Volumetric heat capacity
Volumetric heat capacity (VHC), also termed volume-specific heat capacity, describes the ability of a given volume of a substance to store internal energy while undergoing a given temperature change, but without undergoing a phase transition. It is different from specific heat capacity in that the VHC is a 'per unit volume' measure of the relationship between thermal energy and temperature of a material, while the specific heat is a 'per unit mass' measure (or occasionally per molar quantity of the material). If given a specific heat value of a substance, one can convert it to the VHC by multiplying the specific heat by the density of the substance.
Dulong and Petit predicted in 1818 that the product of solid substance density and specific heat capacity (ρcp) would be constant for all solids. This amounted to a prediction that volumetric heat capacity in solids would be constant. In 1819 they found that volumetric heat capacities were not quite constant, but that the most constant quantity was the heat capacity of solids adjusted by the presumed weight of the atoms of the substance, as defined by Dalton (the Dulong–Petit law). This quantity was proportional to the heat capacity per atomic weight (or per molar mass), which suggested that it is the heat capacity per atom (not per unit of volume) which is closest to being a constant in solids. Eventually (see the discussion in heat capacity) it became clear that heat capacities per particle for all substances in all states are the same, to within a factor of two, so long as temperatures are not in the cryogenic range. For very cold temperatures, heat capacities fall drastically and eventually approach zero as temperature approaches zero.
The heat capacity on a volumetric basis in solid materials at room temperatures and above varies more widely, from about 1.2 MJ/m³K (for example bismuth) to 3.5 MJ/m³K (for example iron), but this is mostly due to differences in the physical size of atoms. See a discussion in atom. Atoms vary greatly in density, with the heaviest often being more dense, and thus are closer to taking up the same average volume in solids than their mass alone would predict. If all atoms were the same size, molar and volumetric heat capacity would be proportional and differ by only a single constant reflecting ratios of the atomic-molar-volume of materials (their atomic density). An additional factor for all types of specific heat capacities (including molar specific heats) then further reflects degrees of freedom available to the atoms composing the substance, at various temperatures.
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