 All right, so we've arrived at this fundamental expression, this important expression that tells us the relationship between chemical potentials in different phases when a system is at equilibrium. In particular, the chemical potential has to be equal in two different phases when the system is at equilibrium. That has a number of important consequences that are useful. So suppose this is not true, suppose chemical potential in some phase is less than the chemical potential in some different phase. Clearly what this statement says is that system is not at equilibrium, but what does that mean? If we have, just to draw a diagram, so we've got two different phases, we've got some molecules of A and molecules of B in phase one, or phase alpha, and we've got some molecules of A and molecules of B in a different phase, maybe a gas above a liquid or any pair of phases you want. If the chemical potential in phase alpha, so let's say this one is alpha and this one is beta, if the chemical potential is lower in this phase than it is in this phase, what that means is since the system is not in equilibrium, it will spontaneously move in one direction or another to achieve equilibrium. So if the chemical potential is lower in the alpha phase, that means molecules from the beta phase, the higher chemical potential phase are spontaneously going to move into the other phase. If this is a gas and this is a liquid, they'll condense. If one's a solid and one's a liquid, it might melt or it might freeze, but molecules are going to spontaneously move to the phase with lower chemical potential. So this means molecules of A in the beta phase will convert to molecules of A in the alpha phase if the alpha phase has lower chemical potential. That should sound familiar. In fact, this statement about chemical potentials, remember chemical potential is partial molar Gibbs free energies, that should sound fairly familiar because we've seen before the molar free energy in phase alpha is equal to the molar free energy in phase beta at equilibrium when we're talking about a single component system. We're just talking about melting or boiling, evaporating or subliming of a single component. Then the chemical potential, I'm sorry, the Gibbs free energy, the molar Gibbs free energy of the two phases are equal on the phase coexistence lines of a phase diagram. That was true about single component systems. This is just the equivalent statement about the partial molar Gibbs free energies, the chemical potentials in a multi-component system. The key difference to keep in mind though is these, again this is for single component. That statement isn't true. In fact, you can't define just a molar Gibbs free energy. We'd say the partial molar Gibbs free energies or the chemical potentials are equal to each other in a multi-component system. So mu i is equal to mu i alpha is equal to mu i beta repeating this statement or the rate at which the free energy changes alpha over here, beta over here. In other words, if I add a molecule of A to the alpha phase and I get the same change in free energy as when I add a molecule to the beta phase, the molecule of A doesn't care which phase it's in. It's perfectly possible for the free energy of one phase, the total free energy of this phase to be greater or less than the free energy of this phase. What matters is not the total free energy, but the rate at which that free energy changes as I add an extra molecule. If it costs one kilojoule per mole data molecule here versus here, then the molecule doesn't care what phase it's in. So remember that for the multi-component system, the reason we compare chemical potentials is because it's the rate of change of the Gibbs free energy that matters. Another important thing to consider about this equilibrium expression is so far we've only talked about two phases at a time. What if we have a system that has three phases in coexistence with each other? Let's say we have a liquid, a gas, and also a solid phase. So three phases in coexistence with each other, again this could be some A's and some B's up in the gas phase, some A's and some B's in the liquid phase, perhaps A's and B's both in the solid phase. So we have multi-component system with three different phases coexisting. Here at the liquid gas interface, this equilibrium requirement requires that chemical potential in the gas phase is equal to the chemical potential in the liquid phase, because the liquid and the gas phases are in coexistence with each other. Down here at the liquid-solid interface, we know that the chemical potential in the liquid, I need to remember to keep the phases as superscripts, chemical potential in the liquid is equal to the chemical potential of the solid. But even though the solid is not in direct contact with the gas, if the gas is equal to the liquid and the liquid itself is equal to the solid, this means that gas and liquid and solid all have the same chemical potential as each other, even though the solid and the gas are not directly in equilibrium with each other. So if we have three phases in equilibrium, it doesn't matter if they're contacting one another, the three chemical potentials must all be equal. That's true if the solid sinks in the liquid as I've drawn in this diagram for substances like water, just for the sake of another example. If I have water as one of my components, when water freezes, of course, it's going to float. So let's say I'm talking about a pond with a layer of solid water on top, water vapor in the air, water liquid beneath the frozen surface of the pond. So in this case, I've got solid liquid, chemical potential of the solid and chemical potential of the liquid are in equilibrium with each other at this lower interface. At the upper interface, the solid is in equilibrium with the gas, and again, because both of them are equal to the solid, they must all be equal to each other. Chemical potential of the gas is the same as the liquid is the same as the solid, and though, in this case, the liquid and the gas are not in direct contact with each other. So when I have multiple phases all in equilibrium with each other, single component, multi-component doesn't matter. Chemical potentials are the same in each one of the phases that are involved. All right, so what else can we say about this statement? One important thing to say to give you a caution or a caveat about a common misconception about this statement. This statement says chemical potential is equal to chemical potential, but you have to remember that it's chemical potential in different phases that are equal to each other. It does not say that the chemical potential of two components are the same as each other. So again, let's take an example like this one. Let's say I have, let's make a concrete example, say I've made a solution of water with a tetrahydrofuran. So both of those solvents are volatile enough, I'll get some water vapor in the air, I'll get some THF vapor in the air. What this statement says is that chemical potential of water in the liquid, if it's in equilibrium, will be equal to the chemical potential of water in the gas phase. If the chemical potential in one of those phases is lower than the other, either water will evaporate or water will condense until the chemical potentials become equal. So the water in the vapor, the liquid in the vapor will be in equilibrium with each other. Same thing for the THF, it will either evaporate or condense as needed to make sure that the chemical potentials are the same in the water in the vapor phase. What is not true is that there must be any relationship between the chemical potentials of water and THF, let's say, in the liquid phase. Chemical potential of water in the liquid, chemical potential of THF in the liquid, are not necessarily equal to each other. In fact, they're not at all equal to each other. In fact, chemical potential in most conditions, chemical potential of water is going to be lower than the chemical potential of THF. So doesn't that contradict this statement? Ask yourself why it doesn't briefly. What I've said over here is that the chemical potential is lower in one place than another than something will happen, a reaction will happen that will cause the thing with higher chemical potential to turn into the thing with lower chemical potential. In general, that's true if the chemical potential of water in the liquid phase is lower than the chemical potential of THF in the liquid phase. What that means is that the free energy of the system would be reduced if THF were to convert itself into H2O, but that's not going to happen. There's no chemical pathway in a solution of just water in THF for THF to react and turn into the product of H2O. That's not a chemical reaction that's going to happen. So because there's no pathway to turn THF into H2O, there's no way for the system to spontaneously reduce its own free energy by removing moles of THF and producing moles of H2O. So this chemical potential requirement, what's true at equilibrium is the chemical potential between two phases is equal to each other because molecules can convert back and forth between those two phases. In general, if the chemical potential will be equal between any two things that can convert back and forth between themselves, but for the things that cannot convert back and forth between themselves, there's no requirement that their chemical potentials be equal.