 So the key point when I'm thinking about activity in electrolytic solutions, solutions that dissolve and dissociate into ions, is to keep in mind the fact that when you dissolve a salt, you get a predictable ratio of the cation to the anion. For example, if I have a simple one-to-one salt like KCl, when I dissolve potassium chloride in solution, I'll get one potassium one chloride ion for each unit of KCl that dissolves in the solution. I will always have the same number of potassium ions in the solution as I have as the number of chloride ions in solution as long as they all came from dissolving KCl. So this is the case for a one-to-one salt like KCl. Let's think about, as we begin to think about activities in thermodynamics of these solutions, I could say if I'm going to be interested in talking about the chemical potential of this salt, I can think of that as the chemical potential of the cation and the chemical potential of the anion added together, in this case cation being potassium chloride being the anion. We know in solutions, I can write the chemical potential of a component as the standard chemical potential plus RT log of the activity. So here's where we're going to start to make connections to the activity of the different components of the solution. So what I've written down here, that's the chemical potential of the cation, perhaps K plus, perhaps sodium, perhaps something else. I can do the same thing for the anion. That would be standard chemical potential of the anion plus RT natural log of the activity of the anion in this solution. So standard state chemical potential of the cation and the anion, I won't dwell too much on what those mean, partially because we can define the standard state to be whatever is most convenient in any problem that we like, and partially because it turns out that quantity is not going to usually matter too much. So what I'll do is I'll combine that cation chemical potential, anion chemical potential in the standard states. I'll combine those two into something that I'll just say standard state chemical potential of my salt. So I've defined that standard state chemical potential of the salt as the sum of these two terms, the two standard state chemical potentials. What I'm really interested in are the terms that involve the activity. So I've got RT natural log of one activity, RT natural log of another activity. If I add these two natural logs together, that's the same as the natural log of their product, activity of cation, activity of anion. So what I'm going to do next is reflect that cations and anions, as I mentioned, always come in a solution in the same ratio, and a one to one salt, they come in in a one to one ratio. So if I were interested in saying what is the activity of potassium ions, go prepare me a solution that's one molar or one molal or whatever concentration we want in potassium ions and nothing else, you can't do it. You can't dissolve potassium ions in a solution without also bringing along some anion, perhaps a chloride anion, perhaps a nitrate anion, but every time you dissolve a cation, it's going to come with an anion paired along with it. So in fact, we can rarely, if ever, measure directly the activity of the bare cations or the bare anions. What we can do, however, is measure this combination of the cation with the anion together. The potential of this solute is some standard state chemical potential plus an RT natural log of this combined activity, and this is the only one we have a real chance of being able to measure in the lab. So what we'll do is say, I'll just rewrite this as natural log of an activity plus minus. That just means the combined, the averaged activity of the cation with the anion. Here I had activity multiplied by activity, so I'll leave this as an activity squared, and the quantity I've just essentially defined by rewriting the equation in this way, this A plus minus, that's something that we'll call the mean ionic activity. It's an activity, not of potassium nor of chloride, but a kind of combination of the two of those. So it's a mean, in particular it's a geometric mean, it's not an arithmetic mean, but it's a mean activity for the two different types of anions. So like I've said, this is how things work if I have a one to one salt, it'll work a little bit differently if I have a salt with a different stoichiometric ratio. So just to see how that works, let's say we have calcium chloride dissolving to calcium and two chloride ions. In that case, if we just repeat these steps, I can say, again, the chemical potential of the salt is the chemical potential of all the cations plus the chemical potential of all the anions, but I've got twice as many of the anions as I have of cations. So I can say chemical potential of the cations plus twice the chemical potential of my chloride ions because they appear in solution twice as much and in a concentration twice as high. So if I rewrite these chemical potentials as the standard state chemical potential plus RT log of activity, I've got one term for the cations and I've got another term for the anions, I need a two out front here. This chemical potential for the anion has a two in front of each of these terms again because of the higher concentration, because of this higher stoichiometric coefficient in the formula. So once again, we can combine the standard state terms and just call the standard state chemical potential of the salt. That in this case is cation plus twice the anion. The more interesting term is if I combine these RT log of cation activity and twice the RT log of anion activity, that's the RT natural log of the product of cation and anion and another anion. So when I go to define the mean ionic activity in this case, here cation times anion was equal to the mean activity squared. In this case, in this 2 to 1 salt, I'll write this as a mean ionic activity cubed. Again, I have three activities, activity of cation and two factors for the anion. So all together I've got an activity cubed, but I've defined in this case cation times the anion squared to be equal to mean ionic activity totally to the third power. So this equation is different than this equation. Like I said, it's different for a 2 to 1 salt. It'll be slightly different for a 3 to 1 salt or a 2 to 3 salt in general. If I can write down the general case. Suppose we have some arbitrary salt that dissociates into some cations and some anions. So let's say the stoichiometric coefficient, the cation, I'm going to use the Greek letter nu for my stoichiometric coefficient. So just like I have k1 cl1 or ca1 cl2, these stoichiometric coefficients, I've got a stoichiometric coefficient nu plus for the cation, nu minus for the anion. It's going to dissolve into some number of cations. And I won't write the charge here. So it's got some amount of positive charge, some amount of negative charge that I haven't specified. But the stoichiometric coefficients are what I need to know. The chemical potential of the salt will be some standard state chemical potential plus RT natural log of a mean ionic activity raised to some power, a power that's related to the stoichiometric coefficients. In this case, that power was 2 because I had 1 potassium and 1 chloride combining to give me a power of 2. If I have 1 calcium and 2 chlorides combining to give me 3 ions in solution, then that power is 3. So this stoichiometric, this combined stoichiometric coefficient is the coefficient from the cation and the coefficient from the anion added together. That's the definition of this exponent. And then the mean ionic activity coefficient, that's defined, the mean ionic activity coefficient raised to this total stoichiometric coefficient is going to be activity of the cation raised to the cation's stoichiometric number times activity of the anion raised to the anion's stoichiometric number. So that entire quantity there lets me define the mean ionic activity coefficient. And if you look at the two examples we've seen, you can see that they both fall under this general definition. Ionic activity squared for the 1 to 1 salt, cubed for the 2 to 1 salt. So this is how we define a mean ionic activity for an electrolytic solution because we have a hard time separating the cations from the anions. We can do a very similar thing for the activity coefficient just as we've done here for the activity. So that's what we'll focus on next.