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Illustration 20.2: Kinetic Theory, Temperature, and Pressure

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In this animation N = nR (i.e., kB = 1). This, then, gives the ideal gas law as PV = NT. The average values shown, < >, are calculated over intervals of one time unit. Restart.

The equipartition theorem says that the temperature of a gas depends on the internal energy of the gas particles. For monatomic particles, the internal energy of a given particle is its kinetic energy (there is 1/2 kBT of energy per degree of freedom, and for a monatomic gas there are 3 degrees of freedom). Thus, for an ideal gas made up of particles of different masses, the average kinetic energy is the same for all the particles. In this animation the yellow particles are 10 times more massive than the light blue ones. How does the kinetic energy of the blue particle (representative of the smaller particles) compare with the kinetic energy of the orange particle (which is representative of the yellow particles)? What would you expect in a comparison of the speeds of the two particles? While the average kinetic energy of the particles should be identical, their average speeds should be different, since they have different masses.

Now triple the temperature.  What happens to the kinetic energy of both the blue particle (representative of the smaller particles) and the orange particle (representative of the yellow particles)? If you triple the temperature, what happens to the kinetic energy? What happens to the speeds of the particles? The average kinetic energy should increase by three, and the speed of the average particle should go up by 1.73, the square root of 3.

Finally, note that <dp/dt>, the average momentum delivered to the walls, is slightly higher than the value of the pressure calculated by the ideal gas law (P = NT/V). This is because the ideal gas law assumes that the particles are "point-like" (point particles), while the animation has particles with a definite radius. This means that they interact with the wall sooner (at the edge of a particle instead of the center) and with each other more often. Thus, the average time between collisions with the wall (Δt) is smaller, making <Δp/Δt> = <dp/dt> bigger than predicted by the ideal gas law. In reality, of course, particles are not "point-like," but the size of the particles is typically much smaller in relation to the size of the container (think about the air molecules in a room), so the "point-like" approximation works well. Now increase the particle size to see what happens when the particles are quite large.

Illustration authored by Anne J. Cox.

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