 Right here. First of all, administrative business. Your prompt set is due today, and that should go directly to Tom, who's sitting right in the front here, if you haven't met him yet. There is a question on one of the problems, and that is the exchange current density for the direct ethanol cell, which some of you just heard a little discussion on, which I think you probably are going to end up with some unreasonable numbers. I heard a lot of different numbers, which is a little surprising. But we're going to get back to that on Thursday. I'm actually going to tell you today why the number that you calculated, if you did it right, should be an unreasonable number, because there's something I've been sort of glossing over so far, ignoring, fudging, if you will, because I didn't want to put in that level of confusion. And somebody actually typed out the whole problem set? Wow. Once a few years ago, I had a professor of mathematics come and ask if he could sit in on my freshman chemistry course. I was, on one hand, very impressed with this. On the other hand, very nervous about this, because typically when you're doing chemistry-type math, you cheat a little bit. And having somebody who knew what you were doing up, sitting in the audience was potentially a problem. Although he was very nice to me, I have to say. When one time I screwed up big time, he didn't tell me in front of the class. He waited till afterwards. Anyhow, he would do the homeworks and the whole thing. He really wanted to learn chemistry. And he turned in his homeworks in tech. They were work processes. And I would just take his, no, it was great. I'd take his homeworks and post them as the answers. You didn't make a mistake. The only time I ever conned him, I said the final exam, and he actually made a mistake on the final exam, and it's so impressed that I could finally find something. Oh, he did the whole thing. In fact, he insisted on doing the lab. He said, you know, I'm a pretty smart guy, and I could sit here and read the book of, you know, that's all it took to learn chemistry. And the only thing to learn chemistry is actually do it. So I wanted to do the lab also. So he was a great guy. He came up at the end of the term and said, you know, he said the difference between freshman chemistry and sort of a freshman calculus course, which he might teach, is that we get to do all these wonderful demonstrations and blow things up and color changes and whatnot. And when he says he's going to do a demonstration in his class, you know, it's some elegant solution to a differential equation. It just doesn't seem to have the same impact. So where goes? Okay, so we have the homework is due. I will, we will, again, if you're having problem with the direct ethanol exchange current density, no problem. We'll talk about that on today a little bit and on Thursday a little bit. The next problem set I'm going to delay. I was going to give it to you today, but I'm going to delay till Thursday. I don't think anybody will be too upset about that. So let's do the following Thursday. And the reason for that is I will be out of town, Friday and Monday. I have to go back to Princeton and do some work there. And I would rather have you have some chance to talk to me after you've looked at the problem set before you turned in. So I'll delay that till Thursday. It'll be due the following Thursday and I'll be around on Tuesday and Wednesday. Okay, one other item of administration. Because of some commuting issues and whatnot between Caltech and Princeton, I need to sneak in two lectures that will not be on the regular Tuesday, Thursday schedule to make this work. And they're going to occur the week of February 6th. And so we will have the normal Tuesday, Thursday lecture that week, but in addition to that, you get a double dose of me on the Tuesday, what's that, the 7th. So we'll do the one and a half hours. We'll take a little break and we'll do another period of time, if you can take it. And then we'll also meet on Wednesday, that week, the 8th and in the afternoon, I believe at 2.45, but I have to double check the time on that and have a lecture then. That means you get out of a week of lecture, so to speak, later on. You don't get out of it because you've had it, but you get a week off. Okay, so now to chemistry, electrochemistry. Again, for those of you, we're switching over to a new board system. So if you see me dancing around here today, that's because we have a new recording system on these boards and we'll see how that works out. If you hear Tom yelling at me, no, don't do that, that's also because of the board system. That's the bad news, the good news is I can use all these different colors of markers now. And the regular markers, they're not electronic anymore. All right, so all term, I keep on reminding you that when you do electrochemistry, you have this unusual problem in chemistry and that is the reaction only happens at the electrode and so we have to get the molecules from way out in solution up to the electrode surface and it's now time to deal with that problem. So we'll spend probably most of this week dealing with that problem. I'm gonna start this off in a very sort of qualitative way and sort of give you a feeling for what's happening and then we'll work the problem correctly after we've kind of jumped through it and made some gross statements and whatnot. But let's start off by first of all saying we're gonna be working in the opposite regime of where we were working when we talked about the Tafel equation. That is obviously in the case of the Tafel equation, the kinetics is the rate limiting thing. That was the whole trick there. Now we wanna go to extreme where as soon as a molecule gets up to the electrode surface, it gets oxidized or reduced. So the rate limiting process now will be the transportation of the molecule up to the surface and things are gonna happen so fast once that molecule hits the surface, we never see that. Now what that's going to lead to is although we normally say that the Nernst equation is an equilibrium statement, which it is, and therefore if you're at equilibrium you wouldn't think any current is flowing, I'd make the claim that the Nernst equation will hold right at the electrode surface under this condition where the kinetics don't come into play. That is, the molecule will come up to the surface, it will collide with the surface and it either will get oxidized or it will not get oxidized. And the only thing that will determine whether it gets oxidized or not oxidized is the potential of the electrode. And so the Nernst equation says that this potential, 10 molecules should be oxidized and 10 should be reduced, that's what'll happen or it'll happen instantaneously. On the other hand, if the Nernst equation says