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Unformatted text preview: Patterned Border Template 1 Purdue University School of Mechanical Engineering ME 501: Statistical Thermodynamics Lecture 26: Elementary Kinetic Theory: The MaxwellBoltzmann Velocity Distribution and Effusion Prof. Robert P. Lucht Room 2204, Mechanical Engineering Building School of Mechanical Engineering Purdue University West Lafayette, Indiana [email protected] , 7654945623 (Phone) November 14, 2011 Patterned Border Template 2 Purdue University School of Mechanical Engineering Lecture Topics • MB translational energy distribution. • MB speed distribution. • MB velocity distribution. • Evaluation of assembly properties from the distribution functions. • The mean free path. • The effusion rate. • Pressure from elementary kinetic theory. • The rate of binary collisions. Patterned Border Template 3 Purdue University School of Mechanical Engineering MaxwellBoltzmann Translational Energy Distribution • The probability that a particle will have a particular energy i tr is given by exp / tr i i B i i tr g k T N f N Z • The probability that the particle will have an energy between i tr and i tr + d i tr is given by: exp / , tr k k B tr tr tr i i i k tr g k T P d Z where tr tr tr tr i k i i d exp / , e x p / exp / tr i B tr tr tr tr tr i i i k k i B k tr tr i B k k tr k T P d g k T Z k T g Z Patterned Border Template 4 Purdue University School of Mechanical Engineering MaxwellBoltzmann Translational Energy Distribution • The term exp[( k tr i tr )/ k B T ] drops out because we can choose an arbitrarily small energy level difference. In differential form, assuming that the energy levels are continuously distributed, this equation becomes is given by • Now introduce the translational energy level distribution function f ( tr ) , defined in terms of dP such that: exp / tr i B tr k T dP dg Z exp / ( ) tr B tr tr tr k T f d d P d g Z • We need to find a relation between d tr and dg . Begin with tr h 2 8 m 2 / 3 ( n 1 2 n 2 2 n 3 2 ) h 2 8 m 2 /3 n 2 Patterned Border Template 5 Purdue University School of Mechanical Engineering MaxwellBoltzmann Translational Energy Distribution • Recall the diagram to the right from Lecture 10. In translational quantum number space, all translational states on the same spherical surface of radius n have the same energy. Because of the very close spacing of the translational energy states, we can regard n, n 1 , n 2 , and n 3 as continuous variables....
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 Fall '10
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 Mechanical Engineering, Kinetic theory, Purdue University, School of Mechanical Engineering

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