Existing textbook expressions for the energy and heat capacity of gases and solids are widely taught in physics courses. However, no such expression exists for a liquid. The reason for this was summarized by Landau as "liquids have no small parameter", and discussed in some detail in Landau & Lifshitz Statistical Physics textbook.
Based on the old idea of J Frenkel, I formulate the problem in the language of phonons, and calculate liquid energy and heat capacity for both classical and quantum cases. The resulting equation relates liquid heat capacity to its relaxation time (or viscosity) with no fitting parameters, and is compared with the experimental data of several liquids, including metallic, noble, molecular and network liquids.
I subsequently discuss the behavior of liquids at higher temperature and our recent idea that contrary to current belief, the supercritical state is not homogeneous in terms of physical properties, and that there is a line (the "Frenkel line") above the critical point separating two physicall distinct states. Crossing the Frenkel line results in qualitative changes of most important properties of the system, including diffusion, viscosity, speed of sound, thermal conductivity and dispersion curves. Unexpectedly, these properties include heat capacity which acquires a new temperature dependence above the Frenkel line, and I discuss recent ideas of how to describe this change theoretically.