everything should be oxidized, then every molecule that hits that surface will be oxidized. So it's just the potential of the electrode now and that means that we're Nernstian, even though we're not in a traditional equilibrium state of no chemical reaction occurring. That's number one. Number two is we have to think about now how we're going to get the molecule from out in solution up to the electrode. So we have at least three possibilities for doing that. First possibility is it might be by migration. So migration I'm talking about the fact that we have a charged particle moving in an electric field and of course I can tell that particle where to go based on the electric field that I'm dealing with. So in other words, there's an electric gradient here, a potential, and there is a charged particle. Possibility number two is diffusion. Of course, that would now involve instead of an electric gradient, a chemical gradient. And now we're just looking at changes in concentration. And the third possibility is convection. It is some sort of active transport. Lots of possibilities here. So for example, I might stir the electrolyte. That would be one way of convecting my molecule from point A to point B. I might apply a thermal gradient of some sort in the electrolyte and get convection due to a moving thermal wave through the material. I might go and build a jet and have my electrolyte shooting through that jet and impinging on the electrode. There's all kinds of ways I could do this. So this looks like one thing, but it's a whole bunch of different things. Of those three general categories, the first one is the only one that's problematic. One on the one hand would think, this is good news. I mean, we're doing electrochemistry. We obviously have electric fields, and therefore potentials, gradients. We're going to have a charged species in at least one oxidation state. So why not use migration to transport our molecules to the electrode surface? So the first problem is, of course, really what I said was a plus. And that is, you need to change the oxidation state of the molecule if you want to do electrochemistry. So I can guarantee that the molecules don't just have one kind of charge on them, but there's at least two different states they're in. In fact, one might be neutral. And of course, that would be a problem. We can't get migration to occur in an uncharged molecule. But even if it is charged, it's changing charge. So the rate that we will be moving molecules either up to the surface or away from the surface will be different depending on exactly what charges we're dealing with. Of course, if we're going to depend on this, we can also be in the unfortunate situation that the potential that we have to put the electrode at in order to do the redox chemistry is such that the electric field is repelling these ions from that area. So migration is going to be very hard to control in the system. It's going to be variable. And that's one thing we don't want. Typically, we like to avoid migration. And this is another reason why we're going to use a serious dose of supporting electrolyte in our system, because the supporting electrolyte gives us a lot of charges in solution, which minimizes the extent that the double layer or the electric field extends into the electrolyte. And so we don't have huge migration terms in that system. So this is something we want to avoid. Diffusion is going to be the one that nine times out of 10 we're going to use for our method of transporting molecules up to the surface. It's more or less built into the system. You're going to sit there with an electrochemical cell. You're going to oxidize, let's say, something at the electrode surface. And in doing that, you will deplete the electrode surface of the molecule in its initial state. And therefore, you've got a built-in chemical gradient from whatever it is out here in the bulk solution to the electrode surface. So you have a beautiful chemical gradient to move things down. And so that's built in. So that's typically what we're going to use. Convection, on the other hand, though, is used. And the classic, probably, example of convection, I've already given you a couple, there are these jet electrode systems and whatnot. But the classic example of convection would be the rotating disk electrode, where one goes and generates an electrode, which is a cylinder, which has a piece of metal. Typically, there's your electrode down here and some kind of a contact that we can rotate around. So we can have a mercury junction in there. We can have brushes in there, something like that. So if we're looking at the bottom of this electrode, then we have an insulating region and a conducting material in the center of that. And this is going to rotate. And as we do that, then, we are going to move material up from the solution like this. And if we do this in a very controlled manner, then we can control the rate that we transport material up to that surface. And then, of course, the rate that it moves across that electrode surface. So it's a good way, as long as we have very good control over rotation, of providing a specific environment that we can define that moves material up to the surface. And we know exactly how much, as we know the hydrodynamics of that system. So in the diffusion case and in this rotating disk case, we have a well-defined system. OK, so how are we going to look at this system? Independent of which one of these three things, actually, you want to use. Yes? So you're using that as your working electrode. Correct. So then how do you incorporate your counter and your reference? OK, so the question is, the rotating disk electrode is a working electrode. How do we get a counter and a reference electrode into this system? They don't not have to be anywhere near this disk electrode or moving with that system. Now why that is, I would like to delay the answer to a little bit later today. We should get to that, I hope. But you could use the standard reference and counter electrodes far away, if you will, from this electrode. By far away, the caveat, remember, is still that the reference electrode is somewhat close to the working electrode, but far enough away, so it does not perturb this nice flow pattern that you want to develop there. So for example, you would not use a Luggins capillary in this system, because the idea there is to get the electrode really close to the working electrode, and that is certainly going to mess up the flow pattern. But typically what you would do is use a counter electrode, which would be a flag electrode. So a flat piece of metal, platinum, say some distance below that, so again, it doesn't mess up this convection. There might be a couple centimeters here. Now why are out here, of course, connected to your