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Statistics for mean free path Look-up Popularity. If there were no other molecules present, an ion would have a constant acceleration until it reached the wall of the container. But because of the presence of the other molecules, it cannot do that; its velocity increases only until it collides with a molecule and loses its momentum.

It starts again to pick up more speed, but then it loses its momentum again. The net effect is that an ion works its way along an erratic path, but with a net motion in the direction of the electric force.

While the field is on, and while the ion is moving along, it is, of course, not in thermal equilibrium, it is trying to get to equilibrium, which is to be sitting at the end of the container. By means of the kinetic theory we can compute the drift velocity. It turns out that with our present mathematical abilities we cannot really compute precisely what will happen, but we can obtain approximate results which exhibit all the essential features. We can find out how things will vary with pressure, with temperature, and so on, but it will not be possible to get precisely the correct numerical factors in front of all the terms.

We shall, therefore, in our derivations, not worry about the precise value of numerical factors. They can be obtained only by a very much more sophisticated mathematical treatment. Before we consider what happens in nonequilibrium situations, we shall need to look a little closer at what goes on in a gas in thermal equilibrium. We shall need to know, for example, what the average time between successive collisions of a molecule is. Any molecule experiences a sequence of collisions with other molecules—in a random way, of course.

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If we double the length of time, there will be twice as many hits. But let us try to make a more convincing argument. If there is equilibrium, nothing is changing on the average with time. A particular particle does not have a collision, wait one minute, and then have another collision.

The times between successive collisions are quite variable. Making this substitution, Eq. The chance is less than one-half that it will have a greater than average time between collisions. The result we have obtained in Eq. We can demonstrate this somewhat surprising fact in the following way. Another way of describing the molecular collisions is to talk not about the time between collisions, but about how far the particle moves between collisions.

## The mean free path for electron conduction in metallic fullerenes

In this chapter we shall be a little careless about what kind of average we mean in any particular case. The various possible averages—the mean, the root-mean-square, etc. Since a detailed analysis is required to obtain the correct numerical factors anyway, we need not worry about which average is required at any particular point. We may also warn the reader that the algebraic symbols we are using for some of the physical quantities e.

The chance that our particle will have a collision is the ratio of the area covered by scattering molecules to the total area, which we have taken to be one. The whole area is not covered, of course, because some molecules are partly hidden behind others. But that is a different subject! So we return to the problem of nonequilibrium states. We want to describe what happens to a molecule, or several molecules, which are different in some way from the large majority of the molecules in a gas. A molecule could be special for any number of reasons: It might be heavier than the background molecules.

It might be a different chemical. It might have an electric charge—i. To list a few: the diffusion of gases, electric currents in batteries, sedimentation, centrifugal separation, etc. What happens to it, in detail , is that it darts around hither and yon as it collides over and over again with other molecules. We say that there is a drift , superposed on its random motion.

We shall discuss later an improved assumption. The starting velocity will take it equally in all directions and will not contribute to any net motion, so we shall not worry further about its initial velocity after a collision.

What is the average value of this part of the velocity? You will notice that the drift velocity is proportional to the force. There is, unfortunately, no generally used name for the constant of proportionality. Different names have been used for each different kind of force. We have from Eq. To get the correct numerical coefficient in Eq.

### Mean free path definition

Without intending to confuse, we should still point out that the arguments have a subtlety which can be appreciated only by a careful and detailed study. To illustrate that there are difficulties, in spite of appearances, we shall make over again the argument which led to Eq. After a collision the particle starts out with a random velocity, but it picks up an additional velocity between collisions, which is equal to the acceleration times the time.

At the beginning of the collision it had zero velocity. This result is wrong and the result in Eq. The fact is that some times are shorter and others are longer than the mean. The error was made in trying to relate by a simple argument the average final velocity to the average velocity itself. This relationship is not simple, so it is best to concentrate on what is wanted: the average velocity itself. The first argument we gave determines the average velocity directly—and correctly!

## Mean free path

But we can perhaps see now why we shall not in general try to get all of the correct numerical coefficients in our elementary derivations! We return now to our simplifying assumption that each collision knocks out all memory of the past motion—that a fresh start is made after each collision. We now apply our results to a special case. Suppose we have a gas in a vessel in which there are also some ions—atoms or molecules with a net electric charge.