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Q & A: collisions and the ideal gas law

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Most recent answer: 05/26/2014
Q:
In every explanation I have read about kinetic molecular theory, the time between collisions the molecule makes with the wall, is based on the molecule traveling the length of the container. How can the collisions between molecules be ignored? Thank you.
- Bruce (age 63)
Cedar Hill, TX
A:

That's a great question. The last time I taught that material in our Physics 213 course, I added a discussion of just that issue.I don't know if it's still included.

The usual derivation starts considers only elastic collisions of the molecules with flat walls. It uses an assumption, usually not really explained, that the positions and velocity directions have uniform random distributions. The direction distribution is needed to get the right average force on the walls, and the position distribution is needed to keep the force nearly uniform over fairly short times.

The collisions between molecules are actually important to justify those assumptions. After many such collisions, almost any starting distribution starts to look uniform.

Why don't the collisions change the result calculated by ignoring them? Look for a brief period, short compared to a typical collision time. All the force on the walls comes from those molecules that are close enough to the walls to make it to a wall in that brief time.What do their positions an velocities look like? The velocities have exactly the same distribution used in the usual argument. The positions are just slices near the walls taken from the usual distribution. So in any small time period collisions between molecules are very rare and the batch of molecules giving the force on the walls looks just like the ones in the collision-free model. So that model gives the right result even though it's not realistic.

Mike W.


(published on 05/26/2014)

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