circuit? And your reference electrode way out of the way. And in fact, it's not so bad if you simply had, as your counter electrode, not a flag, but a sufficiently long wire somewhere else in the cell. Now why that all works, I will get back to, but pragmatically, that's the situation. Of course, if we're going to discuss this, we need a chemical reaction, and so we're going to go to our standard reaction, exciting reaction, an oxidized molecule plus an electrons going to reduce molecule. If all I need to be concerned up, I've jumped a point here when we got to that question, whether we are going to use some kind of active transport, some mass transport, like I've shown you here, or diffusion, or even a migration, then I'll just stipulate that any one of those processes can be characterized by a single constant. It's a pretty good statement. In other words, if I'm looking at diffusion, then there's a diffusion coefficient that tells me something about how a molecule will move down a gradient. If I'm looking at some sort of active mass transport, then there is some other kind of constant, some sort of a mass transport constant that would explain how a molecule moves from the bulk to the solution. But I'm assuming there's one constant, and we'll call that constant m for mass transport independent of what it is. And I have to realize that m might be different from my oxidized molecule than my reduced molecule. So I have an MR and an MO. And this is going to become an important point in a few minutes, because typically we assume that these two numbers are the same, are very, very similar. And that's OK, as long as the oxidized molecule and reduced molecule look about the same. If all I'm doing is pulling an electron out or putting an electron in, nothing else is changing, then probably those two numbers are about the same. But you do get yourself in a situation where maybe when you pull an electron out or put it in, you break a bond. And you make a new molecule that looks very different. And then this assumption can start to break down. And it's easy to forget that that's the case, because we're going to bury these numbers. So keep a track on what's happening there. So we have two coefficients that are going to strive our mass transport. The second comment to make, then, is there is some sort of a rate for bringing molecules up to the electrode surface. So in this particular case, we want to bring oxidized molecules up to the electrode surface and reduce them. And that rate is going to depend on the gradient that's available at the surface. So I'm going to refer to that rate. So I have to write rate all the time as just transport here, new transport. And that would be equal to, for the oxidized molecule, say, the concentration of the oxidized molecule as a function of distance. And I'm interested specifically at the electrode surface for that. So in other words, if I'm carrying out the forward reaction written there, and I have an electrode sitting here in position where the electrode is at x equals 0 and x is getting larger this way, and I map on this axis concentration, then far away from the electrode, I'll be at a concentration, which is the bulk concentration. Remember, I'm using an asterisk to indicate I'm at bulk. And then at the electrode, since I have no charge transfer limitations, depending on what potential I'm at, I will reduce this molecule. The number I reduce just depends on the Nernst equation. And so I will have fewer of these oxidized molecules at the surface. I don't know exactly how many, but I'll have some surface concentration here. And so I predict some kind of a gradient that looks like that, that my molecules are going to move down once the process gets started. So really, what I should be looking at is this derivative. But we won't do that. Instead, let me just say that this transport, to make life simple, is proportional to my mass transport coefficient, again, for the oxidized molecules, since that's what I'm bringing up to the surface, times the difference between the bulk concentration of the oxidized molecule and the concentration at the electrode surface. So I'm simply saying that I can take a difference here in place as a decent approximation of that derivative right there. Now, let's get this in terms of current. So we'll look at current per unit area or current density, since obviously the current will increase with the size of the electrode. And I have been arguing all term that that is simply the rate of the chemical reaction times some coefficients here that are going to get us in the right units. So it's the change in the concentration, in this case oxidized, as a function of time. And that, again, is evaluated right at the electron surface. That's the only area that I'm interested in. And if my current is just this transport rate over here, then I have a situation for a mass transport limited that the current, therefore, is equal to this rate, excuse me, not constant, but rate. And that's going to be equal to coefficient here times the concentration gradient evaluated at the surface. So that's a statement we're going to come back to, yes. This is the Faraday. I've gone and changed my nomenclature, thank you. Yeah, that f is, in fact, the Faraday f. And I'm trying after reviewing what was coming off the board before I decided a regular f may make life a little simpler. So I'm going to switch over to a regular f. That's just Faraday's constant to hopefully help the resolution on the board. And to confuse you a little bit. OK, so what we have then is that the current divided by n Faraday's constant area, just to move all those constants over to one side of the equation, is equal to negative of that mass transfer coefficient times the difference between the bulk concentration and the concentration at the surface. Why a negative sign there? You'll notice this is a positive number that is since I'm carrying out a reduction of the oxidized species, the smallest this number can be is 0. That is, the concentration is the same anywhere. And if any molecules are oxidized at all, then this becomes a more positive number. But it's a reduction reaction. And we've defined reduction reactions as having negative currents associated with them. So I throw in a negative sign here for the sign convention. I could write a similar statement for the reduced molecule that I am going to generate. That is, the current also could be stated to be, and this is the same current, just the mass transfer coefficient times the difference in concentration between the bulk concentration and the solution concentration. Now you'll notice if we are carrying out this particular reaction that I've written over there, and assuming we start with oxidized and we're turning into reduced, then we're making more material