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became close to a constant value, after it had been multiplied by number-
ratio representing the presumed relative atomic weight of the substance.
These atomic weights had shortly before been suggested by Dalton.
In modern terms, Dulong and Petit found that the heat capacity of a

mole of many solid substances is about 3R, where R is the modern constant
called the universal gas constant.

C  3 R

Dulong and Petit were unaware of the relationship with R, since this
constant had not yet been defined from the later kinetic theory of gases.

The value of 3R is about 25 joules per kelvin, and Dulong and Petit
essentially found that this was the heat capacity of crystals, per mole of
atoms they contained.

The modern theory of the heat capacity of solids states that it is due to
lattice vibrations in the solid, and was first derived in crude form from this
assumption by Albert Einstein, in 1907. The Einstein solid model, thus,

gave for the first time a reason why the Dulong-Petit law should be stated
in terms of the classical heat capacities for gases.
Despite its simplicity, Dulong-Petit law offers fairly good

prediction for the specific heat capacity of many solids with
relatively simple crystal structure at high temperatures. This is

because in the classical theory of heat capacity the heat capacity of
solids approaches a maximum of 3R per mole of atoms, due to the

fact that 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
1
equipartition theorem, the average of each quadratic term is ⁄ kT,
2
1
or ⁄ RT per mole (see derivation below). Multiplied by 3 degrees
2
of freedom and the two terms per degree of freedom, this amounts

to 3R per mole heat capacity.

The Dulong-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.

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