 OK, so cyclic voltammetry. OK, why do we want to do this? Do we want to do this? This is a picture I showed you last hour. This is the whole motivation for doing cyclic voltammetry. It's not original with me. It's Professor Bard's idea of how we should motivate things. I really like it a lot. The idea is quite simply this. We already know from everything that we have done that we can generate an analytic solution to the current time problem. And likewise, we have a good solution to the current potential problem. That is supposed to be e1-1-2, right there. We've got a little cut off on this one. And given that this is the reversible case, we can come up with a three-dimensional plot of all possible solutions to the current potential time problem. And for the reversible case, certainly, because there it is right in front of you. And since you've also seen me solve the problem for the irreversible case, there's some other surface with that error function business in it and whatnot. That would give you that. And likewise, there must be a solution where the rate constants for the forward and backward reaction are similar. That is the pseudo-reversible, or quasi-reversible system. You've seen that. So that one exists. And again, we could do it for EC mechanisms and CE and any kind of mechanism you'd want to come up with. And given that surface for whatever the mechanism is, you would have all the information contained in that surface that you could determine electrochemically for that system. So that would be one approach to this problem. But it has two major shortcomings. The first is a laboratory tenacity shortcoming. And that is generating a surface like this, even for the reversible system, with enough resolution that you could be certain, if I picked any random point on here, I would know exactly where I was on that surface. It's going to take a lot of experiments. We have to jump to a lot of different potentials, look over a significant time range, et cetera. And so that's going to be a horrific experiment to learn something like what the halfway potential is for an experiment. The second problem is going to be that all these surfaces, even though they will be subtly different, will look about the same to the casual observer. So you won't be able to look at this surface and say, oh, yeah, that's a reversible surface. You can do it in this case, because I wrote that down there. But otherwise, good luck at doing that. It's very good to come out a little bit more. It should. Yeah, if it comes out labeled like that, you're in good shape. But all these functions you saw that kind of decay with something that looks sort of exponential, you're not going to tell the difference by eye. And so how are we going to get around this problem? And that's the power of cyclic voltammetry. And the idea is quite simple. We are going to take some plane, and we're simply going to slice this complex surface that I've drawn there. And we're going to slice it such that the plane is neither parallel to the current time axis or the current potential axis, because if it was parallel, of course, that would just be one of the chronoamperograms or one of the voltammograms. So we slice it off axis. And in doing that, we get a projection on this plane, which has all three of those parameters somehow convoluted. We'll deconvolute the current, excuse me, automatically, because I'm running something that's parallel to the current axis, if you will, there. But we will have a convolution of the two parameters in this plane down here. That is the potential and the time. So the idea would be that we only need to sample a limited number of those planes to get a feeling for what that surface looks like. So we don't need the whole surface, number one. And by the way, a limited number is not one, I will tell you. It's a number greater than one. I tell you that because in the literature and with my students anyway, there seems to be a large number of people that think that one slice of that complex surface gives you all the information you ever needed. I assume that because they keep on showing me a single cyclic voltammogram and asking me to interpret it, which I can't. You see this in the literature also. People will publish a single cyclic voltammogram and say, OK, we did the experiment. When you did A experiment, you didn't do the experiment. So we need several of these planes, number one. And number two is it turns out fortuitously that there is wonderful pattern recognition. That is the projection on the plane just by looking at it allows you to decipher whether you're in the reversible system or the irreversible system, things like that. So we get around both problems with this technique. So how are we going to do it? Yes? Does this plane go through the current? It does not have to go through the current axis. No, it might. It doesn't have to. Now by the way, the way, well, let me say this. Let's see, that's legible. OK, let me say this first. How are we going to do this? So what we're simply going to do is we're going to apply a different potential wave form than we've been applying so far and monitor the current. That is so far we've been looking at a single step or a double step change of potential. And now we're going to scan our potential. So this is what we're going to do. We have time down here. Although the word time seems to have gotten lost somehow. So this is time. This is potential over here. We're going to start like we always do at time 0 and some initial potential. So that's that point right there. And we're going to scan at some constant rate for some period of time until we get to this point up here, which we're just going to call the lambda point. So lambda potential, lambda in time. And then we're going to scan back in a symmetric way to where we started. That is your cyclic voltamogram right there. How are we going to pick these potentials? Well, we're going to start off with a potential where nothing is happening. No oxidation reduction is occurring. And we're going to scan through this potential regime here where something gets oxidized or reduced, depending. And we'll get past that point. And then we'll pick a return after that event happens. And we'll scan through that event again on our way back. So if we were going forward and we did an oxidation, let's say, when we came back, we'd be doing a reduction. Going through that same region, because we have oxidized stuff there, and it has to be reduced. Mathematically, what are we doing? We're just saying that our potential now is time dependent. It's equal to the initial potential plus omega, which is the scan rate, usually given in millivolts per second, times the time. So that's just the equation for this line right here, up to lambda. And then for times greater than lambda, we have an equation that says we're at the lambda potential, and we scan back down at minus omega t. Again, same scan rate. So you will see in the literature, various notations. But usually, the scan rate, omega, is typically given in millivolts per second. But be careful, because some people like to use other units. Yes? Did I mess them up? They're backwards. That's what happens when you're sitting on a little seat on an airplane and you can't see your screen. OK, thank you. Let's see. So time, what's it, greater than 0 and greater than? Oh, that's good. A physically impossible situation here. So this should be a less than, right? And then we're OK, I think. Time greater than 0 and less than lambda. Time less than 0. Yes, I do. I switched both of them. Thank you. OK. OK, this was in words. I'm not going to stop and correct it right now. But in words, this is time greater