Why does diamond's specific heat capacity deviate from the Dulong-Petit theory at room temperatures? Why does diamond deviate from the prediction of $c_v = 3k_B$ at room temperature more than other solids. Why do we require Debye's theory instead?
 A: Recall that the Debye model is based on a phonon dispersion $\omega = c_s k$ with $c_s$ the speed of sound. The Debye frequency $\omega_D$ can only depend upon the speed of sound and the lattice spacing $a$ so on dimensional grounds $\omega_D \sim c_s / a$. 
In the high temperature limit the Debye model must reproduce the Dulong-Petit law, but how high is high? Again, there are not many parameters in the problem and the only possible energy we can cook up is $\hbar \omega_D$. This gives a Debye temperature $k_BT_D = \hbar \omega_D$ above which the Dulong-Petit rule $3R$ holds fairly accurately and below which it is a poor approximation. But then $T_D \sim \omega_D \sim c_s$ so the Debye temperature is proportional to the speed of sound, which in turn measures how stiff a material is. Diamond is known to be exceptionally stiff (speed of sound is $\sim 12,000\text{m/s}$) and so has a high Debye temperature (around $2000K$, much higher than room temperature) and as such the Dulong-Petit rule is almost always invalid. 
