 So, for the three colligative properties we've considered so far, vapor pressure lowering, freezing point depression, boiling point elevation, those have all been the properties of some solution that's in equilibrium with some different phase. The solution is in equilibrium with its vapor for boiling point elevation and vapor pressure lowering. The solution is in equilibrium with the solid for freezing point depression. The one additional colligative property we need to talk about, the osmotic pressure happens when a solution is in equilibrium with a liquid rather than a solid or a gas. So, I'll draw a picture to show you what I mean by that. So, let's say we have a beaker of, let's see, I'll make this side a solution with some solute molecules dissolved inside this beaker. And I'm going to make that in equilibrium with, so I'm going to connect that beaker to a different beaker. And this beaker has pure solvent in it. So, over here I have pure solvent A, over here I have a solution A with some solute dissolved in it. So, that's actually kind of a silly experimental setup. You can predict what's going to happen immediately here if I have a glass of saltwater connected to a glass of pure water, what's going to happen is the salt is going to diffuse until it fills the entire system. So, that's kind of boring, that's not what we want to study. So what I'm going to do in addition to that is I'm going to add a membrane, a barrier separating these two. If that's a impermeable membrane, if it doesn't let anything pass then the solution stays on this side and the pure solvent stays on this side. But let's consider that this is a, not a perfectly impermeable membrane, but a semipermeable membrane. And in particular it's going to allow solvent molecules to pass back and forth. If these are aqueous solutions water can pass back and forth through the barrier, so it's a membrane that water can pass back and through. But the holes, the pores in the membrane might be large enough that it prevents the solute molecules from passing through. So the solute molecules are going to stay on this side, only the solvent can pass back and forth. So what will still happen, because over here we have pure liquid which has some chemical potential. On the other side we have chemical potential of a solution, which we know chemical potential of the solution is chemical potential of the liquid plus RT log of the activity in that solution. So that quantity, because RT log activity, that's going to be a negative number. The chemical potential on the solution side is going to be lower than on the solvent side. What that means is solvent molecules that have the ability to pass back and forth through this membrane, they can lower their chemical potential by passing through the membrane in that direction. So that's spontaneously what's going to happen. If we wait a little bit and look at the system again, we'll find what has happened is a significant amount of solvent is going to transfer from one side to the other. So there's still a barrier in between. Some solvent has passed through the barrier, I've got less solvent on the right side, more solvent on the left side, and we still have all the solute molecules on the solution side. By doing that, I haven't changed the chemical potential of the solvent on this side, but by diluting the solution, by making the concentration closer to pure solvent, I've moved the activity closer to one, and I've increased the chemical potential on this side. So the chemical potentials are coming closer to being in equilibrium with each other. They wouldn't be in perfectly in equilibrium with each other considering only this concentration effect until I removed all the solvent from this side. But that may not happen because the other factor we have to consider is the pressure. So if this system is open to the atmosphere at atmospheric pressure, let's say there's one atmosphere of pressure pushing down on this side, there's also one atmosphere of pressure pushing down on this side, but because the height of the liquid column is higher on this side, that liquid column has some mass. The pressure exerted by that liquid column on the solvent beneath it is fairly substantial. So there's some additional pressure. We call that pressure, for now let's just give that pressure a label and call it pi. So there's atmospheric pressure of p0 as well as the additional pressure pi that's pushing down on the solvent down here. So the solvent at the bottom of this beaker is under a higher pressure than the solvent at the bottom of this beaker. We also know that the chemical potential goes up when the pressure goes up. In fact we've seen quantitatively how much that happens. The derivative of chemical potential with respect to pressure is equal to molar volume. So if we're talking about chemical potential of A, the change in chemical potential when I change the pressure is the partial molar volume in this case for a multi-component system. So in other words, if I increase the pressure on one side or the other, the change in the chemical potential is going to be, at least for an incompressible solvent, it's going to be the partial molar volume multiplied by that change in pressure. So what does that help us do? If I ask myself what is the equilibrium condition, if the solvent goes in this direction causing the liquid level to rise, the liquid level is going to rise because the chemical potential is lower on this side, but as the chemical potential on this side, as the solvent flows in to the side, number