10_1425_web_Lec_31_KineticTheoryGases

# 10_1425_web_Lec_31_KineticTheoryGases - Kinetic Theory of...

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Kinetic Theory of Gases Physics 1425 Lecture 31 Michael Fowler, UVa

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Bernoulli’s Picture Daniel Bernoulli, in 1738, was the first to understand air pressure in terms of molecules— he visualized them shooting around very rapidly in a closed container, supporting a weight as shown by constantly bouncing off the underside of the piston. Given more room, they would rush in to fill the new space, just as a gas is observed to do. No-one believed him. Applet here.
One Dimensional, One Molecule Gas The molecule roundtrips in time 2 L / v , so it bounces off the piston v /2 L times per sec, each time delivering momentum 2 mv , so the piston will pick up momentum from this “gas” at rate 2 x v /2 L per second. Force from gas on piston: F = rate of change of momentum = 2 / L . An equal opposite force must be supplied from outside to keep the piston at rest. Animation! V v L 1-D gas: molecule bounces between ends of cylinder.

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Molecule in a Two-Dimensional Box Assume perfectly elastic collisions with all walls. The molecule will follow a zigzag path, the time between collisions with the same end, say the end at x = L , is now 2 L / v x , and the momentum transferred per collision is 2 mv x , so the average force on the end is x 2 / L . This will still hold good in three dimensions. a L v x v y 0 x y
N Molecules in an L x L x L Cube Assume first that we have a very large number N of molecules bouncing around, so small that they don’t hit each other, each follows its own zigzag path. The force on the right-hand wall at x = L is just the sum of the forces from each one, so F = mv x 1 2 / L + x 2 2 / L + x 3 2 / L + … + xN 2 / L .

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Bar Means Average The force on the wall depends on the sum We don’t need the individual values v x 1 2 , etc., just the average , written with a bar : So the force on the wall is: 22 2 2 123 x x x xN vv v v + + ++ 2 2 2 x x x xN x v v v N + + = 2 x Nmv F L =
Gas Molecules Have Random Velocities…

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## This note was uploaded on 12/07/2011 for the course PHYSICS 1425 taught by Professor Michaelfowler during the Spring '10 term at UVA.

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10_1425_web_Lec_31_KineticTheoryGases - Kinetic Theory of...

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