Lecture_23 - PHYS 342 Fall 2011 Lecture 23: Kinetic Theory...

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PHYS 342 Fall 2011 Lecture 23: Kinetic Theory of Gases (review) Maxwell-Boltzmann Distribution Ron Reifenberger Birck Nanotechnology Center Purdue University no particles 1 particle 2 particles ………. . N particles Lecture 23
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Historically, (Boyle, 1662; Charles, 1787; Gay- Lussac, 1802) it was known that for many common gasses PV/T = constant This proportionality can be converted to an equation (Clapeyeron, 1834): PV=Nk B T (N=number of gas atoms) or V=nRT (n=number of moles) PV=nRT This is an empirical law; there was no microscopic understanding of why it should be true. k B = 1.38 x 10 -23 J/K; R =8.314 J/K = 0.082 liter atm mol -1 K -1
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Realize that N is a really large number, like 10 23 . It is remarkable that science learned to deal with the properties of enormous numbers of particles BEFORE it discovered how to deal with individual atoms! The reason is that the thermodynamic properties of a system like P and T are AVERAGE values over statistically rge assemblies of particles large assemblies of particles. As a common example, it’s always easier to deal with the average student than it is to deal with any one individual student in this class.
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So let’s study one gas molecule. Confine one air molecule into very narrow tube with moveable piston Moveable For elastic collision, magnitude of velocity does not change, so . . . . ONLY one air piston with area A p molecule = p y (final) –p y (initial) = -m|v y | - (m|v y |) molecule, mass m, velocity +v y , momentum L = -2m|v y | p piston = - p molecule p y =mv y x y t A t = 2L roundtrip ransit time t F av F y |v y | trans t t me t What is F av ? Impulsive Force F = p t
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How to find average force value of N pulses in a time T? t t F o . . . . . 1 2 3 4 5 6 N 0 time time =T Sample F at k discrete values and calculate an average:   k i o o o F p tt NF t F t F t p  1 io av F kT T N t t t   F av = t p
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Evaluating the Average Force and Pressure F av = = p piston 2m|v y | |v = + m|v y | 2 / L |v 2 |v 2 t 2 L /|v y | P=Pressure (one molecule) = = = F av A m|v y | A L m|v y | V PV = 2*KE
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Generalization to 3D: ONE gas molecule in a cylinder with a frictionless piston with area A y m when molecule hits wall, elastic collision • collisions with top and bottom walls produces no change in magnitude or irection of v eg x A direction of v x , e.g.
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This note was uploaded on 12/08/2011 for the course PHYS 342 taught by Professor Staff during the Fall '08 term at Purdue University-West Lafayette.

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Lecture_23 - PHYS 342 Fall 2011 Lecture 23: Kinetic Theory...

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