PS7 - Department of Chemical Engineering ChE 210A...

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Unformatted text preview: Department of Chemical Engineering ChE 210A University of California, Santa Barbara Fall 2011 Problem Set No. 7 Due: Monday 11/21/2011 Objective : To understand microscopic crystals and the third law of thermodynamics, and to understand the properties of the microcanonical and canonical ensembles. 1. Statistical antics : A friendly genie grants you one of the following selfish wishes, without strings attached or side-effects. Which would you pick? (a) to be a millionaire, (b) to be stunningly beautiful / handsome, (c) to have perfect health, (d) to be a talented and famous celebrity, (e) to be an exceptional athlete, (f) to be a powerful CEO or politician, or (g) to never have to work another problem set again. 2. Conceptual problem (2 points). Show that the equation of state for any substance should have the following leading order behavior near g = 0 , G¡g,¢£ = G ¤ ¡¢£ + ¥¡¢£g ¦ where G ¤ ¡¢£ is the pressure at absolute zero and ¥¡¢£ is a density-dependent constant. 3. Conceptual problem (4 points). Consider the following expressions for the canonical microstate and energy distributions discussed in class, ℘ § = exp¡−¨© § £ ª¡g,«, ¬£ ℘ ¡©£ = ­℘ ® ¯ ° ± ²° ® a) Prove that the sum of ℘¡©£ over all values of © must equal one. b) Prove that as g → 0 , ℘ § will be zero for every microstate except the ground state ones. c) Assume that the canonical energy distribution ℘¡©£ can be approximated as a continuous Gaussian or normal distribution. Write an expression for the distribution in terms of 〈©〉 and ³ ´ . Simplify the expression, as much as possible, for the case of a monatomic ideal gas....
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PS7 - Department of Chemical Engineering ChE 210A...

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