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For independent particles with interconverting state families $i$, the equilibrium partition function is a sum $q={\sum }_i{q}_i$ and the family probabilities are $x_i=q_i/q$. Let $u_i$ be the conditional mean one-particle energy in family $i$ and $c_i=d{u}_i/dT$. The total canonical energy variance separates into within-family and between-family contributions, givingThe final positive term is absent if family fractions are held fixed. A frozen ortho-para mixture and an equilibrating mixture can therefore have different heat capacities at the same instantaneous composition.

For a fixed number of non-interacting molecules, $Z_N=q^N/N!$ in the dilute ideal gas limit. The temperature-independent factorial drops out of the internal energy derivative:Differentiate once more at fixed volume to obtainPrimes denote fixed-volume derivatives. If $q$ factorizes into translational, rotational and other independent contributions, their heat capacities add. In particular, the three-dimensional translational factor contributes $3R/2$, while a constant spin degeneracy factor contributes zero. The formula also follows from the positive canonical energy variance, $C_V=\mathrm{Var}(E)/(k_{\mathrm{B}}{T}^2)$.