Einstein specific-heat theory — UPSC CSE 2012 Physics PYQ

The Einstein theory of specific heat of solids gives the expression C_V=\frac{3Nk_Bx^2e^x}{(e^x-1)^2} where x=\frac{\theta_E}{T} with \theta_E as the Einstein temperature.

(i) Mention Einstein's assumptions in deriving it. Also obtain low- and high-temperature limiting expressions for it.

(ii) Give schematic plot of \dfrac{C_V}{3N k_B} versus \dfrac{T}{\theta_E} and comment on the validity of expressions in (i) in comparison with experiments.