at the electrode than we started out with in the bulk. And so this difference is a negative number. So I have two negative numbers. That is, this current and this current are the same number. Everything works fine. Now typically, when I'm doing this experiment, the way I would do it is I would have only oxidized around. And as time went on, I'm making some reduced. So I'm going to make the assumption that at time equals 0, the bulk concentration of the material is 0 when it's the reduced material that I'm talking about. And although you can imagine an experiment where you would intentionally go and pour both the oxidized form and the reduced form of a molecule into an electrolyte, 9 times out of 10. This is how we're going to run the thing. And then you can see in that case, the current that we're dealing with is just minus mR times the concentration of the reduced material at the electrode surface. And look at that. I didn't do what I said I was going to do. Oh, well, I get out of here waiting a little while. Next question we could ask, looking at this statement right here, is what is the absolute largest current that we might get out of this system? And you can see the largest current is going to occur when that's 0. This is whatever you put in. And obviously, if you make that big, you better get a bigger current. But once I've set up my electrochemical cell and put in some concentration of my oxidized molecule, then I will get my largest current here when that value drops to 0. So in that case, we can define a limiting current, i sub l. And we have to divide that again by n, f, and a, which is just this value when every molecule that strikes the surface is instantaneously reduced. So I have no concentration there, or just I've dropped my negative sign, which I did not mean to do. I can't get a bigger current than that there. So to make life simpler, what I'm simply going to do then is I am going to ratio my currents against this limiting current, which will help us get rid of all these little constants that are floating around. So I don't have to write them out all the time. That is, instead of just talking about the current, excuse me, the concentration at the electrode, let me talk about that concentration divided by the bulk concentration. And that is equal to 1 minus the current divided by the limiting current. So all my n, f, and a's disappear, and I have a nice simple statement now. That relates my current to my limiting current in terms of the bulk concentration and the concentration at the electrode surface. Or of course, solving this so I can get back to my surface concentration at the surface. The concentration of oxidized is just the limiting current minus the actual current divided by n. And that's transfer coefficient for the oxides molecule, Faraday's constant times the area. And I'm throwing those guys back in because I'm going to make them disappear in a second. OK. And can you see if I write down here? Is that legible? I noticed when I was writing at the bottom here, people were kind of, OK, well, OK. So let me just point out that what I'm talking about pictorially then, graphically, is again, in the case where the current is the limiting current, if I make a plot of concentration versus position, I'm going from the bulk concentration down to 0. So all I'm really saying is that's the biggest gradient. I can build that at an electrode surface. I can go from some bulk concentration out here down to 0 at the surface. That is, every molecule that strikes the surface, instantaneously gets turned into reduced. So there's never a population over here once the experiment gets started. On the other hand, since I have argued that this is a Nernstien type of system, I might find myself in a situation where, say, I is equal to just one-half of the limiting current. And I can choose that by choosing the potential somehow to be correct. And then I'll generate, again, a situation where I have a gradient, but it will not be as large a gradient. That is, I'll go from my bulk concentration down to some value here at the surface that still leaves a population here. So that raises the question, well, exactly, what is the relationship then between the potential that I've applied this electrode and the currents that I'm talking about here? I'm moving. I am now going to, can I erase these boards now? OK. We're going to move back to this board. I'll erase over here, keeping myself busy. Well, Tom takes a nice picture on the right-hand side. So we have the statement way over there that tells us about the oxidized species. I can write a similar statement about the reduced species. And again, I'm going to assume that all I do is dump the oxidized molecule into solution. So I guess we should keep our wonderful chemical reaction up here. And so at time equals 0, I am assuming that this is at the bulk concentration. That's a great assumption. And maybe not as good an assumption, but reasonable. And that is 0, that you're not going to go and throw any reduced stuff in. And we could handle it if that wasn't the case, but throwing zeroes in makes the arithmetic very nice. So we've already seen then that the concentration of the reduced species is equal to minus i over n f a m r. y minus. Again, the sign convention is going to get us into trouble if we're not careful. And that is, remember that this current is a negative current from over there. And obviously, I don't want my concentration to be a negative number, so I have to throw a negative sign in there to get things to work out. And all these constants down here are obviously positive numbers. And we already have, let me just duplicate it over here, that the, and I should state in here that this is at the electrode surface to concentration, I'm going to duplicate just what I wrote over there, that for the oxidized species, we have the limiting current minus the actual current divided by n f a m r. Now, I've already argued that right at the surface, under this condition where there can be no charge transfer limitation, that the ratio of oxidized to reduced is given by the Nernst equation. Is everybody happy with that one? Not a problem. Good. Over here, oxidized at the surface. Yeah, oh, and zero. Ooh, thank you. This is the beauty now of this new system. I can make a mistake, and it's not recorded for posterity until you've corrected anything. So by the way, if you do see things like that, please point them out, and in particular, point them out before Tom takes a picture, and then I'll look really good. Thank you. All right, so we have the Nernst equation. So we have an electro potential, which is equal to the equilibrium potential under this condition of infinitely fast charge transfer rates. So that's equal to the standard redox potential for the system, minus RT