 So, when we thought about how the free energy changed as we changed the temperature, that led to a lot of interesting conclusions about phase changes, in particular melting and boiling and sublimation. Free energy also changes in response to changes in pressure, so it turns out when we think about how those two quantities are related, it will also lead to some interesting conclusions about phase changes. So, if we first think about how free energy is depending on pressure, we have this fundamental derivative, the derivative of the free energy with respect to pressure at constant temperature is the volume, and we know the volume is always positive and nothing can have a negative volume can't take a negative space, so that slope is always positive. The free energy will always increase as the pressure increases if we do it at constant temperature. So, what that means is if we think about the curve of how free energy depends on pressure, that curve is going to be an increasing function. So, if we then think about how the volume is different for different phases of the material, the free energy of the liquid, the solid, and the gas phase are all going to be different from one another, the volume of a gas, in particular the volume per mole of a gas, that's going to be larger than the volume of the liquid. In fact, under most circumstances, the volume of the gas is quite a bit larger than the liquid, it's orders of magnitude larger than the liquid. Liquid volumes, in general, liquid volumes tend to be roughly the same order of magnitude as the solid, depending on what liquid and solid we're talking about. One may be larger than the other. Most solids have densities larger than their liquids. If you condense a solid into, freeze a solid, freeze a liquid into a solid, it will usually sink in the liquid, so that means the volume taken up by the liquid is going to be larger than the volume taken up by the solid. That's not always true, however. You can probably think of at least one liquid where if you freeze it and make a solid, the solid will float on top of the liquid, and that would be the opposite situation. So for water in particular, water is less dense as an ice cube than it is in a liquid form, so that means it takes up more space as a solid than it is a little liquid. So one of these is going to be larger than the other, but they are significantly smaller than the volume taken up by the gas. So if we, let's say, for a typical substance where the liquid volume is larger than the solid volume, this slope for the liquid is going to be larger than the slope for the solid, so two slopes are close, but not exactly the same. On the other hand, if I draw the same curve for the gas phase, the slope for the gas curve is quite a bit different. That's a positive slope, but it's a large positive slope compared to this relatively moderate slope for the liquid. So this would be what the free energy as a function of pressure curves look like for three different phases. The gas, the liquid, and the solid. For a typical substance whose liquid takes up more volume is less dense than the solid. So this tells us the same type of thing that we saw when thinking about how the free energy changed as a function of temperature. There's three curves, one for each different phase. The most stable phase is going to be the one with the lowest free energy. We know any substance will spontaneously change to the phase with the lower free energy. So the one with the lowest Gibbs free energy will be the stable phase. These will be either unstable or perhaps metastable phases. And then we have at points like these phase transitions, either the free energy of the gas is equal to the free energy of the liquid at this point. We call that point. So at the particular pressure where the gas will condense to a liquid, if we go in this direction, we call that condensation, or if we go in this direction, evaporation. So I can either take a gas, increase the pressure until the point that the condensism becomes a liquid, or I can take a liquid and decrease the pressure until the point at which it evaporates. We don't call it boiling because we haven't done it by raising the temperature, we just call that evaporation. So this pressure at which that takes place, we could call that the pressure of vaporization or more commonly, we'll indicate that as P with an asterisk over top, that's called the vapor pressure of substance. So the vapor pressure is the pressure at which the gas and the liquid are in equilibrium with each other. There's also an equilibrium between liquid and solid. At this point, free energy of the liquid and free energy of the solid are equal to each other. That point, certainly there is a pressure at which that takes place. That one's not as commonly discussed, there's not a terribly common name for that pressure at which we can convert the liquid into the solid. Let me point out one additional feature for these curves, the way we did for temperature based phase transitions, and that is if I'm not going to draw these curves as a straight line, if I'm going to draw them with some slight curvature as I tried to do for this gas curve, what should that curvature be? So this equation tells us that the slopes of these curves are proportional to the volume, but the volumes don't remain constant as we change the pressure. If I change the pressure on either the solid or the liquid or the gas, the molar volume of that solid liquid or gas is going to change. So second derivative of these curves is going to be the first derivative of the first derivative. The first derivative is the volume. So we want to know what is the first derivative of the volume as I change the pressure. That's exactly equal to or at least almost equal to something that we've called the isothermal compressibility. In fact that's equal to negative volume times the isothermal compressibility. Both volume and compressibility are positive numbers, so this quantity because of the negative sign is always going to be a negative quantity. So when we do draw these curves, either we can draw them as a straight line, reflecting the idea that this curvature is actually quite small, it's small compared to the slopes of these curves, particularly for phases like the liquid and solid that are not terribly compressible. But for a phase like the gas, which has a volume that does change quite a bit as we modify the pressure exerted on that gas, the curvature should be slightly negative, should be a concave down curve. So one other thing we can do to understand these free energy versus pressure diagrams is to think about, actually there's a few things we can do. One thing that I won't do, I'll leave you to think about until we have a reason to think about it a little more deeply later on, is to ask yourself what this curve would look like if in fact we had a substance like water whose molar volume was larger in the solid phase than in the liquid phase. That's going to be a little different than this diagram. The other thing we can think of is what would happen in a case where these lines are not placed exactly where I've placed them. We know that the slope of the gas curve is going to be large compared to the slopes of these other two curves. But what if there's a case like, so here's the gas, let's say here's the liquid phase curve like I've drawn here, they're going to cross at some point, but what if before those two curves can manage the cross, maybe the solid phase curve crosses the gas phase curve below where the liquid phase curve is. So in that case, remembering that the stable phase is always the one with the lowest free energy at low pressures below this phase transition pressure, below this pressure the stable phase is the gas, above that pressure the stable phase is the solid. So now, rather than having a gas that condenses to form the liquid phase when we increase the pressure on it, we have a gas that will condense to form the solid phase when we exert pressure on it. Or vice versa, if I have the solid, the most stable phase at high pressure, if I decrease the pressure on that solid, it will evaporate and become a gas without ever passing through the liquid phase. So that's a process we call sublimation when the solid and the gas are in equilibrium with each other. So this pressure at which the solid and the gas have the same free energy as each other, that would be the sublimation pressure. We can still call it a vapor pressure, but now it's the solid phase that's sublimating to become the gas rather than a liquid phase that's evaporating to become a gas. So now we understand both how the free energy changes as a function of temperature as well as how it changes as a function of pressure. So that leads us to the conclusion that we really have to be thinking about how free energy changes as a function of both of these variables at the same time and that in turn will lead to some more interesting conclusions.