 All right, so we've determined the condition for phase equilibrium is that the chemical potential of a component, say component A, has to be the same in two different phases, maybe gas and liquid, maybe liquid and solid, whatever these two different phases are. So the reason that's an important and a useful thing to know is because it allows us to calculate the chemical potential in one phase if we already know it in another phase when the two of them are equal to each other. So let me illustrate what I mean by that. I suppose we have a system with two different phases in equilibrium. I've got A in the liquid phase. So here's a liquid phase, substance A, and in the gas phase in equilibrium with that, I have some gas phase molecules of A. So I'll put a lid on that box so they can be in equilibrium with one another. Molecules are transferring back and forth from the gas phase to the liquid and vice versa in that system's in equilibrium. So we know already how to calculate the chemical potential of a substance in the gas phase. We've seen that that's equal to the standard state chemical potential plus RT natural log of the pressure of that gas relative to some standard pressure. We also know that if the gas is in equilibrium with the liquid, the pressure that's in equilibrium with the gas phase by definition, that's what we mean by the vapor pressure. So this pressure that we're talking about is the vapor pressure of A. So that vapor pressure, if it's water that we're talking about, for example, water vapor in the air in equilibrium with water in the liquid phase, at room temperature, that vapor pressure would be about 24 Torr, 24 Torr worth of water vapor in the air at 100% humidity. So if we know what that vapor pressure is, if we know what the standard pressure is, that might be one atmosphere, one bar. We can define the standard pressure in different ways. But if we know what that standard pressure is, we know what the chemical potential is. Under standard conditions, this tells us how the chemical potential in the gas phase depends on pressure. So if these phases are in equilibrium and therefore the chemical potential in the liquid phase is equal to the chemical potential in the gas phase, then that tells us that the chemical potential in the liquid, which before we didn't know anything about, because the two chemical potentials have to be equal, chemical potential in the liquid is equal to that in the gas. The gas is equal to this expression, standard state chemical potential plus RT natural log of vapor pressure over standard pressure. So that's fine. That does what I told you it would. It allows us to calculate the chemical potential in the liquid phase if we already know what it is in the gas phase. So far, so good. That applies not just to a single component solution, not just to pure water in equilibrium with its vapor in the atmosphere. But I can draw a very similar diagram for a mixture, for a solution. So let's say I have some A and some B in the liquid phase. That system's in equilibrium. There's going to be some partial pressure of A in the vapor phase and some partial pressure of B as well in the vapor phase. I'll repeat the steps I've done here. It's going to look very similar, but with some small subtle differences. In particular, I can say the chemical potential of A in the gas phase, again, is just going to depend on its pressure. It's the standard state chemical potential plus RT natural log of pressure over standard pressure. Here's one of those subtle differences. Because this is not a pure liquid A, the partial pressure of A in equilibrium with that liquid, with that solution, is not going to be the vapor pressure. Partial pressure above a solution is not the same as the partial pressure above a pure liquid. So this pressure may not be, in fact, will not usually be the vapor pressure of A. It's some other pressure. So that's the only difference between this equation. And this one is we're no longer using the vapor pressure. It is still true, however, that the chemical potential of A in the liquid phase in the solution has to be equal to the chemical potential in the gas phase if they're in equilibrium. If A is in equilibrium in these two phases, the chemical potentials are equal. So then I can write the equivalent of this statement, which just says, if I want to know the chemical potential of A in the solution, it's going to be equal to this expression, standard state chemical potential in the gas phase, plus RT natural log of partial pressure over P naught. Very similar expression to this one. In fact, the similarity of those two expressions, the only differences are here I've got a vapor pressure. Here I have a partial pressure, which is not equal to the vapor pressure. On the left, I have chemical potential in the pure liquid phase in this equation, chemical potential in a solution in this expression. That similarity suggests that I can remove some of the terms that I don't particularly care about in this equation by comparing these two equations with each other. In fact, what I'm going to do is I'm going to take this expression minus this expression. So the lower equation minus the upper equation. On the left, that's going to give me chemical potential of A in the solution minus chemical potential of A in the liquid. That's going to be equal to, on the right side, I've got mu naught minus mu naught and RT log pressure over P naught minus RT natural log vapor pressure over P naught. So now we can see there's going to be a fair amount of cancellation. In particular, mu naught cancels mu naught. There's two RTs here, so I can collect some terms together. So I've got chemical potential of A in the solution minus chemical potential of A in the liquid is equal to RT. When I combine these natural logs, RT Ln of this minus Ln of that, that's like Ln of the quotient PA over P naught over PA star over P naught when I combine those natural logs together. And that shows us another cancellation, 1 over P naught in the numerator, 1 over P naught in the denominator. Both cancel. And final version of this, I'll write chemical potential in solution. Let me move chemical potential in the pure liquid. And we'll be moving that to the right side of the equal sign. I'll say that's now equal to positive chemical potential in the liquid plus RT natural log. And now all that's left inside the log is pressure over vapor pressure PA over PA star. Pressure over vapor pressure. So that's an expression we'll come back to and we'll use in the future. What that expression tells us is we can now calculate the chemical potential in a solution. If I know the properties of the solution, doesn't matter what the other solvent is or what the composition of that solution is, I can calculate the chemical potential of component A in that solution as being relative to the chemical potential in the pure liquid phase plus some correction term that just involves the partial pressure above the solution and the vapor pressure above a pure solution. So that's a useful thing to be able to do, calculate the chemical potential in the solution. Notice one very important thing about this expression is that after all this cancellation, there's no more superscript zeros. All of the standard state variables have disappeared. So this expression was somewhat inconvenient because I'd get a different answer if I've chosen to define the standard state as being one atmosphere. I'd have to insert one atmosphere and I'd have a particular chemical potential at standard state here. If I choose to define standard state as equal to one bar, then this value is different and one bar is a little bit different than one atmosphere and so on. So sometimes the work depends on what we choose for our standard state. In this expression, however, the difference between solution chemical potential and pure liquid chemical potential no longer depends on the standard state at all. So that is very convenient. It doesn't matter whether I define the standard state to be one atmosphere, one bar, or some other conditions, I'm gonna get the same result here. And that's, in fact, a pretty common way of removing any dependence on the standard states is to find two expressions that both depend on the standard state in the same way and subtract them from one another. So this difference in chemical potential won't depend on how I calculate the standard state or how I define the standard state. The other thing to notice about this expression is clearly the chemical potential depends on the pressure. So in order to be able to use this expression, I need to know not just what the vapor pressure of my solute or solvent is, but what that pressure would be over a particular solution, in equilibrium with a particular solution. So that's the next step, is to be able to understand and make some predictions about what that partial pressure is above a solution.