over n. We're going to keep that there. Over nf times the log of the concentration of reduced material at the electrode surface divided by the concentration of oxidized material at the electrode surface. OK, now this gets us back to what I was saying at the very beginning of the hour with regard to the problem set and the value of n. In this case, the value of n is pretty straightforward. That is, it's defined by this equation right here. Whatever stoichiometry is needed to make this equation mass balanced, that's the value of n. And I believe for the direct ethanol system, most or perhaps all of you hopefully figured out that that's 12, the way I set up the problem. That same n apparently shows up in the Tafel equation that I wrote before. Tafel himself didn't have a problem with this, because remember, Tafel was looking at the interaction, the reduction of water to make hydrogen and the oxidation of hydrogen to make water. And that can be thought of as a one-step process. So the stoichiometric n and the n in the Tafel equation are the same thing. But as soon as we look at something more sophisticated, we know that there's a reaction mechanism that goes with it. And it might be a multi-step process. So for example, taking that case of ethanol, it is exceptionally unlikely. In fact, I think we can rule out the possibility that when an ethanol molecule strikes an electrode surface, 12 electrons jump into it. So even though that is the stoichiometry of the reaction, there clearly is a set of steps that maybe one electron goes in, and then a second step, maybe two electrons go in. And you might imagine in some very far-fetched system, maybe there's a step where three electrons sort of jump into the molecule, but even that's pushing it. People really haven't seen evidence for anything more than a two-electron-charged transfer in a sort of simultaneous process. When we have a mechanism, of course, kinetically all we can see is the rate limiting step. And so assuming there's a multi-step process, then the n actually that we need to put in for the Tafel equation is the number of electrons in the rate limiting step. Now you, of course, have exactly zero idea of what that might be in the direct ethanol system, and that's why I set up the problem to use the stoichiometric number of electrons, and that's probably why we're getting a really bogus answer. But it is more likely that the n, I don't actually know what the answer is either because the mechanism for direct ethanol oxidation is still a little bit ambiguous, but it's more likely that n equals either one or two in that system and not 12. Because all we see is the rate limiting step. But for this, since there is no kinetics, the n that we put in is the actual stoichiometric n. Okay, well we have an equation here that gives us potential in terms of concentration, we have currents in terms of concentrations here, so we can plug the two into each other and we can come up with a statement that gives us a current potential relationship. So we have that the electrode potential is equal to the standard redox potential, A naught minus RT over n Faraday's constant times the natural log of MO over MR times I over IL minus I. So I have put using the standard nomenclature reduced over oxidized, just like I wrote it over here. So I flipped some things around here. Let me point out to you again that if you are looking at an older derivation of this equation, typically the equation was written, the chemical equation was written in the opposite direction and so you'll either see it written the way I've written it but with a plus sign over here or you'll see the negative sign over here and the quantity in the log flipped over. So don't be confused if you see that. Try now go and make a plot of that where I'm now looking at the electrode potential and current over here. Those of you who are used to doing pH titrations immediately see that I have a sort of sigmoidal shape here. This is right your standard pH titration curve with the logarithmic term and so I'm gonna have something that looks like that in it where this is my limiting current. I'm looking at absolute value of current, yeah. Excuse, thank you. Now, those of you who are less chemically oriented and perhaps more mathematically oriented have noticed that we have a problem with this equation right here and that is things get rather ill-defined when the current is the limiting current. Then my denominator here blows up and so how can I say that that's the limiting current over there? So I really should specify that this equation holds for i less than the limiting current. And then I would just argue what we need to look at, of course, is the limit of this term as i approaches iL. We'll get this value right here and we do that and clearly we can't get a limiting current that's larger than the limiting current. That is the limiting current and therefore this has to plateau at that point. Now let me define a point on this curve where I hit half of the current level of the limit. So this is i equal to i sub L over two and then there's some potential that's associated with that and I will define that potential as E one half or the half wave potential. Now let's look at what that corresponds to so I can plug in for my current up here, half the limiting current. So in the conditions where i equals half the limiting current then I have that my potential is equal to standard redox potential E naught minus RT over NF times the log of the ratio of the mass transfer coefficient for the oxidized divided by the mass transfer coefficient for the reduced plus RT over NF times the log of iL minus i divided by i. Okay so I had taken the log term and I have separated it out into two terms over here and in doing that just cause people seem to like plus signs I flip this term over so I can have a plus sign right there. And now I will say under this condition of having current being half the limiting current my potential is the half wave potential that was my definition of it on this graph over here. So I now have defined the half wave potential and now I'm gonna say well typically the mass transfer coefficient for the oxidized species and the reduced species are very similar and so this term is gonna disappear more often than not that is that MR over MR is approximately one so I'm taking the log of one and so I don't have to be too concerned about that term the way I'm gonna handle that though is I don't want to all right Tom I'm gonna work over here so you have a free shot I don't want to totally ignore that term myself into a little bit of trouble doing that so let me then just define E one half is equal to the standard redox potential minus this term and then I have the statement that the potential of the electrode is equal to the half wave