 So, one concept that needs a little more discussion for solutions than we've given to it in single component solutions is the idea of volume and in particular molar volume. So far, we've defined molar volume for a single component solution for a system with just one substance. Molar volume is just volume divided by moles and that's a way of getting an intensive property out of two extensive properties. The volume is proportional to how much material we have, the moles are proportional to how much material we have. If I take the ratio of those two, I get this intensive molar volume. That concept gets more complicated for solutions for a couple of reasons, some obvious and some more surprising. The first immediate problem is how do I compute this? What volume do I use? What moles do I use? Do I use moles of the solute, moles of the solvent, total number of moles? It's ambiguous how to define this. So it certainly needs a more careful definition if we were to use a molar volume. But it also turns out that even the concept of how to measure this and the values they would get gets a little bit complicated. So suppose as an example to illustrate this, we take 100 milliliters of benzene and prepare a solution by combining that with 300 milliliters of toluene. This happens to be the same 2.8 molar or mole fraction 0.285 solution of benzene and toluene that we've discussed in our previous example. So let's just say I've got a solution containing 100 milliliters of benzene and I'm going to pour that into a larger beaker together with a total volume of 300 milliliters of not benzene but toluene. So I'm going to make a solution by combining those two. The total volume of this solution, if I ask you to tell me what the total volume is when I combine 100 milliliters and 300 milliliters, you might predict 400 milliliters, but you'd be wrong. It turns out if I do a careful enough measurement, I measure a total volume of that solution of a little over 400 milliliters. 300.3 milliliters is the total volume once I mix the benzene and the toluene together. So the important point is that this is not equal to the sum of the two volumes. The volumes are not additive in this particular case. In fact, it's often going to be true that the volumes of the two substances when I combine them together will not add up to the total volume of the solution. That effect gets even more pronounced for solutions of ionic solutes in a polar solvent. So let me give you another example. Let's suppose I want to make a solution that's one molal of NACL in water. So the most straightforward way to do that, if I want a solution that's one molal is maybe take one mol of sodium chloride and dissolve it in one kilogram of water. So if I start with, let's again measure fairly precisely, exactly 1,000 grams, one kilogram of water. That's the mass of the water, the volume of the water. If I ask you what the volume of 1,000 grams would be, your first guess is probably going to be that since water has a density of one gram per milliliter, it's going to be also 1,000 milliliters of water. It turns out, in fact, if we look at the density of water at room temperature at least, the density is not exactly one at room temperature. At 25 Celsius the density is a little less than one, so the volume is going to be a little larger than 1,000 milliliters. So again, if we're being careful, my kilogram of water at 25 Celsius is going to be 1,002.9 milliliters. I'll dissolve in that one mol of sodium chloride. Sodium chloride has a mass, molar mass of 58 grams per milliliter, 58.44 to be a little more precise, grams per mole. So the mass of one mole of sodium chloride is 58.44 grams. If I know the density of solid sodium chloride crystals, I can also figure out the total volume of sodium chloride that I'm going to dissolve into that solution. That volume works out to be 26.9 milliliters, so that's the total volume of the salt crystals that I'm about to dissolve in 1,002.9 milliliters of water. When I dissolve a mole or 26.9 milliliters of salt into 1,002.9 milliliters of water, the volume, the total volume, the solution volume, you might predict would be just adding these two together. As you've probably guessed by now, that's not going to be the right answer, so if I add those two numbers together, that's 1,029.8 milliliters, and that's not what happens. The total solution volume is not 1,029.8. I can't just add the two volumes to get the total volume. If I do that experiment, if I actually mix those two things together and measure the volume, it turns out the volume I get is considerably lower. 1,021.4 milliliters is the volume of that solution. The total solution is less than I'd predict just by adding the two component volumes together. In this case, it's worth a small digression to explain why that happens. In this case, the reason that happens when I dissolve an ionic solute in a polar solvent like water, what happens is the sodium ions have a strong enough electric field, stronger electric field surrounding that positive charge than water does in pure water. The water molecules that are surrounding that sodium ion are pulled closer to the sodium than they would be around a water molecule. In particular, the distances between water molecules themselves get a little bit smaller as they're pulled closer by this electric field. The same thing happens to the positive ends of the water molecule. The hydrogens are pulled more closely around the chlorine than they would be in pure water. The effect of these bear positive and negative charges is to compact the water structure a little bit. That phenomenon is called electrostriction, which is just a way of saying that the electrostatics, the electric field of these ions, pull the waters or constrict the waters, pull them closer together. That's a fairly common phenomenon for ionic salts in polar solvents