Kinetic molecular theory and the gas laws | AP Chemistry | Khan Academy

In other videos, we touched on the notion of kinetic molecular theory, which I'll just shorten as KMT. It's just this idea that if you imagine a container—I'll just draw it in two dimensions here—that it contains some gas. You can imagine the gas as being these particles, where their collective volume is much smaller than the volume of the container. The temperature we're dealing with is related to the average kinetic energy of the particles.

These particles are all moving around, zooming around, and each would have some kinetic energy. Remember, kinetic energy is calculated as ( \frac{mv^2}{2} ), so each of these particles would have some mass and some velocity, but they could all have different velocities for sure. Even if they're the same type of particle, if they're different types of particles, they can have different masses as well. But the average of these kinetic energies across all of these particles is proportional to temperature when measured in Kelvin.

And pressure—remember, pressure is nothing but force per unit area. You can imagine the surface of our container; this could be some type of a cube. So, I can draw it in three dimensions here—there's some area over here. You have your particles (let me do this in a different color). These particles are constantly bouncing off of it, and there's way more particles than what I have drawn here.

So, at any given moment, you're having some particles bouncing off this side of the container—actually, all sides of the container—and these are perfectly elastic collisions. They're preserving kinetic energy, and so they're applying some force collectively on this area. So, the pressure is because of these particle collisions on the surface.

Now, what I want to do in this video is take these ideas that we conceptualize in kinetic molecular theory and understand why the ideal gas law ( PV = nRT ) makes sense when we conceptualize the world. Just a reminder: ( P ) is pressure, ( V ) is volume, ( n ) is the number of moles of whatever gas we're dealing with (the amount of that gas), ( T ) is the temperature in Kelvin, and ( R ) is just the ideal gas constant— that's just whatever constant you're using so that the units all work out together.

So, let's first think about how pressure relates to volume if we were to hold everything else constant. The ideal gas law tells us that pressure times volume is going to be equal to this constant. I could even just write a ( K ) here for a constant, but that would also mean we could divide, let’s say, both sides by ( V ). We could say that pressure is equal to some constant over ( V ).

Another way to think about it is that pressure is proportional to the inverse of volume. You could also write this—if we divide both sides by ( P )—that volume is proportional to the inverse of pressure. Does that make sense from a kinetic molecular theory point of view? Pause this video and think about it.

Well, imagine we have our original cube right over here, and I have the same number of particles. They have the same average kinetic energy, but let's say I were to increase the volume. If I were to make the volume go up—maybe put the exact same number of particles with the same temperature in a larger container—then, at any given moment, you're just going to have fewer bounces of particles off of the container because they just have more room to go in that volume.

Even the surface area of the container is going to be high as well. So, it makes sense that if the volume goes up, the pressure is going to go down. You could think about it the other way: if you make this smaller, that same number of particles with the same average kinetic energy is going to bump into the container that much more often, and that’s going to increase the pressure. So, volume goes down, pressure goes up, and this relationship—that pressure is inversely proportional to volume or vice versa, if you hold everything else constant—that's often known as Boyle's law.

Now, another relationship: what if we were to hold volume and the number of moles constant, and we want to think about the relationship between pressure and temperature? If this is constant, the ideal gas law would say that pressure is going to be proportional to temperature, or that temperature is proportional to pressure. Does that make sense?

Well, let's go back to our original container. If you were to increase the temperature, that means that the average kinetic energy is increased. That means that these particles, when they hit the side of the container, are going to hit it with more velocity. That means that, at any given moment, you're going to have more pressure exerted on the side of the container.

You could go the other way: think about lowering the temperature. If the kinetic energy goes really low, then these particles are just slowly drifting, and the speed with which they are hitting the side of the container is going to go down. So, the pressure would go down. It completely makes sense: if temperature goes up, pressure goes up; if temperature goes down, pressure goes down. This is often known as Gay-Lussac's law.

Now, another relationship—and I'm really just going through all of the combinations over here—what if we were to hold pressure and the number of molecules constant? We’re really looking at the relationship between volume and temperature. Once again, if ( P ), ( n ), and ( R ) are always constant—if those are constant, the ideal gas law would tell us that the volume is proportional to the temperature, once again holding everything else constant.

Well, to think about that, you can go through that same thought experiment we just had. If we increase the temperature, if these things are moving around faster, if you want to have the same amount of force per area on the container—on the side of the container—you’re going to have to increase the volume. So, this relationship, which is completely consistent with kinetic molecular theory, is often known as Charles’s law.

Now, another one is the relationship between volume and the number of moles. If everything else is held constant, the ideal gas law would tell us that volume is going to be proportional to the number of moles of our particle or of our gas that we are dealing with. That makes sense. Because, once again, you're holding everything else constant: you want pressure to be constant, temperature to be constant.

If I were to double the number of particles here, but I don’t want to change the pressure or the temperature, it makes sense that I would have to double the volume. Likewise, if I wanted to double the volume here and I didn't want to change the pressure or the temperature, I would have to put twice as many particles in there. So, I still have a sufficient number of interactions of bouncing of the particles with the sides of the container so that I have sufficient pressure. This notion is called Avogadro's law.

Last but not least, let’s say I have two identical containers. If I have two identical containers—there’s one there, there’s one over here—and actually I'm going to draw that same container a third time. Let’s say over here I have gas 1, and it has some partial—or in this case, it has some pressure due to gas 1. We’re going to assume the volume and the temperatures are the same across all three of these.

Let’s say we have gas 2, and it is exerting pressure too. If I were to take all of the gas in both of them and put them both into this third container, this third container is going to have all of the original gas 1 and all of the original gas 2. But we aren’t changing the volume, and we aren’t changing the temperature.

In any given unit area on the surface of the container, you’re going to get the collisions from particle 1, which would give you ( P_1 ) in that force per unit area, and you’re going to get the collisions from particle 2, which would give you that force per unit area. So, it makes sense that the partial pressures would add up to be equal to the total pressure in the container. This is known as Dalton’s law.

But the whole point of this video is just to appreciate that everything we've talked about with the ideal gas law actually makes a lot of sense. I would argue it makes the most sense when you think about it in terms of kinetic molecular theory.