 Solution

With microstate energies held fixed when differentiating,Since $d\beta /dT=-1/(k_B{T}^2)$,The canonical ensemble fixes volume and particle number, and any other external Hamiltonian parameters. Thus the derivative is $C_V$; pressure-fixed fluctuations would naturally involve enthalpy in a different ensemble.

The translational and two-level energy variance follows because translation and internal states factorize for this dilute classical gas:Differentiating twice yieldsThe terms are independent translational and two-level binomial variances. At $k_{B}T\ll ϵ$, excitation freezes out exponentially and translation dominates. At $k_{B}T\gg ϵ$, the internal variance tends to $N{ϵ}^2/4$ while the translational variance grows as $T^2$; the internal heat capacity consequently tends to zero. Fractional total-energy fluctuations are of order $N^{-1/2}$ for large $N$. The low-temperature statement presumes translation is still nondegenerate; at strictly zero temperature the classical-gas model fails.