like water, but the more general feature, even if we're not dealing with polar solvents or ionic solutes, is that the volume is not an additive property. It is true that when I mix two substances together, their masses are additive. Mass of A and mass of B will add up to give me the total mass. It's also true that I'm not going to destroy any molecules unless there's a chemical reaction taking place. If all I do is mix a solute and a solvent, I can add the number of moles and I'll get the total number of moles. I can add molecules, I get the total number of molecules. I'm not going to destroy any molecules or destroy any mass, but the total volume of the solution is almost never exactly equal to the volume of the two pieces combined. Let's suppose we continue this process. I've added one mole of sodium chloride to a kilogram of water to make my one mole-ol solution and the volume was a little bit surprising, lower than I predicted. What if I continue and add a second mole of sodium chloride to make a two mole-ol solution and so on? Instead of working it out case by case, let me go ahead and put up a graph of the results if I were to do that experimentally. What I've shown here is a graph of the total volume of a solution as a function of the mass of the sodium chloride that I've added to that solution. This diagonal line, the dashed line here, would be my naive prediction if I just add together the volume of the water and the volume of the sodium chloride that I've added, that's this prediction along this straight line. For example, 1029.8 would be what I get if I take 58 grams of sodium chloride and I add it to a kilogram of water, I'd predict something a little less than 1030 milliliters. In fact, what I got is lower than that. I got a point on this line, so one mole, about 58 grams, that's this point right here. The volume of that solution was only 1,021 milliliters. If I do it again, so roughly 116 or 17 grams of sodium chloride, I add it, that's probably this point right here. Again, the volume of the solution has gone up, but it turns out it goes up by a different amount, so let me go ahead and, if I add two moles, the total volume of that solution, actually let me talk about it this way. In going from pure water to the one mole-ol solution, the slope of that curve, so the difference between 1,021.4 and 1,002.9, the difference in volume was 18.5 milliliters. When I added a mole of sodium chloride, I've increased the volume by 18.5 milliliters. You might expect that if I add my second mole of sodium chloride, I'm going to increase the volume by another 18.5 milliliters, not the 26.9 that I'd expect if volumes were additive, but the second mole might contribute the same amount as the first, but that's actually not what happens. When I add my second mole of sodium chloride, the change in the volume turns out to be a little larger. I increased the volume of the solution by 20.5 milliliters. The first mole adds 18.5 milliliters, the second mole adds 20.5 milliliters, and you can see from the second dotted line that if I just extrapolated the trend from the first small amount of sodium chloride, I'd expect this dotted line. In fact, as I add my second mole, and then my third, and then my fourth, and so on, the extra volume keeps getting slightly larger every time. So there's actually several confusing or surprising things about how volume works in these solutions. Number one, volumes are not additive. The total volume of the sodium chloride in the water don't remain constant as I add them together, but as an even more subtle effect, even the change in the volume doesn't remain constant. 18.5 milliliters isn't 26.9, it's not the molar volume, but the volume of the first mole that I add is different than the volume of the second mole, different than the volume of the third mole. So even the changes in volume are not constant. Okay, so that suggests that we need a whole new way of thinking about volumes, instead of just saying the molar volume of water is some value and the molar volume of sodium chloride is some value. I can't just say every mole takes up 26.9 milliliters. It depends whether it's a mole of pure sodium chloride or a mole that's dissolved into pure water or the second mole dissolved into an already one mole-l solution and so on. So it turns out that the most useful way to think about molar volumes in solutions is as the slopes of these curves. So this slope is 18.5 milliliters, this slope is 20.5 milliliters in general. I can say at any point along this curve, the rate of change of the volume as I add a mole of sodium chloride is going to be the molar volume of sodium chloride. So I can define my molar volume in this way, the rate of change of the total solution volume as I add more moles of the solute. So more generally I would say V bar for component I is equal to the rate of change of the total solution volume with respect to the moles of that component. And the name we give to this quantity, since it depends on the conditions, it's different for the first mole and the second mole and so on, is it's not a molar volume, it's a partial molar volume to remind us that what we're actually doing is taking a partial derivative of the volume with respect to the number of moles. So that is the generalization of the more simple single component molar volume to a case where I have more than one component, we need to use this idea of the partial molar volume to describe the volumes of components of solutions. Turns out not only does the volume depend in a complicated way on the concentration of the solution, but other thermodynamic properties also depend on the concentration of the solution and the temperature and pressure and so on. So we'll have not only partial molar volumes, but other properties for which we'll have to define similar properties. So we'll take a look at that next.