 All right, so far we've seen two different equations of state. We know an ideal gas obeys pv equals nrt, or p is equal to rt over the molar volume, both different equivalent ways of describing the ideal gas equation of state. We've also seen the Van der Waals equation of state, which has introduced corrections for the finite size of the molecules as well as the attractive interactions between the molecules and the gas. And the important differences we've seen so far between these are certainly the Van der Waals equation is more accurate than the ideal gas equation of state. The ideal gas equation has the advantage of having been derived sort of from scratch, from quantum mechanics without any assumptions, sorry, it has assumptions built into it, but without any need for parameters like this a and this b. Whereas the Van der Waals equation of state is empirical parameters, has these two empirical parameters a and b. So we can't use the Van der Waals equation of state without knowing the values of a and b, and those usually come from having done some experiments on a particular gas to measure it. But of course, both of these models are only models, they're only approximations. Neither one is perfectly 100% correct in describing the properties of a gas under all different states, all different conditions. So it may not surprise you to hear that there's a range of other models that people have developed that are in pursuit of even more accuracy in describing how the pressure of a gas depends on its volume and its temperature. So we can describe a few of these just to get a taste for what they're like. So the first of these we can discuss is something called the Redlich-Quang equation of state. All of these equations of state have the form of being able to predict the pressure based on the molar volume and the temperature. The Redlich-Quang equation of state in particular, first term looks exactly the same. The effect of the finite volume of the molecules is described well enough by this RT over V bar minus B term that there's no need to adjust or fix that term. The next term looks fairly similar. There's an A constant divided by a V bar and then another V bar, V bar squared. But now with the additional adjustment that I need to account for some effect of the finite molecular volume in the intermolecular interaction term. And there also appears a square root of T in the denominator of this intermolecular interaction term effectively accounting for the fact that those intermolecular interactions may have greater or lesser importance at different temperatures. So things to point out about this equation are it's more accurate than the Van der Waals equation at least under many different circumstances. So it's better in that sense. On the other hand, it's gotten a little bit more complicated. It's both mathematically more complicated, a little more tedious to plug in these extra few terms. And it's also more complicated in the sense that we've begun to lose a little bit of a sense of what these empirical parameters mean. So the corrections like introducing square root of T and introducing the B down here, those are done mainly just because they make the answers better. In part because there's some physical reason for including a square root of T or including the volume in this intermolecular interaction term. But once we start making these more accurate models, our main goal is to make them accurate without being as concerned about what the parameters mean. And we can see that a little bit further if we study another model. It's called the Peng Robinson model. So once again we can predict the pressure from V bar and T. The first term looks exactly the same. We're going to again subtract a term that's due to the intermolecular interactions. That term still has an A on top. It still has some V squareds on the bottom. But now rather than having the temperature show up in the denominator, I've got some extra complications in the denominator. B times the quantity V bar minus B. And in the numerator, things get really complicated. So notice that the temperature dependence in this interaction term has been shifted up into the numerator. And now not only do I have an A constant and a B constant that I need, but I also have this additional constant kappa. So now I've got three parameters rather than just two, alpha and beta, A and B and kappa. I've also got the temperature scaled relative to this other temperature, something that we haven't discussed yet called the critical temperature. So this equation has clearly gotten more complicated. Again, the goal is to make it more accurate. And indeed, it is more accurate than a Van der Waals equation and often more accurate than the Redlich-Quang equation. So in the interest of having the equation be more accurate, we're willing to introduce more parameters, have those parameters be less connected to the physical meaning that they had when we first introduced them, and generally make the equation more complicated. There is, in fact, a whole long list of these equations of state that we could use. There's not just three parameters. There's five parameter and 12 parameter, much more complicated equations of state, depending on how accurate you want to get. But if we take this trend to its natural conclusion, we can say, why bother with physical parameters that still have something to do with intermolecular interactions or size of the molecules or the sphericalness of the molecules? Why not just give up on the physical meaning of the parameters altogether? And we can come up with an equation like the one we call the Virial equation, which says ideal gas was a good start. Pressure is RT over V. That's a good start. But of course, we need some corrections. If we notice that in each of these terms, one way to make the equation better was to introduce a V squared term and then make that polynomial more and more complicated. So we can just say, OK, we've got a correction that looks like 1 over V squared and let's let this constant be however big it needs to be to make this correction correct the pressure. And if that's not good enough, we can include a V bar cubed term and then a V bar to the fourth term and a V bar to the fifth term as many of these as we need. So we've written down essentially an infinite series, keep including correction terms of the order V, V squared, V cubed in the denominator, and then eventually we'll get to a point where the corrections are so small that we don't need to include them anymore. The advantage of this equation is it's in principle infinitely accurate. If we include all infinity of these terms, we can fully describe the pressure as a function of the volume and the temperature. The disadvantage is, of course, it's an infinite series and also that these B and C terms don't any longer have an interpretation that's nearly as clean as saying it's the volume of the molecule or the strength of the interaction between the molecules. So as we move down the list in this direction, we're making the models more and more accurate. On the other hand, if we move up the list in this direction, it makes it easier to do some physical interpretation of the model. If I tell you for two different gases for nitrogen gas and for gaseous benzene, for example, if I give you the value of B for those two different constants, you can make some inferences about those molecules, about which one is larger than the other. If I give you the value of A, you can make inferences about which one is more strongly attracted to itself. But if I give you the coefficients in the virial equation, if I give you the value of C or I give you the value of D, you learn very little about the physical properties of the molecule. All you learn is what values to plug into this equation to calculate the pressure. So if your goal is being able to understand what the model says about the physical properties of the molecule other than pressure, then you might be more interested in Van der Waals or the ideal gas equation of state. But if you want to get three or four or five sig figs worth of precision out of your prediction for the pressure, then you might be more interested in a model down at this end of the scale like the Peng Robinson model or the virial equation of state. So what we'll do next is not spend very much time with any of these additional equations of state, although many of them are very useful in practical situations. But we can stop talking quite so much about pressures and go on to think about additional thermodynamic properties of gases.