 OK, remind you that you have a problem set due today. And at the end of the hour, Thomas decided he'd give you an exam. And nothing to do with it. Remind you that that is due next Tuesday. OK, before we jump into the lecture, I just I want to do a little bit of email with you. Go through my email with your guys present. So the first comment to make is that we were talking about cyclical tamagrams. And I was showing you that there are surface ways, such as platinum, and pointed out to you that you can get these stripping ways, which have sort of a distinctive shape to them when you're looking at something chemisorbed or physisorbed, in some cases, on an electrode surface. And right after I told you that I had this wonderful first-year graduate student, Emily Barton, she sent me some data. Like she knew that had nothing to do with my lecture. But it does, it turns out. And that she has been looking at the reduction of CO2 at a palladium electrode to methanol. To make methanol, she's good. Something like 50% conversion efficiency. That low overpotential. It's going to be wonderful. Anyhow, we're trying to figure out how this works so we can make it even better. And she decided that the thing to do was a stripping experiment. So she's in CO2 and aqueous, and there's some pyridine, which is our catalyst around it, and things like that. Here's just a little cyclic voltamogram to show what the system's like. But what she does is she holds the system at a very negative potential. It's minus 0.9 volts for five minutes and builds up on the surface whatever builds up on the surface. And then she'd like to know something about that. So she ramps the potential at 100 millivolts per second positive and just looks at what's called the stripping technique. So we're only going in one direction. So it's not cyclic voltametry. That is cyclic up there. But this is linear voltametry. And this is called stripping linear voltametry. Because all you want to do is get the stuff to desorb from the surface. And you can see there's a species that desorbs at this potential of plus 0.8, 0.9 volts right there. And you'll notice these are clearly stripping peaks. This is all I really wanted to show you. You notice when you get a stripping peak, there's an asymmetry in the peak. Both sides aren't the same. One side comes down very steeply. And the other is kind of more typical of your cyclic voltametric peak. So you get all these different stripping peaks depending on the first time she does it, she gets a big one. Second time she does a smaller one and smaller one. In other words, she is removing stuff from the surface. So all we have to do now is identify what's coming off at 0.9 volts. And we know what's happening on the surface. That's her next job. So just to give you an example of that. Now the second piece of email is Bruce sent me an email or two, or 10, very useful. He has located this website where you can obtain the software for doing the digital simulation. It's free software. You can just download it. It won't run on a Mac, which is a downside, but it will run on a PC. And we have put that website on our course web page. So if you'd like to download that. Now that website is more than just here's the software. It's actually a wonderful website in that there's links to various papers that explain how the program's developed and the science and the electric chemistry behind it. So if you want to understand what you're doing, it's good. It was a bad idea. Well, it's there. Maybe you don't want to understand what you're doing, but it's there. You don't have to click on it if you don't want to know. It's a very interactive website. It has a lot of tutorial information on it. Very good. And Bruce has used that to look at this dependence of the peak current on the switching potential to play around a little bit. And to also try and ask the there's this link on the website that says the team. So you have to link there to see who wrote this wonderful software. And there's pictures that come out. And the first picture that comes up is a person that says makes the tea. And the next person that comes up is a picture that says makes the coffee. And then the third picture comes up and says writes the code. Now the question is, if you look at these pictures, the tea maker and the coffee maker look like very happy, well-adjusted people, but the code maker is in terrible shape. Anyhow, let's see if I can find, so we're done with Emily here. Let's see if I can find Bruce in here. Then we'll find out how often you do email me. Yeah, it's not too bad. So Bruce ran the digital simulation here, which I suspect means he didn't actually believe me, but had approved for himself. But he's looking at the position of a theoretical peak as a function of where you switch this thing. And you can see that you can get quite a shifting in your peak potential, depending how you do things. So in good case you didn't believe me, there's the proof. I'm right. And we'll get out of email now. Now back to what we were talking about yesterday. And that is, you'll recall that yesterday, I was trying to start to pull everything together by taking the chrono techniques and cyclical tammetry and apply it to a chemically modified electrode, along with some Marcus theory we're going to get to in the whole nine yards. Now, one thing I failed to mention yesterday, although if you ever had the opportunity to hear a lecture, which I guess none of you did by Taube, you would have discovered that he is a wonderful communicator and you could learn a tremendous amount about coordination chemistry by listening to him for even an hour. And he's also an excellent writer. So one thing that's very useful is when he received the Nobel Prize, you get to go to Sweden, all that, and give a lecture. A lecture, and that's all transcribed. That's all written. And so that is publicly available. So I put the PDF of that on the course site. That's worth looking at. And of course, Professor Marcus also got a Nobel Prize and also gave a lecture. And I put the PDF of that on the site. And the interesting thing there is that if you read the original theoretical papers, they're pretty dense. They're interesting. It's not like let's sit down for a half hour and read it. You have to struggle with it. But his Nobel