Despite its simplicity, Dulong–Petit law offers fairly good prediction for the heat capacity of many elementary solids with relatively simple crystal structure at high temperatures. This agreement is because in the classical statistical theory of Ludwig Boltzmann, the heat capacity of solids approaches a maximum of 3R per mole of atoms because full vibrational-mode degrees of freedom amount to 3 degrees of freedom per atom, each corresponding to a quadratic kinetic energy term and a quadratic potential energy term. By the equipartition theorem, the average of each quadratic term is kT, or RT per mole (see derivation below). Multiplied by 3 degrees of freedom and the two terms per degree of freedom, this amounts to 3R per mole heat capacity.
The Dulong&ndash;Petit law fails at room temperatures for light atoms bonded strongly to each other, such as in metallic beryllium and in carbon as diamond. Here, it predicts higher heat capacities than are actually found, with the difference due to higher-energy vibrational modes not being populated at room temperatures in these substances.
In the very low (cryogenic) temperature region, where the quantum mechanical nature of energy storage in all solids manifests itself with larger and larger effect, the law fails for all substances. For crystals under such conditions, the Debye model, an extension of the Einstein theory that accounts for statistical distributions in atomic vibration when there are lower amounts of energy to distribute, works well.