= Solution
For $N$ identical classical noninteracting particles, use $Z_N=Z_1^N/N!$. At fixed $N$,
$$
\ln Z_N=N\ln V+\frac{3N}{2}\ln T+\text{constant},
$$
so the canonical partition-function identities give
$$
p=k_BT\left(\frac{\partial\ln Z_N}{\partial V}\right)_T=\frac{Nk_BT}{V},\qquad
U=k_BT^2\left(\frac{\partial\ln Z_N}{\partial T}\right)_V=\frac32Nk_BT.
$$
Thus, for one <mole>,
$$
\boxed{pV=RT,\qquad C_{V,m}=\frac32R.}
$$
The supplied $T^{3/2}$ dependence includes only three translational degrees of freedom; additional thermally active internal <modes> would change the <heat capacity>.
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