address, which was to a broad audience, actually is an exceptionally good address. And if you want a very readable text on outer-sphere charge transfer, I suggest that article to you. Nice overview gives you some good insight. Now the next comment is that Bruce walked up to me after the lecture yesterday and said, I think in shock, I have never heard anybody give a talk on outer-sphere charge transfer without mentioning the Iron II-III couple. And my response to that is, well, I haven't finished my lecture yet, so just give me a chance. Because of course we're going to do that. It's also the first reaction in Marcus's lecture, by the way. So we will definitely get to that. Now, but to review, what I'm trying to do here is pull together the techniques, apply it to a system, but at the same time, expose you to some new techniques or expansions on the techniques. So we're starting off with this nickel fairy cyanide on nickel system that is the cartoon that I'm showing you here that you saw yesterday, if you were here. A nickel electrode and a layer of microcrystalline, really, nickel fairy cyanide in which these golden balls are nickels locked in the 2 plus oxidation state, the green. I'll say that backwards. The golden balls are irons in the 2-3 oxidation state, and the green are nickels locked in the nickel-2 oxidation state. Nickel's always green. And you have cyanide ligands hooking together carbon on the iron side and nitrogen on the nickel side. And it makes this cubic lattice. And because those cyanides are negatively charged, there's a net excess of negative charge in that lattice, even with the nickel ions and the iron ions. And so there's some more cations. And they sit in this body-centered cubic position in this lattice. And they're whatever supporting electrolyte cations you happen to have around to do that. Now, where you can change the potential of the electrode and in so doing oxidize the irons, but you don't touch the nickels. And of course, when you change the iron oxidation state between 2 and 3, you change the net charge on the lattice. So if we oxidize the irons from 2 to 3, then cations have to leave. And if we reduce the irons from 3 to 2, then cations from solution have to come back in. So we have on this side of the electrode an electronic or electron-based current, a real current. And on this side of the electrode, we have an iron current that we're playing with. And so when I push the button here, I think what happens is the iron goes away. That's good. OK. Now, one other comment to make. I did not tell you too much about the thickness of this layer. What I am showing you here actually is the unit cell for nickel-ferri cyanide. It can't be any thinner than that. That's about 10 angstroms. In fact, the thinnest layers we make are in the order of 100 angstroms and the thickest layers are in the order of 1,000 angstroms. Now, why is that? The reason to do this is that nickel is a fairly inexpensive material, fairly catalytic material, a lousy electrode material for oxidations. Very good for reductions. Why is it lousy for oxidations? Because it would rather make nickel ions than do anything else. So you just corrode your electrode away if you try to do oxidations in aqueous electrolyte. So the idea is we want to modify the surface so that we have a surface that no longer generates nickel ions. And now we have, quote unquote, turned nickel into platinum. We have its catalytic properties and we don't have its corrosion properties. So we want to shut down the nickel ion generation, but we still need to be able to push charge through this layer. And we can do that through the redox events that are occurring at these iron sites, for example. So we need a thick enough layer here that we stop the nickel ion generation. In one unit cell, 10 angstroms won't do that. But 100 angstroms, as long as they don't push the potential to positive, will do that. On the other hand, if I go up to 1,000 angstroms, I can really go to quite positive potentials without any corrosion on the surface. So 100 angstroms to 1,000 angstroms. That is 10 atomic unit cells up to 100 unit cells. That's what we're playing with here. And I showed you last hour that we had achieved our goal of stabilizing the nickel surface because here is 1,400 cyclic voltamograms. We're looking at the iron 2, iron 3 couple there on the surface. And it's stable, relatively speaking, compared to a nickel electrode. Exceptional stable. A little fall off here. 100 millivolts a second. So this is many hours of cycling. In fact, we know it's iron. I told you initially because we saw the redox potential was right for nickel ferricynide. And if you were here yesterday, there's a punch line to that. And you notice that we have a linear scan rate dependence on the peak current. It scales linearly with the scan rate. So we know it's attached to the surface. We also know it's nickel ferricynide because we took IR spectra. And we see cyanide stretches that are diagnostic for nickel ferricynide. And we took absorption spectra by diffuse reflectance off the electrode, which is also diagnostic for nickel ferricynide. So we have spectroscopic data, spectroelectrochemical data actually. And we have regular electrochemistry that says it's nickel ferricynide. This was just pointing out that you could figure out how much nickel ferricynide was there by integrating the area under the curve here. By any of a number of ultra-sophisticated mech techniques such as cut and weigh. We can change some of the ligands around the iron. Well, not some one. After that, it's sort of the lattice falls apart. And here we put a histidine, a big bulky group on, and show that when you do that, although the peak-to-peak separation still stays fairly small, and we can get a redox potential of that. And we know it's on the surface that the peak-width increases. You notice you still have a linear scan rate dependence, but a wider peak. And that peak-width is a diagnostic for nearest neighbor interactions. On the surface, when we have things attached to a surface, they have nearest neighbors. Broadening of the peak means repulsive interactions. Narrowing of the peak is attractive interaction. So you actually can get narrower peaks. In this, in the ideal case, you predict a 90 millivolt divided by