potential plus RT over N Faraday's constant the log of the limiting current minus the actual current divided by the limiting current so there's our relationship so far everything I've done is true no problem right I've left my current dependence in there I've defined an E one half under a specific condition over there and so it's an absolutely true statement and here's where we do the slide of hand if we leave it there everything's fine however what if I say this that the standard redox potential is approximately equal to the half wave potential see that will be a very good statement as long as this term is insignificant compared to this term over here and that's typically what we do and nine times out of 10 it'll be fine and the 10th time you'll have messed up big time because again the molecule has changed dramatically between the oxidized state and the reduced state and it's not diffusing through this to get a feeling for this a typical number for a molecule diffusing through solution is for the diffusion coefficient D that goes with that would be on the order of 10 to the minus five centimeters squared per second if you're walking down Colorado Boulevard and someone walks up to you and says what's the diffusion coefficient for ferricene in acetonitrile and you have no idea what the answer is just say oh it's about 10 to the minus five centimeters squared per second and they won't bother you anymore it's a good number for things in solution okay on the other hand yeah okay now a huge difference because remember the whole assumption behind this is that there is no kinetic limitation and of course in the irreversible reaction there's going to be a kinetic limitation that's the definition of an irreversible reaction so that's going to change things but let me let's consider how it's going to do that but give me a couple minutes to develop that okay so we have to look at that certainly because the assumption is that it's not happening at all here now staying with that diffusion just for one second let me finish that one thought I had to give you a comparison if I'm thinking about a molecule diffusing at room temperature through a solid state matrix then the sort of numbers I'm going to come up with are on the order of 10 to the minus 10 centimeters squared per second okay so you can see in solution if you pick one molecule versus another molecule it might be two times 10 to the minus five centimeters per second and the second molecule might be 10 to the minus six centimeters squared per second or something like that but that is such a small difference compared to what you would have say in the solid state for diffusion that we can consider both these sorts of numbers about the same and not affect this dramatically now there's a question yeah I'm sorry I missed the connection from now these are the same E1 half right it's the situation what happens over here I should have written this out perhaps you may want to record this one I left this a little ambiguous right let me remember my better penmanship here hold on you got the point but for the picture we wanted to smile nice so this term disappears on us this this becomes one under this particular condition right of half the limiting current and therefore you want half is the same there thank you because I did sort of gloss over that point okay so this is fine and then we'll make this assumption it's not a bad assumption except when it's a bad assumption but usually it works you can get away with it okay now that actually brings it yeah the the problem is and you'll appreciate this better than most Bruce that measuring diffusion coefficients isn't easy and so you know one of the reasons that this approximation is made is that we typically cannot measure the diffusion coefficient well enough that we can tell that they're different I mean at the point that you make a measurement and say oh the oxidized species has a different diffusion coefficient than reduced species it has a very different diffusion coefficient so a diffusion coefficient difference of 10% is really not measurable even on a good day it's got to be more on the order of magnitude difference before you get there and of course an order of magnitude does start to have an impact then on a log term but that's what it takes so there's there is a pragmatic reason for doing this and that is until that diffusion coefficient really changes order of magnitude change you're not going to pick up the fact that you want half is different from the standard redox potential and so if if your job if you were assigned the job of making up the table of standard redox potentials and you wouldn't want to do it this way right because it doesn't have enough accuracy to get you know ten significant figures for the standard table that everybody's going to memorize and you want to do it actually by some sort of a direct titration technique where you can get a lot of accuracy and precision in the measurement and the reported number but if your job is just to look at a molecule and say you know what's its redox potential two within say maybe a tenth of a volt then this is going to be a great approximation and typically that's from a pragmatic point of view that's what we're interested in we're not interested in voltages potentials that go beyond that okay so this gets down to the pragmatic issues so based on this discussion the first you know what good is this the first thing you could do with this if you had this set of data if you have this curve right here then quite obviously you could assign a redox potential for a molecule by using this approximation it's easy to find the half wave potential and so one thing this gives us is redox potentials now with all the caveats that I just pointed out and let me point out to you that this is a standard redox potential I think we've got a right standard that is the number I am getting out of this experiment is the redox potential when I have one molar concentrations of oxidized and reduced present and when I have one atmosphere of pressure and when I have room temperature okay now I'll point at you all these equations even though I write t here actually assume temperature so you have to put in the 298 degrees Kelvin that that goes with that to make that statement true but the interesting thing is that I didn't put those concentrations or pressures into the cell in fact I have a cell sitting there that has a lot more oxidized than reduced in it and whatever the pressure is and yet the value I'm getting out is the standard value and when the standard conditions so it's directly comparable to what I would read off of the table okay next thing you'll notice is that if I were to make a plot of the log term I'm going to switch over to log base 10 because that's what most people do and you'll