 So the key point in understanding phase transitions, as we've seen, is the fact that the free energy of a substance depends on its temperature and pressure, and in particular the free energies are different for the solid phase, the liquid phase, and the gas phase. And at any particular temperature and pressure one of these may be lower than the others and that will be the most stable phase. Sometimes two of them are equal and the two phases can coexist. So instead of drawing how the free energy itself depends on temperature and pressure, let's take a different approach and at each value of temperature and pressure we can quantify whether it's the solid or the liquid or the gas that has the lower free energy than the others. So for example, let's say we take a set of conditions that are quite familiar, room temperature and room pressure, and let's say we want to talk about water, H2O. If we have water present in this room at a temperature of 298 Kelvin and a pressure of one atmosphere, the free energy of the liquid phase is going to be lower than the free energy of the gas or the free energy of the solid, and that's why liquid water is the stable phase at room temperature. So at 298 Kelvin and one atmosphere, I'll put a point on that diagram and I'll just label that point and say under those conditions the liquid phase is more stable. In principle, could calculate the free energy of the solid, the free energy of the gas. I can't make those exist for very long in the laboratory because they will spontaneously convert to the phase with the lower free energy. The liquid is the phase with the lower free energy. However, if I start increasing the temperature, when I get to a temperature of 373 Kelvin, the boiling point for water, at that point, one atmosphere and 373 Kelvin, then we'll have liquid and gas coexisting with each other at that boiling point. So at that particular temperature, we know that the free energy of the liquid and the free energy of the gas have the same value. The liquid and the gas phases can coexist under those conditions. Any temperatures below that, so this point and this point are all liquid is the most stable. At this point, I can have liquid coexisting with gas at any temperatures hotter than 373 Kelvin, then the gas is the most stable phase under those conditions. So I've taken a bunch of points, I can label them as liquid being more stable, gas being more stable. I can do the same thing on the other end. I can cool the liquid down when I get down to the freezing point 273 Kelvin. That's the point at which the liquid will freeze and become solid. Below 273, solid is the most stable phase. Above 273, liquid is the more stable phase. At 273, we have solid and liquid coexisting with each other at that phase. So that's all at one atmosphere. And just to make connection to what we've done previously, that's related to the fact that the free energy changes, so liquid, solid, gas. If I plot the value of the free energy, the reason 273 is the melting point is because that's the point at which the solid and liquid free energies become equal. The reason 373 is the boiling point is because that's the temperature at which the liquid becomes equal to the gas. So those two diagrams are connected to one another. The same transition points on the free energy diagrams occur at the same place on this diagram. This diagram that we're building is something we'll call a phase diagram because on this upper diagram, we're no longer interested in the value of G. I don't care whether it's large or small, very large or moderately large. All I care about is which one is larger than the other. So in this range of temperatures, the liquid is lower than the solid and the gas. So I label these points as liquid. All I care about is which phase is lower in free energy than the others. If I draw a curve like this at a different pressure, as we've seen, if I were to do it at higher pressure, the boiling point will have increased. So at larger pressures, the temperature at which I have coexistence is at a larger boiling point. So when I increase the pressure, I also increase the boiling point. If I decrease the pressure, I decrease the boiling point. So this curve has a positive slope. The liquid solid coexistence curve also has a slope, but it's quite steep. So this will focus on this curve at first. So this curve, if I continue decreasing the pressure, the boiling point will decrease. Eventually, these two curves with dissimilar slopes, eventually those two curves are going to have to cross. So I'll postpone until the next lecture talking about what exactly happens when those two curves cross. But for now, we've got the general shape of this phase diagram, which is that there's a region of the phase diagram where the solid is the most stable phase. There's a region of the phase diagram where the liquid is the most stable phase. And there's a region of the phase diagram where the gas is the most stable phase. And any coordinates, if I give you a temperature and a pressure, you can find what portion of the phase diagram we're in. And that will allow you to make a prediction of whether the substance will exist as a solid or liquid or a gas under those conditions. And when we happen to be exactly on the boundary between two of these phases, if we're on the line that separates them, then we have the coexistence between a liquid and a gas, in this case, between a solid and a liquid if we're on this line. So these lines are called coexistence lines or coexistence curves. I'll call that a phase coexistence curve because that is the line along which we're allowed to have coexistence between the solid and liquid phase or the liquid and the gas phase. So the next step will be taking a closer look at what happens when these two curves cross one another down here at low pressure.