 So as one example of what we've named colligative properties, we can look at how the freezing point of a mixture or a solution changes with the concentration of that solution. So as a reminder, this is what, let me finish leaving this diagram, but this is what the temperature composition phase diagram looks like for, in this case it's relatively proportional for sucrose and water. So if I have sucrose on this side, water on this side, I form a eutectic at a relatively low concentration of sucrose in water, and we have this sort of phase diagram. The important point for right now is to observe that this melting point of sucrose, when I add in a little bit of water, the melting point decreases, it becomes lower. Likewise, when I look at the melting point of pure water, zero degrees Celsius, when I dissolve a little bit of sucrose, that multi-point decreases. So in both directions, as I move from pure substance to solution, the freezing point decreases. That's a very generic or general phenomenon. We saw that same sort of behavior for the magnesium-calcium phase diagram, for the lead and tin phase diagram. Nearly every case will show this sort of behavior. So the behavior that's that general, there must be some fundamental reason behind it, and indeed we can identify what that is. The origin of this behavior comes from the fact that as I move from pure solvent to dissolving something in it, I'm going to lower the chemical potential of the solvent. Let me remind you why we know that's true. We've seen previously that if I want to calculate the chemical potential of some substance in a mixture, chemical potential in the solution is related to the chemical potential in the pure liquid by this term RT log of the activity of the substance. And let me sketch what that looks like as a graph. If I plot the chemical potential as a function, in this case, since we want to talk about solidifying or melting a substance as I change the temperature, I'll make this axis be temperature. So remember that what happens to the chemical potential, which is the same thing as the molar free energy, the chemical potential of anything is the partial molar Gibbs free energy. So the Gibbs free energy drops as the temperature increases proportionally to the entropy, like we've talked about when we talked about Gibbs free energy. So there's chemical potential, let's say, of the solid phase. As I heat the solid up, its chemical potential drops. The same thing is true of the liquid. So here's chemical potential of the liquid phase, of the pure liquid phase. And I've drawn that as a steeper curve because the entropy of the liquid is a larger value than the entropy of the solid. So it's falling more quickly than the chemical potential of the solid does. So now think about what happens when I convert this liquid into a solution. I take my pure sucrose or maybe pure water and dissolve a little bit of something into it and make a solution. So as I, if the substance I'm talking about is water, this is the chemical potential of pure water. If I dissolve something in that water to make a sugar water solution or some other solution, the activity of the water will decrease. So in a solution, the activity will be less than one in a solution. It's less active than the pure solvent. So the log of that number that's less than one, that's going to be a negative number. So this entire quantity is negative. So the thing that I'm adding to the chemical potential of the liquid is a negative number. So the chemical potential of the solution is going to be shifted downward relative to the chemical potential of the pure liquid. So before I draw that curve, let me point out how we identify the melting point on these curves. This point where the solid and liquid curves cross, that's the temperature at which the chemical potentials are equal to one another. So that would be the melting point, temperature of fusion. I'll put a superscript zero or a circle on that melting point because that's for the pure liquid, for the substance in its standard state in the pure form, that's the melting point. What we're interested in is why does the melting point decrease as we form a solution. When we form a solution and the chemical potential decreases, then this curve is going to be shifted downward. It's going to have the same shape but just shifted downward. And what that means is the point where these two curves cross is going to shift to a lower value of the temperature. So this melting point or fusion temperature in the solution is less than the melting point of the temperature of fusion in the pure solvent. So qualitatively we understand now anytime, regardless of whether I'm starting with pure sucrose and I make it impure, I've lowered the sucrose's activity. If I start with pure water and I dissolve some sucrose in it, I've made the water less pure and I've lowered its activity. Anytime I lower the activity, I'm going to lower the chemical potential and that's what causes the melting point to decrease in both of these directions. So at least qualitatively, now we understand why dissolving a solute into a solvent will decrease the freezing point. The next step will be to take that a little further and examine this equation and see if we're going to understand quantitatively how much that freezing point will decrease as I prepare a solution.