than 0 and less than lambda, and then the second one is for time greater than lambda as opposed to less than lambda. That's good. Thank you. I'm easily confused. OK, how are we going to solve this problem? The reason that every time we did a chrono-amperometry or a chrono-colometry problem, I took you through it and said, this is how we're going to do it, is we use exactly that approach here. That, for some of you, will be the bad news. The good news for those of you who think this is bad news is it doesn't actually work. Let me explain what I'm saying. OK, so what does work is we start off with fixed second law, of course, and say that we have a diffusion-controlled experiment. So the first thing is when we do our cyclic volt amogram in our three-electrode cell, we aren't going to stir our electrolyte. We don't want any diffusion gradients to be in there to start with. We want everything to be at the bulk concentration and let diffusion gradients build up, as they do, because of the chemistry that we're seeing. I'm going to come back to that point in a few minutes, because that has a practical implication. And for the reversible case, if we're going to start there, like we always do, we'll assume semi-infinite linear diffusion. And we'll have just one new condition, which isn't so much of a new condition. But for the reversible case, we will say that at the electrode surface, the Nernst equation holds. That is, the charge transfer rates are fast compared to diffusion. And so at the electrode, we're always in equilibrium in terms of charge transfer equilibrium. And this part looks about the same. And I use theta there, just like I have so far. And the difference, you'll notice here, is before, this was just E of the electrode, these two terms. And now, the electrode potential is changing with time. So it's E initial plus or minus omega t. Now, that actually is not totally a true statement. I just use a little shorthand there. That is, I should, for the forward scan, the plus sign is exactly right. For the backward scan, I also should add in this omega lambda parameter that I had on term that I had on the last slide. But I'm trying to make a point here, as opposed to a rigorously correct equation, that we're scanning forward and we're scanning backwards. And we now have a time dependence to this situation. And now, normally recall, what we would do to solve this is we would Laplace transform this guy. We would Laplace transform all our boundary conditions. It'll work fine for the semi-infinite linear diffusion business. And we would Laplace transform this. And what we would like to do to be a direct analogy with what we did before with the chronoamperometry is we'd like to come up with a statement that says that the Laplace transform of the concentricity oxidized species is equal to this transform theta times the Laplace transform of the concentration. But we can't do that. Why? Because theta now is not a constant like it was before once I fixed the potential. But it's time dependent. So we run into trouble on doing the Laplace transform. So people wanted a solution to that. And quite interestingly, two separate research groups came up with a solution in 1948. This is not new. And they didn't know, apparently, that each of them was working on the problem. 1948, the fax machines and the email didn't work so well then and whatnot. And so one group, Sevix Group, was working in Czechoslovakia. And the second group, Randall's Group, was in Great Britain. And in the same year, they published papers that are very similar. That layout, the basic background, not only background, but operations of cyclic voltammetry. It wasn't called, you'll notice cyclic voltammetry yet. It was called oscillographic polarography, or same kind of idea here. Cathode ray polarography. They were naming it after the instruments they were using at the time. So the kind of polarography that was available before this time was classic dropping mercury polarography. Also invented in Czechoslovakia in case you're interested at that time, now the Czech Republic. And in that, you would use an x, y, or actually they didn't actually have x, y recorders either in 1948. They had strip chart recorders so that you could plot current versus time. And if you were changing your potential at a given slow rate, you could plot out this polarogram that way. Well, to do this, you're scanning a lot faster. And a strip chart recorder can't keep up with it, so you have to use an oscilloscope. So the novel aspect, actually, for these guys, you can see it from the title, was that they're using an oscilloscope as the recording medium for this experiment. It's a high speed, quote unquote, high speed experiment compared to anything that had ever been done in electrochemistry before that, with the exception of an obvious step, which is a very high speed experiment. And so they both focus on that. But in fact, we don't care that they're using an oscilloscope. The real value of these two papers is they work out the mathematics for a reversible reaction under a triangular wave potential. And the approach goes as follows. You have this time-dependent Nernstien boundary condition now, and what you're going to do then is break it into a time-independent piece and a time-dependent piece, S of t. And this is why I introduced the concept of this time-dependent function in our double-step chronoamperometry. This is so far exactly what I did for our double-step chronoamperometry, where we have our time-independent piece, which just depends on the initial potential and the redox potential. And then our time-dependent piece, which has the scan rate omega in it, the time, obviously, and these other terms are just thrown in there because it's convenient to keep all that together. So we have an exponential decay with time for this time-dependent piece, and everything else is as it was before. Now, by the way, even though they're both published in 1948, Savic won the race. He published first, earlier in the year, just. However, as I tell you in a minute, coming in first isn't always best. So it turns out when you do this, you still have to Laplace transform this S of t function. If you don't like Laplace transforms, you still have an integral that's going to have that in there. And once you use that as a boundary condition, with fixed second law, you find there is no analytical solution to the integral you're going to develop. That is, you can't do the Laplace transform. And so both these gentlemen came to the conclusion separately as, well, we don't have an analytical solution. So we'll just generate a series as an approximation, and we'll come up with a solution that way. So again, no max, no PCs, no IBM mainframes, no nothing. Maybe an abacus. They had, by hand, pick as many terms as you can until you think it converges. I hope it converges, and work it out by hand solution to the problem. And they couldn't work out all the various mechanistic cases this way, just because it was too much work. So they start with the obvious reversible case and work that one out. And I believe also the irreversible case. And they come up with a solution. And although Sebek gets the award for doing this and coming up with this all this first by a few months, he made an arithmetic error in his series when he was plugging the numbers in. And one of the diagnostics you'll find in a cyclical tamagram is the separation for a reversible case between the oxidation peak and the reduction peak. And because of this error, he came up with 80 millivolts peak to peak separation. It's the wrong number. It turns out it doesn't affect life at all. But I'll explain why that is a little