one, the solution will get less dilute, sorry, more dilute, less concentrated. That will increase the chemical potential over here. Also the pressure will rise, which will also increase the chemical potential over here. So eventually the chemical potential on this side will equal the chemical potential on this side and then will be in equilibrium. When the chemical potential is equal in two different parts of a system, then that system's at equilibrium. So if we consider not just the concentration effects, but also the pressure effects, what must be true at equilibrium is on this side, the chemical potential of this liquid at pressure p0 plus pi, as well as the contribution from the activity of the solution on this side, that is all the chemical potential over here on the left. It's going to be equal to the chemical potential on the right, which is just the chemical potential of the solvent in liquid state at just atmospheric pressure, without this extra pressure that we might as well go ahead and name the osmotic pressure. When this system reaches equilibrium, this pressure caused by this osmosis effect is going to cancel out the reduction in chemical potential due to the solution. So number one, this process of the solvent moving through the semipermeable membrane from the pure solvent side to the solution side, that process is called osmosis. The osmotic pressure is specifically the pressure needed to stabilize the system under these circumstances, the additional pressure needed to balance the chemical potential so that the side with elevated pressure and concentration of some solute is equal to the chemical potential of the pure liquid. So that's our condition for being in equilibrium. If I rearrange that equation a little bit to say chemical potential at elevated pressure and if I move this chemical potential of the pure liquid at standard pressure over to the left side and on the right side I'll keep minus RT, or I'll move this RT log A over to the right side so I've got minus RT log of activity, I'm just rearranged this equation to get this one. This difference in chemical potentials, how much did the chemical potential change when I went from standard pressure up to an elevated pressure, P0 plus the osmotic pressure pi? Well, that's exactly what this expression tells us. The chemical potential is going to change by this much, partial molar volume times the change in pressure. So the left side becomes partial molar volume of the solvent multiplied by the difference in pressure, P0 plus pi minus P0 just leaves me with the osmotic pressure pi. That's all equal to minus RT log activity. So there's a perfectly valid expression that we can use for osmotic pressure. If we solve that expression for the osmotic pressure, I've got minus RT over partial molar volume multiplied by log of activity of the solvent. So if we would like to know how high will this column of liquid rise, how much extra pressure will be generated over here on this side, the answer to that question depends on the temperature, depends on the partial molar volume of the solvent, depends on the activity of the solution over here on this side. As usual, activity is not always the most convenient property to work with. So if we make our dilute solution approximation, as we've done it before, just as a reminder, log activity, that's about the same thing as log of mole fraction in an ideal solution, which is log of one minus mole fraction of the other component, the solute, and log of one minus x, that's about the same thing as negative x. So using that series of approximations, I can write minus log activity as positive mole fraction of solute. So I've got an RT over partial molar volume, and that's a different expression for the osmotic pressure, RT over partial molar volume times mole fraction. This is now mole fraction of solute rather than activity of solvent. And as we've done with our other colligative properties, if we don't like using mole fraction in a dilute solution again, I can say that mole fraction of B, that's moles of B over moles of A and moles of B together. And if the solution is dilute, which we've already assumed it is, that's going to be moles of B over moles of A, where I'm neglecting this small number of moles of B relative to the number of moles of A. And so that will allow me to rewrite this equation as osmotic pressure is RT moles of B over moles of A and a partial molar volume. But now think about what this denominator looks like. Moles of solvent times partial molar volume of solvent, again in a dilute solution. If this were pure solvent, moles times molar volume, that would just be the volume of the solution. So again, in dilute solution that's going to be about the same thing as RT over volume of the solution, again multiplied by moles of the solute B. And now moles of B divided by the total volume of the solution, that's exactly what we think of as a concentration in molarity. So the last rewriting of this equation we can do would be to say, moles divided by volume, that's a concentration in molarity, concentration of the solute. So the most simplified version of this expression, which has made a few levels of this dilute solution approximation, we can say that the osmotic pressure is molarity times gas constant times the temperature. So again, as with other colligative property, we've got more accurate versions of the expression and then a series of approximations leading to one that's relatively simple. So let's work an example of this and we'll do that in the next video.