 So let's look a little closer at this distinction, the important distinction between the condition for phase equilibrium where the chemical potential of a component in two different phases has to be equal for the two different phases. So alpha and bay here are two different phases, but that's different than saying, for example, chemical potential of A and chemical potential of B, two different components. Those two are not necessarily the same as each other. So it's important to remember the difference between those two statements. Chemical potential is equal across two phases. It's not equal across two different compounds, two different substances. But there is actually a connection between the chemical potentials or at least the changes in chemical potentials for two different substances in a system at equilibrium. So we can learn a little bit more about that connection between the two chemical potentials. They're not equal to each other, but they're related in a different way. So I guess we'll start by saying, start by recalling the fundamental equation for Dg. In particular, not the first version of the fundamental expression that we've seen, but now for a multi-component system, we have to remember that there's a term that involves chemical potentials. So it's minus SDT plus VDP reflecting the change in temperature and the change in pressure, but also some terms that look like chemical potential times DN that reflects the change in the Gibbs free energy if I'm changing the amounts of any of the components in the system. So at constant temperature and pressure to make that a little easier to think about, if I restrict myself only to think about constant T and P, then the DT and the DP terms go to zero, and the only term I have to worry about is these terms that involve chemical potentials. This is telling me how the Gibbs free energy of the system is changing if I'm changing the amounts of any of the components. So for example, I could be changing the composition. I could be diluting a solution or making a solution more concentrated, changing the relative amounts of the two components or perhaps more than two components, or somewhat easier to think about, I could be changing the total extent of the system. Let's say I have some amount of each component. I have some number of moles of component one, component two, component three. If I were to double the amounts of all those components, I'm going to just double the size of the system. I haven't changed the concentration. I haven't changed any intensive properties, but by doubling all the amounts, let's say each of the moles turns into twice as many moles, or each of the moles turns into three times as many moles, or whatever multiplying factor I want. If I change all of the amounts by the same ratio, then I haven't changed the intensive properties. I've just changed the extent of the system. So that's what we can think of lambda as is how much of this, what multiplier, what extent I've changed the system. So let's think, if we do this, the change in the number of moles when I change the extent of the system is going to be however many moles I originally had multiplied by the change in lambda. So if I change from one, change from lambda equals one to some different value, the amount by which I changed lambda multiplied by the original number of moles tells me how much the moles of each component are changing. And notice the change is only in the multiplier. The initial amounts, the ratios of the initial amounts stay fixed. The other thing I can say that lambda affects is the free energy. Free energy, remember, is an extensive property. So if I double the size of the system, I'm going to double the free energy. Likewise, if I change lambda by a smaller amount, then that's going to induce some change in the free energy. That's just the initial change in energy multiplied by the change in this multiplier. So those two statements that tell me how to calculate the change in the free energy, if I know d lambda, and how to calculate the change in each one of my molecule numbers, if I know d lambda, I can put those into this expression. I have a dg and I have a dn. So I'll use those two terms to rewrite this as dg is equal to g times d lambda. And mu times dn is equal to mu times n times d lambda. I have to sum that over all the components. And in that sum, the only thing that depends on the component number is the chemical potential of each component and the number of moles of each component. So I can pull that d lambda out of the sum. Once I've done that, there's a d lambda on the left and a d lambda on the right that cancel. So I've learned that g is equal to the sum of chemical potential times moles. So that's already an important conclusion. That's the fact that we're going to come back to relatively soon. Not the change in g, not the differential change in g, but the absolute amount of Gibbs energy is the product of the chemical potential multiplied by the number of moles of each component. Each substance has its own partial molar Gibbs free energy. If I multiply those partial molar Gibbs free energies by the amount of each component I have, and I add them all up, that gives me the total Gibbs energy of the system. So that's useful enough, but let's continue to see what we can learn from that statement. We normally talk about dg rather than g, so let's go ahead and take the differential of this expression that we've just obtained. If g is equal to mu times n, sum of the overall components, dg is equal to mu times dn plus dmu times n. Again, each of these needs to be summed over all the components. So that's just the differential of this expression. But now we're almost to the point where we have reached the payoff that we're looking for, the thing that's going to be called this Gibbs-Duhem equation. We have this expression for dg, and we have a similar looking expression for dg, this one right here. This one also has a term in it that looks like sum of mu times dn. Here's another term that looks like dg is equal to sum of mu times dn. This expression says dg is equal to the sum mu times dn plus a few terms that involve t and p. This expression says dg is equal to that same sum plus something different. So in other words, this term must be equal to those two terms. So if I write down that conclusion that I've just drawn, sum of n times dmu, summed overall components in our system is equal to minus s dt plus v dp. So that turns out to be an important observation, again one that we'll come back to multiple times. It's important enough that it gets a name. That is the Gibbs-Duhem equation. And let's see, I'll point out a couple of things about that expression. This is the more general form of the Gibbs-Duhem equation. Sometimes we use that expression at constant temperature and pressure if dt is equal to zero and dp is equal to zero because we're not changing temperature and pressure, then that simplifies and we can just say sum of moles times changes in free energy is equal to zero. So that's a more specialized but useful version of the Gibbs-Duhem expression. So again, at constant temperature and pressure, the sum of n times dmu is equal to zero. What does that mean? This has been a fairly abstract algebra filled discussion. What does it mean to say the sum of moles times changes in chemical potential is equal to zero or is equal to some combination of the temperature and pressure? Let's just think about what that looks like for a binary system, a solution, let's say, with two components. So I just have a and b, two different components to worry about. If I write down what this sum looks like, sum of n times dmu, that's going to look like moles of a times change in chemical potential of a plus moles of b times change in chemical potential of b. And that's equal to perhaps zero if we're at constant temperature pressure, perhaps minus sdt plus vdp if we're allowing the temperature and pressure to change. But what that tells us now is when I make a change to the system, perhaps I only change the composition, perhaps I concentrate or dilute the solution without changing temperature and pressure, then this sum of moles and change in chemical potential has to sum to zero. That means that when the chemical potential of the solvent changes as I change the composition of the solution, there must be a corresponding change in the chemical potential of the solute such that these two terms sum to zero. Or if the way I affected some change in the chemical potential is by changing the pressure of the system, for example, then still there's some relationship between the amount by which the chemical potential of the solute and the amount by which the chemical potential of the solvent have to change. They together have to add up to whatever I've done to the temperature and pressure. So it's not true that the chemical potentials are equal, but the Gibbs-Duhem expression tells us something about the relationship between any changes to the chemical potential in the different components of the system as I modify the system in some way.