recall that that just changes my natural log by a factor of 2.3 so we'll take that into account in fact I'll make it explicit I'll put the 2.3 there if I make a plot of the log of this current term versus potential then that should be a straight line according to the equations I've written here and I should have a slope of RT over N Faraday's number and that will equal 0.059 millivolts over N when T is temperature now the 2.3 has to be that's why I did it in those excuse me yes all right the point Bruce is making is the 2.3 is incorporated in this number here the fifth if you don't plot just log thank you Bruce you'll be off by a factor of 2.3 what the log of this term versus E is linear right according to this equation and that's the slope of it right 59 millivolts oh I excuse me well ah thank you whoa yes I was being so clever here writing in a vault so I wouldn't confuse anybody and then I wrote millivolts next to it yeah or just to be explicit here 59 millivolts over N okay yeah this gets back to our taffle and our equation a question a moment ago first thing about this of course if I don't for some reason know what N is I can back in out of this okay assuming that I'm certain that I'm diffusion limited or mass transport limited in general okay if I do know what N is then I can use this as a test to see if in fact I meet the condition of being mass transport limited and typically I would know what N is okay so in other words if I make that plot and it's not a straight line with that slope I'm not mass transport limited okay what'll actually happen if you look at that plot that I'm suggesting you make there that's something very similar to the taffle plot right in the taffle plot is log of current versus the over potential okay and the over potential in this number scale with each other right so in the taffle equation I'm going to expect a similar straight line dependence fine instead of being mass transport limited I am charged transfer limited in taffle we'll have a slope that's equal to RT here's my ambiguous NF I'm gonna put a little alpha down there I have to throw another alpha in okay so I have a different slope first of all it'll differ by this term alpha which is the symmetry of the of the barrier but remember that's probably 0.5 and and second of all now that's why I put the alpha down there it will be affected by this number N okay so you can see in the case of direct ethanol taking as an example if I am mass transfer limited then N equals 12 the stoichiometric value over here if I am charged transfer limited then N is either going to be one or two depending on exactly what the mechanism is and so that makes a huge difference in this slope okay which probably was the problem there's the solution of the problem but experimentally what does this mean it means that if you make that plot you can immediately tell whether you're in the taffle regime or you're in the diffusion limited regime now of course there can be situations where you're not in either regime but again the slope will not be 59 millivolts per decade of electron there so you'll find out and finally what you can do with this is you if you develop this whole curve then you can back this limiting current out and so you can make plots of concentration both concentration versus limiting current and from that you can come up with that mass transfer coefficient and again given the limit to which you can determine a concentration and given the limit to which you can record a current you're not going to see 10% changes in this number okay now let's go back any questions on this it's totally dependent on it really and we're just looking at it in other words so I must what I'm saying by all this and I think I'm you can take a pretty picture the right-hand side top what I'm saying is that the reason we have a current at that electrode is is because of the existence of that concentration gradient and the bigger that concentration gradient is the bigger the current is so it's the only parameter affecting the system assuming you did not have a linear gradient there now it's going to turn out as I will show you one second it's going to be very hard to get away from this kind of gradient okay or yes two is a base ten log yeah there's a factor of 2.3 in there I wrote this all out in terms of the natural log because typically when we're doing kinetics we're taking derivatives and so we're dealing with natural logs but that log rhythm originally comes remember from the statement of the Nernst equation and when Nernst stated the Nernst equation he used a base ten log so for different things we have this tradition of using the natural versus the base ten log and that 59 millivolts which keeps on popping up is always based on a base ten log okay I'm going to leave where I can clear out this right-hand side leave that left-hand side up there can use that a little bit more let's go back to the rotating disc electrode which is a germane in light of what kind of diffusion gradient do you have now I would suggest to you that it would appear that the two extremes you could get to on the one hand you can have an absolutely quiet solution and you'd have a beautiful diffusion gradient in there and on the other hand you could have this rotating disc situation we're actively transporting material up to electrode and you would think that you would establish a very different gradient in those two cases that those should be sort of night and day examples but in fact the way you operate your rotating disc draw that here's our disc again and it's rotating is you make that surface as flat as possible in fact it's a polished surface and the idea is that we will have laminar flow of the solution over here and so what that means is if I were to break my solution down into little differential elements that I would have a situation like this where I have these elements and the first element if you will sticks to the electrode it moves with the electrode the next one slightly displace and of course I can I'm a lot of make these small enough that this is true okay so that is the assumption for laminar flow very smooth and it turns out that all the hydrodynamic equations that one develops for a rotating disc electrode are under this assumption so what can go wrong with that assumption well if you don't have a very smooth surface or if you simply rotate too fast then you'll get turbulence building up in the solution and this situation will no longer exist this turns out to be a really important thing if it decides that of doing electrochemistry you decide to build airplanes right you don't want turbulence going over the wings and so there's been a lot of engineering work done on hydrodynamics for airplanes boats things like that that tell you when the turbulence will set in under