 So, as one last commentary on the Maxwell-Boltzmann speed distribution for molecules, let me point out that this distribution of speeds of molecules, so we know roughly what that distribution looks like. If I were to plot that probability distribution, it's somewhat asymmetrical with a longer tail on the high speed side than the low speed side, mainly because that distribution needs to reach zero by the time we have zero velocity. The shape of that distribution and the average speeds of the molecules depend on several variables. They depend on things like the mass, because, for example, if we remember what the root mean square speed of a molecule in a gas is, depends on the mass, so molecules with larger masses because M is in the denominator inside the square root, they have smaller root mean square speeds. So that means heavy molecules have slower root mean square or average or most probable velocities and lighter molecules have faster velocities. So that's why, for example, when you breathe helium from a helium balloon, your voice sounds high-pitched is because the molecules of helium are moving much faster than the molecules of oxygen and nitrogen in the air that you're normally breathing. So certainly the mass has an effect on the distribution of the speeds of the molecules, but also the temperature has an effect as well, so temperature shows up in the distribution as well. And from the most simplistic point of view, when the temperature goes up, you expect the molecules to move faster. You expect the root mean square and the average and most probable speeds to be higher because T is in the numerator of the square root, but there's another somewhat more subtle and complicated effect as well that we can see if I plot what this Maxwell-Boltzmann distribution looks like for several different temperatures. If I plot it at one temperature, the temperature T I've written here looks like this. If I do it again at a, let's say, a colder temperature, then the distribution is going to look like something like this. It still has the same shape. It's still asymmetric. It's got a bigger tail on the large speed side than the small speed side. The peak in the distribution has shifted down to lower speeds, like we'd expect, because I lowered the temperature. This cold temperature is less than the temperature for this curve. Then I've lowered the most probable velocity to a lower value. So I've shifted it lower on the speed axis. Likewise, the root mean square and the average speed will be lower as well. If, on the other hand, I do this for a relatively hot temperature, one that's larger than this temperature in the middle, then what this distribution is going to look like is something like this. The maximum is going to be shifted to larger speeds, as we'd expect. This curve has to get lower because since this is a graph of probability, the areas under each of these curves should be exactly the same. I have 100% probability of having some speed. What this shows us is at cold temperatures, I have a relatively narrow band of speeds at which I'm likely to find the molecules. At intermediate temperatures, that range of speeds at which I'm likely to find the molecule has increased. At hot temperatures, the range has increased considerably. Not only should you think about increasing the temperature as increasing the average speed of the molecules, but you should also think of it as increasing the range of speeds that those molecules can have. That's one very important way to think about temperature, not just changing the magnitude of the velocities, but changing the width of that distribution as well. If you think about temperature in that way, it will help us as we understand several different thermodynamic effects and how they depend on temperature in the future.