 So, we have an expression for the osmotic pressure, which is a fairly simple equation to use, but we can work an example, see how it works, and also provide some discussion for what the values in that equation mean. So, for this example, let's say that we have a solution that is 0.1 molar in some solute. It might be a sugar water solution, 0.1 molar sugar water solution. And if our temperature is 298 Kelvin, so our sugar water solution is at room temperature, that's enough information for us now to calculate the osmotic pressure. So the osmotic pressure of that solution is easy to calculate. C times r times t, 0.1 moles per liter is our concentration. The gas constant turns out not to be convenient to use the value of the gas constant joules per mole Kelvin, because what we're going to want out on the left side when we're done is a pressure with units of pressure, units of atmosphere, for example. So it's going to be more convenient if we use a value like 0.08206 liter atmospheres per mole Kelvin. And if I multiply that by the temperature 298 Kelvin, now Kelvin will cancel, moles will cancel, liters will cancel, and I'm left with just units of atmospheres. And if I work that math out, 0.1 times this value of the gas constant times 298, that works out to a pressure of 2.4 atmospheres. So as a numerical calculation, that's not a difficult calculation. The osmotic pressure for a 0.1 molar sucrose solution is 2.4 atmospheres. What does that mean physically? What it means is that if I have two beakers, one with a pure water solvent, and one with 0.1 molar solution, 0.1 molar in some solute, and water as a solvent, that means that the height of this column will be enough that it's generating a full 2.4 atmospheres worth of pressure. Either we can think of it as the height of this column of solution, force generated by its mass, mgh divided by the area, that pressure is equal to 2.4 atmospheres. We can think of 2.4 atmospheres, if I convert that to units of tor, if I multiply by 760 to get units of tor. That value, I can think of it as 1900 tor, or the more old fashioned name for the tor unit is a millimeter of mercury. So this column will weigh as much as a column of 1.9 meters, almost 2 meters high column of mercury, or since mercury is much denser than water, if I were to have that column of water rather than mercury, the density of water is I think 12 or 13 times larger than the density of mercury is more dense than water, so that corresponds to a column of about 25 meters of water. So that illustrates that osmotic pressure is actually a pretty strong force, the pressure with which pure solvent will attempt to pass through this semipermeable membrane to dilute this .1 molar solution will be enough to support a column of water that is a full 25 meters high, so that's much larger than just the size of two beakers in the lab, for example. And that's perhaps not all that technologically interesting phenomenon, it happens all the time in biological systems for example, but we usually don't want to dilute a solution by allowing water or solvent to pass through a semipermeable membrane, if we really wanted to dilute this solution, all we would have to do is just pour some extra solvent into that half of the beaker. Often where this becomes more interesting is in the reverse case, let me go back to a case before the osmosis had happened, so here's two beakers separated by a semipermeable membrane, in this case I'm going to let the heights of the solution be equal at the start, aqueous solution .1 molar in some solute, before osmosis happens this is what the system looked like, if I allow osmosis to happen the solvent will drop and the solution side will rise, if I want to prevent that from happening I need to push down with an additional pressure equal to the osmotic pressure on this side, if I push down with pressure harder on this side than this side, if that osmotic pressure, if that pressure is equal to the osmotic pressure of 2.4 atmospheres then I can keep these two sides in equilibrium. If I push less hard than that, if the atmospheric pressure here is p0 and over here I push with p0 plus pi, I'm in equilibrium, if I push with only atmospheric pressure or atmospheric pressure plus a smaller amount then osmosis will occur, but what if I push with the pressure greater than p0 plus the osmotic pressure, if I push not with 2.4 atmospheres but with 2.5 or 3 atmospheres, I can actually cause solvent to flow the other way. That's contrary to the direction thermodynamics tells us it will spontaneously flow, so I have to provide energy for that to happen, that's why I have to exert so much pressure more than 2.5 atmospheres to cause that to happen, but if the pressure on this side is greater than p0 plus pi, what will happen is solvent will flow in this direction. That process, because it's the opposite of osmosis, is called reverse osmosis. So if you exert a pressure greater than the osmotic pressure, you can cause the solvent to flow from the concentrated side to the diluted side, reverse osmosis. So that's useful, for example, in purifying seawater, in many places around the world with shortages of freshwater and in abundance of seawater like many islands, they purify some of their water by this process of reverse osmosis. They take seawater, concentrate it in some ions, exert a pressure and we can force pure water to flow through a semiperbium membrane to get pure solvent water. Water has a concentration greater than 0.1 molar, it's also not an ideal solution, so the osmotic pressure isn't exactly concentration, we'd have to go back to an equation that used activity to get an accurate prediction for the osmotic pressure required to purify seawater. That value turns out to be 27 atmospheres. If you want to purify seawater, you need to exert 27 atmospheres of pressure to squeeze it through a semipermeable membrane, but in fact that can be an economical way to purify water in some circumstances. So that's the summary, not just of the calculations for osmotic pressure which are relatively simple but some of the physical picture of what's going on. We've now covered four different colligative properties, so the next thing we'll do is compare and contrast them in a little bit of a summary.