Classical ideal gas in a homogeneous soft trap ($Z_1=V{I}_n(kT{)}^{3/2+3/(2n)}$)

For the potential $U=r^{2n}/V^{2n/3}$, rescaling $r=V^{1/3}(kT{)}^{1/(2n)}u$ in the classical one-particle partition function gives the formula above. The parameter-conjugate force is $P=NkT/V$, the internal energy is $(3/2+3/(2n))NkT$, and $C_P-C_V=Nk$. The kinetic term contributes $3NkT/2$, while the potential term contributes $3NkT/(2n)$. For indistinguishable particles the $N!$ factor changes free energy and entropy but not these force and heat-capacity formulas.