 So we know what the root mean square velocity of a gas is for an ideal gas. It's a square root of 3KT over m. This quantity of a root mean square velocity is a representative velocity, but it's not the same thing as an average velocity. In particular, what we really want to know is the average velocity or average speed of a molecule. We want to do something a little bit different. So the average, let's go ahead and talk about velocities. If I did want to calculate the average velocity of a gas molecule, what I would need to do is something like take the individual velocities, multiply by the probability of having that velocity, and integrate that over all the possibilities. So that's how to calculate an average. Some fraction of molecules are moving at 100 meters per second. Some fraction are moving at 200 meters per second in one direction or another direction. Combine all those together, and we get the average velocity. So that's really what we're interested in doing if we want to know the average velocity of a molecule. Luckily, with Boltzmann's help, we know a lot about how to calculate probabilities. So we can write down what the probability is that a molecule has a particular velocity. We know that that's just e to the minus energy over KT divided by a partition function. And the energies, in particular, since we're treating these gas molecules as having nothing other than kinetic energy, their energy is 1 half m v squared. So probability that a molecule has a velocity v is e to the minus 1 half mass times that velocity squared divided by KT all divided by a partition function. So we need to know what the partition function is. The partition function is also e to the minus energy over KT, the Boltzmann factor, but integrated over all possible values of the velocity. So again, let me distinguish here between velocities and speeds. So just to be clear, when I write v with a arrow on top, that's talking about a velocity vector. That has not only some magnitude, but also a direction in x, y, and z. Speed, as opposed to velocity, we can think of that as just the magnitude of the velocity vector. So a molecule can be moving with a speed of 500 meters per second, or it can be moving with a velocity of 500 meters per second in a particular direction. So the partition function we need to think about is this one, so I'm integrating over all velocities. So I'm integrating over vx, vy, and vz. So this is really a triple integral. Likewise, the velocity squared, this quantity, the velocity squared, that's vx squared plus vy squared plus vz squared for the magnitude of the velocity in three dimensions. So that actually points out that this integral is not that difficult. This triple integral over dvx, dvy, dvz. Let me go ahead and write that out. So this d cubed v is the same thing as saying dvx, dvy, dvz. But since my v squared is just the sum of the vx squared, vy squared, vz squared, these integrals look like integral of minus 1 half mv squared over kt in the x direction integrated over dvx. The possible values of vx range from negative infinity to infinity. I can be moving strongly in the negative direction or quickly in the positive direction. Velocities have a direction as well as a magnitude. That's the vx component. There's going to be a similar component for vy. A Gaussian, e to the minus 1 half vy squared over kt integrated over vy and an identical one with vz instead of vx or vy. So I've broken my integral up into three different integrals. Luckily, each of those integrals has the exact same result as the others. This Gaussian integral, negative infinity to infinity, gives the same result as this one and this one. A Gaussian integral of a Gaussian from negative infinity to infinity is square root of pi over the coefficient that multiplies the variable. So everything that's not the v squared is the coefficient. So I've got a 1 half m over kt, 1 half m over kt. And I have that same term three times 1 from each of these three one-dimensional integrals. So if I clean that up a little bit, the fraction in the fraction, if I resolve that, the 2 in the denominator of the denominator moves back up to the top. So I've got what looks like, oh, and there's also, it's m over kt. So there's a 2 and a kt in the denominator of the denominator. So I've got 2 pi kt that moves up to the numerator. m stays in the denominator. Instead of raising it to the 1 half power, I do that three times. So it's all raised to the 3 hat's power. So that looks a little reminiscent of a result we've had before for the thermal de Broglie wavelength. But it's not the same thing. Don't get the two of those confused. This partition function, 2 pi kt over m raised to the 3 half's power, that tells us the partition function or the denominator we need here to calculate the distribution of velocities. So if I go back to this expression and use my newly found q, the probability that a molecule has a particular velocity is 1 over q, 1 over this quantity. So I'll write that upside down. m over 2 pi kt to the 3 half's multiplied by this exponential, e to the minus energy 1 of mv squared divided by kt. So that tells us if we want to calculate what is the probability the molecule has a velocity of 500 meters per second in a particular x, y, z direction, I compute v squared. I know the mass and the temperature. I plug everything into this expression. This expression may be a little surprising if we think about what that means. So I'll try to graph this expression, which I'm going to have a hard time doing in three dimensions. So let's start in one dimension. If I consider only, let's say, the vx component of the velocity, and graph what this Gaussian looks like, e to the minus velocity squared multiplied by some constants, that's, of course, a Gaussian. So what that tells us is it's most likely the highest probability. So what I've graphed here is the probability of vx as a function of vx. That tells us the most likely velocity is 0. It's more probable that the molecule is standing still in the x direction. That's more probable than moving positively in the x direction or negatively in the x direction. So that seems a little bit surprising, right? If I extend that to two dimensions, so that I can almost draw reasonably. If I draw how the probability depends on vx and vy as a function of vx and vy, let's say, let me, so that's a Gaussian e to the minus vx squared and vy squared multiplied by some constants. So that's a Gaussian that's going to look like this. It's a Gaussian in two dimensions. So to give it some 3D shape, I've got a hill, a Gaussian hill centered on the origin. So again, 0, vx equals 0, vy equals 0, is the value that has the largest probability. So again, it seems strange that the most likely velocity that this molecule has is 0. But that's exactly what Boltzmann's telling us. And it turns out to be true. And the confusion is just in how we're interpreting that result. And I won't attempt to draw that three dimensional function because I'd need a fourth direction to plot what the probability looks like. So we found one surprising thing, the most probable velocity. It looks like it's that the molecule is stationary. The other surprising thing is that if I were to calculate the average velocity, the problem we started out wondering about. How do I calculate the average velocity of this molecule? The average of vx, because I have equal probability of moving 100 meters per second to the left and 100 meters per second to the right, the two sides of this curve cancel each other out. So if I calculate the average velocity, I'm going to get 0. If I plug this result back into here and do the integral, I would find out that the average velocity is 0 because of the symmetry of being equally likely to move to the left and to the right. Again, that's not a surprising result. Once we understand that that's the question we asked, average velocity, which has a direction, the two directions cancel each other out. Likewise, in two directions, in two dimensions, plus y and minus y cancel each other out. Same thing in three dimensions. Turns out the question we should have been asking all along is what's the probability of having a molecular speed, the magnitude of the vector? And the answer to that question, so what we've calculated is the distribution of molecular velocities, which is important. But to get the real intuitive sense for how quickly these molecules are moving and how likely each speed is, we need to be asking the question about the speed of the molecules rather than the velocity of the molecules. So the next step is to convert this distribution of velocities, which is the result we've obtained here, into a distribution of molecular speeds instead. So that's coming up next.