 Okay, let's take a minute and do a summary of all the different colligative properties we've talked about so far. So what I've put up on the board here is a summary of all the equations we've seen so far for the four different colligative properties, vapor pressure lowering, freezing point depression, boiling point elevation, and osmotic pressure. So there's a few things to point out here. Number one, first of all, freezing point depression, boiling point elevation, those are almost the same thing, just one mirrored for freezing point versus boiling point. So each one of these expressions in these second and third rows is exactly the same. I've just replaced fusion with vaporization for freezing point and boiling point. We have a freezing point depression constant or a boiling point elevation constant. The only qualitative difference between those two is boiling point gets elevated, freezing point gets depressed. Comparing all four of the colligative properties, they started out with the assumption that the chemical potential in the solution is equal to the chemical potential in some other phase if that solution is in equilibrium with the phase. Maybe it's evaporating or boiling, maybe it's freezing to form a solid, or maybe in an osmotic pressure setup, it's in equilibrium with the pure solvent. So from that starting point, we were able to derive expressions for how the size of that colligative property, vapor pressure, freezing point, boiling point, osmotic pressure, how that depends on the activity of the solvent in each case. Each one of these expressions involves the activity of the solvent. So these are the most fundamental versions of the expressions. It's not always convenient to use activity. So that's why we ended up deriving a different version of these expressions. In dilute solution, we can make assumptions about the activity of the solvent being related to the mole fraction of solvent and then solute. So these expressions all involve mole fraction, but notice that's mole fraction of solute, not mole fraction of solvent. So that begins to sound more like a description of how much of the small trace concentration, trace amount of material there is in the solution. That begins to sound like a concentration. We can think about concentration certainly in units of mole fraction if we want, but often it's more convenient to think about concentrations in units of molality or perhaps molarity. This is concentration of B. So in the final column, we have this most simplified version of this expression where we've often combined a bunch of constants into one constant, and we've also converted mole fraction into perhaps a more convenient unit, although mole fraction is usually not that inconvenient a unit to use. So we have a hierarchy of solutions, of sets of equations to use. If you happen to know the activity of your solvent, these equations are most accurate, somewhat less accurate are these expressions that are expressed in terms of the mole fraction and then less accurate still are the simplest versions of these equations. So depending on the circumstance, depending on the colligative property you're interested in, we have lots of different varieties of these equations to use. One final thing I'll point out before we move on and away from colligative properties is this name colligative properties. What all these properties have in common, as you can see in any one of these equations, the size of the modification of some property of the solution is proportional to the concentration of the solute, mole fraction of solute, molality of solute, concentration of solute, multiplied by some things that are properties of the solvent. That might be hiding inside one of these coefficients, but boiling point of the solvent, heat of vaporization of the solvent, those are all properties of the solvent itself. So the magnitude depends on how much solute I have, but not which solute, and depends on which solvent I have because I need to know the properties of that solvent. So those features proportional to the amount of solute, but not the identity. Proportional depends on the type of solvent. That's what makes something a colligative property.