Physics Quiz 831 #4: Thermodynamics and Statistical Mechanics, Quizzes of Statistical mechanics

Download Physics Quiz 831 #4: Thermodynamics and Statistical Mechanics and more Quizzes Statistical mechanics in PDF only on Docsity! Physics 831 Practice Quiz #4 - Wednesday, Nov. 28 YOUR NAME: FUN FACTS TO KNOW AND TELL ∫ ∞ 0 dx xn−1 ex − 1 = Γ(n)ζ(n), ∫ ∞ 0 dx xn−1 ex + 1 = Γ(n)ζ(n) [ 1− (1/2)n−1 ] , ζ(n) ≡ ∞∑ m=1 m−n, Γ(n) ≡ (n− 1)!, ζ(3/2) = 2.612375..., ζ(2) = π2 6 , ζ(3) = 1.20205..., ζ(4) = π4 90 ,∫ ∞ −∞ dx e−x 2/2 = √ 2π, ∫ ∞ 0 dx xne−x = n! 1. (2 pts each) Consider a single electron that can be either spin ↑ or ↓, with the two energies being  or −. (a) What is the average energy when T = 0? (b) What is the average energy when T →∞? (c) What is the specific heat when T = 0? (d) What is the specific heat when T →∞? (e) What is the entropy when T = 0? (f) What is the entropy when T →∞? 2. (10 pts) Beginning with the fundamental thermodynamic relation, and the definition of the specific heat, TdS = dE + PdV − µdN, CV = T ∂S ∂T ∣∣∣∣ N,V , derive the relation: − 1 T ∂CV ∂N ∣∣∣∣ T,V = ∂2µ ∂T 2 ∣∣∣∣∣ V,T