using the printed convention $Z_N=Z_1^N$, the free energy isThis convention treats the particles as labeled. For indistinguishable classical particles, $Z_N=Z_1^N/N!$ adds $kT\log N!$ to $F$. It changes the entropy by $-k\log N!$ but leaves the following force, energy and heat capacities unchanged. This normalization qualification is required to distinguish the printed expression from the usual Gibbs-corrected gas free energy.

The generalized force conjugate to the external parameter $V$ is $P=-(\partial F/\partial V{)}_T$, henceIt is the same equation of state as a hard-wall ideal gas, although $V$ now dilates a soft confining potential rather than being the occupied geometric volume.

Writing $L=\log V+a\log (kT)+\log I_n$, thermodynamic differentiation givesThe mean kinetic energy is $3NkT/2$ and the trap potential contributes $3NkT/(2n)$. At fixed $P$, $V=NkT/P$, so ${\partial }_{T}L{|}_P=(a+1)/T$. ConsequentlyThe same result follows from enthalpy $E+PV=(a+1)NkT$. These identities describe the classical ideal gas in a homogeneous soft trap.