 Alright, so our next step is to get a little more quantitative about predicting the ideality or the non-ideality of electrolytic solutes in solutions and how we can use the ionic strength to do that. So if we're talking about the ideality or non-ideality of a solute, the quantity we're interested in is the activity coefficient and perhaps the activity coefficient, the mean ionic activity coefficient for the entire solution, since cations and anions always come in pairs or in a particular stoichiometric ratio, or we can think of the activity coefficient for individual ions, maybe a sodium ion or a calcium ion. So we expect that the activity coefficient should go down further away from one, meaning the solution is getting less and less ideal as the concentration of the solution increases. So dilute solutions, if they're extremely dilute, they're going to behave relatively ideally. As the concentration increases, it's going to get less and less ideal and the activity coefficient is going to drop further below one. The other thing we expect qualitatively is going to happen is as the charge of an ion increases, let me put absolute value signs around that, so it doesn't really matter whether it's a cation or an anion, but a divalent anion like a calcium will be less ideal than a monovalent ion like a sodium. So the activity coefficient is going to decrease. This solution is going to get less ideal as the charge of the ion increases. So likewise a trivalent anion will behave less ideally than a divalent anion. So those two things we expect should be true, and in fact, if we look for a relationship that describes exactly how that works, we get this result which is called the Debye-Huckel limiting law. So this result, we'll consider in a little bit more detail, is an equation that allows us to predict using the ionic strength, using the charge of an ion, exactly what the activity coefficient is going to be. There's a constant in this expression. This constant, let me give you a value for that. That has a value of 1.172 in units of square root of kilograms over square root of moles. So that looks like an unusual unit, but that turns out to be the correct unit. That's the unit we need in order to cancel the units of molality of the unit strength which are inside this square root sign. It's also common to see this expression written in terms of a common log, a base 10 log, rather than a natural log. If you do prefer to use that expression as a base 10 logarithm, I'll write that one with an A prime for that constant. That constant, if you're using base 10 log, that's of course related to this constant if I divide this by the log, natural log of 10. I'll get this value 0.509. You might see this equation written with a natural log or a base 10 log with ionic strength in molarity or ionic strength in molality, but if it's in molality with a natural log, this is the constant we'll end up using. So we can think of this where this constant comes from in perhaps a couple different ways. We could think of it as being derived empirically. Imagine that I did some experiments and calculated how the activity coefficient depended on concentration. For a dilute solution, it's going to behave ideally, and the activity coefficient is going to be 1. The more concentrated the solution gets, the less ideal the solution is going to become. So for maybe a sodium chloride solution, it decreases like this. For a calcium chloride solution, because the charge of the calcium is larger, it's going to decrease a little bit faster. So that's what the curve is going to look like for calcium chloride. So that's maybe something of the sort that we would get by making experimental measurements. If I measure the, if I express the activity coefficient in terms of ionic strength rather than molality, what we'll find is if I plot those two curves now versus ionic strength, remember the ionic strength of a calcium chloride solution is larger than its concentration, so that's going to shift this curve over to the right. So the sodium chloride solution, the calcium chloride solution will be much more similar to each other, and in particular, at low concentrations, I can approximate them both with the same line. So the pink line I've drawn here is the prediction given by this Debye-Huckel limiting law, which says that all solutions, doesn't matter what it is, will have activities that depend on their ionic strength in a particular way. So the ionic strength is one way of taking account of the fact that the more strongly charged ions decrease their activity faster. So the other way of thinking of where these constants come from, the first way was from experiments we can just derive the slope of this equation. The second way is we can actually derive this equation not empirically but theoretically. We can write down some assumptions about how ions behave in solution and arrive at this equation completely theoretically. That will take a fair amount of work, enough work that if you did that work back in 1923 and your name was either Debye or Huckel, then you'd be quite famous. So we're going to postpone that for just a little bit, and first let's work an example or two using this equation just to make sure we understand how it works. So let's say we have, I think I'm going to use a 0.01 molal calcium chloride solution. Now using nothing other than the concentration, we can use the Debye-Huckel limiting law to predict what is going to be the activity coefficient, the mean ionic activity coefficient of that system. As a way of predicting how ideally or not ideally it should behave. So we'll do that step by step. First we can do it for each one of the individual ions in the solution. So let's say for the cation first, for calcium. In order to calculate the activity coefficient for calcium, we need to know this constant, we need to know the charge on the calcium ion, which is 2, and we need to know the ionic strength. So for both calcium and for chloride the ionic strength is going to be one half concentration of cation times ionic charge squared and concentration of anion to chlorides times 0.01 molar gives me 0.02 molar times ionic charge squared, 0.04 plus 0.02 halved, that ionic strength is going to give me 0.03 molal, which is very similar to an example we did in a previous video lecture. So that's the ionic strength. The log of the activity coefficient for calcium according to the Debye-Huckel limiting law is going to be negative 1.172 square root of kilograms per square root of moles. Charge on this particular ion, calcium is 2, that gets squared, multiply all that by the square root of the ionic strength, square root of 0.03 moles per kilogram, and now we can see square root of moles, cancel square root of moles, square root of kilograms in the denominator, cancel square root of kilograms in the numerator, so that was the reason for our strange looking units on the coefficient A, and when I negative 1.172 times 4 times the square root of 0.03 works out to be negative 0.81 unitless because all these units have canceled, which means after taking E to both sides, the activity coefficient for the cation is E to the negative 0.81, which is equal to 0.44. So we see that our cation, the calcium cations have an activity coefficient of 0.44 as predicted by the Debye-Huckel limiting law. If we do the same thing now for the anion, our chloride anion, the charge is different, that's negative 1. The ionic strength is exactly the same, so I'm multiplying by the, still by the square root of 0.03 mol. That arithmetic works out to be smaller by a factor of 4. The only difference in the arithmetic is I've turned this 4 into a 1. So instead of negative 0.81, I have negative 0.2, which means that the activity coefficient for the anion is E to the negative 0.2, which is 0.82. So again, the anion because it's singly charged is considerably less non-ideal than the cation, which is more strongly charged. We're not done. That's the activity of the cation, the activity of the anion. What we're interested in is the mean ionic activity. So gamma plus minus to the third, because I have a total of three ions in my salt, is activity of the cation to the one, activity of the anion to the two, because I have a one to two salt, one calcium ion to chloride ions. So now I combine these two single ion activities, 0.44 for the cation, 0.82 for the anion, so 0.44 to the one, 0.82 squared. That gives me gamma plus minus cubed. Gamma plus minus is that quantity. So this works out to be, I don't know what that one works out to be, but if I take the one third root of it, one third power of it, the third root of it, gamma plus minus works out to be 0.67 when I do that arithmetic. So that is the answer we were looking for. A 0.01 molal solution of calcium chloride, as we've used in several examples up to this point, has an activity coefficient, a mean ionic activity coefficient of 0.67, predicted theoretically by the Debye-Hookle limiting law. So that turns out to be in relatively good agreement at these concentrations with the experimental measurements that we would make, and we've made that prediction purely with this Debye-Hookle limiting law. So we can use this equation now both in a practical sense. We can predict activities or how ideal a solution will behave at a variety of concentrations using nothing other than the concentration itself, as long as this Debye-Hookle limiting law is valid. To understand better where this equation is useful, where it's not useful, where it comes from, it's also helpful to figure out where it comes from by deriving it from first principle. So we can consider that next.