Equilibrium heat capacity of interconverting states ($C_V=N{\sum }_i{x}_i{c}_i+N{\mathrm{Var}}_i(u_i)/(k_{\mathrm{B}}{T}^2)$)

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.