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# Van der Waals equation

## Video transcript

- [Instructor] So we've talked at some length about the Ideal Gas Law, where if we assume that if a gas in a container is behaving like an ideal gas, and the key assumptions there are that there's no forces between the actual gas molecules and that the volume of the gas molecules is negligible, especially when we think about it relative to the container, well in that case we can assume we're dealing with something that's kind of ideal, and we could apply the Ideal Gas Law, that the pressure times the volume is going to be equal to the number of moles times ideal gas law, times ideal gas constant, times temperature measured in Kelvin. But we've also talked about in other videos is that well, real gases don't behave that, and especially as they approach their condensation point, real gases aren't always, you can't always make these assumptions about them. And so what we're going to explore in this video is how we might be able to adapt the Ideal Gas Law. So, it's a better description of what happens with real gases, and in particular, we are going to explore the Van der Waals equation, named for Van der Waals, same guy who came up with the idea of Van der Waals forces, one of those attractive forces that make it hard or make real gases start to defy assumption number one a little bit, and what Van der Waals went about doing is, saying well let me see if I can modify this P part right over here, and this V part, so it better describes real gases. And to understand the logic of the Van der Waals equation, which we're going to see in a second, let's think about how we might want to modify P, and then we'll think about how we'll want to modify V. So let's give ourselves a container. Let's put some gas molecules in that container. So I've got some gas molecules here. And remember, pressure is force per unit area, and the pressure here is gonna come from the collisions between the gas, the collisions from the gas molecules bouncing off of the walls of the container. Every time they do that, they're gonna exert a little bit of force on that area. And so, you know, that's why the more temperature you have, these things are gonna move around faster, you're gonna have more frequent collisions, the more molecules you have, you're gonna have more things to bounce around, and so those are things that if you have more of you're gonna increase the pressure, especially if you're holding your volume constant. Now this is assuming an ideal gas, but let's think about how things would change if we had a real gas, where you do have, let's say attractive forces between them. Let's say they're Van der Waals forces. So if you have attractive forces between the actual gas molecules, we will also assume that the molecules are not attracted to the walls of the container, which isn't always the case. Sometimes they're more attracted to the walls of the container. But let's assume that they're not attracted to the walls of the container. Well, the molecules of the middle, they're gonna be attracting, we're gonna assume that they're uniformly distributed about the volume. The ones in the middle, they're gonna be pulled in every direction, and so there's not gonna be really a significant net force, but the ones near the side of the container, so let's say a molecule right over here, well, that's gonna have most of its fellow molecules are going to, in this case, be to its left, and so those are the ones that it's gonna be attracted to, so in this, and there's not gonna be too many or any in this case, to the right of it, and so it's going to have a net inward, a net inward force. Similarly this one up here will have a net inward force. This one right over here would have a net inward force, and the closer we get to the center, it's gonna start diminishing a little bit, but as you can see, because of this attractive force, these things are more likely to maybe gravitate a little bit more towards the center than to ricochet off the side. They still will ricochet off the side, these things are buzzing around, super chaotics, you know, everywhere bouncing, but if there's some small, attractive force, it might be a little bit more likely to move towards the center than to bounce off the walls of the container. And so as we talked about in a previous video, it's reasonable that if you hold everything constant, that the real pressure, the pressure real is going to be less than the ideal pressure because of, in this case, this attractive force. But, Van der Waals, said, well can I quantify this difference. Well, what if we say the real pressure, we know it's gonna be less than the ideal pressure because of the attractive forces. The real pressure is going to be equal to the ideal pressure, minus, and we try to figure out some expression that helps us take into account these net forces, especially from things that are closer to the boundaries. Well, for every gas it's gonna be different, depending on how they interact with each other, so let's just put a little constant in there and say, alright, well for any one of these molecules, the net inward force is gonna be proportional to just how many molecules it has in the container. So, especially the density. So it's going to be, and the density is a number of moles divided by the volume, and that would be for any one of these molecules, but obviously we have many molecules and the number of molecules is also going to be related to the density, so you'd want to multiply times the density again. So you could view this as, okay, we're kind of taking it into consideration any one molecule, but then we have a bunch of molecules that are all gonna have a little bit of an inward force to different degrees, and we have this proportionality factor that helps us take into consideration the type of gas that we might be dealing with. And so, you have this notion that the real pressure could be equal to the ideal pressure minus a times n over V squared. All, we can algebraically manipulate this, add this to both sides of the equation, and we could get that the real pressure plus a times n over V squared is equal to the ideal pressure. Now, what's interesting about this is, Van der Waals said, well if I assume that this is ideal pressure, and this is ideal volume, well maybe I could just replace the P sub i, the ideal pressure, with this thing right over here. And that's what he does, so we're gonna slowly reconstruct the Van der Waals equation. So instead of that, he has P sub the real pressure, plus, and this looks like this wacky stuff, hard to understand, but hopefully this makes a little bit of sense that look, the real pressure is gonna be less than the ideal pressure, it's gonna be the ideal pressure minus something, or you can say that the ideal pressure is gonna be the real pressure plus something. And that plus something is related to the density of the actual molecules, and there's gonna some constant that's given for that gas. And, I've only seen positive ones here, which imply an attractive force, but, if you had a repulsive, but I won't even get into that, I've only seen positive a values used for actual gases, so they're really talking about attractive forces there. And then we're gonna do something about the volume. We're gonna do something about the volume right over here. So we're gonna have the real, the volume sub real, we're gonna add or subtract something, and then this is going to be equal to nRT. Now, let's think about how we'd want to adjust real volume relative to ideal volume. So let me draw another container really quickly. So, in this container, I'm gonna challenge assumption two and we're gonna assume that these molecules, that their volume is not negligible. And so in this case or situation, holding everything else constant, if you want the same pressure, and everything else is constant, your real volume, in order to have the same pressure, is gonna have to be larger. The real volume, V sub r, needs to be greater than the ideal volume. Why does it have to be greater? Well, in order to have, if you don't make the volume greater, these things are gonna bump into each other a lot more, 'cause they're each taking up a lot more space than you would've assumed, than you would've assumed under the negligible volume assumption. And so if they're bumping into each other a lot more and ricocheting off of each a lot more, if you held the volume constant, you would have more pressure, but if you want to hold the pressure constant, well you've got to make the volume bigger, the real volume has to be greater than the ideal volume. And by how much? Well, the real volume could be equal to the ideal volume, and you just need to make space for the volume of the actual molecules. So the ideal volume plus space for the actual molecules. And so, we could multiply some constant times the number of molecules measured in moles. And this b would be related to, essentially the size of the molecules. You can even think of it as kind of a bigness factor. You can think of this a bit as an attractive factor, and this is a bigness factor. And once again, algebraically manipulating this, you would get the real volume minus the volume of our actual molecules, is equal to the ideal volume. And we can go back and substitute that in. This whole thing is the ideal volume, so the ideal volume is equal to the real volume minus some proportionality constant, times the number of molecules measured in moles. And this thing that I have just generated, this is the Van der Waals equation. And the reason why I went through this little, hopefully conceptual experiment thinking through it, is that, at first when you see, it looks really, really daunting until you realize that this first thing, this first thing right over here, that's just the ideal pressure, and then the second thing is the ideal volume, and then we make adjustments between the ideal and the real based on, okay, maybe we have some forces between the particles, and yeah, we have to take into account the actual volume of the particles. And once again, the Van der Waals equation, it is not perfect, but it's the one that is typically given as a, a next version to get a little bit more real than the Ideal Gas Law. You can continue to modify this, you can do computer models, you can do all sorts of things that get you even more exact, but the Van der Waals equation is a good start.