later on. But there was an error in there. But nonetheless, he laid it out. Randall did it correctly, came up with a number closer to 60 millivolts peak to peak separation. And the reason it doesn't matter is that experimentally, you probably can't tell the difference between 60 millivolts peak to peak and 80 millivolts peak to peak. So even though there's technically a math error in there, it doesn't make a big problem. And of course, Savage didn't see this in his actual experiment. He did experiments with his oscilloscope because he could not distinguish between a 60 versus 80 millivolts peak to peak separation. So he was within his experimental error. So that's it. That's about as far as you can get, 1948. And then nothing happens after that. Why? Because you need all this expense of large equipment to do this experiment. This is a specialist experiment. This is like the guy today who goes and does a femtosecond laser experiment. It's not just any person in this room can do it. Bruce is the only one that can get it right. So lots of equipment. And it has to be a good day. Usually you get it right. Lots of equipment, lots of money, lots of expertise. You don't just plug it in and run the experiment and get the answer. You need all this state-of-the-art electronics. Number one. Number two is we don't have truly a mathematical framework for doing this. We can work out the reversible case, but the whole world is not reversible. What do we do when we run into something else? So we're stuck. There's nothing else happening. So this was a novel experiment that a couple experts in the field could do. And that's the way it stayed for a long, long time, 1948, down to 1964, essentially. In 1964, or actually early 1960s, integrated circuits became available. And you could build things with integrated circuits. You could buy them and build things. And that allowed you to build two things. One is it allowed you to build computers, digital computers, these large mainframes, these things where each campus, if they were lucky, had one. And it filled up a room about this size that had to be carefully air-conditioned. And no one was allowed near it, because if you looked at it the wrong way, it broke. But you could take these cute little cards and card punch them with code. Unfortunately, remember this. And one line per card. And then they could get fed into the computer. And then you could find out that you mistyped one of those cards about halfway through the stack. And you could find it and redo it and resubmit again. And then a few months later, you finally got the thing to run. And you could run, though, these codes that were very impressive. And in other words, instead of having to do a series by hand, you could now tell your computer, do this numerical integration with lots of terms and do it very precisely. That became possible, not easy, but possible at that point. And the other thing that became possible is you could now build good current followers and current to voltage converters. So now this measurement of current, which, remember, has always been a limitation, anybody can do it. By your current to voltage converter, high resolution amp meter, if you will, digital amp meter. Plug it in. It wasn't called digital then, but the way you go. So you can build a potentiostat, and you can sell it a little while after that. You didn't have to have a whole room full of vacuum tubes and oscilloscopes and all kinds of strange things to build a potentiostat that was very reliable. So in 1964, approximately, the University of Wisconsin gets their mainframe and starts, the professors start using it. And there's two people in the chemistry department, a professor by the name of Shane and his graduate student, Nicholson, who use this computer to work out the theory of cyclical tammetry. Now, for a wide variety of mechanistic cases, and they publish a paper in analytical chemistry, theory of stationary electrode polarography, they're no longer enamored, you'll notice with the fact that there might be an oscilloscope involved. They understand what the problem is and it's the calculation that's the problem. They publish this paper and cyclical tammetry, based on that paper and the availability that have good potentiostats, which really didn't happen in terms of a widely available to about 1970. First PARs, yeah. Well, the 173 is later, but the first PAR was the 170, I believe it was. And that came out right around 1970. PAR stands for Princeton Applied Research, company that was right down the street from us until about, I don't know, about 10 years ago now, when they decided that the property value in Princeton could not justify building new potentiostats, so they moved, I believe it was to Virginia or Kentucky, I can't remember, and not at that point. And then they were bought out by Perkin and Elmer more recently, just in the last five years or so. No, yeah, they were still in Princeton, though, when EG and G owned them, yeah. They had gone down EG. They, actually, the 273, which everybody uses, was EG and G. So not a lot of people complain about it, but everybody used it, so EG and G was okay still. It was really wonderful, by the way, when they were right down the street from us, because whenever our potentiostat broke, we'd just walk into their shop and say, fix it. They're no longer a company other than Perkin-Elmer, and Perkin-Elmer isn't known for their pretentious stats. We'll leave it at that. Okay, there are a variety of other companies now, though, that make these systems. Okay, so what did Nicholson and Shane do? Now, let me point something out before I tell you what Nicholson and Shane did exactly. Has nothing to do with chemistry, but this seems to be an important part aspect of electrochemistry. It's just a sociological thing that I've observed. So Shane, again, was the professor there, and he went off to become, eventually, the president of the University of Wisconsin, and Nicholson, the graduate student, went off to head the NSF. So you see, that worked out well. And now think about the gentleman who wrote your textbook, Bard and Faulkner. And Professor Bard, you probably are aware, was the editor. You might have remembered this of Jacks. It's one of his pastimes, and Professor Faulkner, of course, has become the president of the University of Texas at Austin, is now retiring from that position. And so there does seem to be this trend, and it goes all the way back to the very beginning there, that if you're gonna do state-of-the-art electrochemistry, you're gonna become a bureaucrat. Okay. Now, but before that happened, what did they do? Okay, if you read through this paper, this is taken right out of that analytical chemistry paper, you're not actually supposed to have to read this table. Although everybody should copy it down real quickly. No. We will put this PowerPoint on the web, and probably the most useful thing to do is, I gave you the reference to the paper on the prior page there. You can pull the paper down yourself, but this is a table out of the paper. Now, what they've done is they've come up with mechanisms, different theoretical mechanisms, and they give a number, Roman numeral here, mechanism three through eight. Mechanism one is the reversible mechanism. Mechanism two is the totally irreversible mechanism. And then you can see, for example, mechanism three here, this is a chemical step, reversible chemical step, Z going to the oxidized, reversibly followed by a reversible charge transfer. So, mechanism three is a CE mechanism, C reversible, E reversible mechanism. Mechanism four is again a reversible chemical step followed by an irreversible charge transfer. We