# PS1 - 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 #1 Due: Friday, October 2, 2009 Objective: To become familiar with the thermodynamic entropy, its derivatives, and its connection to microscopic, molecular properties. Helpful reminders: Take heed of the following combinatorics formulas and approximations: ways to pick g objects from G , order matters ¡ ¢ £ ¤ G! ¥G ¦ g§! ways to pick g objects from G , order doesn’t matter ¨ £ ¢ ¤ © G g ª ¤ G! g!¥G ¦ g§! ways to pick g objects from G , with replacement G ¢ Stirling’s approximation for factorials lnG! « G lnG ¦ G 1. Statistical antics : After you put the finishing touches on your first perfectly-worked problem set, what’s the first thing that you’re looking forward to exploring in Santa Barbara: (1) outdoorsy activities (hiking, biking, etc.), (2) beach, (3) restaurants / nightlife, (4) the arts (concerts, theater, visual arts, etc), (5) shopping, or (6) historic sites. 2. Fundamentals problem (2 points). Simple functional forms are often used to correlate thermodynamic properties. In one set of experiments, it is found that a pure substance obeys the following heat capacity and equation of state relations: ¬/G ¤ ­® ¯ ° ± ¡ ¤ ²®³ ´ where ³ ¤ G/µ and ² , ­ , and ° ± are G-, ¬-, and µ-independent constants. The first of these expressions invokes the so-called constant heat capacity approximation. Find the underlying entropy function ¶¥¬,µ,G§ , up to an ¬-, and µ-independent constant. Be sure to consider that ¶ must have proper extensive behavior. 3. Conceptual problem (3 points). In class, we discussed several properties of the entropy function. In the following you will create rough scaling arguments for the origins of these properties by using the entropy’s connection to the density of states. a) Show that g G ¡ ¢ lnΩ is extensive, i.e., scales linearly with the size of a system. One way to do this is to consider scaling a system by duplicating it £ times over. If the original system is macroscopic in size, what can be said about the interfacial interactions between the different copies of the system, relative to the total energy? Write down the final density of states in terms of the single-system one, and show that...
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PS1 - University of California Santa Barbara Department of...

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