 So Raoult's law tells us for a solution, a mixture of two liquids, the partial pressure of one component above that solution can be calculated if we know the composition of the solution, in particular the mole fraction of that component, and the vapor pressure of the pure component when it's a pure liquid. So that's Raoult's law. That tells us everything we need to know to calculate the pressure, total pressure above that solution. So for an example, let's consider a mixture of two liquids, two solvents, isopropanol and water. Let's say I tell you that the mole fraction of isopropanol is 70%, and I'll also tell you the vapor pressure of isopropanol at temperature, so I'll specify the temperature, 25 degrees Celsius. At this temperature, the vapor pressure of isopropanol is 45 tour, and the vapor pressure of water is a different number, 24 tour. So that's enough information. In fact, I can't give you any more information than that. And I can't also give you, for example, the pressure of that system. We know that's true because the Gibbs phase rule tells us if we have two components, isopropanol and water, and we have two phases, the liquid and gas coexisting with one another, two minus two plus two, that's two degrees of freedom. So I can give you the composition of the solution by providing one of the mole fractions. I can give you the temperature, but I can't independently specify the mole fraction of water. That number has to add up to 100%. Mole fraction of water has to be 0.3. I can't independently tell you the total pressure above that solution because that's going to be determined from these other properties. So that's the question is, what would the pressure above that solution be? Before we dive in and answer that question, let me instead do this essentially the same example, but with rounder numbers so that we can focus on the big picture of what's going on. So I'll do that over here. Let's say instead of real solvents like isopropanol and water, let's say a system with solvent A has a vapor pressure of 400 Torr, solvent B has a vapor pressure of 200 Torr. So as I said, nice round numbers. Now I'm going to make a solution that has mole fraction of A and mole fraction of B both equal to 50%. So half A and half B as mole fractions. So that's enough information already to calculate the pressure. So we can calculate the pressure, partial pressure of A as mole fraction times vapor pressure. So 0.5 times the vapor pressure, so that works out to 200 Torr for which we don't need a very fancy calculator. Half of 400 is 200, likewise for component B the partial pressure of B above that mixture is going to be its mole fraction times its vapor pressure or in this case 0.5 times 200 Torr gives me just 100 Torr. So so far so good. I can add those two numbers together. The total pressure above the solution, in other words the pressure of A and the pressure of B added together, 200 and 100 add up to give me 300 Torr. So if I sketch a picture of what's going on in that system, so I've got A and B in a solution, they're in equilibrium with the vapor. So I have some partial pressure of A, some partial pressure of B. The partial pressure of A is greater than the partial pressure of B. So more A has escaped into the vapor phase than B has escaped into the vapor phase. But Raoult's law tells us exactly how much of each have escaped into the vapor phase under our in equilibrium with the liquid and the total pressure of A and B together above that solution is 300 Torr. So so far so good. We can do essentially that exact same process for our real world example. The pressure of isopropanol, partial pressure is going to be its mole fraction times the vapor pressure of isopropanol. So 0.7 times 45 Torr. So I do need a calculator for that works out to be around 32 Torr. Partial pressure of water, again if it were pure water at room temperature the vapor pressure is 24 Torr but in this mixed solution since it's only 30% water as a mole ratio, 30% of 24 Torr works out to about 7 Torr. And so when I add those together the total pressure above that solution 32 and 7, total pressure is 39 Torr. So that's the numerical example for this particular case if I have a rubbing alcohol or a disinfecting solution made up of isopropanol and water in this mixture. We can predict that the vapor pressure is going to be about 39 Torr if the solution is ideal if the solution obeys Raoult's law. That means a couple of different things. I can, we've done essentially the same problem twice. I'm going to do it one more time for the general case. If I don't even know what the solvents are, I can still use Raoult's law to say partial pressure of A is its mole fraction times its vapor pressure, partial pressure of B if it's a binary solution, it's going to be mole fraction of B times its vapor pressure but in a binary solution the mole fraction of B is 1 minus the mole fraction of A. Then the total pressure is going to be the sum of these two. It's going to be xA vapor pressure of A plus, let me expand this out and write it as vapor pressure of B minus mole fraction of A vapor pressure of B. Or I can clean that expression up a little bit and write it instead as mole fraction of B taking this term first plus mole fraction of A multiplied by a PA star and multiplied by a negative PB star. So that expression, that tells us how the total pressure above the solution depends on the composition, depends on the mole fraction of component A. If I draw a graph of what that looks like it will begin to make some qualitative sense. So I can draw a graph of mole fraction and total pressure. That's a linear function, it's a constant plus my variable times some other constants. We know that when the mole fraction is zero, the smallest I can make the mole fraction is zero. Both according to this equation when I plug in a zero here that whole second term goes away and the total pressure is just PB star. In this example that was 200 tour so I'll put that about here on the graph. So PB star, when the mole fraction is zero that makes intuitive physical sense as well. If the mole fraction of A is zero the solution is not a solution, it's pure liquid B and the pressure above that solution is just the vapor pressure of B. On the other hand, the largest I can make the mole fraction is 100% A and in that case either mathematically when I make mole fraction of A equal to one then this one times negative PB star will cancel the PB star and all I'm left with is PA star. Or just thinking about it physically, when the mole fraction of A is one I don't have a solution I have a pure liquid A system and the vapor pressure of that system will be whatever the vapor pressure, the pressure above that system will be whatever the vapor pressure of A is. So I'll draw that higher in this case just because in this example I had vapor pressure of A being larger than the vapor pressure of B. In between is where things get interesting. If I have a 50-50 solution or 80% A or a 20% A solution if I have some composition and solution that's not pure liquid A or pure liquid B the total pressure above the solution is just a straight line connecting those two points. And the way we can think about what that straight line is, why it's a straight line, Raoult's law tells us it's a linear function but this is the partial pressure, the pressure of each individual component. Pressure of A is a linear function that rises from zero up to the full vapor pressure as the mole fraction goes from zero up to 100%. Likewise the partial pressure of B is going to rise from zero up to its full vapor pressure but in that case it's going to rise in the opposite direction. So this graph is the graph of the partial pressure of A, this one is the graph of the partial pressure of B and when I sum those two things together partial pressure of A plus partial pressure of B that's what gives me the total pressure. They're both straight lines, when I add those two straight lines together I get this straight line that represents the total pressure above the solution. So numerically, intuitively, graphically, all work out to the same thing, Raoult's law tells us that the pressure above the solution, the pressure of this solution varies linearly between the vapor pressure of B and the vapor pressure of A depending on what the concentration of those solutions are. Now where things get a little more interesting is when I think about not the mole fraction of the solution, so the mole fractions we've been using, 70% isopropanol or 5050A and B, those mole fractions are the mole fractions in the liquid phase. If I think about what the mole fraction in the vapor phase is, your first guess might be that that should be the same but that's clearly not true. I've got twice in this 5050 liquid mixture, I ended up with twice as many molecules in the vapor phase, A molecules as I do, B molecules in the vapor phase. So when we start to think about what the mole fractions are in the vapor phase, things will get a little bit more interesting and that's what we'll explore coming up next.