keep on going down here. Here we have the EC type mechanism. Right here, reversible and reversible. Here's another EC reversible, irreversible. Another EC here, et cetera. And we go through eight. And then there are a couple more, another there. These last two are catalytic mechanisms. As you'll notice, we take oxidized here. We reduce it to reduced. That's a C step. We then have an E step here, the reduced interacting with some of the species that regenerates the oxidized species. So we do not consume oxidized in this process. It's catalytic. So this is today probably what we would call mediated charge transfer. That is, O species gets reduced at the electrode. It goes off and reduces the Z species in the solution, which for some reason doesn't get reduced at the electrode. Gets converted back to O and away we go again. So catalytic cyclic process. A couple more mechanisms have been added in the intervening years. Probably the most famous is the ECE mechanism. But basically it's all right here. In order to, if you pull down this paper and you want to read it, life becomes much simpler if you go to this table and memorize what these Roman numerals mean. You need to know that one is the reversible case. You just need to know that. And that eight is the catalytic case and that the reversible EC mechanism is four or five, rather, and whatnot. Once you've done that, the paper becomes decipherable. So what do they do? First they write down all the mechanisms that they believe are important. The next thing they do is they write down the appropriate fixed second law format. So you'll notice, for example, we not only have our standard diffusion term there, but since we have a chemical component, we have some kinetic terms in there for this particular one. They do it for the oxidized. They do it for the reduced. And if there's another species, a Z species, they write it out for that. And you go through that. And you should see here differential equations if you squint so that you can see them. They look pretty similar to what we've already written on the board. There's nothing new here. They write out initial conditions, which are the same initial conditions that we consider for chrono-amperometry. There's nothing new there. And then boundary conditions. And exactly the same kinds of boundary conditions we've talked about before at the beginning of this lecture. And then based on that, for the different mechanisms, they come up with differential equations and integrate them. So this is what you have to do. They've thrown out a lot of the constants and whatnot. This is the heart of what you have to integrate if you want to solve mechanism three right there. And they have a function in here. Chi, you'll notice, of T, which is this time-dependent thing. And you can do some things to simplify these integrals. But in the end, you end up with a set of integrals out here that you can't solve analytically. And so they come up with a series solution down here. And they throw that into their IBM mainframe and away you go. And they come up with solutions. Now, the solutions aren't presented like equations like the Cotrell equation and whatnot. They're presented as tables in this manuscript. So here's case one, the reversible case. So this is what you get. There's the function right there. They happen to multiply by a square root of pi to make life simpler when you go to plug it in to get the current out. Because remember that's square root of pi, square root of diffusion coefficients is always in there. But they have picked a definition of E1 half, which is the potential halfway in between two cyclic voltumetric ways of oxidation and reduction as a point to start things off. And so they reference everything to that. So this is the electric potential referenced against that. And it's the number of electrons in the charge transfer step. And for these different potentials, you come up with a function here that isn't the current. But within a few constants is the current. And so you could plot that out. And you'd have a cyclic voltamogram. And they did this for semi-infinite linear diffusion. And they point out down here. For example, if you have a spherical electrode, which would be very important then because most people would be doing this at a mercury drop electrode, that you have a correction to make. And it's all down here. And we've talked about that before. OK, so there's the solution right there. Yeah, by the way, Bruce was kind enough to go and do some research on coutrelle for us. The other day, when I said I had no idea, and there was two shocking things about that. I don't know if you looked at that. But number one is coutrelle was late 1800s. And number two is coutrelle is the same coutrelle that you get coutrelle foundation grants for, which are. He was a very successful industrial chemist. And the coutrelle equation is just hanging in there for some reason. I don't know. I didn't know anything to do with electric chemistry. But yeah, I mean, he was very good at it. And so I guess the equations are the same. But a little further research. It wasn't there. OK, I didn't read that carefully, so it was a grad. OK, that makes sense. I just see the neat pictures showing how the electrostatic precipitator actually worked. The smoke's coming out. You turn the thing on and no smoke. That was very good. So that's the same thing as you're doing in chronograph. Industrial chemistry, see? OK, now, so what did I just say on that last transparency, bringing us back to cyclic voltammetry? What I said was, if you take that table of chi values, which is what's listed in that table, you can generate a cyclic voltammogram by simply taking that chi value, multiplying it by the normal things that we multiply by here, diffusion coefficient, and this signal, which I defined a few before. But that's just the NF over RT times the scan rate. So if we were to go and plot that out, though chi values come out of that, that is the solution to that series. The chi value here is, you take those various series that I just blew through because we really don't care exactly what they are. But those chi values, those series are potential, dependent, or time dependent, depending how you want to think about it. And those are the solutions. So from that, we can convert the chi value into a current, I have t into a current, and we have a potential. And so we have an initial potential. And again, we're going to pick that so that there's no chemistry happening at that point. We're going to scan, say, let's make this an oxidation. We'll scan forward. At some point, we'll start to oxidize solution species. We'll generate a peak according to that thing. We'll scan back and generate something like this. So this is our lambda point that we're talking about, in that where we turn the thing around. Not the world's best cyclical tomogram, but if it was actually following those chi values, it wouldn't look like this because that's not very reversible looking, but something like this. Now, a couple of points to make about this before we go any further. The first is, this is normally how we plot this. We say this is a plot of current versus potential. And it is very important to remember that that potential, let's say in the forward going direction, is equal to this. And so I could call this a potential axis like I always do, but I could equally well call this a time axis. That this axis has both components on it. Why is that important to bear in mind? That is important to bear in mind because, first of all, there are sometimes things that happen in an electrochemical cell that don't have anything to do with you scanning the potential. That