ps4rev - Statistical Physics I(8.044 Spring 2009 Assignment...

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Statistical Physics I (8.044) Spring 2009 Assignment 4 February 25, 2009 Due Thursday March 5, 2009 Please remember to put your name and section number at the top of what you turn in. Readings By now you should have completed your reading of the Notes on Probability. The core readings for Chapter III of 8.044 are: Chapters 1, 2 and 3.1-3.7 of C.W.Adkins Equilibrium Thermodynamics . I am also requiring that you read Chapter 1 of Baierlein because later on you will be reading this book, and it is good that you read its introductory chapter. Baierlein assumes that you have seen thermodynamics elsewhere, which in your case is Adkins. Problem Set 4 1. Kinetic Energy of Molecules in a Gas (16 points) In lecture, I told you that the probability density for the x -component of the velocity of a molecule with mass m in a gas with temperature T is p ( v x ) = 1 2 πσ 2 exp b v 2 x 2 σ 2 B where σ = r k B T/m . From this, we worked out that the probability density for the x -component of the kinetic energy KE x = v 2 x / (2 m ) is p (KE x ) = 1 πmσ 2 KE x exp ± KE x 2 ² for KE x 0 and 0 for KE x < 0. p (KE x ) is the probability density for the energy of the molecules in a one-dimensional gas. Throughout this problem you should remember that you may do integrals using MatLab or Mathematica if you wish. (a) Compute a KE x A . (Express your result in terms of the temperature T , rather than σ .) (b) The energy of a molecule in a two-dimensional gas is E 2 d KE x +KE y . Compute the probability density for this quantity, via evaluating a convolution integral. Use your result to compute a E 2 d A . (Express your result in terms of the temper- ature T , rather than σ .) 1
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(c) The energy of a molecule in a three-dimensional gas is E 3 d = KE x + KE y + KE z . Compute the probability density for this quantity, via evaluating another convolution integral. Use your result to compute a E 3 d A . (Express your result in terms of the temperature T , rather than σ .) (d) There is also another way to evaluate the E 2 d or E 3 d probability density. Lets try it out for E 3 d . The probability density for the three random variables v x , v y , and v z is p ( v x ,v y ,v z ) = 1 (2 πσ 2 ) 3 / 2 exp b v 2 x + v 2 y + v 2 z 2 σ 2 B .
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This note was uploaded on 09/17/2009 for the course PHYSICS 8.044 taught by Professor Krishnarajagopal during the Spring '09 term at MIT.

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ps4rev - Statistical Physics I(8.044 Spring 2009 Assignment...

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