Statistical Mechanics Exam, December 13, 2007: Thermodynamics and Equations of State, Exams of Statistical mechanics

Download Statistical Mechanics Exam, December 13, 2007: Thermodynamics and Equations of State and more Exams Statistical mechanics in PDF only on Docsity! Statistical Mechanics Subject Exam, Dec. 13, 2007 SECRET STUDENT NUMBER: STUDNUMBER 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! LONG ANSWER SECTION 1. (15 pts) Beginning with the fundamental thermodynamic relation, and the definition for CP , TdS = dE + PdV − µdN, CP = T ∂S ∂T ∣∣∣∣ N,P , derive the equality ∂CP ∂P ∣∣∣∣ T,N = −T ∂ 2V ∂T 2 ∣∣∣∣∣ P,N . SECRET STUDENT NUMBER: STUDNUMBER 2. (15 pts) Consider the equation of state, P (ρ, T ) = ρTeρ/ρ0 − aρ 2 ρ0 . Solve for the critical density ρc in terms of ρ0 and a.