n, the number of electrons peak-to-peak, not to peak-to-peak, to me, half full-width at half max. Got that right? And then broader here, you see we have a couple hundred millivolts as the repulsive interactions. Narrower, again, would be attractive interactions. And it was pointed out that since you have a lattice on the surface, a crystalline lattice, that you could very well believe that one molecule changing would affect the others. That is, there are nearest neighbor interactions. I pointed out to you that on this prior scheme, that actually is 90 millivolts right there, which means there's either no nearest neighbor interactions or that the attractive interactions and the repulsive interactions exactly cancel out. And me, being a naive electrochemist, figured 90 millivolts, Occam's razor, no interactions. And you, being sophisticated chemists, told me you're out of your mind. It's a lattice. Of course, the interactions cancel out, and you're right. This was the last topic we started to get into last hour. And that is, if we take one electrode with nickel-free cyanide on the surface and we put it in a variety of electrolytes with the same anion, it's nitrate in this case, but different cations using the alkali cations, we get a very different response. We spent a lot of time studying this, it has lots of implications. The first thing you notice about this response is as you go down the periodic table here, the peak half-wave potential is shifting. And it's just more positive going from lithium to cesium. And we talked about that a little bit as being related perhaps to the size of the cation that's involved, although there's some interesting discussions you can get into about that. The other thing you notice here is obviously the wave shape has changed. And so here in the potassium and sodium cases, you're seeing an ideal wave shape for a chemically modified electrode. In fact, I go so far to say that nobody has ever seen anything more ideal on a surface than that. That's about as ideal as you can get. Zero peak-to-peak separation, 90 millivolts as the width of the peaks, a little bit of baseline distortion down here. It's not completely flat, but it's a wonderful cyclical tanogram. Linear scan rate dependence up to at least 1 volt per second. This is a textbook example, but you go into lithium ions or cesium ions, and that's more typically what you see. There are things happening here, and they could be attributed just based on this data either to nearest neighbor interactions that are changing things or to sluggish kinetics. And as you'll see today, we can make an argument for both those things happening, actually. Before you do that, if you simply go and you plot the redox potential of the iron system as a function of the ionic radius of the various cation, you see this wonderful linear relationship, which to us suggested that something happening here was a solid state phenomena. However, there is another interpretation, and that is that if you plotted this instead of against ionic radius, hydrated radius, then you'll get a straight line going in the other direction. And so there is a question about what the role of water is. That is, you know when those cations are going in or out in solution, they have a certain number of waters in their first and second coronation spheres. But you also must realize that when they go into that lattice, they can't take all those waters with them. There's not enough room in there to take all the waters. Well, maybe cesium, cesium's supposed to have one water, so I could probably bring that if it wanted to. But lithium's supposed to have 12 waters that it likes to carry around with it. That's not going to fit in. Even sodium with six waters is not going to fit in. So what's the role that the water is playing? And one thing I suggest it is, one interpretation this data might be that, well, when the waters go into the lattice, they have to leave. And so there's a dehydration step, if you will. And there's a free energy associated with that. And that would get fed into this free energy. And so maybe this whole shift is just a hydration, dehydration energy term added onto the normal redox potential term that we're looking at. That's one interpretation. Another interpretation is the lattice, the ions go into the lattice. This is the solid state interpretation I'm showing you here. And the lattice has to expand or contract somewhat to accommodate these ions. And how does the lattice do that? Well, to do that, the metal to cyanide bond distances obviously have to change. So if they change in the crystal field around the ions would change. And one would expect to see maybe spectroscopic differences due to a change in 10DQ of the crystal field or electrochemical differences due to a shift in the energy of the orbitals that contain the electron that we're taking in and out, which would be the T2G orbital in this octahedral iron complex. I will tell you, we look for spectroscopic changes. Now this is a non-trivial experiment because remember, it's 1,000 angstroms of ill-defined material sitting on a nickel electrode. So it's a diffuse reflectance experiment. And we could not detect a spectroscopic shift. But we also concluded that the shift in 10DQ would be sufficiently small. Probably that it would be really hard to pick up. And it was poor spectroscopy that we could do. And so we weren't too upset by the fact that we couldn't see that. One could still, again, interpret its redox potential in terms of movement of that T2G level. So let's see if we can get a handle on what might be happening here. We have two options. I just lasted this up at you last hour at the very end. So what do we have here? One of the early experiments we did to try and get a handle on this was essentially a chronocoolometry experiment. Not only essentially, but there is a chronocoolometry experiment. So we're doing two things. We're stepping based on our cyclic volt amogram from 0 volts where nothing is happening. And all the irons are in the reduced state to about 0.7 volts where we've stepped all the way through the cyclic volts metric wave and everything is oxidized. And we're doing this in sodium ions so that we know exactly what the potentials are. And that's the well-behaved system, remember. And then we step