is, time passes and other kinds of chemistry happen. So it is quite possible that the change you're seeing in this cycle of ultamogram is just because the time is changing, not because the potential is changing. And that's always worth bearing in mind. The second point to make is I said we need to take a slice of that three-dimensional plane and look at several of those, a three-dimensional shape and look at several of the planes that pass through that. And the way we can do that is simply by changing this sigma term right here, which it means really changing the scan rate there omega. And you'll notice that it's really the scan rate to the 1 half power that we're dealing with here. And so to make a significant change in the current here, you have to make a significant change in the scan rate because you're going to take the square root of it. That is, it is true for any two scan rates, you are picking, selecting two different planes to transact this thing. But if the scan rates are too small, too close together, even though the planes mathematically aren't parallel, it won't matter. They're functionally parallel. So you want a big range of these scan rates. And what this equation is telling us is as we increase the scan rate without worrying about the specific of it right now, we expect the current to go up as the square root again of the scan rate. So if we picked a second scan rate, with a higher omega, we expect that to get bigger like that. Once you have an equation like this, of course you can do things like take the derivative of current with respect to time or potential. And the way this is set up, you do it with respect to time. And then find out where the maxima is by saying that to 0, right? And if you do that and put it back in, then you find out that you can determine the peak current for the first peak. So in other words, we have two peaks in this thing, obviously. But the way I drew this, this is the first peak I came to by this equation right here. And you see again, now explicitly, it's easy to see that this peak current is a square root of the scan rate dependent. Likewise, I pointed out to you that this peak to peak separation is an important parameter. And we could get that out of this equation also, again, by plugging in different values for our chi over here and realizing that all the less presented is time. It's also potential by this simple linear transformation. And if we did that, then the way this is just right out of Nicholson and Shane's paper, again, they reference it to the halfway potential. And that halfway potential, again, being exactly in between the two peaks. So if we go halfway in between these two peaks, right here approximately, we have the halfway potential. They reference it to that. And so they're measuring, in this case, from the peak. And it doesn't matter which peak, the upward going one or the downward going one, to the halfway potential. And they say that they show that that is equal to the difference between the peak potential and a term that should look very much like the volumetric term that we saw before. And again, assuming that the diffusion coefficient for the oxidized species and the reduced species are about the same, this log term will disappear on you more or less. And we find that the halfway potential then is just the redox potential in that case. So we can read the redox potential off from the halfway potential in a reversible system. So the diffusion coefficients aren't changing on us. And we also find, and you'll notice that's approximately right there, squiggly line, that from the halfway potential to the peak is 28.5 millivolts divided by the number of electrons that transfers, or about 57, right? 57 millivolts peak to peak separation approximately. Big approximately. Again, for the reversible case. Now before we go any further and analyze that, the one thing that these tables of numbers, like I was just showing you, or these equations don't do is give you any physical feeling for what in the world is happening here. So what's actually happening? You draw another cyclical tamagram over here, see if I can do a better job. We've got a lot of notation on the one over there. It's supposed to be symmetric. It's getting there. So if we go halfway in between, if this is reversible, then we have E1 half, which is approximately equal to the standard redox potential for the system. So I start off at this potential over here, let's say, right there, where I started to draw. That's my E initial. And I've picked that potential because I've gone there and I've observed that there's no current flowing at that point other than a non-fair day of current, which falls off very quickly. So I know nothing's getting oxidized or reduced at that point. And I start scanning in this direction. And as I approach the redox potential for the system, I start to observe a current. You'll notice the current on sets before I hit the redox potential. OK, why is that? Doesn't the redox potential tell me where the delta G passes through 0? So why do I get current prior to that? Quite a bit of current. Anybody want to guess? Yeah, right. The redox potential, remember, you have a 50-50 concentration of oxidized and reduced. And in your standard cyclic voltimetric, experiment, we just had the oxidized molecule around. And so we have that Nernstien concentration term. That shifts things, the concentration over potential. So we start to get a current here. And as we increase the potential, the current increases. And this is, of course, just the concept that we have a rate constant for this process, if you will, and that it's over potential dependent. So as we kick out the potential, things go faster because we have a bigger rate constant. Eventually, we get to a point up here where the rate that we are oxidizing the molecule at, and, oops, eraser right here in front of my nose. Set this up. I'm doing a very different reaction than what we've been doing all term, so I think I should write it down. There we go. I'm oxidizing it instead of reducing it because I can draw more easily that way. Yes, I said that all. So I'm doing an oxidation here. So I should say this is all reduced. And I said that thank you backwards. So I'm starting with, totally, I have this stuff around and not this. See, in this case, the bulk concentration of that is 0. CR equals the bulk concentration. OK. So which way is tough on that? I'm using the IUPAC convention here. OK, we've got that all straightened out. I shouldn't just throw these things on the board. This is what IUPAC says you're supposed to do. And since I'm a young dashing guy that follows all the IUPAC rules, there it is. And that is simply why I picked a reduced species just so I could go up first under IUPAC rules. OK, so I'm going to oxidize it. At this point, the rate of oxidation exceeds the rate of diffusion. And so I start to consume all the material that's at the electrode surface. So there's fewer molecules to oxidize. And so the current falls down. And eventually I get to a point out here where I have a diffusion-limited current. This is the current given by the Cattrell equation for a big potential step. That is, every molecule as soon as it hits the surface is oxidized. And so what I'm simply looking at here is the rate that molecules can be brought up to the electrode surface. And somewhere out here, once I've hit that, ideally, I say, OK, I've got that. Turn around and scan back the other way. And as I do that, now I have a lot of oxidized species near the electrode surface and very little reduced. So at lambda, my oxidized concentration, whatever it is, exists. And my reduced concentration is approximately