 So let's look at a simple hydrocarbon. Let's say that I have liquid cyclohexane, very simple and common solvent. Here it's just pure liquid cyclohexane, number of molecules per liter, and here is cyclohexane where I've tried to solute it in water. Very low solubility. If I take those two numbers and plug that into the equation from the last concept, I will get that the delta G of moving from liquid to the aqueous phase is roughly plus 7 kcal. In our scale that's a large number and it's large because we are comparing it to RT in the denominator or KT if you're a physicist. KT is roughly 0.6 kcal per mole at room temperature. 6 divided by 0.6 is roughly 10 or minus 10 in this case and that's e raised to the minus 10 is going to be the relation between these two concentrations and that's why this number is very low. So that was plain and simple. Fun. But I would like to get slightly more things here. Wouldn't it be great if, in addition to delta G, I could also get my delta H and delta S so that I can study the entropy specifically. It turns out that I can. But I'm going to need to use some of my equations. In the last lecture we spent a lot of time on F equals E minus TS. I need a new pen there. F equals E minus TS. There was also another equation I used to define the thermodynamic definition of temperature. Do you remember what that one was? That was that the temperature was the derivative of the energy with respect to the entropy. But as I mentioned, today is a chemistry day. So we're going to replace that. We're going to say G equals not E but H minus TS. And then you're going to need to believe me that this will just translate in an analog fashion or better yet. Maybe prove it. That would be DG, yes. We're going to use that equation in a second. In principle you could think, can't I just use that equation? There's a T here. What if I take the derivative with respect to T? That would be minus S. Well, you don't know what the temperature dependence of H is yet, at least. So I'm going to use the same approach as I did in the last lecture and study these so-called differentials. So let's look at the very small change in G here and see what happens. So I should be able to write DG equals DH minus DTS minus SDT. And now things start to look very nice because if I take this definition of temperature up here, it literally, oh sorry, so DG there, bad Eric, it should be H because E there corresponds to H there. I found that before. This number literally says that DH equals TDS, right? But if DH equals TDS, that means that this term should vanish. And now I can divide these side by minus DT and then get that S, the entropy, equals minus DG DT. And that in particular means if I now do exactly the same experiment, calculate the concentrations, but I repeat that at say three different temperatures and take the numerical derivative, the slope of that line, that means that I can get the entropy from the slope of the line. I already had G, I got the entropy from the temperature dependence, I obviously know what temperature I made, did they experiment that, that means that I can solve for H. So then we can get all these components. Let's try to draw that in a small scheme between all the possible different phases this can exist in and see if we can learn something from it.