back. So we do the oxidation down here. And we do the reduction up here. We start at 0.7. We step back to 0. So it's sort of a double-step experiment but done in two single steps as shown here. And the points are the actual data from those steps. Now we were a little bit concerned about this experiment because remember, we start off with an unstable nickel electrode, and we claim we're stabilizing it. But how do we know we're totally stabilizing it? Maybe 1% or 0.1% or 0.01% of the current is going to make nickel ions still. And that would mess up this curve, obviously. We'd have two processes happening simultaneously and we have a problem. So we took advantage of the fact that we could do the spectral electrochemistry by this diffuse reflectance off the surface. And we did it by reflectance also. And so you see exactly the same experiment, but instead of monitoring the charge, we're monitoring the change in the surface reflectivity at a wavelength that tells us whether the irons in the 2 or the 3-oxidation state. And again, you get the dots. And what you might be able to see is that the dots and over here and the dots over here are, I said the dots, but that's wrong. It's a solid line. I'm sorry. The solid lines over here and the solid lines over here are identical. So all of the charge it would appear is being used just to oxidize or reduce the irons on the surface. We don't have a secondary process going on. OK. So this shows you that within probably 5%, it doesn't show that you're using it. Actually, we figured it was within probably 1%. We could detect a change. We had done a fair amount of this, doing steps here and looking at spectral changes. So we thought we're good to 1%. So yeah, if it's happening, and it probably is happening at some low, low level, to be honest. But it's less than 1%. And of course, from the point of view of trying to analyze this data, these solid curves, a process that is only 1% of that is OK. It's within the air of our analysis. And in fact, actually, in the end, we ended up doing a whole analysis based on the reflectivity data. So whether or not it's sensitive or not is unclear, but that did not have a corrosion component in it. So what do we have there? Well, we have a surface that, by cyclic voltammetry, looks reversible. I just showed you the data. It looks ideal and reversible and non-diffusive, linear scan rate dependence. So what are we going to fit it to? Well, we're going to fit it to the integrated control equation. Why are we going to do that? Well, first of all, because everybody else fits their chemically modified surfaces to the integrated control equation, so that seems like a good place to start. But there seems to be a problem with this, because by cyclic voltammetry, I have been arguing there is not a diffusional component. That's why we don't have the scan rate to the 1-1-1-2. We don't have the peak-to-peak separation. All it is is the molecules are pasted on the surface, and they get oxidized. And it's the kinetics of that oxidation that controls everything. And yet, I'm telling you what I want to fit it to is a diffusion-limited equation. Could be diffusion of the cations, but I'm only using one cation here, and I have one molar I will tell you cations around, so they don't have to diffuse too far. And everybody sees this, even for systems that aren't starting off as anionic systems that you need a cation to diffuse into. Now, there is. You're absolutely right. There's got to be a component that is diffusing the cations. Assuming the cations, by the way, are diffusing. For example, when I do the oxidation step, I expel cations from the surface. They really don't have to diffuse. They're already there. And once they're out of solution, I can't monitor them anymore. They're diffusing out there, but I don't see that. The reduction cations have to move into the surface, and so that could be diffusion-limited. So if it was some sort of process that was cation limiting, I'd expect some difference between these two curves. And in fact, they're the mirror image of each other. They superimpose very nicely. So it can't be that. So what else could it be? If there's a dehydration step, then there's a rate constant that goes with that. And that would have a very different dependence than a diffusion. I'm saying that fixed second law should hold, as opposed to a first-order rate constant. And you know those integrate out differently. Yes, it's your multiple length. You basically have got it, yeah. So there is a virtual diffusion process going on here. Not a real one. That is, let's see if I can get back to my first little model picture there. Remember, this is a lot thicker than I'm showing you here. But what's the first thing that happens? Well, there's an iron down here near the surface, and it obviously accepts an electron. And that's not diffusion-limited or anything like that. That's just related to the kinetics of charge transferred to the interface. But now I have an iron over here. How does that iron get oxidized or reduced? That is, I guess in this particular case, this particular iron happens to be five angstroms away from the surface, but it's more than five angstroms actually away from this iron. Straight distance to the surface. So there could be some sort of a tunneling process. But tunneling doesn't really go very quickly over more than about 10 angstroms. And so how do the irons out here get oxidized or reduced? So what has to happen? There has to be a self-exchange process. That is, it's not electrons that are jumping out and oxidizing or reducing something way up here. This iron gets oxidized, and then there's a self-exchange between this iron and that iron over there within five angstroms. That's OK. And now that iron gets reduced, and that one's oxidized. Now another electron come out and re-reduce that. And then this one goes, and there's a self-exchange from here to here. And we go through, and so you can see if you were looking at this surface, and if you had eyes, which actually you do, that could see the difference between an iron 2 and an iron 3. They're different colors, so you could see that. But you have to see that now at the atomic level. If you could see the color of a single atom, you would see essentially a wave of irons that would appear like moving