Dulong and Petit's law

According to Dulong and Petit's law the product of atomic weight and specific heat for all the elements in its solid state is always constant and its value is approximately 6.4.
This law is used to determine the atomic weight of substance.

Dulong and Petit’s law from Kinetic theory

According to the law of equi-partition of energy, the average kinetic energy of translation associated with each degree of freedom is 1/2kT.
If the oscillatory motion of atom is simple harmonic then in each oscillation the average kinetic energy = Average potential energy.

Total energy associated with each degree of freedom of atom

= 1/2 kT + 1/2 kT = kT
Since there are three degrees of freedom for oscillatory motion of each atom

So the total energy of each atom = 3kT

If we consider the 1 gm-atom of any solid at absolute temperature T

So the number of atoms in 1 gm-atom of gas = N, where N is Avogadro number

Total energy of 1 gm of solid

U = N * 3 kT = 3NkT [ R = Nk ]

U = 3RT

dU/dT = 3R

Since dU/dT is the atomic heat of solid at constant volume i.e., C_v

So, C_v = 3R

Since R = 1.98 cal/gm-atom/°C

So, C_v = 3 * 1.98 = 5.94 cal/gm-atom/°C

Atomic heat of solid = 6 cal cal/gm-atom/°C (approximate)

Thus it agrees with Dulong and Petit's law

C, B, Si like non-metallic elements have value of atomic heat different from 6.4 at normal temperature (about 6.0).
But above 500°C their value tends to about 6.4.
At absolute zero temperature the value of atomic heat of all the elements tends to zero.

It is clear from the following figure

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