Editing Specific heat capacity (section)

===Solid phase===
The theoretical maximum heat capacity for larger and larger multi-atomic gases at higher temperatures, also approaches the Dulong–Petit limit of 3''R'', so long as this is calculated per mole of atoms, not molecules. The reason is that gases with very large molecules, in theory have almost the same high-temperature heat capacity as solids, lacking only the (small) heat capacity contribution that comes from potential energy that cannot be stored between separate molecules in a gas.

The Dulong–Petit limit results from the [[equipartition theorem]], and as such is only valid in the classical limit of a [[microstate continuum]], which is a high temperature limit. For light and non-metallic elements, as well as most of the common molecular solids based on carbon compounds at [[standard ambient temperature and pressure|standard ambient temperature]], quantum effects may also play an important role, as they do in multi-atomic gases. These effects usually combine to give heat capacities lower than 3''R'' per mole of ''atoms'' in the solid, although in molecular solids, heat capacities calculated ''per mole of molecules'' in molecular solids may be more than 3''R''. For example, the heat capacity of water ice at the melting point is about 4.6''R'' per mole of molecules, but only 1.5''R'' per mole of atoms. The lower than 3''R'' number "per atom" (as is the case with diamond and beryllium) results from the “freezing out” of possible vibration modes for light atoms at suitably low temperatures, just as in many low-mass-atom gases at room temperatures. Because of high crystal binding energies, these effects are seen in solids more often than liquids: for example the heat capacity of liquid water is twice that of ice at near the same temperature, and is again close to the 3''R'' per mole of atoms of the Dulong–Petit theoretical maximum.

For a more modern and precise analysis of the heat capacities of solids, especially at low temperatures, it is useful to use the idea of [[phonons]]. See [[Debye model]].