176-2nd-midterm-03-27-09

176-2nd-midterm-03-27-09 - Print your name clearly:...

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Print your name clearly: Signature: “I agree to neither give nor receive aid during this quiz.” Second Midterm Exam for Physics 176 Professor Greenside Friday, March 27, 2009 This exam is closed book and will last the entire class period. Please note the following: 1. Only true-false and multiple choice questions should be answered on the exam itself. 2. All other questions should be answered on the extra blank pages. If you need extra pages during the exam, let me know. 3. Please write your name and the problem number at the top of each extra page. 4. Please write clearly. If I can not easily understand your answer, you will lose credit. 5. Please feel free to ask me questions during the exam if the wording of a problem is not clear. The following formulas may be useful: dU = TdS - PdV + μ dN, U = Q + W, E = - ln( Z ) ∂β . C V = 1 kT 2 E 2 - ( E ) 2 · . (1) F = U - TS, dF = - SdT - PdV + μ dN, F = - kT ln( Z ) . (2) D ( v ) = 4 π m 2 πkT · 3 / 2 v 2 e - mv 2 / (2 kT ) , v = r 8 kT πm , v 2 = 3 kT m . (3) Z 0 e - αx 2 dx = π 2 α - 1 / 2 , Z 0 x e - αx 2 dx = 1 2 α - 1 , (4) coth( x ) = cosh( x ) sinh( x ) = e x + e - x e x - e - x , Z π 0 e a cos( θ ) sin( θ ) = 2 a sinh( a ) . (5) e x = 1 + 1 1! x + 1 2! x 2 + 1 3! x 3 + . . . , 1 1 + x 1 - x. (6) Problems That Require Writing Please write your answers to the following problems on extra blank sheets of paper. Also make sure to write your name and the problem number at the top of each sheet. In this part of the exam, you need to justify all of your answers to get full credit. 1
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1. (8 points) Consider a small system in contact with a large reservoir whose constant temperature is T , and assume that the system and reservoir consist of only one type of particle. Assume that the small system can exchange heat and exchange particles with the reservoir so neither the energy of the system nor the number of particles in the system is conserved. Using an argument similar to how the Boltzmann factor was derived and clearly stating your assumptions and approximations, derive an expression for the probability for the small system to be in a state s that has energy E ( s ) and that has N ( s ) particles . Your expression will look like an “exponential of something” over a quantity that looks like a partition function. 2.
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This note was uploaded on 10/20/2011 for the course PHYSICS 176 taught by Professor Behringer during the Spring '08 term at Duke.

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176-2nd-midterm-03-27-09 - Print your name clearly:...

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