 So we understand a fair amount about phase diagrams now, how to use the Clapeyron equation or the Clausius-Clapeyron equation to predict the positions of these phase coexistence lines, the fact that the liquid gas coexistence line stops when we enter the region of supercritical fluids, stops at the critical point. These are all for a pressure-temperature phase diagram. Pressure and temperature are often the most convenient variables to use, but not always. So we can think also about what these phase diagrams look like in different coordinates, like the pressure and the volume. So to do that, what we're going to do is we'll imagine what happens as we move from one region in the phase diagram, let's say a gas, and convert it to a liquid. But rather than changing the temperature to do that, we'll do that by changing the temperature. So we're going to imagine taking an excursion from a gas phase, increasing the pressure while keeping the temperature constant, increasing the pressure until we have a liquid phase. So the first step is just to make sure we understand what that involves, compressing a gas to condense it. But let's draw some detailed pictures so that we can talk about the pressure and the volume of that system. So we're going to start. So I've got my gas in a piston. I'm starting it out in the gas phase. So this is my first diagram. I'll call this label A on my diagram. We're starting out in the gas phase. I have a gas inside some container. I'm exerting some pressure on it. I'm in the gas phase, which means that pressure must be below the vapor pressure. If the vapor pressure is the pressure at which this gas will condense into the liquid phase, if I'm exerting less pressure than that on it, it's in the gas phase. I can imagine climbing up, increasing the pressure until I get to the phase coexistence line. So my next point is going to be just as I get to the phase coexistence line. I'll do that by increasing the pressure on this piston. When I get to the phase coexistence line, the pressure is equal to the vapor pressure. So I've increased the pressure, decreased the volume of this gas. I'm at the coexistence line, so I have gas coexisting with liquid. So not only do I have gas in this box, as I just reach the phase coexistence line, I've just started to get my first droplet of liquid forming in the incentives container. I have liquid coexisting with gas. So this would be my second diagram B. That's going to be as I reach the phase coexistence line. But now imagine what happens if I continue increasing the pressure. I can't immediately jump across the coexistence line and enter the liquid phase, because I still have gas molecules. In order to convert the whole system into liquid, that has to happen bit by bit. There must be some intermediate stage at which I've turned many of the gas molecules into liquid, but not yet all of them. So I've got liquid down at the bottom of the container, gas above it. I still have coexistence between the liquid phase and the gas phase. So for this system, the pressure that is being exerted on this system must still be equal to the vapor pressure, the only pressure at which liquid and gas are allowed to coexist when the system is at this temperature. So here's a different visual representation of the system, a physically different set of conditions, but the volume has decreased, but the pressure hasn't changed. So on the pressure-temperature phase diagram, we're still sitting at the original temperature and at the vapor pressure, but we have two different ways to draw the system that can both be represented on this phase coexistence line. And that can continue all the way until... So I have quite a bit of liquid. The piston is just above the surface of the liquid with just a few molecules of gas remaining. But again, because I have liquid coexisting with gas, I have both phases in the system at the same time, even this system must still be at the vapor pressure. It's only once I make the system a little bit smaller, condense those last few molecules of gas into the liquid phase, at that point the system will be fully liquid. So everything beneath the surface of that piston is liquid, I'll call that label E. Now the pressure can increase beyond the vapor pressure to higher values. We can imagine what happens as I increase the pressure. I can increase the pressure from maybe its initial 24-tor, is the vapor pressure water, or some higher value for other gases. I've reached the vapor pressure, if I increase the pressure even further, the liquid will shrink a little bit, but it won't shrink very much. So as I continue to increase the pressure, I won't be able to compress the liquid by nearly as much as I can change the volume of a gas. So essentially the system will look like this, getting only a slight amount smaller as I continue to increase the pressure. So that is what it takes to get up into the liquid phase over here. So we now have a decent understanding of what it means to compress this gas, sit on the phase coexistence line for a while as I finish the compression of the gas down into the liquid phase, and then continue up into the liquid phase. What does all that look like in pressure volume coordinates rather than pressure temperature coordinates? So we've talked about the pressure changing, and we've seen that the volume is also changing as I perform this process, so I can plot what that looks like on the pressure volume curve. In general, the volume decreases as I go through this process. The pressure does something interesting. The pressure increases from some initial value up to the vapor pressure, then it stays at the vapor pressure for a while before it increases. So on these P and molar volume coordinates, initially the system was a gas, and it's going to behave like a gas, perhaps an ideal gas. So at large volumes, low pressures, this is part of PV equals nRT, pressure is proportional to one over volume curve. So initially the system is just behaving like perhaps an ideal gas. At some point, the pressure will have increased until I get to the vapor pressure. When the volume has decreased and the pressure has increased until I get to the vapor pressure, then for labels B and C and D, the volume keeps decreasing, decreases quite a bit, but the pressure remains constant. The pressure remains constant at the vapor