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Thermodynamics part 2: Ideal gas law

To begin, Sal solves a constant temperature problem using PV=PV. Then he relates temperature to kinetic energy of a gas. In the second half of the video, he derives the ideal gas law. Created by Sal Khan.

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Video transcript

Welcome back. In the last video, I told you that pressure times volume is a constant. That if you increase the pressure-- or if you increase the volume, you're going to decrease the pressure. Hopefully, you got an intuitive sense why. Or likewise, if you squeezed the balloon, or the box, and there are no openings there, then the pressure within the box would increase. With that said, let's see if we can do a couple of fairly typical problems that you'll see. So let's say that I have a box, or a balloon, or something, and it has a volume, and so let me call this the initial volume. My initial volume is 50 cubic meters, and my initial pressure is 500 pascals. Just so you remember, what's a pascal? That's 500 newtons per meter cubed. I take that box, or balloon, or whatever, and I compress it down to 20 meters cubed. So I compress it, so I squeeze it-- that was the first example that I gave last time. It was the same container, and I squeeze it down to 20 meters cubed. What's going to be the new pressure? You should immediately have an intuition-- what happens when you squeeze a balloon? It becomes harder to do it. What's going to be the new pressure? It's definitely going to be higher-- when you decrease the volume, the pressure increases are inversely related. The pressure's going to go up, and let's see if we can calculate it. We know that P1 times v1 is equal to some constant, and since we have no aggregate change in energy-- I'm just telling you that the box is squeezed, I'm not telling you whether it did any work, or anything like that-- the same constant is going to be equal to the new pressure times the new volume, which is equal to P2 times V2. You could just have the general relationship: P1 times V1 is equal to P2 times V2, assuming that no work was done, and there was no exchange of energy from outside of the system. In most of these cases, when you see this on an exam, that is the case. The old pressure was 500 pascals times 50 meters cubed. One thing to keep in mind, because this equivalence is not equal, and we're not saying it has to equal some necessary absolute number-- for example, we don't know exactly what this K is, although we could figure it out right now-- as long as you're using one unit for pressure on this side, and one unit for volume on this side, you just have to use the same units. We could have done this same exact problem the exact same way, if instead of meters cubed, they said liters, as long as we had liters here. You just have to make sure you're using the same units on both sides. In this case, we have 500 pascals as the pressure, and the volume is 50 meters cubed. That's going to be equal to the new pressure, P2, times the new volume, 20 meters cubed. Let's see what we can do: we can divide both sides by 10, so we can take the 10 out of there, and we could divide both sides by 2, so that becomes a 250. We we get 250 times 5 is equal to P2, and so P2 is equal to 1250 pascals, and if we kept with the units, you would have seen that. When I decreased the volume by roughly 60%, I have the pressure actually increased by 2 1/2, so that gels with what we talked about before. Let's add another variable into this mix-- let's talk about temperature. Like pressure, volume, work, and a lot of concepts that we talk about in physics, temperature is something that you probably are at least reasonably familiar with. How do you view temperature? A high temperature means something is hot, and a low temperature means something is cold, and I think that also gives you intuition that a higher temperature object has more energy. The sun has more energy than an ice cube-- I think that's fair enough. I think you also have the sense that-- what would have more energy? A 100 degree cup of tea, or a 100 degree barrel of tea. I want to make them equivalent in terms of what they're holding. I think you have a sense. Even though they're the same temperature, they're both pretty warm-- let's say this is 100 degrees Celsius, so they're both boiling-- that the barrel, because there's more of it, is going to have more energy. It's equally hot, and there's just more molecules there. That's what temperature is. Temperature, in general, is a measure roughly equal to some constant times the kinetic energy-- the average kinetic energy-- per molecule. So the average kinetic energy of the system divided by the total number of molecules we have. Another way we could talk about is, temperature is essentially energy per molecule. So something that has a lot of molecules, where N is the number of molecules. Another way we could view this is that the kinetic energy of the system is going to be equal to the number of molecules times the temperature. This is just a constant-- times 1 over K, but we don't even know what this is, so we could say that's still a constant-- so the kinetic energy of the system is going to be equal to some constant times the number of particles times temperature. We don't know what this is, and we're going to figure this out later. This is another interesting concept. We said that pressure times volume is proportional to the kinetic energy of the system-- the aggregate, if you take all of the molecules and combine their kinetic energies. These aren't the same K's-- I could put another constant here and call that K1. And we also know that the kinetic energy of the system is equal to some other constant times the number of molecules I have times the temperature. If you think about it, you could also say that this is proportional to this, and this is proportional to this. You could say that pressure times volume is proportional to the number-- and these are all different proportional constants, and we'll figure out this exact constant later-- so we could say that pressure times volume is proportional to molecules we have, times temperature. And we said that we can view temperature as energy per molecule. Another way we could say is that if this constant is constant, which is by definition, and the number of molecules is constant-- we have PV over temperature. Pressure times volume over temperature is going to be equal to something times the number of molecules, so we could say that's some other constant, like k4. This is another interesting thing to think about: we said pressure times volume is equal to pressure times volume, and now we added temperature into the mix. We could say P1 times V1 over T1 is equal to P2 times V2 over T2. Does this make sense to you? What happens if I have another box, and I have my particles bouncing around like always. I have some volume, and some amount of pressure-- what happens when the temperature goes up? What am I saying? I'm saying that the average kinetic energy per molecule is going to go up, so they're going to bounce against the walls more. If they bounce against the walls more, the pressure's going to go up, assuming volume stays flat. Another way you could think about it-- let's say the temperature goes up, and the pressure stays flat. So what did I have to do? I just said if the temperature goes up, the average kinetic energy of each molecule-- they'll bounce more. In order to make them bounce against the sides of the walls as often, I'd have to increase the volume. If you hold pressure constant, the only way you can do that is by increasing the volume while you increase the temperature. Let's keep this in mind, and we will use this to solve some pretty typical problems in the next video.