 In our discussions of phase changes so far, we've made good use of the fact that the derivative of the Gibbs free energy with a factor of temperature at constant pressure is negative entropy, which is a negative number, must be less than zero. So that means Gibbs free energy drops as the temperature increases. And that told us something about, for example, the phase change between solid and liquid. Solids will eventually melt and become liquid. If we go to a high enough temperature, there's always going to be some temperature where the liquid becomes more stable than the solid. But that's true not just for the solid-liquid transition, but also for transitions between liquids and gases, and even between solids and gases. So we previously made use of the fact that the entropy of the liquid is larger than the entropy of the solid. But what's also true is that the entropy of the gas, being not a condensed phase, the molecules are much further apart. They occupy a larger volume. So the entropy of the gas is, in turn, larger than the entropy of the liquid. In fact, usually the entropy of a gas is quite a bit larger than the liquid. Several orders of magnitude larger than the liquid under most conditions. So if we use that relationship between these entropies, we can use that as the relationship between the free energies and how they change as the temperature changes. So if we draw this relationship between the free energy of a substance and the temperature, those are negatively sloping curves. I'll draw a gently negatively sloping curve for the solid, a slightly more negatively sloping curve for the liquid, reflecting this double greater than sign a significantly more negatively sloping curve for the gas. So solid has a free energy that decreases as I raise its temperature at constant pressure. The liquid also decreases, but at a steeper rate, the gas also decreases, but at a much steeper rate. And in fact, those slopes of those curves, the slope of this curve is negative the entropy of the solid. The slope of this curve, that slope, is negative the entropy of the gas. This is negative the entropy of the liquid. So that tells us a few things. As we've seen before, if I go to high enough temperatures, the liquid will always have a free energy that drops below that of the solid. So that temperature, we call the melting point or the temperature of fusion. But also, there's always going to come a temperature where the gas, even if it started much higher than the other, the free energies of the solid and liquid, eventually there's a temperature at which the gas will become more stable than liquid, and that's the temperature we call the temperature of vaporization or the boiling point. So because the slope of the gas curve is the most steeply negative of all three of these curves, if we go to high enough temperature, the gas will always be the most stable form at the very highest temperatures. Likewise, we can find temperatures where the solid curve crosses the liquid curve and where the liquid curve crosses the gas curve, or vice versa. That raises a question though. So the way I've drawn this diagram, as I take a substance and I heat it, the solid will become liquid, eventually the liquid will become gas. The substance will melt, and then at a higher temperature, it will boil. That doesn't tell us anything about sublimation. So in this diagram, solid becomes liquid, becomes gas, as I raise the temperature. That doesn't help us understand the phase transition of solid turning directly to gas, process of sublimation. So we can also understand sublimation with a diagram like this. So maybe that's an interesting challenge for you to stop. Pause the video for a second and ask yourself, what would this diagram look like for a transition where the solid converts directly to gas without turning into a liquid in between? How would these curves be different? So I'll let you pause the video and think about that for one second. And then I'll continue and tell you how that works. For a substance that sublimates the free energy as a function of temperature, it's very much the same. The solid still has a gently downward sloping free energy. The gas still has a steeply downward sloping free energy. But the liquid, with its intermediate slope, is somewhere above. So if I start at low temperatures and I ask myself as I go along, which substance has the lowest free energy? Is it the solid? Is it the liquid which is up here somewhere, or is it the gas which is up here somewhere? At coldest temperatures, it's always the solid. As I increase the temperature, the solid remains the lowest until I get to this point at which the gas has become lower free energy than the solid. So that would be the sublimation temperature. And the reason the substance sublimes rather than melting and then boiling is because the liquid never has the lowest free energy. These curves do cross each other. There is a temperature where the solid and the liquid have the same free energy. There's also a temperature where the gas and the liquid have the same free energy. But we don't call those melting points and boiling points because these free energies are larger than some other phase. Either the solid or the gas is always the lowest free energy phase of the material. So it goes directly from solid to gas. The liquid is never the most stable phase under these conditions for that particular substance. So depending on where this liquid line lies, we can either melt and then boil, or we can sublime directly. There's one other feature of these curves that I'll point out. So far I've drawn these free energy versus temperature curves as fairly straight lines, although not quite ruler straight. It's interesting to ask ourselves what should the slopes of those lines be. They're not actually going to be constant slopes. We know that the free energy changes with respect to temperature is the negative entropy. That's the slopes of these lines. The slope of these lines is entropy. But the entropy is not a constant. The entropy depends on the temperature. So if I were to ask myself, should these lines be straight lines with constant entropy, or is the entropy increasing or decreasing, which would make these lines curve downward or curve upward, then I'm asking myself not what the derivative of g is with respect to t, but the second derivative, the curvature of these lines. So first derivative of g with respect to t is negative entropy. So the second derivative of g with respect to t is the derivative of negative entropy with respect to t, or let's say negative ds dt at constant p. ds dt at constant p is one of the thermodynamic relationships we have run across before. The ds dt at constant p is just the heat capacity divided by temperature, the constant pressure heat capacity divided by temperature. So if I'm doing these derivatives, so the second derivative of g with respect to t at constant p, that's negative heat capacity divided by temperature. Temperature is, of course, a positive quantity. The absolute temperature is always a positive number. Heat capacity is also a positive number. So this ratio is a positive number. The negative sign guarantees that that quantity is always going to be negative. So what that means is, if I'm drawing these curves accurately, if I were drawing them to scale and with proper numbers for a particular substance, the lines may be straight or very close to straight. But if they have a curvature, they're slightly downward curving. So the curvature of these lines is negative. So perhaps I could draw the lines as I have tried to do very slightly, either as a straight line or a slightly downward curving line. These lines have a slight negative curvature to them. So if you, you can draw them straight because usually they're relatively straight, but any curvature they do have is negative. Be sure not to draw the lines as, so do not draw the lines with an upward curvature. That would definitely be incorrect. That would be applying that the signs are different than they are in the actual case. So, let me put an X through this so we know that is not the correct answer. What we've used, this one simple relationship to tell us now is the order in which solids will convert to liquid and then gas or perhaps directly to gas as we raise the temperature. Because the entropy of the gas is bigger than the liquid, which is bigger than that of the solid. These diagrams tell us that if we're at equilibrium, one of these phases will be lower or perhaps two of the phases will coexist. Except we know that systems are not always in equilibrium. We've talked about reversible versus irreversible processes, how we can do processes in equilibrium or we can maybe be doing them too fast for them to remain in equilibrium. So there are certainly some cases where the systems move away from equilibrium. And it turns out thermodynamics can also tell us something about those non-equilibrium processes. These same diagrams will tell us what happens when a system moves away from equilibrium. So we'll talk about that next.