out from the surface as it oxidized. The irons wouldn't be moving, but the 2 to 3 would just move out in a wave. And it would look very much like a control type process would be the theory. So we have a virtual diffusion associated with charge transfer at a chemically modified surface that's thicker than a monolayer. And we're thicker than a monolayer here. So we expect it. I just argued to fit the diffusion equation. And there are lots and lots of examples in the literature of chemically modified surfaces that are on the order of a micron in thickness that do that. They look like the integrated Coutral equation. Yes. So the CV doesn't show the diffusion. Why? So the issue is an issue of time scales. The CV goes slow enough that you are always charged transfer limited. You never pump enough electrons out there that you have to wait for the self-exchange to occur. So there is no virtual diffusion process. But here we dump a ton of charge into the surface very quickly, surface layer, and we have to wait for the self-exchange to occur. That is, if you had a process that had an extremely fast self-exchange rate constant, then again, the control equation should not hold. But very ferrocyanide has an intermediate self-exchange rate constant, and so you expect it to hold. And for surfaces, as they get thicker, things tend to be further apart. So you expect the self-exchange to go slowly, and so this virtual diffusion works. So now having argued that this is the case, I go and I fit my data to the Coutral. So I'm making plots of charge versus to the 1 half or reflectance versus to the 1 half, and one's for the oxidation, one's for the reduction. And it doesn't work, you'll notice. That is, I can nicely fit the early time data, but when I do that, the late time data falls off the curve. And if I wanted to instead fit the late time data quite obviously, I couldn't get the early time data on that same curve. There was a lesson in this, by the way. I said that very carefully because actually, I think Bruce, this might have been before you were at Brookhaven. Very early in this game, I went out to Brookhaven and was telling all about this exciting chemistry, and I gave my talk. And I said, it holds at early time, but not at late time. And Carol Kreuz came up to me at the end of the talk and correctly and politely in private corrected me and said, if you have an equation that you're trying to use as a model, it either holds or it doesn't hold. It's not early time versus late time. So this doesn't hold. The model is wrong somehow. So I said, oh, I know what's wrong, actually. In fact, Royce-Marie had published a paper right around the time we were doing this saying, even though all these people have fit their data to the control equation for these chemically modified electrodes, it really shouldn't work because the control equation assumes semi-infinite linear diffusion. And this certainly is not an infinite diffusion system because there's a limited amount of material on the surface. And once it's gone, it's gone. You don't have this reservoir of stuff continually coming in. And remember, our diffusion gradient I've been talking about is based on that. So Royce said, what you really should do is use finite diffusion, not semi-infinite. And the argument was that the people were getting away with it because they had rather thick layers on the surface and they were only oxidizing or reducing part of the layer for this. And so the condition held. But if you were going to oxidize the whole thing, there was not a chance it was going to work. And so he comes along and he comes up with a new equation. That is, he does the normal thing, changes, takes a fixed second law, changes the boundary condition. So it's finite diffusion boundary conditions. Does the Laplace thing. I'm going to Laplace this. And he comes up with this equation, which I changed it to be in terms of current. We integrated it to get into charge. And then I've changed it to be in terms of reflectivity here. But this is the equation. And you'll notice we switch over from a t to the 1 half dependence to an exponential dependence when we do that. So pretty big difference, as one goes about doing that. Or d is our normal diffusion coefficient. The little d down here is the thickness of the layer. And the pi's are in there and everything else is sort of as you expect. And so I said, oh, we're going to fit it to that data and it's going to work perfectly. It's about as good a fit as the one I'm showing you over here. It does not fit. So now I'm very confused. The system by cyclotometry is reversible. And I know it's reversible because we can do those cyclotamograms I showed you for 1,400 times with very little change. So it is reversible. So even if you don't believe the diagnostics, it's reversible. And yet it doesn't fit all the standard equations. Further, if you go instead of plotting it this way, you go and you plot it this way. What you find out, this is actually, see this dash line right here, these dash lines? These are actually the fit to this equation. So the solid line, again, is the data. The dash line is the fit to the equation. And the second confusing thing here is that the discrepancy is in the same direction, whether you're doing an oxidation or reduction. You would think if you were off in one direction for an oxidation, you'd go off in the other direction for a reduction. Because what could change? Diffusion coefficients could change. So one would be big and one would be small. You can't imagine one would be big and one would be bigger. Concentrations maybe could change. But they change when they get bigger and they get smaller. They have to balance each other out. You can't have an overshoot in the same direction for both the oxidation and the reduction. So you go through all the things that are hidden in here, and you can't find any suspect, really. What you're left with is the diffusion coefficient. So you say qualitatively, what this looks like is the diffusion coefficient is changing. It's not a constant, but it's a time-dependent parameter. And it can change the same way. Now, since it's time-dependent, not potential-dependent, it could change in the same direction, whether you're doing an oxidation step or a reduction step. So that's why you could always