Derive the equation for the molar specific heat capacity of solids.

(N/A) In solids, atoms oscillate about their mean position.
Let the number of atoms in $1$ mole of a solid be $N_{A}$.
According to the law of equipartition of energy, the energy associated with the oscillation of atoms in one dimension is $2 \times \frac{1}{2} k_{B} T = k_{B} T$.
Therefore, in three dimensions, the average energy of an atom is $3 k_{B} T$.
The total energy of $1$ mole of solid is $U = 3 k_{B} T \times N_{A} = 3 RT$ (since $k_{B} N_{A} = R$).
From the first law of thermodynamics, $\Delta Q = \Delta U + \Delta W$.
Since $\Delta W = P \Delta V$ and for solids $\Delta V \approx 0$, we have $\Delta Q = \Delta U$.
The molar specific heat of a solid is $C = \frac{\Delta Q}{\Delta T} = \frac{d(3 RT)}{dT} = 3R$.
Substituting $R = 8.31 \ J \ mol^{-1} \ K^{-1}$, we get $C = 3 \times 8.31 = 24.93 \ J \ mol^{-1} \ K^{-1}$.

Substance	Specific Heat $(J \ kg^{-1} \ K^{-1})$	Molar Specific Heat $(J \ mol^{-1} \ K^{-1})$
Aluminium	$900.0$	$24.4$
Carbon	$506.5$	$6.1$
Copper	$386.4$	$24.5$
Lead	$127.7$	$26.5$
Silver	$236.1$	$25.5$
Tungsten	$134.4$	$24.9$