The Physics Philes, lesson 128: Kinetic-Molecular Model of Ideal Gases, Part 3

We’ve just got a couple more topics to talk about before we can move on from the kinetic-molecular model of ideal gases. So let’s get going!

First, let’s talk a little more about molecular speeds. Just to jog your memory, last week we determined the average translational kinetic energy of a gas molecule and the average kinetic translational kinetic energy per mole of gas. Those equations were, respectively,

and

We can use those equations to find what is called the root-mean-square speed, or the rms speed:

This value follows directly from our equations for the translational kinetic energy. As you can see, at any given temperature T, gas molecules of different masses can have the same kinetic energy but different rms speeds. For example, nitrogen molecules have a molar mass of 28 g/mol. Oxygen, by contrast, has a molar mass of 32 g/mol. From our rms speed equation, we find that nitrogen molecules in the air move more quickly than the oxygen molecules. This also explains why the most abundant element in the universe, hydrogen, is almost completely absent from Earth’s atmosphere. Hydrogen has a molar mass of only 2 g/mol. According to our rms speed equation, this means that hydrogen molecules move very, very fast. Faster, in fact, than the escape velocity of Earth. So hydrogen molecules escape into space, while heavier molecules like nitrogen and oxygen are trapped. Science!

The last topic in this section on the kinetic-molecular model of ideal gases has to do with collisions between molecules. So far we’ve been thinking about gas molecules hitting the wall of a container. However, gas molecules frequently hit each other. It’s true! So, you know…let’s think about that.

For this thought exercise we can’t consider gas molecules as point particles. If they were points, they would never collide. We need to think of these molecules in more realistic terms. Let’s assume that molecules are rigid spheres with a radius r. There are N number of these spherical molecules in some volume V. Let’s assume, for now, that only one molecule is moving. When this moving molecule collides with another molecule, the distance between the tow centers is 2r.

Now, let’s draw a cylinder with a radius 2r with its access in the direction of the velocity of the moving molecule. Let’s say that the moving molecule collides with a stationary molecule, whose center is inside our cylinder. During a very short interval of time that we’ll call dt, a molecule traveling with a speed v with travel a distance v dt. During this very short time interval, the moving molecule collides with any molecule in a cylinder with a volume of radius 2r and length v dt, or

There are a certain number of molecules per volume (N/V), so the number of molecules in our cylinder, denoted by dN, is

To find the number of collisions per unit of time, we get the differentials on the same side of the equation:

This equation assumes that only one molecule is moving, which isn’t very realistic. It turns out, though, that all we have to do is multiply the above equation by the square root of two to model more frequent collisions:

From this equation, we can find the average time between collisions. Just flip the above expression! This is called the mean free time:

We can use this mean free time to find the average distance traveled between collisions, called the mean free path. All we have to do is multiply the velocity by the mean free time:

We denote the mean free path by the Greek letter lambda.

Notice that the mean free path is inversely proportional to the number molecules per volume as well as inversely proportional to the cross-sectional area of a molecule. That means the more molecules there are and the bigger those molecules are, the shorter the mean distance between collisions. It doesn’t, however, depend at all on the velocity of the molecules.

The mean free path can also be related back to the ideal gas equation. Remember, the ideal gas equation is pV = NkT. So the mean free path written in terms of the ideal gas equation ends up being

Think about what this means. If the temperature is increased at a constant pressure, the gas expands and the average distance between the molecules increases, thus the mean free path increases. If the pressure is increased at a constant temperature, the gas is squeezed together and the mean free path decreases. Just like we would expect.

We’ve done it! We’ve completed the section on the kinetic-molecular model of ideal gases! Good job, everyone. Next week we’ll start discussing something called specific heat.

Featured image credit: Colllin Messer via Flickr