overshoot, because your diffusion coefficient isn't what you think it is up here. It's smaller. You put it too big a number in. So you go and take that data. You take this equation. You just say, let the diffusion coefficient be time-dependent and just fit this curve with a time-dependent diffusion coefficient. You're not worried about the functional form, the significance of the functional form. But just give me something. And if you do that, it turns out the data fits very nicely. If you let the diffusion coefficient equal the diffusion coefficient, you would expect divided by this quantity 1 plus t quantity squared. And if you now take that and put that back into this finite diffusion business and come up with some kind of fancy equation like that, just substituting in, that is the points for all four of these curves. And you can see we have a very good fit. Now the question is, what does that mean? What does a time-dependent diffusion coefficient mean? The first thing it meant was that Andy Bacarsley had a walk out of the chemistry building at Princeton University over to the engineering school, because who in the world has heard of a time-dependent diffusion coefficient, except for an engineer? Chemists don't know about such things as a rule. So I did that. And this wonderful chemical engineer said, well, there's this wonderful textbook by Crank that will teach you everything you ever wanted to know about diffusion, which is unfortunately true. And it's sitting in the engineering department. If you ever need to know anything about diffusion, get Crank's book. It covers every situation, including the possibility that you might have a time-dependent diffusion coefficient. Now Crank tells you that, of course, when you have a time-dependent diffusion coefficient, there's a reason for it. It's not time. There's some physical parameter that's changing with time and gives you this apparent behavior. And Crank points out, for example, that concentration might be something. If you have a time-dependent concentration, then that could do it. But of course, we don't here, because we've already taken the concentration into account separately. We know about the concentration. And he points out, because it's an engineering text, that if there is a time-dependent thermal gradient going through your system, that that will do this. But of course, we don't have that either. And you go through the various possibilities that Crank suggests, and you throw every one of them out. It's not applicable to this system. So all you know right now is you have a time-dependent diffusion coefficient. Say, OK, what might cause that? So you go back and you think about the chemistry. And you go back to this idea that it's perhaps a naked ion going in and out that loses its water. Maybe not naked, that's too extreme. But loses its waters, or some of its waters, it goes in and out. And when that ion gets in the lattice, the lattice has to accommodate its size. When you start thinking about models like this, when I have an iron in the three-oxidation state, then I am going to be having a certain number of ions in there and a certain lattice size. When I have an iron in the two-oxidation state, I'm going to have a different number of ions in there in a different lattice size. So these circles are supposed to indicate that that lattice parameter is actually expanding and contracting as I carry out the redox process. So in addition to an electron that I have to pop into the system to go from iron two to iron three, since cations are going to leave when I do that, solvent's going to leave, and there might be a dehydration effect on this whole thing. On the other hand, when I oxidize the surface and an electron leaves, then the cation also has to leave. Solvent could come in to fill up empty space then, and there could be a hydration effect of the cation. Now, I will not presumably see the solvent coming in as a dynamically limiting event, because there's 55 molar water out there, and it will come in, and it's already there. And so I'm never going to pick that up. That's going to happen way too fast. I'm not going to pick up the hydration of the cation as a limiting event, because the cation shoots out the surface, it's out in water, and it hydrates rather at its own time scale, and I can't see that, because I'm not doing any redox chemistry. In theory, if I have to push-solve it out here, I might see that. Cations coming in, if they have to dehydrate before they can get in, I might see that process. That could take some time, and so I could have trouble shoving them in. And so I might pick up those sorts of things. Well, what this argues for, since the oxidation and reduction are very reversible looking, that is, they superimpose over each other, there aren't different processes happening here. So I argued over here the only limitation I could have is the movement of a cation or the movement of the charge. But over here, I could have the movement of the cation, the movement of the charge, or the dehydration of the cation. So I can now argue either the dehydration of the cation is extremely fast, and therefore I don't see it, or it's, for other reasons, unimportant. Therefore, that cannot be the reason that I'm seeing this shift with different cation in the redox potential. So it really must be the cation itself, not the dehydration, dehydration of the cation that's affecting the shift in redox potential. In addition, I could argue, why would I have this time dependence? Well, as this thing is oxidizing and reducing, that lattice is literally expanding and shrinking. So a given cation, depending when it leaves the surface, and this is where what you were saying comes in, samples a different environment. That is, a cation leaving early on has this big open lattice, say, and it has a certain effect of viscosity, if you will, for the cation moving through that lattice. And it sees a certain diffusion coefficient for the cation now. And that diffusion coefficient for the cation is coupled to the diffusion coefficient for the electron, because I can't do one without the other. On the other hand, a cation that's leaving later on, maybe after the lattice has shrunk somewhat, sees a different effect of viscosity for that lattice. It has more trouble getting out, a