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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 Thi ti lit b t d t This proportionality can be converted to an equation (Clapeyeron, 1834): PV=Nk B T (N=number of gas atoms) or PV=nRT (n=number of moles) This is an empirical law; there was no microscopic d t di f h it h ld b t 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 i h h i f b f 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 large assemblies of particles large assemblies of particles. As a common example, it’s always easier to deal ith th t d t th it i t d l ith 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 i t ith (fi l) (i iti l) For elastic collision, magnitude of velocity does not change, so . . . . ONLY one air l l 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 t F A t = 2L |v | roundtrip transit time F av y |v y | p t What is F av ? Impulsive Force F = 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: 1 k i i o o o F p t t N F t F t F t p F a v k T T N t t t p F av = t
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Evaluating the Average Force and Pressure F av = = t p piston 2m|v y | 2 L /|v | = + m|v y | 2 / L F m|v | 2 m|v | 2 /|v y | P=Pressure (one molecule) = = = av A m|v y | A L m|v y | V PV = 2*KE
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