pressure. Eventually, I get to a point where I've condensed all the gas into liquid, and now the pressure can start to rise again, but because the liquid is fairly incompressible, the pressure can increase a lot, and the volume will only decrease by a little. So the slope of this line is quite steep, representing the very small compressibility of the liquid phase, much steeper than the curve for the gas phase. So here's a curve. We can call this curve an isotherm, because the whole thing happened at a single temperature. The temperature didn't change. So this will be an isotherm at one particular temperature of a gas being compressed and turning into a liquid. So over here we have liquid. Over here we have gas. In this region we have liquid and gas coexisting. We can draw an isotherm at a different temperature. I suppose we do that at a temperature that's larger, closer to, but not yet at the critical point. The picture is going to look very similar. The main difference is going to be when I do that compression at a higher temperature, the vapor pressure is larger. When the temperature increases, the vapor pressure will increase as well. So there's going to be a gas phase portion of this curve, but it's going to increase up to a higher value, a higher. So this is the vapor pressure under different conditions, the different vapor pressure, which is higher. The pressure has to go up to a higher value, remains constant for a while before it converts to a liquid and can be compressed only a little bit as the pressure increases. We can continue doing, we can draw curves at many of these different temperatures. Something special happens, of course, when we get to the critical point. At the critical point, there's no longer any phase change. If I'm above the critical temperature, if I take a gas under these conditions and increase the pressure, there's no phase change. There's no flat region where gas and liquid coexist. It's just a fluid. The curve just continues to increase fairly monotonically. And it's a supercritical fluid at large volumes or at small volumes, but at relatively high temperatures. For temperatures above the critical temperature. If I'm exactly at the critical temperature, if I try to do this at exactly the critical temperature, what's going to happen is it will become horizontal for just a tiny amount of time before it begins to increase. So this small region right there where the curve becomes horizontal and then starts to increase is the equivalent of these liquid gas coexistence region. I have liquid and gas coexisting only at a single point rather than across a range of volumes and pressures. So I can draw a whole family of curves with varying amounts of flat liquid gas coexistence portions, each of which is a different isotherm. So I should label this one as the isotherm at the critical temperature where it becomes horizontal for exactly one point on the graph. But if I really want to talk about this as a phase diagram, I'll have to draw a boundary and I'll draw a better picture in just a second between the region that I can describe as liquid, the region that I can describe as gas and the region that liquid and gas coexist and the region that describes the supercritical fluid. So if I draw a better version of that same diagram that draws just the regions for the different phases rather than the isotherms themselves, that's going to have this general slope. I can have liquid phase at small molar volumes. I can have a gas phase at large molar volumes. There's a region in between that is a liquid and gas coexistence region. That's composed of all these horizontal lines where liquid and gas can coexist. And then above this point, above this isotherm, we have the supercritical fluid region. Anything above the critical isotherm is a supercritical fluid. So some important features of this phase diagram to point out. Number one, we have liquid and gas that are two distinct phases, but above the critical temperature, so there's some critical pressure, there's some critical volume associated with the top of this curve, pressures above this point at temperatures higher than the critical temperature, we just have a supercritical fluid. We also have these horizontal lines that I've drawn in the phase coexistence region. Those are called tie lines. Look back at this curve to understand what those mean. If I have a system at, let's say, these conditions, this molar volume and this pressure, the vapor pressure, in our visual depiction of the system, I have some system like system C. I have liquid and gas coexisting. I can have liquid and gas coexisting over a range of different volumes, and that represents these various points on this tie line. So we call it a tie line because it ties together the liquid system and the gaseous system, both of which can exist at the vapor pressure, or I can have them exist in any arbitrary combination. A lot of liquid and a little gas, a lot of gas and a little liquid, those are all tied together with this tie line. The other important thing to recognize about this pressure volume phase diagram is the phase coexistence curves on a pressure temperature diagram. We understand what those are. If I'm anywhere along one of these lines at this point or this point or this point, I have two phases in coexistence. There's a solid gas coexistence line, a solid liquid coexistence line, a liquid gas coexistence line. In a pressure volume phase diagram, notice I don't have a phase coexistence line. I have a phase coexistence region. Anywhere in this region that's filled by these tie lines, I can have liquid and gas coexisting. So it's not mandatory that a coexistence curve be a line. When plotted as a function of pressure and volume, I have a whole phase coexistence region. So that's some details of what these pressure volume phase diagrams look like. They're qualitatively quite a bit different than a pressure temperature phase diagram, but they're just as useful under different circumstances. We began by talking about these gases as if they were ideal gases. But of course, we have better models, better equations of state for describing gases than just the ideal gas. And it turns out we can learn a fair bit about the phase behavior of gases and liquids and supercritical fluids if we use better equations of state to describe their behavior.