different diffusion coefficient. And so I expect my diffusion coefficient to reduce with time. Now, again, the particular functional form I have here doesn't mean a thing. It just fits the data. But that would account for everything, that the lattice has to accommodate the cation that's coming in and going out. And I have this expansion and contraction. Now, that's a very nice story, but it falls in the realm of a fairy tale. That is, it is totally consistent with the data that I have shown you here. And we came up with all of that from the data I'm showing you here. But there probably are lots of other models that are totally consistent with that data. You don't want to really go and stake your life on that whole picture based on a couple of chrono-ampograms. It doesn't really tell you much about structure. It is convincing, though, that the dependence is not due to hydration, dehydration. That, in fact, our guess that we should make the plot versus ionic radius of the lattice of the cation was correct. And that what we are seeing there for when we see a shift in that redox potential is moving around at the T2G levels of the electrons. So for those of you just that need a reminder, if I have iron in a ferrocyanide environment, then the d-orbital split into this 3-2 pattern. And if it's iron-2, we have a low spin configuration. It looks like that. And so all the chemistry I've been showing you has to do with removal of one of these electrons. And so this orbital can move around as a function of the cation that's present up or down. They're not going to shift in the redox potential. I might, again, have seen a spectroscopic change. I didn't. I think it's because of the quality of our experiment. But in fact, it might turn out that this orbital moves up and down, and this one moves up and down also. And then spectroscopically, I don't see it, but I could still see it electrochemically. And in fact, that's what we think is happening. OK, why don't we think that? Yeah? Let me go back. The viscosity has to get higher as you lose cation. It doesn't have to then get lower as you put them. It has to change. Let's see what that way. It gets complicated in that the pictures are very clear in your mind. I'm hopefully going to get to this. You know, when it's fully oxidized or fully reduced. But most of the states have a mixed environment. And there are ways in a mixed environment I can have the viscosity change go in the same direction whether cations are going in or out. And I will show you that. Yeah, that's another problem, but I can handle that one. OK, but first I should convince you that my fairy tale is reality. There's the proof. What are you looking at? That's part of the proof, actually. We decided what we needed to do was we needed to do diffraction, X-ray diffraction experiments in the electrochemical cell dynamically. OK, this is an experiment done by Jonathan Chun. This is a real tour de force. This is not easy to do. First of all, X-rays don't like to go through water. Sort of hydrophobic things that get absorbed and don't make it out of your cell. And second of all, we have a very thin layer of nickel-ferry cyanide here, even though we bulked it up as much as we could. And third of all, you have to have it in a cell. And so you have to have an X-ray transparent cell. Then you have to be able to fit this all in your diffractometer. And remember, the thing about a diffractometer is it rotates. Looking at an angle down here versus a scattering intensity. And there's all these wires and whatnot that can get tangled up. This was so Jonathan built the cell. He made it work. And what I'm showing you is just one of the reflection peaks. There's not that many reflection peaks, to be honest. We typically could pick up three or four. But this is just one, because we're doing a dynamic experiment. So we don't have time. We're going to step our potential. And we're going to look at just how one reflection peak shifts. We don't have time to look at them all. We can do them separately. They all shift the same way. So we don't look at the right thing. But here's the idea. This is a scattering peak when you had the iron in the reduced form. Here it is when you have it in the oxidized form. So in other words, I am telling you there's the direct evidence. The lattice parameter is expanding and contracting as I oxidize and reduce it. And in this particular case, two peaks there, broadened out, is in a form where it's sort of a 50-50 mixture. Why do you need to do it? Like that? Yeah. Well, we did that first, of course. And then what we found was, I'm not showing you the diffraction patterns here, they weren't all known. First of all, people, not too surprisingly, hadn't made every one of those cation ones, because who would bother? There's a reason to make the cesium one. If you ever decide to go into the nuclear waste industry, you'll find out that things like zinc and nickel-ferrous cyanide are very important. So you have your spent fuel rods, right? And you want to pull the radiocesium out of the rod. And so it turns out you make a column of typically zinc ferrous cyanide. But nickel-ferrous cyanide has also been used. And you use that as your support material. And you pour this nasty acid glowing in the dark stuff down the column. And the cesium ions have an amazing affinity for these nickel-ferrous cyanide-prussian blue type of lattices. And they just go in there, and the rest of the stuff goes through. And then you take this rod now. The problem with the affinity is it's so high that it's not like you're going to elude it now and go and collect it and get all your cesium 137 easily. You could have, but why bother? It's just waste. And it's 1950 or something like that. And so you just take the whole column and you ship it off to Idaho, right? There's somewhere around there. Is it Idaho? Washington, Idaho. No, no, it was all up in the Washington and Idaho. Washington has all the tanks. Yeah, you put them in the tanks. Yeah. So I was shocked. We were doing this work. And about five years into this work, I'm shocked to read in sceny news scientists at, I guess it was Battelle Northwest, are concerned that ferrous cyanide may explode. What? Ferrous cyanide doesn't explode. So it turns out they've got these tanks full of