Ideal Bose-gas heat capacity below condensation ($C_V=\frac{15}{4}N{k}_B\frac{\zeta (5/2)}{\zeta (3/2)}(T/T_c{)}^{3/2}$)

For quadratic three-dimensional dispersion below condensation, the excited-state thermal energy is proportional to $T^{5/2}$. Differentiating at fixed particle number and volume gives the displayed heat capacity, with limiting value approximately $1.926N{k}_B$ at $T_c^{-}$. The macroscopic ground occupation carries no excitation energy in the shifted reference. The exponent differs from the linear-phonon result of an interacting gas.