 All right, so our pressure volume phase diagrams tell us about the difference between liquids and gases with a phase coexistence region in between and supercritical fluid at high temperatures. We can understand those phase diagrams quite a bit better. It turns out if we use the Van der Waals equation of state rather than the ideal gas equation of state, it turns out the Van der Waals equation of state will help us understand not just the properties of gases, what we're used to thinking of the Van der Waals equation is telling us about, but also indeed something about liquids in this phase coexistence region and supercritical substances as well. So remember that we have several different ways of thinking about the Van der Waals equation. These are all just different rearranged versions of the same equation. We have a version like this that looks a lot like pressure times the volume equals RT. So it's reminiscent of the ideal gas law. We have this version that's the one that we would want to use if we want to solve for pressure as a function of V and T. That describes pressure as a function. So that's the isotherms that we'll graph in just a second that draw a graph for pressure as it depends on volume at some temperature. But we also have this way of thinking about it that's going to turn out to be useful where we've rearranged it to look like a cubic expression in volume that would be what we would need to use if we wanted to solve for the molar volume. So let's start by thinking about this one, this cubic expression. What that means is if in fact we want to think about the function V bar, the molar volume as a function of P and T, to solve this cubic equation. A cubic equation has three solutions, has three roots. So what that means is sometimes we'll solve for, if I give you a pressure and a temperature, turns out sometimes you can find three solutions to that equation, three different values of the molar volume that solve those conditions equally well. So sometimes we have three real solutions, or as you remember from Algebra sometimes those three solutions might only be one real solution, which means that we have a pair of imaginary solutions as well. So thinking only about the real solutions we can solve for the molar volume and sometimes get one answer or sometimes get three answers. So what does that mean? That means if I give you a pressure, if I say here's a pressure and I give you a temperature, perhaps you can come up with three molar volumes or perhaps you can come up with one and we'll see what that looks like on this graph. So what we need to do is draw some of these P as a function of, this graph is set up for P as a function of V. So if we use this function to draw the graphs, let's draw some curves at a variety of temperatures, draw some isotherms on this curve. So the first, because it's easiest, we'll think about an isotherm that's at a temperature well above the critical temperature. If we're at a very high temperature, so this term that involves the T is quite large, I can make that one large compared to this one. So this term becomes negligible and the pressure is going to look essentially just like 1 over V. It behaves a lot like an ideal gas at very high temperatures. So there's an isotherm for how the pressure depends on volume at high temperatures. And that is what we'd expect to see from an ideal gas. If I give you a pressure and say find a point on this isotherm that has this pressure, you can find exactly one point on this curve and read down and obtain the volume. So there's one solution at that pressure and that temperature. But this is a cubic function in V, or if we look at it here, there's a 1 over V and a 1 over V squared. What that means is if I graph this expression at lower temperatures, I can get equations that look like this. So that oscillatory function might be a little bit reminiscent of a cubic. It goes down and then back up and then down again. That's an example of the type of curve we might get by plotting this graph at a lower temperature where both the negative term from attractive interactions and this first term are significant. So notice that the way I've drawn this curve intersects the liquid and gas region of this phase coexistence curve. What it also means is that if I provide you a pressure, let's say this pressure, and a temperature, let's say the temperature of this isotherm that's below the critical temperature, if I say find me a point on this graph that has this value for the pressure, you can find me several values. You can find me three, in fact, values where on this isotherm, the pressure is equal to the pressure that I gave you. So that's the case where there's three solutions. If I were to give you this pressure, here's a pressure, here's a different pressure. If I give you this pressure, there's only one solution. If I give you this pressure, there may be three solutions. So that's what it means to have this cubic equation maybe have three or one solutions on this graph. We can draw a few more cases in between. At high temperatures, the graph might begin to wiggle a little bit without actually executing any minima and maximum. Just begins to look not quite like an ideal gas. Turns out there's a particular curve exactly at the critical temperature where it, as we've seen before, where it just kisses the top of this phase coexistence region, becomes horizontal just for exactly one point. undergoes an inflection point, changes from being concave up, up to this point to concave down, just beyond that point. So the critical temperature has a lot of interesting features along that isotherm. So