PS8 - University of California Santa Barbara, Department of...

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Unformatted text preview: University of California Santa Barbara, Department of Chemical Engineering ChE 210A: Thermodynamics and Statistical Mechanics Problem set #8 Due: Wednesday, November 25, 2009 Objective: To understand the properties of and develop models using the canonical ensemble in classical systems. 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). Consider the classical canonical partition function. a) Show that, for any system at any state, the total free energy and entropy can be written as the sum of an ideal gas part and an excess part, gG¡,¢,£¤ ¥ g ¦§ G¡,¢, £¤ ¨ g ©ª G¡,¢,£¤ and «G¡,¢,£¤ ¥ « ¦§ G¡,¢,£¤ ¨ « ©ª G¡,¢,£¤ . Derive general expressions for these two excess quantities in terms of the configurational partition function. b) “Hard-spheres” are among the simplest models of solids, liquids, and granular systems, and have a long history in statistical mechanics. These are molecules of diameter ¬ that interact with the following pairwise potential energy function: ­®¯ °± ² ¥ ³ ∞ ¯ °± ´ ¬ ¯ °± µ ¬ In other words, hard spheres only experience an excluded volume interaction. Note that the total potential energy for a given configuration is given by ¶ ¥ ∑ ­G °·± |¸ ° ¹ ¸ ± |¤ , where º and » are particle indices. Why do g ©ª /¡ and « ©ª for hard spheres depend only on the density, independent of temperature? (This is not a detailed derivation, but rather a consideration of the form of the partition function)....
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PS8 - University of California Santa Barbara, Department of...

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