Question

Work out the details of the derivation of Eq. (5-42) for the Debeye Cv

[Book: An Introduction to Statistical Thermodynamics; Terrell L. Hill

ISBN-13: 978-0-486-65242-9

ISBN-10: 0-486-65242-4]

Answer 1

C_v = 3 NK partial differential/partial differential T (T D (u)) D (u) = 3/U^3 integral^u_o x^3 dx/e^x - 1 as x = h v/K T = A/T {because A = h v/K} dx = -A/T^2 d T & u = h v_m/K T = B/T { because B = h v_m/K} D (T) = 3 T^3/B^3 integral^B/T_o A^3/T^3 (e^A/T - T) (-A/T^2 d T) D (T) = -3 T^3/B^3 integral^B/T_o A^4/T^5 (e^A/T - 1) d T partial differential D (T)/partial differential T = -9 T^2/B^3 integral^B/T_o A^4/T^5 (e^A/T - 1) d T - 3 T^3/B^3 times B^4/