Question Video: Determining Particle Distribution for Given Brownian Motion Physics

A very small object moves with Brownian motion as it passes through a region containing gas particles. The path of the object is shown in the diagram. Which of the following most correctly represents the distribution of gas particles in the region?

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Video Transcript

A very small object moves with Brownian motion as it passes through a region containing gas particles. The path of the object is shown in the diagram. Which of the following most correctly represents the distribution of gas particles in the region?

Here, in these five answer options, we see five possibilities for the way that gas particles are distributed in this region. The idea is that this object and the gas particles occupy the same region and that the reason this object’s path changes direction, we see, several different times is due to collisions with gas particles. So for example, this small object first collided with a gas particle here, sending it off in this direction, then it collided with a gas particle here, sending it this way, and so on.

If we sketch in a gas particle at each point where the object’s path changes direction, we start to get a sense of how the gas particles, sketched here in blue, are distributed all throughout this region. Specifically, we’re getting a sense for how the particles are spaced. Notice, for example, that answer choices (A) and (B) show particles with different average spacings between them. In answer choice (A), the particles are slightly closer together than they are in answer choice (B). This difference will be reflected in the type of path that our very small object would follow.

For a less-dense region of gas particles like that shown in option (B), the path of our small object as it moves through these gas particles would change direction less frequently. That is, the arrows that show the uninterrupted motion of this very small object would increase in length for a less-dense region of gas particles. On the other hand, for a more densely occupied region, like we see in choice (D), we would expect the path of our very small object to change that much more frequently. These arrows would decrease in length.

What we’re discovering is that the average length of these arrows here, showing the different segments of our small object’s path, will correspond to the average spacing between particles in our gas. The length of the arrows in this region are shorter than they would be if our gas was populated with the density shown in answer option (B). But then, they’re longer than they would be if our gas was as dense as shown in answer choice (D).

What we find is that answer option (A), which is in a sense a midpoint between choices (B) and (D), shows a gas density that agrees well with the density of particles that would have to be in this region to create the path of our very small object shown.

Now, let’s consider options (C) and (E), which we can see show regions of varying gas density. If answer choice (C) was correct, we would expect the average straight-line distance traveled by our very small object to be greater on the left-hand side of this region and shorter on the right-hand side. And the reverse of that is true for answer option (E). If this option showed the way that gas particles are distributed in the region of interest, then we would expect our small object to change direction frequently as it moves through the left side of this region and infrequently as it moves through the right side.

This is because since particles are more dense in the left, there will be more collisions per unit time, while those collisions will be rarer on the right, which is less densely populated. However, as we look at the actual path followed by our object of interest, we see that though the length of these different line segments do vary, they’re certainly not all the same, on average the lengths of these arrows seem to be about the same on the left-hand side of this region as they are on the right. This suggests a fairly uniform density of gas molecules all throughout the region. That, therefore, eliminates answer choices (E) and (C) from contention.

And we’ve already seen that choices (B) and (D) are not great fits either. This is because their densities do not suggest the type of path that is actually followed by our small object. For our answer then, we choose option (A) as the choice that most correctly represents the distribution of gas particles in the region of interest.

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