 All right, so with what we know about free energies at this point and spontaneity and thermodynamic relationships, we're at a point where we can put all those pieces of information together and start to learn something about phase transitions. So changes of phase between two different phases of matter. For example, let's say ice and water, the solid and the liquid forms of water. We all have plenty of experience with both liquid and solid water, of course. And we know that at least at room temperature and room pressure, we're at 298 Kelvin in one atmosphere. We know it's a spontaneous process that solid ice will melt at room temperature, right? So that process is spontaneous. We know that a spontaneous process at constant temperature and room pressure will have a Gibbs free energy change that's less than zero. So when this process takes place, solid water turns into liquid water. Solid water turns into liquid water at these conditions. That free energy change must be negative for a spontaneous process, actually not less than or equal to because it's spontaneous and not equilibrium. It will be strictly negative, it will be a negative free energy change. So that tells us for sure that the Gibbs free energy of the solid is bigger than the Gibbs free energy of the liquid. When the solid turns into a liquid, the Gibbs free energy is becoming smaller. So the Gibbs free energy change is negative. So we know this fact about the relative free energies of ice and water at room temperature and room pressure. What about substances that we have less experience with? If I were to hand you a substance that you don't know, the melting point, for example, could you predict whether it would melt or whether it would spontaneously freeze at a particular temperature? We want to be able to predict what phase transitions are going to happen spontaneously. So to answer questions like that, we need to start thinking about what we know about free energies and how free energies change as we change the temperature. So that's the thermodynamic relationship. We'd like to know about. So remember the fundamental equation for the Gibbs free energy. So that's a plus sign, that's a minus sign. So the fundamental equation for Dg is minus Tds plus Vdp. So that tells us immediately, actually that's not correct. That would be a different fundamental relationship. Dg is equal to minus SdT plus Vdp, the fundamental variables, natural variables for Gibbs free energy, your temperature and pressure. So there's the correct fundamental equation for Dg. That reminds us that Dg, Dt at constant P, derivative of G with respect to T, holding this term constant is negative S. So there's a thermodynamic relationship that we know. When I change the temperature, suppose I raise the temperature of an object at constant pressure, its Gibbs free energy is going to change as negative S. We know something about entropy. We know entropy is always positive. Third law tells us the entropy is zero if we're at zero Kelvin. But at any temperature above zero Kelvin, the entropy will have increased above the value of zero. So entropy is always positive. So that tells us that Dg, Dt at constant P, that's always going to be a negative number. Free energy is always going to decrease as I raise the temperature. When the temperature goes up, Gibbs free energy is going to go down if I do that at constant pressure. So let's see if we can understand what that means about the relative free energies of solids and liquids. I draw a graph now of how the free energy is changing. As I change the temperature, we've just discovered the slope of that graph is negative S. The slope of that graph is a negative number. If I draw a graph for how the free energy of, let's say this is the solid, it's decreasing. As the temperature increases, the free energy is decreasing. I'll now put another line on that graph for the liquid. And to know how to draw that line for the liquid relative to this line for the solid, we need to know that the entropy of a liquid compared to the entropy of the solid, the liquid is more disordered. The solid is an orderly crystalline arrangement of the molecules. The liquid is a condensed but still disordered arrangement of the molecules. So the entropy of the liquid is bigger than the entropy of the solid. That means when I draw this line for the liquid, it's going to decrease at a faster rate. So there's a line for the free energy of the liquid dropping faster as a function of temperature than the solid does. So all that we've determined, so shallow slope for the solid, steeper slope for the liquid. We've determined that just by knowing this thermodynamic relationship and the fact that the entropy has to be positive. This tells us a couple of things immediately. The slopes of these two lines are different ones bigger than the other. Those two lines always have to cross each other at some point. Doesn't matter that at some conditions the free energy of the liquid is bigger than the free energy of the solid. If I go to high enough temperatures, these lines will eventually cross. And beyond that point, the free energy of the solid will be bigger than the free energy of the liquid, just like was true for water at room temperature. So water is above the point where the lines cross. Free energy of the solid is bigger than the free energy of the liquid. For every substance, the lines will eventually cross at high enough temperatures, the liquid will always have a lower free energy than the solid. The liquid will be more stable than the solid, and the solid will spontaneously convert to the liquid at high enough temperatures. There's also guaranteed to be a point where those two curves cross. That point, of course, is the melting point or the freezing point. More formally, we call that the temperature fusion, but that's the point where the solid and the liquid have the same free energy. If I had asked you to define the melting point, you'd say that's the temperature at which the solid converts into a liquid. As we raise the temperature, but in a thermodynamic, from a thermodynamic point of view, at the melting point, the difference in free energies between the solid and the liquid is zero. The free energy of the liquid is equal to the free energy of the solid. The difference in free energy between those two quantities is equal to zero. Those are all saying the same thing. Another equivalent statement is to say that the solid and the liquid are in equilibrium with one another. This is the only temperature at a particular pressure where the solid and the liquid can coexist, can be in equilibrium with each other. Because that is the single temperature at which the curves cross and at which their two free energies are equal to one another. So that's in a more thermodynamic sense, the definition of the melting point is the temperature at which the solid and the liquid have the same free energy as each other. So what can we do with that information? We've in fact learned something not just about solids and liquids, about melting, which is the example we've used right here. But so the delta G when solid converts to liquid, that delta G is equal to zero if we're talking about melting. But we could just as easily be talking about boiling. The delta G of vaporization would be equal to zero at the boiling point, at the temperature of vaporization. We could also be talking about the free energy of sublimation. So any one of these free energy changes for any particular phase change at the particular temperature where those two phases are in equilibrium, more generally I could say the free energy change is equal to zero for some arbitrary phase change. We'll use this Greek letter phi to indicate an arbitrary phase change, regardless of whether we're talking about melting or boiling or sublimation. The free energy change during that phase change is equal to zero when we're at the temperature where those two substances are in equilibrium with each other. So we've learned something useful about phase changes now. If we combine that with a little bit more thermodynamic knowledge, we can start to make some predictions about what temperatures substances will melt or boil or sublimate at, and that'll be next.