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Unformatted text preview: Department of Chemical Engineering ChE 210A University of California, Santa Barbara Fall 2011 Problem Set No. 8 Due: 12/02/2011 (Friday) Objective : To understand the properties of and develop classical models using the canonical, isothermal-isobaric, and grand canonical ensembles. To understand chemical equilibrium. 1. Statistical antics : What was your favorite statistical antics question this quarter? (a) kind of break after a problem set, (b) superpowers, (c) 22 nd century travel bureau, (d) costume party, (e) most influential future scientific or technological advance, (f) murder mystery character, or (g) wish from a genie. 2. Conceptual problem (2 points). Consider the grand canonical ensemble. Show that the average number of particles is given by the following expression, where g = expG¡¢£ : 〈¤〉 = g ¥ lnΞ ¥g 3. Conceptual problem (4 points). Consider a single-component system held at constant ¦,§ in the isothermal-isobaric ensemble. In this ensemble, the volume and energy of the system fluctuate. Show that the compressibility ¨ © relates to the average volume ⟨ª⟩ and the fluctuations in volume, « ¬ = ⟨ª ⟩ − ⟨ª⟩ . How do the fluctuations in volume scale with system size ¤ ? Hint: you may want to consider the distribution ℘Gª£ . 4. Applied problem (6 points). In this problem, you will derive the van der Waals equation of state. A van der Waals fluid makes the following approximations to enable the exact determination of the configurational integral: (1) a particle isn’t influenced by other particles in a detailed way, but instead sees an average potential energy field due to a...
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