Question

Problem 3.8. Starting with the result of Problem 3.5, calculate the heat capac- ity of an Einstein solid in the low-temperature limit. Sketch the predicted heat capacity as a function of temperature. (Note: Measurements of heat capacities of actual solids at low temperatures do not confirm the prediction that you will make in this problem. A more accurate model of solids at low temperatures is presented in Section 7.5.)

Answer 1

Given data The Expression for energy of Einstein solid as a function of temperature is U = N E e^-E/KT Here U is the total energy of the solid, N is the Number of Oscillators of the solid. E is the unit of the energy. K is the Boltzmann constant T is the absolute temperature The heat capacity at constant volume is Given by the following formula e_v = (partial differential u/partial differential T)_XY Here, v is the volume of the solid. \r\n Hence, to get equation of heat capacity, difference the equation U = N E e^-E/KT with Respect to T. Partialdifferential U/partial differential T = N E e^-E/KT partial differential/partial differential T (- E/KT) = N E e^-E/KT (E/KT^2) = NE^2/KT^2 e^-E/KT Substitute NE^2/KT^2 e^-E/KT for (partial differential U/partial differential T)_NY in C_v = (partial differential U/partial differential T)_Ny C_v = N_k(E/KT)^2 e^-E/KT Expand the Exponential function. C_v = NK (E/KT)^2[1 - E/KT + (E/KT)^2/2! - (E/KT)^2/3! + ellipsis] = NK[(E/KT)^2 - (E/KT)^3 + 1/2! (E/KT)^4 - 1/3! (E/KT)^5 + ellipsis]