zero times some lambda. And I start scanning back. And so now I have a lot of oxidized molecules. And so again, I start to get a reduction current now. And it happens before E1-1-2 or E-redox because the Nernst equation is now playing in the other direction for me. And I've flipped the fraction around in the concentration over potential term. And again, eventually I get to a point where I consume those molecules. I run out of those. And the current starts to fall off because I run out of those as the concentration drops off because my rate constant now for the reduction has gotten so large. And I get back to a point where there are no longer any molecules to reduce at the surface. And I can start the process over again. So physically, we're looking at that interplay between the rate constant and the diffusion process. You'll get the concentration at the surface. There'll be a slight change right in the foot here. But the concentration at the surface will be dominated by what you're actually doing there because you make more very rapidly. But there are subtle changes that do take place if you change your initial conditions. This brings up actually a very important point. Most people, including myself, when they run a stick of ultamogram, go and they run a few of them, maybe 10 or so. And things settle down nicely and say, oh, that's a nice stick of ultamogram. That's the one I'll show to my research advisor or publisher, things like that. The boundary conditions that led to this table over here were that you start off with something like this. Now on the second stick of ultamogram, you're not actually at this condition anymore. You have various species left over for the first one. There's some diffusion gradients in there and whatnot. And by the third, it's even a little different and whatnot. And so really, if you were rigorously applying Nicholson and Shane's equations to your stick of ultamogram, it should be the first stick of ultamogram. And you want to stop and mix up your electrolyte somehow in between each CV in order to make sure everything was at the bulk concentration and whatnot. That is a lot of people would, for example, including myself, do multiple stick of ultamograms and apply this equation for the peak current. Not exactly rigorously correct. It turns out the effect is small enough that you probably won't detect it when you make this mistake. But it's something to be aware of. Nicholson and Shane actually in their paper point out that if you do multiple stick of ultamograms, it takes about 50 of them. And after that, you have established diffusion gradients that you are not going to wash out unless you stop and mix up the electrolyte. So probably if you do two or three, it's not going to matter, but it is an effect. You don't have the conditions that these equations were calculated for when you do multiple stick of ultamograms. But I don't know anybody who doesn't do that. That's something to think about. Next point to make about this, there are an awful lot of people. And it's in the literature. And it's even in text that are primary sources. How to do stick of ultamograms and whatnot. That will tell you if you have a reversible system and you want to prove that it's reversible and figure out what the redox potential is, all you need to do is take a single cyclic voltamogram and you'll get the e1-1.5, that'll give you e redox, and that if these two peaks are 60 millivolts apart, then divided by n, the number of electrons, then you know it's reversible. So one cv will do it. Except for the part about e1-1.5 equaling e redox, everything else is false in that statement. Yes? Yeah, what? Yeah, well, the interplay between diffusion and the rate constant, right? Right, if the rate constant is extremely large, that's the case that I'm drawing here in the table we're looking at here. Yeah, you have totally consumed your, this is the process of totally consuming the material by the electrode. By the time you get out here, you are totally diffusion controlled. But there is a non-Feride component that gives rise to a little bit of baseline, but I'm not even drawing that here. I'm assuming that's going to be small, yeah? No, it goes to the diffusion, limited current. Just like the control equation. At a very, very positive potential, you have molecules coming in. When you say eventually, yes, the control equation tells us that at infinite time, it does go to zero, but that's a long time. So yes, mathematically speaking, you're correct. But in fact, the control equation also tells us that for reasonable times, there will be this t to the 1 half dependence of the current on time, and it'll be out there as a small current for a long time. By the way, one of the things that we will have to deal with and come back to is if we want to measure p currents, we have to know how to determine the baseline. That's a big issue here, so we'll come back to that. We're not quite ready for that. I want to stay for a moment with this 60 millivolts thing since so many people think that that's what's supposed to happen. First of all, even if you look at the original paper and just read the first few pages of it, it shouldn't be 60 millivolts. It should be 57 millivolts, but Nicholson and Shane actually don't say it should be 57 millivolts. That's if you just read and stop, selectively read the paper. This is actually right here. There's an interesting way of looking at a cyclic volt amogram. This is their cyclic volt amogram out of the original paper. And we don't normally, this is important, we don't normally look at it this way. Maybe this will answer the question we were just talking about. But they have used down here, they're just using their functions. They haven't multiplied by NFA business and whatnot because it's not a real molecule. But they're using a time axis down here, not a potential axis. And of course, on a time axis, there is no time reversal. There can be potential reversal, but time keeps going. So what they have done is they've unfolded this. So that's what you're looking at there. And they're simply showing you what happens under several conditions. One is if you let the time go on and you'll notice you do get the Cattrell equation out here. And that will eventually damp out, but that's number one. And number two is that depending where you pick your lambda potential, you will impact on this return wave. I'm going to say a lot more about baseline, but you'll notice for this peak right up here, all these curves are the same for these three curves, four curves, one doesn't have a return. That quite obviously you could get the peak current by just reading it off of this. It's 0.4 or whatever is on this thing. But for the return current, it's this difference here, which depends on a lot of things, like what your lambda potential is and a million other things. This is what Nicholson and Shea say. They say table two. They run their simulation, their cyclical tammetry, for different lambda potentials. And they determine just like I showed you on that last slide that the peak-to-peak separation, this is half the peak-to-peak separation, varies depending on where you switch the electro potential. Now it turns out, you'll notice, as you get further and further away from the redox potential, if you get way out there, it's not very much anymore. But it does vary. And it varies pretty respectably as you do that. So they never said 60 millivolts anywhere. This 57 was just one of their intermediate kind of numbers that they've used here. But if you can get to the point where you can vary it well beyond the redox potential, you'll notice that you're approaching 60 millivolts. And