 Okay, so here's a recap of what we know about phase transitions so far. In particular, when we change the temperature of a substance, we can heat it from the solid phase. Its free energy will decrease slowly until it becomes equal to the value of the liquid phase. At that point, the liquid phase is lower because it's dropping faster. It will continue to drop until the point at which the gas free energy has a lower free energy than that of the liquid. So we go through solid to liquid to gas. If we draw what that looks like on the phase diagram, we start solid. We cross the solid-liquid boundary. We enter the liquid phase. We continue heating until we boil as we cross the liquid gas boundary. So moving from solid to liquid to gas on this free energy curve is like moving from solid to liquid to gas on this phase diagram. If we do that at a different pressure, a somewhat lower pressure, on the phase diagram, the way that shows up is because we're at a lower pressure, the boiling point has decreased. So instead of boiling at this temperature, we've boiled at this temperature instead. So the boiling point decreased as the pressure decreased. On the free energy diagram, what that looks like is each one of these curves decreases, but the gas curve in particular fell by a lot further than the other curves. So instead of the boiling point being here, the boiling point decreased to a lower value. So again, there's a connection between drawing a line on the phase diagram, drawing a curve on these free energy diagrams. But as we've noticed, these two curves are approaching each other. Eventually the two curves are going to have to meet and understand what goes on there. But if we drop the pressure to a point where we want to understand what's happening right here, solid changes to become the liquid. But at this point, according to the two curves we've drawn on this curve, the solid and liquid free energies are equal to each other. So anywhere along this curve, the free energy of the solid is equal to the free energy of the liquid. Anywhere along this curve, the free energy of the liquid is equal to the free energy of the gas. This point is on both of those curves. So at that point, the free energy of the solid is still equal to the free energy of the liquid because we're on the solid-liquid coexistence curve. And that free energy of the liquid is equal to the free energy of the gas because we're on the liquid-gas coexistence curve. So all three of those phases are in equilibrium with each other at the point that I've drawn right here. If we draw what that looks like on a free energy diagram, what that corresponds to, we've lowered the pressure some more. So these curves have dropped even further. The gas curve in particular has dropped by a lot. So it's dropped all the way down to the point where it now intersects the solid curve and the liquid curve. All those curves cross at one point. Instead of having two different phase transition temperatures, the vaporization temperature of the boiling point has continued to decrease down to the point where it matches the melting point. So the melting point and the boiling point are occurring at the same point. They coexist with each other. Solid, liquid, and gas phases all coexist with each other. That point is called the triple point because we have solid, liquid, and gas phases all coexisting with each other at that point. We can then ask what goes on as we continue even further at lower pressures still. Let's go down to yet lower pressures and see what happens if I draw. If I start in the solid phase and increase the temperature and see what happens to the solid under these conditions. In this case, I guess we'll start with the free energy curves. Lower pressure, so each one of these curves has continued to drop. The solid curve is lower. The liquid curve is lower. The gas curve is lower. But now, solid, gas, liquid. Now notice that the liquid phase curve is never the one that has the lowest free energy. If I take the solid and I increase the temperature, the most stable phase starts out being the solid, and then switches over to the gas. So what we have here is if I continue to draw boundaries on my phase diagram, the solid region of the phase diagram is all of the region up to this solid liquid coexistence line or up to this new line that I've drawn. And when I cross this line, the thing that happens as I cross that line is the solid phase has become equal and free energy to the gaseous phase. So it's in equilibrium with the gas phase, and the liquid phase is not involved anymore. So at this point, I have a solid gas coexistence curve rather than a solid liquid or a liquid gas coexistence curve. So that's the process that we call sublimation. So rather than melting a solid to form a liquid or boiling a liquid to form a gas, if the solid transforms directly into the gas without going through the liquid phase in between, we call that sublimation. And the reason that happens is because the liquid state free energy is now high enough that that's never the most stable phase. Only the solid or the gas have a chance of being the most stable phase at these low temperatures. So this begins to point out the feature that we can sometimes have just one phase. Sometimes we're on a coexistence line where we have two phases in equilibrium, solid liquid, liquid gas, solid gas. Or sometimes, like at this triple point, we have three phases in coexistence with each other. And those different locations on the phase diagram, single phase, double phase, or triple phase regions of the phase diagram have important thermodynamic differences as well, so we'll cover that next.