The van der Waals equation | Khan Academy

We have so far spent many videos talking about the ideal gas law: that pressure times volume is equal to the number of moles times the ideal gas constant times temperature measured in Kelvin. What we're going to do in this video is attempt to modify the ideal gas law to try to take into account when we're dealing with real gases. Gases where the volume of the actual particles are worth considering; that we don't just say they're negligible compared to the volume of the container. Intermolecular forces are something that we would like to take into consideration.

So let's think about how we could modify this. To help us a little bit, I'm just going to solve for p. So, I'm going to divide both sides by volume. We can say pressure is equal to the number of moles times the ideal gas constant times temperature measured in Kelvin divided by volume.

So first, how would we adjust this if we want to take into account the actual volume in which the molecules can move around? Well, if we wanted to do that, we would replace this volume right over here with this volume minus the volume of the actual particles. So, what's the volume of the actual particles going to be? Well, it's going to be the number of particles times some constant based on how large each of those particles are, maybe on average. Let's just call that b.

So we could view this as a modified ideal gas law equation where now, all of a sudden, we are taking into account the fact that these particles have some real volume to them. But, of course, we also know it's not just about the volume of the particles; we also need to adjust for the intermolecular forces between the particles. We know that in many cases, those intermolecular forces are attractive forces and so they would take away from the pressure. Therefore, we need some term that accounts for that—a term that accounts for taking away the pressure due to intermolecular forces.

Now I know what some of y'all are thinking: do we always subtract? Might not there be some situations in which we actually have repulsive forces between particles and it would actually add to the pressure? There could be scenarios like that; you could imagine if they all have a strong negative charge, they want to get away from each other as far as they can, and that could actually add to the pressure. But in that situation, we could subtract a negative, and then that would be additive.

Now, how can we take this into consideration? We know from Coulomb's law that the force between two charged particles is going to be proportional to the charge on one particle times the charge on the other particle divided by the distance squared. Now, obviously, if we're dealing with a lot of particles in a container, we're not going to be able to think about the forces between any two particles.

But one way to think about it is in terms of how concentrated the particles are generally. So we're trying to think of a term that takes into account the intermolecular forces or how much we're reducing the pressure because of those intermolecular forces. Maybe that can be proportional to not just the concentration of the particles—and that would be the number of particles divided by the volume—but that times itself because we're talking about the interaction between two particles at a time, very similar to what we see in Coulomb's law.

Because the end of these really are just Coulomb forces. So this thing right over here is going to be proportional to the concentration times itself. Or we could maybe call this some constant for the proportionality times n over v squared, where a would depend on the attractive forces between gas particles.

What we have just constructed, and let me rewrite it again, this ideal gas equation—and actually, let me put this orange term back on the left-hand side—so if I write it this way: that pressure plus a times n over v squared is equal to n rt over the volume of our container minus the number of molecules we have times some constant b based on how large on average those molecules, or those particles, are.

This right over here is a pretty good compensating equation for when we're dealing with more real gases—ones that have intermolecular forces and one where the actual particles have volume. This actually does a pretty good job, and there's a name for it: it's called the van der Waals equation.

There are many different ways you might see it. You could see it written like this, or we could try to take this blue part and get it on the left-hand side so it really looks like what we saw at the top, where there it would be written as, and I'll write it actually this way: pressure plus some constant times the density squared—let me close that parenthesis—times the volume minus the number of molecules times some constant is going to be equal to nRT.

All of this looks really complicated, but at the end of the day, it is just our ideal gas law modified for intermolecular forces and the actual volume of the particles.