so people read that paper and said, oh, it's 60 millivolts, some people, just based on those numbers right down there. Now by the way, you'll notice these numbers don't exactly follow a simple trend. And that's because there's a Monte Carlo simulation in here to get to those solutions, those integrals. And so to the extent that the approximation of the integral is good, there's a little bit error in here. And so those numbers are bouncing around somewhere around 60. So people pick 60, if you take all those numbers and average them, that's what you get, 57. But that's about 60. So some people said, OK, look at a table I said 60 millivolts. OK, sort of, depending exactly what you're doing. It could be 69 or 70. And it could be closer to 60 or 57. And then other people came along and they saw written out 60 millivolts, not from this paper, but from the people that had read this paper and then written about it. And they said, 60 millivolts. Where does 60 millivolts come from? The only number like 60 millivolts that we know in electrochemistry is in the Nernst equation. It's 59 millivolts divided by n. So they concluded that the people that wrote 60 millivolts really were just meant 59, but they were rounding to 60 because what's a millivolt between friends? And so you get this number 60, which is supposed to be an approximation of 59 from the Nernst equation. Well, it's got nothing to do with the Nernst equation. It comes from this table. And so you shouldn't walk in anybody's office ever and say, this is not a reversible system because the peak to peak separation on the one cyclic voltamogram I did is not 60 millivolts. First of all, it shouldn't be 60 millivolts if it's a reversible system. It will depend on exactly where you switch your potential, number two. And number three is, even in reversible systems, we tend to have, and even with good reference electrodes in doing things more or less correctly, we tend to have some residual uncompensated resistance. In the way uncompensated resistance plays out in a cyclic voltamogram, the IR drop, is it increases your peak to peak separation. So about the best I've ever seen is 70 millivolts peak to peak. I will tell you in my personal experience. And often for reversible system, you'll see 100, 120 millivolts peak to peak. It's still reversible. It's just that you have a little more resistance. You picked an unfortunate land to point. Lots of things happen that add up. Yes? So then Nicholson had that later paper where he constructs the working curves for the kinetics versus peak splitting. Right. Well, actually, it's in here also. But that's in the same thing. There's more, but yeah. And so do you think that has much validity? The kinetics versus to determine the mechanism versus peak splitting is because all of those diagnostics, these things that we look at as a function of scan rate dependence, which are in this paper, are diagnostics based on scan rate, not on absolutes. So in other words, independent of the exact numbers, if the plot of whatever the diagnostic is versus scan rate has the functional form that they say it should, then you can be pretty certain that it's the mechanism they say it should be. And they're very clear on that in the paper. But they never said it, but it's out there. It's out there in writing over and over again. And that brings up this. This is what you're talking about. This is later in the paper. There are three diagnostic plots. Nicholson and Shane, they work out all this math. They present their eight tables of numbers. They say, now, what are we going to do to make sense out of this? And this comes back to the second point I made at the beginning of the hour. And that is the beauty of cyclical tammetry is that you have pattern recognition capability. And as they noticed, when they plotted out these various mechanisms that the cyclical tammograms to the eye look different, and in particular, they look different as a function of scan rate. And so you can look at the scan rate dependence and come up with a mechanism, as opposed to that three-dimensional shape where good luck. And they pull out three different diagnostics in the paper. And they're shown by these three graphs here. So the first one is something called their current function. And although I've used omega everywhere to symbolize scan rate, they use a capital V. So this is versus scan rate, versus scan rate, versus scan rate. So they say current function versus scan rate. What is the current function? Nicholson and Shane's current function is simply a peak current divided by scan rate to the 1 half power. Why do they do that? The scan rate to the 1 half power, you'll notice, is popping up in all these equations that we're talking about. That dependence, if you go back and look at it, falls right out of fixed second law. That is the diffusional dependence on the molecules. They're interested in kinetics. They're not interested in diffusion. So they would like to divide out, if you will, the impact of diffusion. Get rid of that as a player in this game. And so they suggest we use the current function, which is the peak current, which has both diffusional and kinetic components in it, and divide that by the square root of the scan rate. And we will have then just a kinetic phenomena. So if we look at that, for example, if you see right here, there's a little Roman numeral 1, which you all recognize now as the reversible case. And you see it is independent of scan rate, which is exactly what you expect if you have a system that has infinitely fast charge transfer kinetics. So there is no charge transfer kinetics dependence. And you just divide it out. The dependence on diffusion, you expect no dependence. And so you have a straight line. You will notice that mechanism 2 is also a parallel line. And as we're just saying, the actual magnitude here can be problematic, whether it's potential or current actually, it's the shape. Is it a line? Is it a curve? That's the issue here. And so you cannot tell the difference by the current function between mechanism 1 and mechanism 2. Well, what is mechanism 2? Mechanism 2, since you've all memorized them now, is the purely irreversible case. This one is usually pretty easy to see in that in the purely irreversible case, there is no return wave. You can go in one direction at a reasonable rate constant. So you see that. But you can't go the other direction at a reasonable rate constant. You don't see it at all. So even though the current function doesn't distinguish, here's your pattern recognition. No return wave, linear scan rate dependence. It's irreversible. And then you can go to mechanism 3 right here. Remember, that's a CE mechanism. And 4 was also a CE mechanism. And then we have 7 and 8 up here, which were the catalytic, electric catalytic mechanisms. And in between here, with these things that jump from one curve to the other, we have your EC mechanism. So you have very different curve shapes here. It's easy to figure out things going up, going down, and staying horizontal. Use your the mechanism. Now, Nicholson and Shane say, it is not fair to say, oh, my favorite diagnostic is the current function. And therefore, I'll just make that plot and ignore the other two plots. They say, if you want to know the mechanism, then you've got to do all three. So because there is some ambiguity here. For example, it might be hard to distinguish these two mechanisms from each other. They're both up and going curves, things like that. The second mechanism that they suggest you look at is down here, which is the ratio of the anodic current to the cathodic current, peak current, this would be. So measure that anodic