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Chapter 13 | Temperature, Kinetic Theory, and the Gas Laws 557
Figure 13.21 Gas in a box exerts an outward pressure on its walls. A molecule colliding with a rigid wall has the direction of its velocity and momentum in the -direction reversed. This direction is perpendicular to the wall. The components of its velocity momentum in the - and
-directions are not changed, which means there is no force parallel to the wall.
If the molecule’s velocity changes in the -direction, its momentum changes from to . Thus, its change in
momentum is . The force exerted on the molecule is given by
(13.45)
There is no force between the wall and the molecule until the molecule hits the wall. During the short time of the collision, the force between the molecule and wall is relatively large. We are looking for an average force; we take to be the average time between collisions of the molecule with this wall. It is the time it would take the molecule to go across the box and back (a distance at a speed of . Thus , and the expression for the force becomes
(13.46)
This force is due to one molecule. We multiply by the number of molecules and use their average squared velocity to find the force
Because the velocities are random, their average components in all directions are the same:
(13.48)
(13.49)
(13.50)
(13.51)
Thus,
or
(13.47)
where the bar over a quantity means its average value. We would like to have the force in terms of the speed , rather than the -component of the velocity. We note that the total velocity squared is the sum of the squares of its components, so that
