 So now we understand a little bit about ideal gases and not just PV equals NKT or PV equals NRT, as you knew about ideal gases from general chemistry, but now the whole process of starting from assuming no potential energy, doing some quantum mechanics to dissolve for the 3D particle in a box energies, using those to get a partition function and thermodynamic connection formulas to get energy and pressure for an ideal gas. So that's the full process we've considered for an ideal gas. But of course real-world gases, real gases are not ideal, they don't behave ideally. In particular they don't obey the two different assumptions we've made about ideal gases which are that they have no intermolecular interactions, the potential energy is zero, and also that they have no molecular volume that they don't take up any space. So when we get to be able to talk about real-world gases we need to do a little better than the ideal gas model. So one approach might be to go back to the beginning and correct our assumptions and say, well rather than V equals zero, no potential energy, we can say that potential energy is something non-zero, a better estimate for what the potential energy of a real gas would be and then repeat these steps, do some more quantum mechanics, convert that to a partition function, turn that into thermodynamics. And we can do that, that's that's possible to come up with more sophisticated models by improving the potential energy like this, but that's fairly difficult work. So we can also consider a somewhat easier approach which is to jump to a later stage in this process and look at something like the partition function and say what would this partition function look like instead if we didn't have zero molecular volume and zero for the intermolecular interactions. So that's what we can do, we can consider, here's a diagram of a box containing a bunch of gas molecules and instead of assuming that molecules have no volume, don't take up any space, we can now allow them to have a finite molecular volume. So instead of molecules being point particles, we can have each of these molecules be finitely sized objects that take up some space. In fact, if we've got let's say this is n molecules in some volume v, I can say each one of these molecules itself takes up some little bitty volume b, lowercase b represents the volume of one of these molecules. So the effect that that has is that instead of v being the volume of the container, actually the empty space in that container is v minus the volume of the that's taken up by the individual molecule. So if I have n of these molecules each of which takes up a volume v then it might seem reasonable to instead of using v for the total empty volume of this box to use the excluded volume instead or v minus the excluded volume, total volume of the box minus the volume taken up by n molecules each of which has this finite molecular volume v. We can also allow for there to be intermolecular interactions. So if I allow these molecules to interact with one another, let's say those interactions will normally be favorable interaction. So each of these molecules is interacting with the ones nearby being attracted to it, for example, and I can say let's let the total energy of the system be not zero but some total amount of energy, which would be n times the energy of each individual molecule on average. And if I say that the energy of one of these molecules I can I could just say since it's an attractive interaction, the molecules are being attracted by many of their nearby neighbors, I can just make that a negative constant. I could say just like I chose some constant b to represent the size of an individual molecule, I can choose a constant a to represent the energy, but I can do a little better than that. I know that the amount of this interaction, because it interacts with each of its neighbors, especially the nearby neighbors, that's going to be proportional to the number of molecules in the box. And it's also going to be inversely proportional to the volume of the box. So essentially what I've done here is I've made that energy proportional to the number density, the number of molecules per unit volume. And the reason I know that the proportionalities must be in this form is if I were to take this box, if I were to double the number of molecules in the box, each one of these molecules would then be interacting with twice as many nearby molecules, so it's proportional to the number of molecules. Or I could leave the number of molecules the same, and if I shrink the size of the box by half, then each region of space is occupied still by twice as many molecules. So if I have the volume, the interactions are still going to double, because there's twice as many nearby molecules to interact with. So I made the assumption that the interaction energy is proportional to, in some constant, negative proportional to the density of molecules in this box. And then if I multiply that by n, so minus a n squared over v, would be the total amount of energy, the non-zero amount of potential energy in this box. And then what I can say is that the partition function, so instead of the ideal gas partition function, I can say that the partition function looks a lot like this, but with some modifications to acknowledge the fact that I have some finite molecule volume and some intermolecular interactions. So these first two terms, the 1 over n factorial, that accounts for the fact that the molecules are indistinguishable, that's still the same. The 2 pi mkT over h squared to the 3n over 2, there's no reason to change anything there. The term that used to look like volume raised to the nth power, volume is now v minus nb. So if I raise v minus nb to the nth power, and I have one more correction to make, which is to include the fact that I have this interaction and partition functions, remember, are sums of Boltzmann factors. So if each of these molecules involves some energy, the total partition function is going to involve a term that looks like e to the minus total energy over kT. But the total energy looks like minus a n squared over v, so this negative sign and that negative sign cancel each other, and I have a n squared over v, and I have to divide additionally by kT in the Boltzmann factor. So that's e to the minus total energy minus a n squared over v divided by kT. So this would be a new partition function. I've gone in and I've made some ad hoc adjustments to the ideal gas partition function, which I could derive from quantum mechanics, acknowledging these things that I know are true about real world gas molecules. So that model is called the Van der Waals model. So rather than treating the molecules as an ideal gas, if we treat the molecules using the Van der Waals model, then that's using this partition function here. So there's a couple things we should point out about this partition function, about the Van der Waals model, namely that it is not derived theoretically. It's an ad hoc model. I've looked at what this ideal gas partition function that I know where that one came from, from doing quantum mechanics, and I've just made some corrections to it that I think are reasonable ways to correct for the fact that the molecules have some volume and interact with each other. So it's no longer a quantum mechanical model. It's no longer theoretically, as theoretically motivated. And the other thing to notice about this model is that it has these constants that appear, the B constant and the A constant. I have no way of knowing what those are or what they should be other than the assumptions I've built into the model, namely that B represents the size, the finite size of the molecules and A represents the strength of these intermolecular interactions. So certainly if I have one gas that interacts more strongly than another, I would expect the A value of the A constant to be larger for the strongly interacting gas. If I have one gas whose molecules I expect to be larger than another, I'd expect its value of B to be larger. But these are essentially just empirical constants. I can choose the values of A and B so that this model predicts the properties of the gas reasonably well. So it's an empirical ad hoc model much less theoretically motivated than the ideal gas model. And that empirical nature of the model, namely that I need to use experimental values to determine what A and B should be, brings up the fact that we are interested in knowing what are the experimental properties of this gas. Now that I have a partition function, I can use those to obtain an energy or a pressure just like we did for the ideal gas using the thermodynamic connection formulas. So that's the next thing we'll do in the next video lecture.