 Hi, I'm Stephen Nesheva, and I am here to tell you a little bit about a Maxwell probability density function. And the Maxwell probability density function is all about the speeds of a molecule, as opposed to the velocity components. Let me just say something about that. I've got a molecule here and it's going with some velocity components in the x, y, and z direction. If I want to know the speed, I take the square of each one of those components, add them up, take the square root, and now we have a speed. Now, because of the way this is defined, speed can only be a positive number, and so the relevant range for speed, of course, is zero out to some large number. And I've drawn here what the shape of that probability density looks like. It's symbolized as f with the sub m for Maxwell, and it looks something like that. There are a few points on this curve that bear special note. The most probable speed, which would be the speed at which that peak occurs, has a special name and is called C star. There's another number, which I'm going to exaggerate how far out it goes, just so I can fit it here. It would be the average speed, that's a little bit farther out. And a third number of interest would be, sorry, that's the average speed, and then C itself is called the root mean square speed, RMS speed. That's a little bit farther, that's even higher. And we could talk about even higher moments, the cubed root of the cubic cubed speed, meaning cubed of the speed would be even farther out. So that's how that goes. I'm not going to draw it because it makes it a little bit more complicated, but if you can just follow my hand, if we had a hotter gas, if the gas were hotter, it would tend to have a larger wing out here, because it would be more likely to find that gas with higher speeds than it would be for this whole gas, let's say. Okay, so that's that part. There's a condition of normalization that we impose to create this probability density. That idea there is that the whole area under that curve is one. And mathematically, that's given by that formula. Now, notice that we integrate only from zero to infinity, just that that's because we start off with speeds at zero. And I'm just integrating the Maxwell function with respect to the speed, and it's going to all add up to one. Okay, what's the form of the Maxwell probability density? Well, there's a normalization factor that kind of determines how high that curve is. We multiply by v squared, and then we multiply by that Gaussian term, e to the minus d v squared. And both d and that normalization constants, they both depend on the temperature, and they depend on what gas you're dealing with. Let's see, moments, well, we can talk about the mean speed. How would we get that? Well, I integrate. I take the speed times the probability of finding that speed at family, this probability density, integrate over all the speeds. That has a special name. I've already introduced it. That's the mean speed that appeared right there. I didn't mention here, though, the intervals, it must be going from zero to infinity because that's the range of speed. How about the mean of the square of the speed? That means the mean. That inside means the square of the speed. Well, I integrate from zero to infinity, but I multiply v squared times the Maxwell probability density. And now, the mean of the square is, therefore, it would be called c squared. If I take the square root of that quantity, I get c, which is the rms speed, that's the quantity right there. And so on for higher moments. The other thing is that it's useful to use the Maxwell probability density to get probabilities of finding molecules within certain intervals of speeds, let's say between v and v prime. How would that work? Well, if I'm interested, for example, in finding the probability of finding a molecule between the root mean square speed plus or minus some amount, 5 meters per second, let's say I would go plus 5 meters per second above that minus 5 meters per second below that and evaluate the area under that curve. That in general is kind of a difficult integral to do, but one could always do an approximation which is like get the midpoint height and multiply by that width, and that would give you an approximation to the probability of finding a particle in that range of speeds.