 So we understand now that the free energy changes in response both to the temperature and the pressure and remembering that when the free energies are equal, we have coexistence between two different phases. So going back to the diagram of free energy gives free energy as a function of temperature. As we heat a substance, it will always change from the solid to the liquid to the vapor or perhaps directly from the solid to the vapor passing through melting points and boiling points along the way. So we determine that information from thinking about the temperature dependence of the free energy. But each one of these curves, this curve for the free energy of the solid, this curve for the free energy of the liquid, this curve for the free energy of the gas, those are all curves that explain how they vary with temperature, but they're all for a fixed constant pressure. So let's say these curves are all at a pressure of one atmosphere. If we imagine doing the same thing as we change the pressure, draw curves at a different pressure. If we change the pressure when the pressure goes up, because the volume is positive, the free energy will go up and the free energy will go up by an amount proportional to the volume of the phase. So remembering that the volume of the gas is much bigger than the volumes of the liquids and solid phases, we can redraw these curves at a different pressure. If I increase the pressure to two atmospheres or ten atmospheres or some higher pressure, all of these free energy curves are going to increase. The one that's going to increase the most is the one with the highest volume, the gas phase. So this curve for the gas phase is going to increase by quite a bit. The curves, so that curve has gone up. Likewise the curve for the liquid phase is also going to increase, but it's not going to increase by as much. So let me take away this arrow so my diagram is a little cleaner. So here's the curve for the liquid phase. Here's the curve for the gas phase, both of which have increased, but the liquid increased by a little, the gas increased by a lot. There's still a phase transition, but notice that the phase transition is now at a different temperature. So these curves are at some increased pressure bigger than the initial pressure. At that increased pressure, because the gas phase, free energy, increased by more than a liquid phase, the result of that is always that the temperature of vaporization, the boiling point, will be shifted to a higher value. So what we've just seen is the boiling point will increase as the pressure increases. We can ask ourselves what happens down here at the melting point rather than the boiling point. So again, the free energy of the solid as I increase the pressure will increase, making this curve rise. In general, the increase of those two curves is roughly the same as each other. So those curves are going to cross at some different point that the melting point might shift upwards or might shift downwards, depending on which of these two curves increases by more. If one curve increases by more than the other, it'll shift the melting point in one direction. Curve increases by more, it'll shift in the other direction. So we can't say as a blanket statement, like we can for boiling, whether the melting point is going to always increase or always decrease. We'll be able to say a little more about that in a little bit when we look in more detail at how much these phase transition temperatures shift when we change the pressure. But the main conclusion for right now is we can draw these curves. We can do the same thing just like we've done for an increase in pressure. We could also do something similar for a decrease in pressure. I won't redraw those curves, but we know the boiling point will decrease if I decrease the pressure. If I don't boil water at one atmosphere of pressure, but if I boil it at a half an atmosphere, then the boiling point will decrease. So these two statements, you may actually, if you think about it, have some practical experience with or at least have heard about. Taking this first one first. There's a fairly common situation in which we do increase the pressure above one atmosphere in order to change the boiling point of water in particular. If you've ever cooked anything in a pressure cooker or an instant pot, for example, the whole purpose of a pressure cooker is to seal the lid on tightly, boil the water, create a lot of gas, steam in the vapor phase, increasing the pressure above one atmosphere so that the boiling point of water is increased above 100 degrees Celsius or 212 Fahrenheit. So that allows the food to cook much faster at that increased pressure. So a pressure cooker is a practical example of how we increase the boiling point when we increase the pressure. An example where you might have experience to decrease in the boiling point when you decrease the pressure would be at altitude. If you go hiking or camping, let's say on a mountain that's at a relatively high altitude or even try to boil water when you're on an airplane flight. When they boil the coffee on that airplane flight, the pressure, if the pressure is below one atmosphere, the boiling point will be lower. So if you've ever tried to cook mac and cheese around the campfire at high altitude, you may have noticed that the macaroni needs to cook for a little longer and that's because the water is boiling at a temperature below the normal boiling point of 100 degrees Celsius. So we are familiar with these effects, but that's not just a peculiarity of water. Every substance that goes through a boiling transition like this, simply because gases take more volume than liquids do. They will have boiling points that are elevated compared to the normal boiling point if we do the boiling at high pressure. So we can, next thing we can do is investigate that in a little more detail for both the solid to liquid transition freezing, the liquid to gas transition boiling, as well as the solid to gas transition if there is one sublimation and ask ourselves how those boiling points change with pressure and in particular what phases are more stable under which conditions.