MBVelocityDistributionLect26ME501F2011

MBVelocityDistributionLect26ME501F2011 - Patterned Border...

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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 Maxwell-Boltzmann Velocity Distribution and Effusion Prof. Robert P. Lucht Room 2204, Mechanical Engineering Building School of Mechanical Engineering Purdue University West Lafayette, Indiana Lucht@purdue.edu , 765-494-5623 (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 Maxwell-Boltzmann 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 Maxwell-Boltzmann 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 Maxwell-Boltzmann 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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MBVelocityDistributionLect26ME501F2011 - Patterned Border...

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