 Okay, so we've talked about the activity of an electro catalytic solution in terms of the combined or the mean ionic activity, being a combination of the activities of the cation and the anion raised to the stoichiometric coefficients with which they come into solution. A 1 to 1 solute like sodium chloride would have these coefficients being 1 and 1. A 1 to 2 salt like calcium chloride would have 1 for the cation and 2 for the chloride. That raises the question not just of activity, but how do we think about activity coefficient for an electrolytic solution as well. So let's remind ourselves what we know about activity coefficients. We know that the activity in general, the activity of a particular component of a solution can be thought of as the activity coefficient times perhaps the mole fraction depending on what standard state we're using for our solution. If our standard state for the component is the pure liquid or pure substance, then we can consider the mole fraction of 1 to be the standard state and that's what we'd use for the pure substance standard state. Often for solutes in a solution, we don't use that as our standard state. We might use the molality, sorry not m, but b for molality. A 1 molal standard state, so if we're using a 1 molal standard state, then the standard state would be a 1 molal concentration solution. So the activity would be 1 when we're in a 1 molal solution. So this is actually more common for electrolytic solutes. So if we imagine replacing these activities with their expressions using an activity coefficient and a concentration, we can rewrite the mean ionic activity as the activity of the cation. So that would be activity of the cation as an activity coefficient times the molality. The activity of the cation times the molality of the cation all raised to this new plus stoichiometric coefficient and then likewise for the anion, activity of the anion is its activity coefficient, that should have been a gamma, activity coefficient for the cation over here, activity coefficient for the anion over here, molality of the anion all raised to the stoichiometric coefficient for the anion. If we go ahead and write this as, go ahead and take these exponents. So I've got gamma for the cation raised to the new plus power, B plus to the new plus, gamma minus to the new minus, B minus to the new minus. And then what I'll do is I'll collect the gamma terms together, collect the molality terms together to obtain gamma plus new plus, gamma minus to the new minus, and then the molality terms with each other here. So now I've got something that looks more like an activity is equal to some activity coefficients multiplied by some combination of the molalities. So if I rewrite all of this as a new term, the combined activity coefficient for the cation and the anion raised to some total power, combined molality raised to some total power, what I've just done is I've defined these two terms to be my new mean ionic activity coefficient. Combine the combination of these two terms to be this new mean ionic molality. So those definitions, so mean ionic molality is the molality of the cation to its stoichiometric coefficient times the molality of the anion raised to its stoichiometric coefficient, essentially the same definition as we have for activities. Likewise for the mean ionic activity coefficient, gamma plus minus raised to the new total stoichiometric coefficient power is equal to activity coefficient for the cation raised to its stoichiometric coefficient activity coefficient for the anion, sorry activity coefficient should be a gamma raised to its stoichiometric coefficient. So as I say, this term is called a mean ionic molality. It's an average in a particular way or B plus minus that is, is called the mean ionic molality. It's an average in a particular way of the molalities of the cation and the molalities of the anion because they always come in a solution in the same ratio. Likewise, gamma plus minus, we call the mean ionic activity coefficient, again defined essentially in the exact same way as we did for activity, again because these cation and anion arrive in solution in the same ratio. And as before, that new without a subscript is just the sum of the stoichiometric coefficients for the cation and for the anion. All right, so that's all we have in terms of definitions, but those definitions are a little bit abstract to give them some meaning to see what they mean. Let's plug some numbers in and see what they say about an actual electrolytic solution. So let's say we have a .01 molal solution. I'm going to take .01 moles and to make it at least a little bit interesting, we'll take a not one to one salt, a one to two salt like calcium chloride in a kilogram of water. So by definition, that's a .01 molal solution. So I could say that's going to be .01 molal in calcium chloride, or since we're going to need to know the molality of the cation and the molality of the anion, because it's a one to two salt, I've got twice as much chloride as I have calcium. It's going to be .01 molal in calcium ions, but twice as much .02 molal in chloride ions. So