[UPSC CSE 2019] Thermodynamics (Q52, 20M)

Einsteins molar specific heat capacity of a solid is given by C_V = 3R\left(\frac{\theta_E}{T}\right)^2\frac{e^{\theta_E/T}}{\left(e^{\theta_E/T}-1\right)^2}, where \theta_E=\frac{\hbar\omega}{k_B}. Obtain the expressions for the cases:

(i) when T \gg \theta_E

(ii) when T \ll \theta_E What is the discrepancy of Einstein model to explain the variation of specific heat capacities of solids with the temperature? The molar specific heat capacity of a solid at constant volume is 2.77\ \mathrm{J\,K^{-1}} at 36.8\ \mathrm{K}. Determine the Debye temperature of the solid.