Obtain the molar specific heat of a solid by using the law of equipartition of energy.

(C) The law of equipartition of energy can be used to predict the molar specific heat capacities of solids. Consider a solid consisting of $N$ atoms,each vibrating about its mean position.
An oscillator in one dimension has an average energy $= 2 \times \frac{1}{2} k_{B} T = k_{B} T$,where $k_{B}$ is the Boltzmann constant.
In three dimensions,the average energy $= 3 k_{B} T$.
Therefore,for one mole of a solid,the total internal energy $U$ is the average energy multiplied by the number of atoms in one mole $(N_{A})$:
$U = 3 k_{B} T \times N_{A}$
Since $k_{B} N_{A} = R$ (the universal gas constant),we have:
$U = 3 RT$
To find the molar specific heat capacity $C$,we differentiate $U$ with respect to temperature $T$:
$C = \frac{dU}{dT} = \frac{d}{dT}(3 RT) = 3R$
According to the first law of thermodynamics,$\Delta Q = \Delta U + \Delta W$. For a solid,the change in volume $\Delta V$ is negligible,so the work done $\Delta W = P \Delta V \approx 0$. Thus,$\Delta Q = \Delta U$. Therefore,the molar specific heat capacity of a solid is $C = 3R$.