 So, a phase diagram actually tells us several important things. One thing, quite obviously, tells us is where the solid versus liquid versus gas phases are more stable. If you know what temperature and what pressure you're at, you can read off of the phase diagram and determine whether you're in the solid or liquid or gas phase or whether you're at some coexistence line where more than one phase is coexisting at the same time. The other useful thing we can learn from a phase diagram is details of the phase transitions, the melting points, the boiling points, the sublimation temperatures. So if I give you a pressure, for example, you can read off of the phase diagram the point on the liquid vapor coexistence line that occurs at that pressure and that tells you the boiling point at that particular pressure. So phase diagram allows us to predict the boiling point as a function of pressure or vice versa. It's interesting therefore to ask ourselves mathematically in detail what is that connection between the temperature of a phase transition and the pressure of a phase transition. As we've seen, as we increase the pressure, the boiling point increases, so the slope of this curve is positive. We haven't really talked about the slope of these other portions of the curve. Should they be positive? Should they be negative? Even for the boiling point coexistence curve, exactly what is that slope? How many degrees should we expect the boiling point to rise when we increase the pressure by a certain number of atmospheres? So we're asking by how much should, as we look at this curve, by how much should the boiling point change when I change the pressure or vice versa? The slope of that curve would be dp dt. So we're interested in knowing what is that slope. So how to go about determining mathematically what that slope should be? One place to start would be to start with the one thing we know about these coexistence curves. By definition, on the coexistence curve when the liquid and gas are coexisting with each other, the two substances have the same free energy as each other. Likewise over here on the solid liquid phase coexistence line, the solid liquid have the same free energy as each other, same Gibbs energy as each other. So on the coexistence curves of any sort, the free energy of one phase is equal to the free energy of another phase. And I'll write the two phases as alpha and beta so that they can stand for solid or liquid or gas, any coexistence curve that we're interested in. We're interested not just in what the free energy is at this particular coexistence, point on the coexistence curve, but if I travel along that coexistence curve to somewhere else, I'm interested in this ratio of how much the pressure changes of the temperature changes. So I can also say if I modify where I am on that curve, the free energy of the liquid will change as I move along that curve, the free energy of the gas will change as I move along that curve, but they have to remain equal to each other. So whatever the change in the liquid free energy was, it's going to be matched by the change in the gas phase free energy. So the change in Gibbs free energy in phase alpha is going to match the change in the Gibbs free energy in phase beta. Mathematically, the way to think about that is I've just taken the differential of these two equations, dG is equal to dG. But we know something about the differential of the Gibbs free energy. The fundamental equation tells us that the Gibbs free energy dG is equal to minus SDT plus V dP. So on the left side, I have dG is equal to this. On the right side, I have dG is equal to the same fundamental equation. The only difference is the left side is for phase alpha. The right side is for phase beta. So the entropy here is the entropy of alpha, and these are all molar entropies, molar volumes. The molar volume is for phase alpha on the left, phase beta on the right. I don't need to put subscripts on the temperature and the pressure, because as I move along this curve, I'm changing the temperature and the pressure of both the liquid and the gas phase simultaneously. So the change in the temperature and pressure I'm talking about, those are really the pressure of the phase transition and the temperature of the phase transition. So let's rearrange this a little bit, and let's see. Let's get the entropies on the left side. So I'll bring the minus s beta dt over to the left and call it positive s beta. Combine it with the minus s alpha, multiplied by the same dt. On the right side, I'll bring the volumes over the right side. So I have a vb, v beta, dp, and I have a v alpha that becomes minus v alpha when I move it to the right side. All right, so that's now starting to look pretty close to what I'm looking for. I'm looking for dp divided by dt. So let's bring the dt over to this side and the volumes over to the other side. So what we're left with is s beta minus s alpha, divided by the volume difference between the volumes v beta minus v alpha. And on the right side, I've got the dp. And when I bring the dt over, it's in the denominator. So dp divided by dt. If we think about what this fraction means, molar entropy of beta minus molar entropy of alpha, that's just the difference in the molar entropy when I perform the phase change. When I convert phase alpha to beta, maybe I'm taking liquid converting it to gas, or solid converting it to liquid, or maybe the other way around, liquid to solid. But when I convert phase alpha into phase beta, the entropy changes by this amount, entropy of the final state beta minus entropy of the initial state alpha. So I can rewrite the numerator as the change in the molar entropy for this phase transition. So my phase transition means converting alpha into beta. In the denominator, molar volume of beta minus molar volume of alpha, that's the change in the molar volume when I perform the phase transition. All right, so that's actually what we're looking for. That is how much the pressure changes in response to the temperature as I move along this coexistence curve. I guarantee I was going to remain along this coexistence curve by guaranteeing that the free energies remained equal to each other as I modified both phases. We can actually do one step better than this, however. For this next step, we're going to recall something that we already know about phase changes. The temperature of any particular phase change we have seen is equal to delta H of that phase change divided by delta S of that phase change. So what that means is if I rearrange this equation, delta S is equal to delta H divided by the temperature of the phase change. And I'm going to use that expression for the entropy, the change in the molar entropy of the phase change, to rewrite the numerator here. So I can say dp dt is equal to, instead of writing dS, I'll write delta H divided by t. And I've still got a delta V in the denominator, a change in the molar volume of that phase change. So I'll go ahead and stick phase change labels on the left-hand side as well. And now what that's telling us is if I want to know the slope of one of these curves, it doesn't matter whether it's the solid liquid curve, liquid gas curve, any of these curves, dp dt, the slope of that curve is going to be equal to the molar enthalpy of that phase change, maybe the molar enthalpy of vaporization or of fusion or sublimation divided by the temperature at which it occurs and the change in the molar volume when it occurs. So now we can do quantitative numerical examples to predict the slopes of these curves and get a more detailed picture of what these phase diagrams are going to look like. And we'll do an example or two of that next.