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Unformatted text preview: Department of Chemical Engineering ChE 210A University of California, Santa Barbara Fall 2011 Problem Set No. 2 Due: Friday, 10/10/11 Objective : To become familiar with the thermodynamic entropy, its derivatives, and its connection to microscopic, molecular properties; to understand the nature of molecular interactions; and to understand the principle of entropy maximization. 1. Statistical antics. If you could have only one of these superpowers, which would it be? (1) teleportation – you can instantly transport yourself to anywhere, (2) telekinesis – you can move small objects with your mind, (3) mind reading, (4) psychic ability – you can foresee events in the future, and (5) invisibility. 2. Conceptual problem (3 points). Consider the following system of two compartments of an ideal gas which are linked by a moveable, insulating, impermeable wall: Thus the compartments can exchange volume, while the total volume g G = g ¡ + g ¢ is conserved. a) Assume that each box can be subdivided into very small cells of volume £ ; each cell serves as one particular location where one or more ideal gas particles can be placed. Find an expression for the density of states of the system in terms of £,¤ ¡ ,¤ ¢ , g G and ¥ ¡ ¦= § ¨ § © ª . Neglect energies. b) Find an expression for the value of ¥ ¡ , and hence g ¡ , that maximizes the density of states. c) What happens if the volume is just slightly different than its value at the density of states maximum? Consider another value of g ¡ , given by g ¡ = 0.9999g ¡ ∗ where g ¡ ∗ is the value found in part b. Determine the base-10 logarithm of the ratio of the number of microstates at the two volumes, lo«Ω(0.9999g ¡ ∗ )/Ω(g ¡ ∗ ) . Take ¤ ¡ and ¤ ¢ to be 3 × 10 ¢¬ and 4 × 10 ¢¬ . Given the principle of equal a priori probabilities, what does this result imply for the frequency with which the volume 0.9999g ¡ ∗ will be seen, relative to g ¡ ∗ ?...
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This note was uploaded on 12/29/2011 for the course CHE 210a taught by Professor Staff during the Fall '08 term at UCSB.
- Fall '08