what conditions it'll set in and systems are rated with a number called the Reynolds number that comes out of these equations that tells you at what point you'll come up with turbulent flow and so we're going to always assume that that were below the Reynolds number indicated rate in these systems I happen to mention to some of you yesterday that professor Lewis at one point where we were having a problem actually with a kinetics in a system that we wanted to study using a rotating disc system and he said well if we could just rotate fast enough then we could get over this problem and so his proposal was to go and get a VW bug these are the old bugs not the new bugs engine was his proposal and strap it to the wall and he decided that that could be geared down and we could go at 2,000 was his calculation rpm with the right gear box on that thing and you're going to have two problems with that we never did it number one is you will exceed the Reynolds number of this solution if you do that so it doesn't matter that you can rotate at 2,000 rpm second of those VW engines didn't exactly run super smoothly so you will exceed the Reynolds number quickly because of the vibrations that you're going to do some to yourself so you want to have a situation that looks like that okay now which means that you can get up to something around a thousand rpm though if you have a nice nicely polished electrode and everything is perfectly flat there what does this mean in terms of everything we've been talking about here well if this last little layer of solution is really moving with the electrode then that means that this flow pattern I was telling you about earlier does not encroach on that last layer that is you can think of material as being transported actively up to the electrode until it gets the last layer more or less and then when it gets to that last layer it's diffusing and if that's the case and assuming that last layer is a thickness delta associated with it then our mass transport number is simply the diffusion coefficient for the molecule divided by the thickness of the layer and you can see since I've now let you make the thickness that layer anything you want this delta I can get myself into this regime as long as I'm not in a turbulent region that is if I'm having a situation where I am not mass transport limited I can simply make my layer thinner and eventually get to a point where mass transport limited so if I take the equations that we wrote down here and simply substitute in this for the mass transport coefficient and likewise this for the mass transport coefficient of the product species the reduced species then I have the equations that describe a rotating disc electrode this curve I've shown you here is the curve that one gets out of a rotating disc electrode and one can pull out the halfway potential and the redox potential from such an experiment as well as determine the limiting current now the one flexibility that you have here is that I can obviously rotate that electrode at different rates and what is that doing that is affecting the thickness of that final layer so I can adjust delta by rotating at different rates up till this turbulent boundary by doing that I'm adjusting the limiting current so plots of a limiting current versus the rotation rate approximately it turns out that this boundary happens to go as a rotation rate to the square root or square root excuse me a rotation rate will tell me what the diffusion coefficient is for the system that was my point number three on this board if instead of figuring out diffusion coefficients and redox potentials what I'm interested in is concentrations and I haven't really stated this but one should recognize that the very first application of electrochemistry independent of the technique it has been electro analytical chemistry where the question simply is how much of this stuff do I have in this solution electrochemistry is really good at that that's where your pH electrode comes from right and again I might do that by monitoring a limiting current and we have equations on the board that will tell me concentration as a function of limiting current so I want to know how much of this stuff is around if I happen to know all of the coefficients that I need down here then I can tell you what that limiting current is going to be for a typical system rotating disk system a number for m on the order of ten of the minus two centimeters per second would be typical and then I can calculate a limiting current based on that and what I'm going to do is I'm going to assume an area here my a term of one square centimeter which is a little bit big I will admit for a rotating disc electrode but it's a correct order of magnitude they're not square millimeters or things like that about a tenth of a square centimeter is reasonable I can calculate then a limiting current for that that would tell me that my ball concentration of ten of the minus nine molar is is quite observable so I should be able to see nano molar concentrations and rotating disc electrodes one of the more sensitive techniques that one has available assuming things like you can put the material on solution you have enough supporting electrolyte around etc etc okay questions right then then the next step in all of this is to do this problem correctly okay that is to make life simple here where shouldn't I be standing doesn't make life simple writing more on the board today to make life simple here I have taken a derivative a gradient and just said it's a difference between two concentrations and that's not quite right and I've assumed that I always have a zero concentration out in the bulk solution and that I'm always a mass transfer limiting that charge transfer kinetics never gets into the situation of those assumptions the last one's a pretty good one that is I can get myself in that situation by either picking the right molecule with that does fast charge transfer kinetics and the right electrode that mates with that or by going to a sufficiently high potential that the mass of the charge transfer is is limiting so I can always get myself in that regime the other assumptions may not be quite that valid so what we need to do then is go back and work this problem out correctly using an actual flux gradient and that will lead us then to the chrono-amperometry experiment where we do a potential step and we'll set it up initially so it's a potential step that is sufficiently big so that as soon as the molecule hits the electrode it gets oxidized or reduced to always mass transport limited unlike that what we worked out here we're going to specifically say the transport of molecules up to the electrode is by diffusion that's it and then we have chrono-amperometry okay so we'll pick up with that on Thursday