# Why is the Dulong–Petit law not applicable to the elements Be, B, C, or Si?

According to the Dulong–Petit law, the molar heat capacity of a solid element is approximately $3R$ (where $R$ is the gas constant $\pu{8.314 J K-1 mol-1}$.)

This is useful for calculating the atomic mass of an element when its specific heat capacity (i.e. heat capacity per unit mass) is given.

However, the Dulong–Petit law is said to not apply to the solid elements Be, B, C, and Si. Why is this so?

The deviation is quantum mechanical in nature. The Dulong–Petit Law is exact only if all vibrational modes are fully activated, in which case equipartition theory can be used. As every atom in a solid can be considered to be a three-dimensional harmonic oscillator, the contribution to the heat capacity is $3k_\mathrm{B}$ for one atom, or $3R$ for one mole.
$$k_\mathrm{B}T \gg \hbar\omega$$
For most elements – except the elements listed in the question – room temperature is high enough to come close to full excitation. For the elements in question, the vibrational frequency $\omega$ is large, particularly because of the strong bonds and relatively small masses ($\omega \propto \sqrt{k/m}$.)