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# Ideal gas equation: PV = nRT

Intuition behind the ideal gas equation: PV=nRT. Created by Sal Khan.
Video transcript
Let's say I have a balloon. And in that balloon I have a bunch of particles bouncing around. They're gas particles, so they're floating freely. And they each have some velocity, some kinetic energy. And what I care about, let me just draw a few more, what I care about is the pressure that is exerted on the surface of the balloon. So I care about the pressure. And what's pressure? It's force per area. So the area here, you can think of it as the inside surface of the balloon. And what's going to apply force to that? Well any given moment-- I only drew six particles here, but in a real balloon you would have gazillions of particles, and we could talk about how large, but more particles than you can really probably imagine-- but at any given moment, some of those particles are bouncing off the wall of the container. That particle is bouncing there, this particle is bouncing there, this guy's bouncing like that. And when they bounce, they apply force to the container. An outward force, that's what keeps the balloon blown up. So think about what the pressure is going to be dependent on. So first of all, the faster these particles move, the higher the pressure. Slower particles, you're going to be bouncing into the container less, and when you do bounce into the container, it's going to be less of a ricochet, or less of a change in momentum. So slower particles, you're going to have pressure go down. Now, it's practically impossible to measure the kinetic energy, or the velocity, or the direction of each individual particle. Especially when you have gazillions of them in a balloon. So we do is we think of the average energy of the particles. And the average energy of the particles, you might say oh, Sal is about to introduce us to a new concept. It's a new way of looking at probably a very familiar concept to you. And that's temperature. Temperature can and should be viewed as the average energy of the particles in the system. So I'll put a little squiggly line, because there's a lot of ways to think about it. Average energy. And mostly kinetic energy, because these particles are moving and bouncing. The higher the temperature, the faster that these particles move. And the more that they're going to bounce into the side of the container. But temperature is average energy. It tells us energy per particle. So obviously, if we only had one particle in there with super high temperature, that's going to have less pressure than if we have a million particles in there. Let me draw that. If I have, let's take two cases right here. One is, I have a bunch of particles with a certain temperature, moving in their different directions. And the other example, I have one particle. And maybe they have the same temperature. That on average, they have the same kinetic energy. The kinetic energy per particle is the same. Clearly, this one is going to be applying more pressure to its container, because at any given moment more of these particles are going to be bouncing off the side than in this example. This guy's going to bounce, bam, then going to go and move, bounce, bam. So he's going to be applying less pressure, even though his temperature might be the same. Because temperature is kinetic energy, or you can view it as kinetic energy per particles. Or it's a way of looking at kinetic energy per particle. So if we wanted to look at the total energy in the system, we would want to multiply the temperature times the number of particles. And just since we're dealing on the molecular scale, the number of particles can often be represented as moles. Remember, moles is just a number of particles. So we're saying that that pressure-- well, I'll say it's proportional, so it's equal to some constant, let's call that R. Because we've got to make all the units work out in the end. I mean temperature is in Kelvin but we eventually want to get back to joules. So let's just say it's equal to some constant, or it's proportional to temperature times the number of particles. And we can do that a bunch of ways. But let's think of that in moles. If I say there are 5 mole particles there, you know that's 5 times 6 times 10 to the 23 particles. So, this is the number of particles. This is the temperature. And this is just some constant. Now, what else is the pressure dependent on? We gave these two examples. Obviously, it is dependent on the temperature; the faster each of these particles move, the higher pressure we'll have. It's also dependent on the number of particles, the more particles we have, the more pressure we'll have. What about the size of the container? The volume of the container. If we took this example, but we shrunk the container somehow, maybe by pressing on the outside. So if this container looked like this, but we still had the same four particles in it, with the same average kinetic energy, or the same temperature. So the number of particles stays the same, the temperature is the same, but the volume has gone down. Now, these guys are going to bump into the sides of the container more frequently and there's less area. So at any given moment, you have more force and less area. So when you have more force and less area, your pressure is going to go up. So when the volume went down, your pressure went up. So we could say that pressure is inversely proportional to volume. So let's think about that. Let's put that into our equation. We said that pressure is proportional-- and I'm just saying some proportionality constant, let's call that R, to the number of particles times the temperature, this gives us the total energy. And it's inversely proportional to the volume. And if we multiply both sides of this times the volume, we get the pressure times the volume is proportional to the number of particles times the temperature. So PV is equal to RnT. And just to switch this around a little bit, so it's in a form that you're more likely to see in your chemistry book, if we just switch the n and the R term. You get pressure times volume is equal to n, the number of particles you have, times some constant times temperature. And this right here is the ideal gas equation. Hopefully, it makes some sense to you. When they say ideal gas, it's based on this little mental exercise I did to come up with this. I made some implicit assumptions when I did this. One is I assumed that we're dealing with an ideal gas. And so you say what, Sal, is an ideal gas? An ideal gas is one where the molecules are not too concerned with each other. They're just concerned with their own kinetic energy and bouncing off the wall. So they don't attract or repel each other. Let's say they attracted each other, then as you increased the number of particles maybe they'd want to not go to the side. Maybe they'd all gravitate towards the center a little bit more if they did attract each other. And if they did that, they would bounce into the walls less and the pressure would be a little bit lower. So we're assuming that they don't attract or repel each other. And we're also assuming that the actual volume of the individual particles are inconsequential. Which is a pretty good assumption, because they're pretty small. Although, if you start putting a ton of particles into a certain volume, then at some point, especially if they're big molecules, it'll start to matter in terms of their size. But we're assuming for the purposes of our little mental exercise that the molecules have inconsequential volumes and they don't attract or repel each other. And in that situation, we can apply the ideal gas equation right here. Now, we've established the ideal gas equation. But you're like, well what's R, how do I deal with it, and how do I do math problems, and solve chemistry problems with it? And how do the units all work out? We'll do all of that in the next video where we'll solve a ton of equations, or a ton of exercises with the ideal gas equation. The important takeaway from this video is just to have the intuition as to why this actually does make sense. And frankly, once you have this intution, you should never forget it. You should be able to maybe even derive it on your own.