 So when we're describing the properties of a solution, usually the most important property we need to talk about is the concentration of that solution. Concentration is just a measure of the relative amounts of the solute in the solvent. A concentrated solution will have a lot of solute, a less concentrated or a dilute solution will have relatively little solute compared to the amount of solvent. But since there's different ways of measuring the amount of solute, or more particularly the amount of solvent, that leads to lots of different definitions for how to measure concentration. So for example, if we choose to measure the amount of solute in terms of number of moles, so let's say I dissolve some number of moles of A in some amount of moles of B, if I think about the concentration as moles of the solute divided by the total moles of the solution, that's the measure we call mole fraction, and total moles of course would be moles of the solute plus moles of the solvent. If I have a binary solution, a solution with two components, just A and B, then a total number of moles is A and B. If I have a solution with more than one component, I just need to include of course all the moles of all the components in the solution. So that we would call a mole fraction. Notice that mole fractions will be unitless, moles divided by moles, there's no units in that quantity. We usually use the variable X to denote mole fraction, but there's other ways of measuring how much solvent I have or how much solution I have rather than moles. Probably a more common measure of concentration, more familiar perhaps, is the molarity of a solution, and we measure the molarity of a solution by describing the number of moles of solute dissolved in a total volume of the solution. So it might be easier to measure, certainly possible to measure the volume of the solution. We may or may not know the total number of moles of all the components of the solution, or we'd have to do a fair amount of work to figure that out. So it's easier to measure the volume perhaps. So if we know how much solute is in a certain volume, that's the molarity of the solution. We can use the notation A in square brackets to denote molarity. In some cases that can be confusing the square brackets, sometimes you might see concentration in molarity written as C sub A for the concentration of A. That usually denotes molarity. Notice the units here, in this case moles divided by volume. So if we have, those units are typically going to be moles divided by, always going to be moles over some volume. If it's moles divided by liters, we denote that as molar. So one mole per liter, we say that that's a one molar solution. Two moles per liter would be a two molar solution. So molarity, these units of molar, describes the molarity of the solution. We use capital M for that unit. The third important, there's many, many units of concentration. The third one that turns out to be more convenient, perhaps less familiar for you at this point, is the molality. And as we'll see as we go forward, molality often turns out to be more convenient than molarity, mainly because we were going to replace the volume with a mass. And rather than talking about the volume of solution, we'll talk about the mass of the solvent. Just the solvent without the other solute or solutes in the solution. So when we define concentration, again, as amount of solute divided by some measure of the solution of the solvent, in this case, the mass of the solvent, we'll call that molality. That could be written either as little m sub a for molality. That tends to be a little bit confusing because even in this single equation, we've already got m serving another purpose, mass of the solvent. Here, m means mass. It's a little confusing to have mass, meaning molality. Also, so I won't tend to use that notation. I prefer b sub a to describe the molality of the solution. The units here, moles of solute divided by mass of the solvent. If we express it in terms of moles per kilogram, then rather than capital M, denoting the molarity of the solution, we use a lowercase m. Again, a lowercase m to denote the molality of the solution. So if I say a solution has a molality of one mole per kilogram, I could also just say that that is a one molal solution. Its concentration is one mole per kilogram. So the reason molality is more convenient, there are several reasons, molality is more convenient than molarity. For one, mass is typically a little bit easier to measure than volume. Just put the solvent on a balance before you dissolve the solute and you know how much the mass of the solvent is that you've used to prepare the solution. Measuring volume is certainly possible, but it requires using volumetric flasks or something like that that's typically slightly less convenient. More importantly, mass is a conserved quantity. If I dissolve some solute in a solvent, I haven't changed the mass of the solvent that's in that solution. Volume, on the other hand, is not conserved. The volume of the solution can change in response to thermodynamic properties, or even in response to dissolving the solute in it. So volume is inconvenient in an experimental sense from that point of view. And also, it turns out that if we have a complicated solution with lots of components in it, it's much more convenient to talk about only the amount of solvent we use to prepare the solution without having to consider or analyze or discover all the different components that are in an unknown solution. So it's often more convenient just to talk about the solvent, rather than the whole solution as a whole. So, like I've said, there's many other measures of concentration. These are probably the most common in physical chemistry, and we'll frequently have occasion to convert back and forth between these two. So that's always possible, although you may sometimes need some extra information. For example, to convert back and forth between mass and volume, you might need a density to convert between moles and mass. You might need a molar mass and so on. So just to make sure, that's clear. Let's work an example where let's say we have a solution of benzene and toluene. I've mixed these two organic solvents. They're highly miscible with one another. I can combine them in whatever proportions I like. Let's say that I've prepared a solution in which the mole fraction of benzene is 28.5%. 