peak current, measure the cathodic peak current, ratio them, look at that versus scan rate. A lot of the mechanisms here you'll notice fall right on that horizontal line where the anodic and cathodic peak current is the same. Reversible, that's obviously going to be the case. For totally irreversible, it's not worth making the plot because you can't make it because you only have one current. But there are some mechanisms here where you get some very distinctive curves, the EC type mechanisms for that ratio of peak currents. And then finally, the one we started to talk about a little bit, perhaps the most complex one, is this plot of peak to peak separation versus scan rate. And actually, the way they do that is it's the difference in peak to peak separation, actually divided by 2, but that doesn't matter, divided by the log of the scan rate. And again, they are pulling out the diffusional component. Again, here's why they're doing that. And these curves are really neat. These are really wild curves. And it would be hard to mistake one of those curves from another one, you'll notice. And a lot of people say, if you're only going to do one, even though you shouldn't do that, this is the one you should do. Yeah? Something wrong. Yes, they're taking it far away from, yeah, they're going way out here. So what could be wrong? Yeah, if you're going to do one, you should do all three. What could be fatal about just doing that one? There are two things. One is there is a lambda dependence. And it's different depending on the different mechanisms. That plot I just showed you was for the reversible mechanism. That's number one. And number two is none of their plots take into account any sort of IR drop in the system. And that has the biggest impact on peak to peak separation. So you could be fooled to some extent by having some IR drop in there that's uncompensated. Probably an uncompensated IR drop would never give you a curve that looks like that curve. You'd be pretty certain about that. But some of these other curves that are slowly changing with scan rate. As the scan rate goes up, the current goes up. As the current goes up, i times r goes up. And so there's a bigger peak to peak separation just from that. This is the heart of the matter right here, those three plots, those diagnostic plots. Now, in order to get those plots, not only do you have to be able to read the peak to peak separation, but for a lot of these plots, that is these two, you need to be able to get good values for the peak currents. So you have to have good baselines. So we need to talk about baseline. And I think what I will do is, ah, before I talk about baseline, I forgot about this, very, very important. So there's an aspect of Nicholson and Shane's paper that is sort of legalistic. Recently, I bought a ball of string. And it has a warning on it. It does. This is not a joke. This ball of twine has a warning on it that if I use the twine to attach something to the roof of my car and I'm driving down the freeway, then if things don't work out right, then I might get some air underneath, then the thing may take off like a wing and smash through somebody's window or something like that. Now, do I really need this on a ball of string? I mean, there are some string manufacturers somewhere who think either they did or they're concerned about someone suing them because they tied it to their car, and no one told them that. If they don't tie it down really tight, it might fly off when you're going 60 miles an hour. And so I got interested in this. And I looked at other manufacturers of balls of string. They all have the same warning on there now. They haven't covered that one yet, but I'm sure someday. That's like McDonald's didn't warn you about their coffee until they got sued because someone managed to spill their coffee on them and burn themselves. And now you have a warning on every cup of coffee that it's hot in. And if you spill it on yourself, you might burn yourself. So there is an aspect of Nicholson and Shane's paper sort of like this, even though it's only 1964 and we weren't so thoughtful about our legal aspects then. And they make it very clear. Number one, you should use all three diagnostics. That is, if you don't use all three diagnostics and you come up with the wrong mechanism, don't blame them. And more importantly, because this is one that people regularly violate, number two here, Nicholson and Shane specifically say, you must go over three orders of magnitude in scan rate in order to make any one of their diagnostics valid. Now you will notice, if I go back to those plots, that there's a log scale down here on all these plots. That is easy to do on a computer. Then it's very hard to do in the laboratory. But they're saying the problem is that although it's easy to see the difference between these lines, you don't see a difference between these curves if the scan rate range isn't big enough. You could confuse, for example, if you took this one out here, if you just had this part out here, you couldn't tell between that and a flat line. So you don't know if it's mechanism 8 or mechanism 2. So they're saying you need to go over. They've looked at these plots again. If you don't go over three orders of magnitude, forget it. Now, it is exceptionally rare to see somebody go over three orders of magnitude in a publication to come up with their Nicholson and Shane mechanism. What does that leave us? OK, first comment is there really is no excuse for a single cyclic multamogram as you're diagnostic. That's just not good enough evidence for anything. When my students come into my office with a single cyclic multamogram, which only happens once, they just see me laugh. However, they're not going to come in with three orders of magnitude because if you think about what happens there, most machines easily go up to about 500 millivolts a second or maybe one volt a second. And after that, you start to have to work a little harder because you need fast current to voltage converters and good data collection and things like that, or an oscilloscope, heaven forbid. And so people usually don't want to go fast about one volt per second. So to get three orders of magnitude, that means they have to go down to 10 millivolts per second for the other scan. And if you think about it, a typical scan window might be, well, at least let's say it's a half a volt in one direction. So you're going over a whole volt since it's cyclic and it's 10 millivolts a second. And who has that kind of time? On top of which, it is likely over that period of time that your lab mate will slam a beaker down on the table and mess up everything because no longer diffusion control. So typically, you will see 20 or 50 millivolts per second up to maybe about 500 millivolts per second. So about two orders of magnitude. So if you want to take Nicholson and chain to court over that, if you get the wrong mechanism, you'll lose because they have their warning in the paper. But it's probably OK. You'll probably get the right answer with two orders of magnitude. But one order of magnitude saying I did the 50 and I did the 100 and the 200, I don't think so. You've got to get at least two orders of magnitude in there. This is a very, very important point. The next point is the baseline issue. And what I think we should do, this is the other reason, the other way you can get thrown out of my office. There's only two ways. You show up with a single single tammogram and the other is don't do a separate baseline scan. So we'll talk about that in a few minutes. But I think what we should do now is take a break and we will reconvene at 2.45 for our next lecture.