 Hello, I'm Stephen Nesheva and here to tell you a little bit about Boltzmann probability densities. So what's the Boltzmann probability density? Well, it's a curve that tells you the likelihood of finding a particle, a gas particle going at a certain velocity component. What's a velocity component? Well, it could be going in the x direction. We would be talking about that or in the z direction or in the y direction. I just imagined a particle going, you know, have some combination of those velocity components. The idea behind this curve, the density function itself is given by this symbol f sub b. And the idea is that it's most likely to be going at this zero actually in the x or y in the x direction. And then it's got these wings saying that it might be going really fast to the right or really fast to the left. What if the temperature of this gas that we're thinking about were higher, let's say, well, you'd expect the wings to get bigger and something like that. Now, I've drawn this in such a way that the total area under the curve is still one. That's why I had to squash it down a little bit. And the reason for that is that the Boltzmann density is what we call normalized. That is to say, the chances of finding it going some speed has to all add up to one. So that's mathematically what this first condition is. It's a normalization condition. The area under the curve has to integrate out to one, so that's one. There is a particular form to the Boltzmann probability density, and it's given by this. There's a normalization constant that determines how high it is, and sort of the shape of it is given by this Gaussian function. So n and d are constants that pertain to a particular gas at a particular temperature. So it's a Gaussian. Other things that we can do with this function is we can use it to calculate what are called the moments. So, for example, the first moment is just the average of that velocity component, what is the average velocity component. And the way you get at it mathematically is to say I'm going to integrate from minus infinity to plus infinity, because that's the range of Bx. I'm going to take Bx times that Boltzmann probability density, and the number that I get out of it will be just the average x component velocity. What if I wanted the mean of the square of that quantity? Well, that's designated this way. It's the mean of the square of Bx. That's very simple. You just take that same interval that we just did, put a square in front of the Bx, still multiplying by FB and a Greek over all the space. Same thing goes for the third moment, the fourth moment, et cetera. We just add greater exponents there. Finally, one thing that we can, it's useful to use these probability densities for is to actually calculate probabilities. The probability, let's say, of going between one speed and some other speed. Graphically, what it is, is it's the area between two different speeds that you might be interested in. Let's suppose I was interested in the speed that this molecule is going between that point here, which we'll call Bx, and that point there, which I'll call Bx prime. What's the probability that I'm going to find a molecule going in those speeds? Well, it's just given by that area, which, as we've already said, since the total area is one, this has to be a fraction. I can multiply it by 100 to get a percentage. Yeah, and usually this is a little bit maybe difficult to get exactly, but an approximate way of doing it would be go to the midpoint, take the height, and then multiply it by the width. The height, which is Fb at that particular point times the width, which would be the interval that you're interested in. That's going to be about equal to that probability. Okay.