0.285 is the mole fraction of benzene. So clearly the mole fraction of toluene is the remaining 71.5%. I will need to tell you the density of that solution. So the density of a 28.5 mole percent benzene toluene solution turns out to be 0.865 grams per milliliter. That's the density of the solution that I've prepared. Let's say I don't want to know the mole fraction of benzene though. Let's say I want to know the molarity. What is the molarity of benzene in that solution? So I need to convert from mole fraction to molarity. So easiest way to do that is just to start with listing what we know about moles and masses and volumes, et cetera, of the different pieces of the solution. So we've got benzene in the solution. We've got toluene in the solution. Let's start with, since we've been given mole fraction, one thing we certainly know is for every total, if I had a total of one mole of molecules, 0.285 of those would be benzene. That's exactly what the mole fraction tells us. 0.285 over one is the mole fraction of that solution. So it doesn't guarantee I have exactly one mole. But for every one mole that I have, 0.285 of them are benzenes. And the remainder of them, 0.715, in order to make them add up to one, have to be toluenes. So here's a sample solution, one mole consisting of 0.285 moles of benzene, 0.715 moles of toluene. We can, as an intermediate step, talk about masses. So to get from moles to masses, of course, we can multiply by molar mass. If I multiply benzene, this number of moles by the molar mass, 72.1 grams per mole, 0.285 moles. Multiply by that molar mass, moles will cancel, and I'll end up with a number of grams. And I've already done that calculator work. That works out to be 22.3 grams of benzene. I can do the same thing with toluene. Toluene has a different molar mass. So the mass of toluene in that solution is 65.9 grams. The total mass, now let's say I want to know the total mass of that solution, mass is conserved. If I mix 22.3 grams of benzene and 65.9 grams of toluene, add those two numbers up, 90, 88.2 grams total in that solution. So it's not a concentration unit that we've talked about. But if, for example, I wanted to know not the mole fraction, but the mass fraction of benzene, now I have the information I would need to calculate that. But we don't want to know mole fraction. We don't want to know mass fraction. We want to know molarity. In order to know molarity, I need to know moles of the solute divided by not mass, but volume of the solution. So I want to know moles of benzene divided by volume of the entire solution, which is a number I don't know yet. How do I get from mass of the solution to volume of the solution? Well, of course, I would use the density. If I, the density is 0.865 grams per milliliter, so if I multiply by 0.865 grams underneath 1 milliliter, so 88.2 grams for the total solution divided by 0.865, that gives me 102 milliliters. So now I've been able to determine knowing nothing other than these two facts that my hypothetical one mole, which is 28 and 1 half percent benzene, 71 and 1 half percent toluene, must have a volume of 102 milliliters or 0.102 liters. So now the concentration of benzene moles over total volume, so 0.285 moles of benzene over, if I wanted a molarity, I'll use the volume in liters. So 0.285 divided by 0.102 gives me 2.8, and now my units are moles per liter, or I could say that's a 2.80 molar solution of benzene. So that's just a quick example to show you that if we have mole fraction, we can obtain concentration. Of course, we could have gone the other way as well. I could also obtain molality. I can convert molarity and molality. It's possible to convert any of these concentration units from one to another. Sometimes we need extra information. In this case, I needed to know the density of the solution in order to convert back and forth between masses or moles and volume of the solution. But that's a relatively common problem the face is to be given one concentration units and prefer that it be in another one. Now that we understand a little about how to describe concentration units, convert back and forth between them, we'll start to use them in thermodynamic properties of these solutions. In particular, I'll point out right now that each of these concentration units is an extensive property divided by an extensive property, total amount of solvent or solution. So when I have an extensive property divided by an extensive property, the concentration will work out to be an intensive property. That's convenient. That's in part what led me to say it doesn't matter how much of the solution I assume I have, the concentration, this intensive property, is not going to depend on the total amount of solution I have. So I could have started with two moles or 10 moles or any number of moles I wanted to. Here, I'd obtain the same intensive concentration at the end. But unfortunately, when we start talking about multi-component solutions with more than one component in them, the idea of what's intensive and what's extensive gets a little more complicated and needs a little more discussion. So that's what we'll do next.