 So we have seen that the probability that a molecule has a particular velocity can be written in terms of this Boltzmann factor for the energy, the kinetic energy of the gas divided by its partition function. But we've discovered that we're not terribly interested in the probability of the velocity of a vector. What we're much more interested in is knowing the probability of the molecules moving with a certain speed. For example, I might be interested in how many molecules or what fraction of molecules in the air in this room are moving with a velocity of 200 meters per second. But I don't really care whether they're moving at 200 meters per second in this direction or this direction or that direction or this direction. So we'd rather have a distribution of molecular speeds rather than molecular velocities. So we can start with this expression and remind ourselves that this probability, this is the probability, if I were to integrate it, the integration variables would be three different velocity dimensions, vx, vy, and vz. So if I rewrite this expression out, just writing the differential terms for the variables that I would need to integrate with respect to, just tacking on a dvx, dvy, dvz onto the end of both sides of that expression, that'll be useful because it lets me point out that these are Cartesian variables, and the speed variable, the magnitude of the velocity that we're interested in, that's like the magnitude in spherical polar coordinates. So if we convert Cartesian coordinates into polar coordinates, so just like in Cartesian coordinates x, y, z, the spherical polar equivalent would be r squared sine theta, dr d theta d phi. Same thing here. When I convert velocities in the x, y, z dimensions into velocities in Cartesian coordinates, I would need v squared sine theta dv d theta d phi, where the theta and phi coordinates are the angles that the velocity vector has. And let me point out, since it actually affects the result that we will get, the choice of variables that I'm using, so these are my Cartesian x, y, z coordinates, if I have some point in space that I want to represent in polar coordinates, if I want to represent that point, then v is the, or r if we were in position coordinates, v if we're in velocity coordinates, that's the magnitude of the vector. The value theta that I'm using is the angle by which the vector is bent down away from the z-axis, and the angle of phi that I'm using is the rotation of that projection into the x, y plane away from the x-axis. So depending on whether you've taken math courses or physics courses or chemistry courses, most recently this choice of theta and phi may or may not be what you're used to, and math courses very often these definitions are flipped, but in physics and chemistry courses it's very common to take the theta variable as the angle bending down away from the z-axis and phi to be the azimuthal angle that rotates around the z-axis. So just to clarify that those are the angles we're using, so in these polar coordinates then x, y, z becomes r squared or v squared sine theta dv d theta d phi. So the rest of this remains the same. So I've got m over 2 pi kt to the 3 halves, e to the minus 1 half mv squared over kt. Still is a velocity distribution. That's the probability that a molecule has this particular v and theta and phi. And again, I'm not interested in the theta and phi. I don't care what direction a molecule is interested in. I only care what speed it's moving with. So what I want to do in order to obtain the probability distribution for just the speed is I want to get rid of the thetas and the phi's. I want to integrate them away. So I have this expression if I integrate away the theta and phi variables. Just reminding us what the limits of those integration are in case you're used to the opposite convention. I want to take the integral of this thing, note that I'm only doing two integrals. I'm not doing three. I'm doing the theta integral and I'm doing the phi integral. I'm leaving the v part alone. So I want to do this integral of m over 2 pi kt to the 3 halves and Gaussian times v squared sine theta. So I have all three differential terms here, dv, d theta, d phi. But I want to integrate only two of them. And what that will give me after integrating away the theta and phi, the theta and phi variables will go away. I'll still be left with the dv term that I didn't integrate away. And that's telling me the probability of having a particular magnitude of a velocity, the probability of having a particular speed. And luckily these integrals are not terribly difficult. The phi integral is the easiest one. There's no phi's anywhere inside this integral. So d phi integrated from 0 to 2 pi is just 2 pi. Theta, I do have this sine theta term. We can do the integral and find that sine theta integrated from 0 to pi is 2. That's not a terrible integral, difficult integral to do. You can check that on your own. Or the shortcut to not having to bother with that integral is by integrating over all theta and phi values. Essentially what we're doing is we're calculating the surface area of a sphere. So the surface area of a sphere with unit radius would be 4 pi. So by integrating all possible angles, the number of places on that sphere is 4 pi. So that's a shortcut to doing those two particular integrals, the theta and phi integrals. What we have left after that, we still have this m over 2 pi kT to the 3 halves. We have a v squared that I'll pull forwards a little bit. Make it precede. This Gaussian e to the minus 1 half mv squared over kT. dv, because I haven't integrated away the velocities. So what we found is that the probability of having a particular speed is some constants times v squared e to the minus v squared with some additional constants up in the exponent. So this is what we were after that's a little more useful than a velocity probability distribution. This is the probability distribution for molecular speeds instead of velocities. It's useful enough that it gets a name. That's called the Maxwell-Boltzmann distribution. That's because it's essentially a combination of two different things. Boltzmann tells us how to calculate probabilities of anything, e to the minus energy over kT. And Maxwell's contribution was applying this to the molecular speeds. But if we want to know quantitatively the probability for having a particular speed or a range of speeds, then this is the probability distribution that we would need to use. I'll point out what this function looks like. So notice that now we're talking about speeds instead of velocities. The lowest the speed can be is zero. We can't have a speed in the negative direction because speeds don't have directions. They're not velocities. So we can have speeds anywhere from zero up to infinity. And if I plot v squared times a Gaussian instead of having a maximum at the origin, having the v squared makes it have small values at the origin. So the graph looks something like this. It's got a longer tail at large speed than it does at small speed where it has to die by the time it reaches the origin because v equals zero kills the function. So what we see is that for small speeds, if I ask what's the probability that a molecule of gas in the air, nitrogen gas, let's say, with a particular mass at a particular temperature, what's the probability that it has a speed of 5 or 10 or 20 meters per second? That's going to be a relatively small number. The probability is small that it's moving slowly. It's also going to have a relatively small probability of moving very quickly. If I ask what's the probability that it's moving at 5,000 meters per second or 10,000 meters per second, that's way off on the large speed tail of this distribution. So I have small probability of having a small velocity, small probability of having a very high speed, and there are some intermediate ranges of speeds where I have a significant probability. The reason for those two small probabilities are different. The curve dies. This part of the curve is behaving like e to the minus v squared, roughly, like a decaying Gaussian. So the reason that the speed is not found to be with large values very often is because of this Boltzmann factor. Boltzmann says molecules with very high speeds have very high kinetic energies and that's just not very likely to happen. On the other hand, over here, on this side of the curve, it's behaving roughly as v squared before the exponential kicks in and begins to kill it. So the reason that the molecules don't have small speeds very often is because of this term, and that's essentially came from the integration factor in spherical polar coordinates. So the reason we don't find gas molecules moving at 5 or 10 meters per second very often isn't because Boltzmann says that's not likely. Boltzmann is perfectly happy to let molecules have small speeds, but it's because there's relatively few ways to have that speed. A molecule moving with 10 meters per second, the surface area of that sphere with radius 10 meters per second is much smaller by this factor v squared than a surface area of a sphere with radius 100 meters per second or 200 meters per second. So the larger the velocity, the more ways there are to have velocities with that speed, but also the larger the speed, the more energy it has. So the sweet spot in the middle is when the speeds are large enough that there's many different velocities consistent with that speed, but small enough that the Boltzmann penalty for having that speed is not too much. Combine those two things together and we have the Maxwell-Boltzmann distribution. What we can use that distribution to do now is to go back and calculate things like the average speed for molecules in a gas at a particular temperature, and that's the sort of problem we'll tackle next.