let's see what we can understand about the liquid gas coexistence regions and the liquid and gas phase behavior from these Van der Waals curves. In cases where there are three real solutions, in cases like this one, where I've said, find me a pressure. Find me a volume, either this volume, or this volume, along this isotherm that has this pressure. There were three solutions. The reason the Van der Waals equation is predicting three solutions, turns out it would be better if it didn't predict three solutions. It would be better if it predicted an infinite number of solutions. If you remember what the actual isotherms look like, not for a Van der Waals gas, but for an actual substance, the isotherms look like this. We have gas like behavior up until the vapor pressure, so I can identify this particular pressure with the vapor pressure at this isotherm. Up to the vapor pressure, the volume will drop without increasing the pressure as the gas is compressed and condensed into a liquid. And then once we've converted it all into a liquid, it'll begin to rise very steeply as I compress the mostly incompressible liquid. So what the Van der Waals equation is attempting to do is to describe this behavior where it undergoes a discontinuity in the slope, becomes horizontal, another discontinuity in the slope, and behaves like a gas. But a cubic equation has no hope of actually ever being horizontal for a long stretch of time like this. So it does the best it can, and the way to do that is change from negative slope to positive slope so that it can begin decreasing again over here. So that points out a few things. Number one, the Van der Waals isotherm is sort of attempting to predict these tie lines in the region where it undergoes what we call these phase loops. So the name for this region where it oscillates below the vapor pressure and then back above the vapor pressure before it continues decreasing as a gas should. Those are called phase loops. Notice that in some region inside this liquid gas coexistence region, in some range of volumes, the Van der Waals equation predicts that the pressure actually increases as we increase the volume. So in this region up to about here, the Van der Waals equation is predicting that the slope of this curve is positive. Pressure is increasing when we increase the volume. If you think about it, that's not the way anything physical behaves. If I have a liquid and I increase its volume, the pressure drops. Or maybe the easier way to think about it is I increase the pressure on it, its volume will decrease. Likewise, for gas, if I increase the pressure, the volume will decrease. So that's not correct. In fact, we know what should, so this is for Van der Waals. What should happen is dp dv, if I do that at constant temperature along one of these isotherms, that's related to the isothermal compressibility. It's negative v times the isothermal compressibility and no substance has a negative isothermal compressibility. So this negative sign means that slope should always be negative. The real substance will always decrease in volume as I increase the pressure and vice versa. So that's the way it should behave. The Van der Waals behavior in this region right here is doing something incorrect. But again, that's just the best a cubic equation can do at trying to make a horizontal line. It has to go down, then up, then down again. All right, what else can we learn from these Van der Waals isotherms? We've talked about the critical temperature. The critical temperature is the point at which the equation becomes horizontal for exactly one moment. Instead of exhibiting a minimum and a maximum at a different temperature, those isotherms might have minimum and maximum. They get closer together, closer and closer. As the minimum and maximum get closer together and meet at one point, that point is the critical point. And the temperature at which that happens is the critical temperature. The supercritical fluid region, again, the supercritical fluid is any fluid with a temperature greater than the critical temperature. So any one of these isotherms at a temperature above the critical temperature, we call a supercritical fluid. And one more thing to point out about these curves is the way I've constructed the curve, it's relatively easy to see that the Van der Waals isotherm shows the overlap between the coexistence line and it drops below, rises above that coexistence line. If I were to give you, for example, a Van der Waals isotherm where I hadn't drawn the phase coexistence region, you might not know where to draw the horizontal line. Does it increase to here before cutting across? Does it increase to here before cutting across? And the thing to know about that is the position you draw the tie line connecting the gas phase behavior to the liquid phase behavior is at the point where the two portions of the phase loop have exactly equal area to one another. So that process is called the Maxwell Equal Area Construction. So you can, in fact, build the whole phase diagram from scratch using the Van der Waals isotherms. If you know the A and B constant for Van der Waals gas plot lots of these isotherms, for each one of these isotherms that exhibits phase loops, find where to draw a line that leaves an equal area below and above the curve. And that's where you draw the tie line and that helps you find the boundaries of your phase coexistence region. So turns out that not just the Van der Waals equation is not just an equation that describes gases, it does a pretty good job at describing liquids and tells us when liquids coexist with gases as well.