 Alright, so if we're going to get quantitative about studying chemical reactions, we need to introduce a little bit of notation. And in particular, the most important thing to be able to talk about is to be able to quantify how much or how far a reaction has to precede it. And the variable that we use to do that is a variable that we'll call the extent of reaction. And conventionally, we use the variable xc to represent the extent of reaction. So, because my handwriting is relatively bad, I'll point out that's the Greek letter xc. This is sort of the most unfortunate of the Greek letters. Everybody draws it a little bit differently. It's hard to draw in the first place. Everybody pronounces it a little bit differently. You might pronounce it xi or z or xc, but this is the Greek letter that we're stuck with conventionally as the extent of reaction. Because it's difficult to draw and difficult to say, many people just call this Greek letter squiggle. We sort of universally decided that xi is not a good name for this Greek letter. Often we use the name squiggle for this Greek variable. So if you hear me describing the variable squiggle, that's the one I'm talking about. But to illustrate what the extent of reaction means, we'll use an example. What it is is the number of times that a reaction has preceded. So if I use this example of H2 and Br2 reacting to form two molecules of HBr, let me give you some initial conditions. Suppose that initially I have two moles of H2 and three moles of Br2. And I have no HBr initially. When the reaction proceeds, I can tell you how much H2 and Br2 I've reacted and how much HBr I've produced, those values are all coupled to one another by the stoichiometry. Every time I react one molecule of H2, I have to also consume one molecule of Br2 and I'll produce exactly two molecules of HBr. So in terms of the extent of reaction, I might have lost some number of moles of H2. I must have lost the same number of moles of Br2 and I'll have gained exactly twice as many moles of HBr. So that's the purpose we use extent of reaction for. It's just the number of times a reaction has preceded, which is coupled to the amount of reactants that I've lost and the amount of product that I've gained. So after the reaction has preceded to this extent, I've got two minus C moles of H2, three minus C moles of Br2 and a total of two C moles of HBr. So notice that in this table, since I have units of moles for my initial amounts of H2 and Br2, I'm using my extent of reaction also in units of moles. As with any quantity that represents the number of molecules, you can measure the number of molecules either as a literal count of the number of molecules or as a number of moles and just use Apogater's number to convert back and forth between them. The other thing to notice about this is both in this amount of change in the amount of reactants and products, as well as in this final equation, notice here these coefficients negative one, negative one, positive two that appear in front of the extent of reaction. Two things about those coefficients. Number one, they match the stoichiometric coefficients in the reaction. Again, because one mole and one mole turn into two moles, I lose one mole, lose one mole, gain two moles. So those extent of reaction is preceded by these stoichiometric coefficients. Notice also the signs. Reactants are consumed in a reaction. Products are created in a reaction. So when I'm losing with a negative sign, reactants I'm gaining with a positive sign product. So the signs of these changes as well are determined by the stoichiometry of the reaction and which molecules are reactants and which molecules are products. So in order to do chemistry like this but treat it more quantitatively, more mathematically, it turns out there's an easier way to think about this reaction rather than our normal chemical way of viewing a reaction with an arrow converting reactants to products. And that's to think about it a little more algebraically. So it might seem strange at first glance but we can also think of this reaction as saying a molecule of H2 and a molecule of Br2 is equal to two molecules of HBr in the sense that these can be combined and generate these or vice versa. If I, now that I've written this as a mathematical seeming equation rather than a chemical equation, if I manipulate it algebraically, let's say I move this H2 and Br2 over to the other side of the equal sign, then what I'll have is this strange looking equation. Negative H2, negative Br2, and positive two HBr sum together to get zero. What that means when we interpret this equation as a chemical equation is that I can lose an H2 and lose a Br2 at the same time as I gain two HBr's and that's okay. I haven't created or destroyed any atoms in that process. The advantage of writing the chemical reaction in this way is that now I can write this a little more generally. Notice also the stoichiometric coefficients that I pointed out in these expressions are exactly the same over here. I've got negative one copies of an H2 because H2 is reactant and its stoichiometric coefficient is one. Negative one Br2 because it's a reactant and its stoichiometric coefficient is one. Positive two HBr's because it's a product and its stoichiometric coefficient is two. That means I can write any chemical reaction as a sum of stoichiometric coefficients multiplied by the species. Negative one H2's, negative one Br2's, positive two HBr's all added up. Those are the stoichiometric coefficients of the species multiplying by the species themselves. So S sub i here is the name of one of my chemical species. New sub i are the stoichiometric coefficients and remember stoichiometric coefficient isn't just the one or the one or the two, it's one with a negative sign, negative sign for any reactants, positive sign for any products. This is a nice compact way of writing down a chemical reaction in a form that will let us manipulate it more mathematically. If I want to write the equivalent of this statement in this sort of reaction notation, number of moles of H2 after my reaction is the initial amount minus one times the extent of reaction in this case or the initial amount plus the stoichiometric coefficient negative one multiplying the extent of reaction. So in the general case I can say the amount of any reactant at some point at the end of a reaction, during a reaction, the middle of a reaction is however much I had to start with and not plus stoichiometric coefficient times the extent of reaction. If I've done the reaction one mole of times, then the number of each species I have is however much I had initially plus one mole multiplied by the stoichiometric coefficient. Again, losing reactants, gaining products. That's all good. This tells me the change in the number of moles of any component. A very useful form that we will find for that reaction is when the reaction proceeds a little bit more or a little bit less, how much does the number of moles of a particular species change? It's changing when the reaction goes a little more or a little less is the extent of reaction. Every time the reaction goes one time I generate some number of moles of each product and consume some number of moles of each reactant. So whether we think about this as an infinitesimally small change in the number of moles or you can in fact think of this as taking the differential of this expression, the change in number of moles, the initial amount doesn't change, the only thing that's changing is the extent of reaction. So the change in number of moles is equal to stoichiometric coefficient multiplied by Dx, the differentially small change in the extent of reaction. So this is an equation that we'll use again in the future that will help us understand when the reaction shifts a little forward, shifts a little backwards, quantitatively how much am I changing the amount of each species in that reaction? And then being able to use this reaction notation in a mathematical sense will help us understand chemical equilibrium in a quantitative way.