this is the cation concentration, this is the anion concentration. If I also tell you that let's say I've measured the activity of the solute in that solution, I can't measure the activity of the calcium and the activity of the chloride separately, but if I measure the activity of the solute by say how much the vapor pressure of the solution is depressed, I can tell you the mean ionic activity of the solute as a whole. And if I tell you that we've measured that to be .01.06 on a molal scale. So I'm using a standard state, a molal standard state. So my activity is going to have units of molal. Number .01.06, that looks relatively close to the concentration of the solution, but remember this solution has put three ions in the solution for each, three moles of ions for each mole of calcium chloride that it dissolved in the solution. So that's actually considerably lower than we would have expected for a .01 molal solution of this salt. The question though is how ideal or not ideal is that solution? That's the question that's best answered by talking about the activity coefficient. The activity coefficient is near one, then it's a very ideal solution. If the activity coefficient is far away from one, it's very non-ideal. So the way we answer that question is by solving for the value of the mean ionic activity coefficient. So we have the mean ionic activity. We can calculate the mean ionic molality. We'll use that to compute the mean ionic activity coefficient. So first step would be what is the mean ionic molality? I'll go back to the definition we've written down here. Mean ionic molality is the cation molality raised to its stoichiometric coefficient and likewise for the anion. In our case, what that means is the mean ionic molality cubed. I've got one cation and two anions entering solution. So the total number of ions is 3. Molality of the cation is 0.01 raised to the 1 power because I have actually let me do this slightly differently to make it more general. In general, for this 1 to 2 salt, let's say I have molality of B for my total solution. Let's call that value total molality of the salt. The cation in this case is going to be exactly the same number. The cation molality and the total salt molality are the same number. Again, that's raised to the first power. The activity of the, I'm sorry, the molality of the anion because it's a 1 to 2 salt. I have two chloride ions. That's double the molality of the solution. Molality of the anion is twice as big as the molality of the solution. I'm raising that to the second power because I have two anions entering solution. So 2B quantity squared is 4B squared multiplied by an extra B. So this statement is always going to be true for any 1 to 2 salt. The mean ionic molality cubed is going to be equal to four times the molality of the original solution quantity cubed. So this 2 here and the 2 here match each other and those are both due to the stoichiometric coefficient being 2 for my chloride ions. All right, so now I think we're ready to say, to solve for the mean ionic activity coefficient, rearranging this equation. I can say that the mean ionic activity coefficient is the mean ionic activity divided by the mean ionic molality. So I could raise all those to the third power or I can just take the one third root of the entire equation at this point and say, get rid of each of these powers of new and I'd say the mean ionic activity coefficient is the mean ionic activity divided by the mean ionic molality. So now I have enough to insert numbers into that equation. The mean ionic activity is this value, 0.0106, that we were given or perhaps obtained experimentally. The mean ionic molality, mean ionic molality cubed is this quantity 4B cubed. 4 times the original concentration, 0.01 moles per kilogram cubed. But that's B cubed. I have to take the one third root, one third power of that quantity. So 3 raised to the one third power, that just has my original concentration but that's going to, I'm also going to need the one third power of this 4 down here in the denominator. But if I use a calculator to determine what that's equal to, unit wise this moles per kilogram cancels the moles per kilogram raised to the third power and then the one third power. So my result is unitless as it should be. And as a result what I get is 0.298. So numerically that's my answer, the mean ionic activity coefficient in this solution determined from the activity is 0.298. What that means physically is this solution is very far from ideal. This numerical value is well below 1. So the ions in this solution are behaving only 30% as ideally as they, only 30% ideally. I can't say just from this result alone whether it's the calcium ions or the chloride ions that are contributing more to this non-ideality. In fact generally it's true that the more strongly charged calcium ions are going to be more non-ideal than the singly charged chloride ions. And in fact that general statement that more strongly charged ions tend to contribute more to the non-ideality of these solutions than lesser charged ions leads to an idea that we'll consider in the next lecture.