 Alright, so the Gibbs phase rule tells us that the number of degrees of freedom we have the flexibility to choose is equal to the number of components of a solution of a system minus the number of phases coexisting in equilibrium in that system plus two. So let's work a few examples and make sure that makes sense to us. We can first do a couple of examples where we already know the answer. In the case of a single component system, we just have one component, so c equals 1. Then if I just plug c equals 1 into this equation, 1 plus 2 is equal to 3 minus 5. And that's the same result we've gotten previously for the number of degrees of freedom accessible to us in a single component system. So the Gibbs phase rule is a more generalized version of the rule we've used previously for single phases. If we want to talk about multiple component solutions or systems, the two examples we considered by hand in the previous lecture. The first of those was a gaseous system with a composition something like air, a mixture of nitrogen and oxygen. So that was a two component single phase system. The number of degrees of freedom is two components minus one phase plus two. So two minus one plus two, that equals three. And that's the result we convinced ourselves was reasonable for that system. For example, we could specify the temperature and the pressure and the composition of that mixture. But once we've specified three variables, I can't also specify the mole fraction of oxygen. The other example we thought about in the previous lecture was carbonated water. A mixture of CO2 and H2O in the liquid and gas phases coexisting. So the picture we drew in that case was liquid, water, dissolved CO2, water and CO2 both in the gas phase. That's again a two component system, CO2 and H2O. Two phases coexisting with each other. So the number of degrees of freedom, components minus phases plus two, that gives me two. So we have only two variables we can specify. Again, that matches what we convinced ourselves was the case in that example. I can dissolve a certain concentration of CO2. I can set the temperature to whatever I want. But then the pressures of the vapor phase components will be determined by the phase equilibrium. So, for example, I can set temperature and concentration of CO2. I could put the system under whatever pressure I want. I can choose the temperature, I can choose the pressure. But once I've determined the pressure, if the pressure in the vapor phase is higher than the vapor pressure of CO2, it will dissolve into the liquid. If the pressure is higher than the vapor pressure of H2O, it will dissolve. It will condense down into the liquid phase. So I can choose two variables, but once I've chosen those two, the other ones will be determined for me. To work a few examples we haven't considered yet, let's do one in the solid phase. Let's take a mixture of two compounds in the solid phase. So, for example, brass is a mixture of copper and zinc. How many degrees of freedom am I allowed to specify for a chunk of brass? So that's two components, copper and zinc, one phase. I'm just talking about the solid, so there's only a single phase. So, components minus phases plus two, two minus one plus two is again equal to three. That's the same math as in this situation. So, is that reasonable? Can I think of three different variables that I could specify for a sample of solid brass? I can specify the mole fraction of copper in that sample, and then I can take that sample, and I can heat it to whatever temperature I want, I can put it under some amount of pressure. There's no contradiction between doing all those three things at the same time. But again, I couldn't also independently set the mole fraction of the other component. Once I choose the mole fraction of copper, mole fraction of zinc is determined for me. Let's take as a multi-phase system instead of liquid and vapor phase. Let's do one, actually first let's do a solution with only a single phase. So let's say pure liquid, so there's no vapor in this case, just a container caning liquid. And let's do, I want to eventually consider a solution that we could oversaturate. So let's take a solute like sugar, sucrose, I'll dissolve some sugar in water to make a sugar water solution. So that's two components, single phase. Again, like every time we've considered two components, single phase, 2 minus 1 plus 2 is going to work out to three different variables that we could independently specify. And again, that makes sense in this case. I can choose a composition variable. I can choose the concentration of sucrose. I can make a one molar solution of sucrose. I can make a half molar solution of sucrose. I can place that sample at any temperature I want and any pressure I want within a certain range of possible values. So that's very much like the single phase systems we've considered in the gas phase or the solid phase. But now, let's take the concentration of that sucrose solution to a point where it's saturated. I'll keep dissolving sucrose in my water until it's so concentrated that the sucrose begins to precipitate. So I'm recording this in South Carolina where we like our sweet tea. So the way to make sweet tea is dissolve so much water that it precipitates and sits on the bottom of the solution. So that's saturated solution. Now I've got two components and two different phases in coexistence. Water and sucrose are the two components. The two phases are the liquid phase and the solid phase. So now the number of degrees of freedom, 2 minus 2 plus 2, that works out to two degrees of freedom. That's a different answer than I had for this sub-saturated solution, the solution below the saturation concentration. So which variable have I lost? Why does it make sense that I can no longer specify all three of these thermodynamic variables at the same time? I could specify the temperature and the pressure. I can certainly take my sugar water solution and heat it up or cool it down, make it any range of temperatures. I can put it under some pressure. There's nothing to stop me from doing that. But what I can't specify any longer is the concentration of sugar. So I can only specify temperature and pressure. If I try to increase the concentration of sugar to even more concentrated concentrations, if I add more sugar into this solution, then what's going to happen is it won't stay in solution. It will just precipitate out, keeping the concentration of the solution at the saturation concentration. Likewise, if I try to reduce the concentration up here in the liquid phase by removing some sucrose molecules from the liquid phase, because of the equilibrium between these two phases, some solid molecules will just dissolve and replace the monos I've taken out of the liquid solution. So the sucrose concentration is fixed at the saturation concentration at this particular temperature and pressure. So the Gibbs phase rule is correctly telling us that we've lost a degree of freedom in this case. And as one final example, just to take something even more complicated to show you that you don't need just simple two-phase cases, you don't need two-component solutions, let's make an example that's even more complicated. Let's say we have a liquid in equilibrium with vapor, let's make it a two-component solution. So I've got, who knows, water and propanol in this liquid solution. Both of those are volatile solvents and they'll have vapor up in the gas phase. But now into this two-component, two-phase solution, let's also dissolve a solute. So let's dissolve a third component, maybe sugar or salt or something else, that's in the liquid phase. If it's salt, it's a non-volatile solute, so there won't be any up in the vapor phase. But it will, if I'm at saturation conditions, precipitate down into the solid phase. So now I've got three components, water, propanol, and salt, or three different components. In three different phases, vapor phase, liquid phase, solid phase, so that tells me the number of degrees of freedom. In this case, it's three components minus three phases plus two. I still only, despite the complicated setup of the system, I've only got two degrees of freedom that I'm allowed to specify according to the Gibbs phase rule. So does that make sense? Why can't I specify more than just two degrees of freedom? As an example, I can dissolve, I can certainly set the mole fraction of component A in the liquid phase. So I can make a 50-50 mixture of the two solvents if I want to. I can heat or cool that system down to whatever temperature I'm interested in. But I can't choose the mole fraction of C since I'm at saturation, then this is going to be the saturation concentration. I can't choose the amount of A in the vapor phase. At this particular temperature, the vapor pressure of A is some number. At this particular temperature, the vapor pressure of B is some number. So those two pressures are going to add to some total pressure in the gas phase. So I can't choose the pressure to apply on the gas phase if I try to make it some different value than either more molecules of A or B will evaporate to raise the pressure or A and B will condense to lower the pressure. So once I've chosen those two degrees of freedom or any two degrees of freedom that I try to choose independently, all the other ones will be determined. So it doesn't matter how complicated the system is. It doesn't matter how many components, how many phases. It doesn't matter if some components contain all the phases, some contain only one phase, some contain only one of the components. Some phases may contain a subset of the components. As complicated as a system, as you can construct, the Gibbs phase will tell you how many different variables you can try to control at the same time.