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Unformatted text preview: Statistical Physics I (8.044) Spring 2009 Assignment 10 April 29, 2009 Due May 6, 2009 Please remember to put your name and section number at the top of what you turn in. Readings The reading for Chapter VIII of 8.044 is Baierlein Chapter 7. At some point after you have read Chapter 7, you should reread Chapter 10. When you originally read chapter 10, you did not yet know what μ was; now, you do. The reading for Chapter IX of 8.044 is Baierlein Chapters 8 and 9. These are the final reading assignments for 8.044. I will post a Problem Set 11 next week, with problems that will help you understand Chapter IX of 8.044 and prepare for this component of the final exam. You will not turn in Problem Set 11, however. This Problem Set (10) is the last one that you will turn in for grading. Problem Set 10 1. Chemical Potential and Various Thermodynamic Potentials (15 points) (a) We defined the chemical potential via the Helmholtz free energy: μ = parenleftBigg ∂F ∂N parenrightBigg V,T . (1) Once we are talking about a system in which N can vary (like either the upper or the lower gasfilled volume in the example in the chemical potential lecture), the natural variables for F are T , V and N . And, dF = SdT pdV + μdN . (2) Derive equations analogous to (1) relating μ to each of E (internal energy), H (enthalpy) and G (Gibbs free energy). And, write expressions for dE , dH and dG analogous to (2). [Don’t be surprised if you find his part of the problem easy. It is.] (b) In part (a), you should have concluded that G is a function of T , P , and N . But, G is extensive, and N is the only one of these three variables that is extensive. Therefore, G is proportional to N . What is the proportionality constant? 1 (c) Lets check your result by making sure that it holds for the monoatomic ideal gas, whose Helmholtz free energy is F ( T,V,N ) = Nk B T bracketleftBigg log parenleftbigg V N parenrightbigg + 3 2 log parenleftBigg 2 πmk B T h 2 parenrightBigg + 1 bracketrightBigg . Evaluate μ , P and S by taking suitable derivatives of F . [Note: of course, we already evaluated P and S this way in lecture, early in Chapter VI.] Evaluate the thermodynamic potentials E ( S,V,N ), H ( S,P,N ) and G ( T,P,N ). Show that your equations analogous to (1) from part (a) hold for this example. Show that your result from part (b) holds for this example. 2. Chemical Potential as a Lagrange Multiplier (10 points) This problem extends Problem 5 of Problem Set 7. Begin by rereading that problem and its solution in its entirety. You should keep that problem and its solution in front of you as you do the present problem. Below, I will refer to Problem 5 of Problem Set 7 as “the old problem”....
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 Spring '09
 KrishnaRajagopal
 Physics, Thermodynamics, old problem

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