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# PS7 - 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 #7 Due: Wednesday, November 18, 2009 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 : You win free tickets to a box seat at Carnegie Hall. Which performance do you choose? (1) symphony, (2) stage theater, (3) a musical, (4) opera, (5) solo music artist (classical or contemporary), (6) ballet, or (7) scalp the tickets. 2. Fundamentals problem (2 points) . Consider the microscopic model of monatomic crystals discussed in class. Let the positions of all of the atoms at any instant be described by g G , a vector with 3¡ components. Similarly, let the positions of all of the atoms at their energy- minimized lattice positions be g ¢ G . We might approximate the potential energy about the minimum in a harmonic manner, as: £¤g G ¥ ¦ £¤g ¢ G ¥ § ¨|g G © g ¢ G | ª where |«| denotes the length of a vector « and ¨ is a constant. Assume the potential energy at the minimum is zero, £¤g ¢ G ¥ ¬ 0 . Here, you will show that the value of the intensive constant- volume heat capacity in the classical approximation is always 3­ ® , independent of ¯ . a) For the moment, neglect kinetic energies and let ° ¬ £¤g G ¥ . This will allow us to compute excess properties, i.e., those above and beyond that for an ideal gas. Show that the number of classical microstates is proportional to ° ±G ª ⁄ . Hint: envision a...
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PS7 - University of California Santa Barbara Department of...

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