cesium, zinc, ferrous cyanide. And they were getting kind of hot over time due to the nuclear decay. And they were concerned that the tanks might rupture. They're trying to figure out what to do. So the solution was, now it turns out that the tanks are leaky. That's another issue. That's right. Nothing to be too concerned about. And besides, never mind. So some of these have been done, but not all. So we made them all. And it also turns out getting a sample that's crystal enough to get any kind of diffraction off of is a little tricky. So we made all these. And we measured the lattice parameter for the various cations around, both in the reduced state and in the oxidized state. And as Bruce said, why didn't you do this? It changes. And it changes by about a tenth of an angstrom. And in the lattice parameter field, that's huge. Big change. So this is an amazing thing. That little thin layer on the surface is oxidized and reduced it thousands of times. It's expanding, contracting, expanding. It's a solid. And it keeps doing that. And you get away with this because it's so thin that it's flexible. But the thing is, they're expanding and contracting very reversibly. And of course, after you do this, though, you have to go and do the experiment I showed you first to see it in action and actually see the dynamics. Because there is a question. Well, there were a lot of questions. Just because you got this book, Carly, do you really think that's happening in the electrochemical cell? But probably the more interesting question is does this thing expand linearly as the electrons are going out? Or is it just sitting there, and then at some magic point does it expand? What's the relationship between the mechanics here, if you will? And the electrochemistry. And it turns out it's linearly correlated, we know, from that experiment. And what do you mean by linearly correlated? I mean that. I mean, I would have included that if I had to run the outside edge, that either if you're expanding it, it works right in because the. Yeah. There's a way for it. I don't know. I can't actually tell if it happens on the outside edge or the inside edge, actually, because, of course, the x-ray doesn't tell me that. But there is essentially a wave front moving through it. That is this effective diffusion model holds in that sense. Now, let's see if I can handle, though, the problem of how does this viscosity effect going to work out. But let me do that in a secondary context. And that is, hey, you have these cations and you're sensitive to them, so why don't you build a cation sensor? In fact, it's easy to show, I pointed out to you, CZM and nickel-free cyanide are just made for each other that you can detect really small amounts of CZM ions, nanomolar CZM ions. And it turns out there is no good electrochemical technique or other kind of technique, actually, unless you happen to have radiocesium, of detecting CZM ions. So CZM ions is something you wanted to detect. This would be, by far, the best way to do it by these cyclical tamagrams. And you could do it in the presence of other cations. So let's say you are a professor of chemistry and you were married to a woman who's a professor of pathology. She's an immunologist, a viral immunologist. And she likes to look at viruses infecting cells, human cells, and killing them. HIV is her favorite one. So my wife is one of the discoverers of HIV. But that's not what she was working with here. This was something a little less potent, like herpes. That would make your life miserable forever. And so what they do is they take a cell and they expose it to the virus. The virus does this nasty thing. And when the cell dies, it lices. It ruptures open. And so by counting the number of lices per second, they could follow kinetics, essentially, of the viral process. Now, of course, you have to be able to monitor the lysing. And so the way they used to do that in the old days was they first would feed the cells chromium as chromate. It turns out cells, for reasons I don't totally understand, like to eat chromate. It's kind of like chocolate bars. They gobble it up. And if the chromium they use is radiochromium, then later on when the cell lices and it gets spit out, you can analyze it and see how much is released. So they had a problem, though, that chromate, unlike chocolate, is not very healthy for a cell. Well, maybe chocolate's supposed to be good for you these days, right, if it's dark chocolate. So there was a fair amount of lyses that occurred just because you put the small oxidizing agent inside your cell. And so they had this thing called spontaneous release that could go out to about 30% before the virus ever did anything. And that was a little problem. So I proposed that we could detect cesium at the same sort of level of sensitivity as they were radiochemically detecting chromium. And so why don't we just feed the cells cesium? And it turns out a cell can't tell the difference between cesium and potassium. So they incorporate a certain amount of potassium. There's active transport. So we could sneak the cesium in there a little bit. And then when the cell liced, everything was fine. And it turns out the cesium is not as bad for cell as chromate. There's about 1% spontaneous release. But you have some anyway because cells do die. So how are we going to do this? We're going to run cyclical hemograms in the media that has the cells. Because the cesium that's in the cell stays in the cell. And we can't detect it. And when the cell lices comes out, we detect it. And then the media has lots of salt and things in it to make the cells happy. So we have plenty of supporting electrolyte and water and whatnot. So the idea was supposed to be, here is your electrode primarily in sodium ions, sodium nitrate, at CV, and just a few cesium ions around. And then you add in more cesium ions. And you see the sodium peak go down and the cesium wave go up. And this is dropping down. And the cesium wave over here is moving up. Now a couple interesting things about this. The first one is, the first time we did the experiment, we had no cesium over here, just the pure sodium wave. And then we saw all this change like I'm showing you. But after that, we always kept picking up a cesium wave, even when we said we're using pure sodium nitrate in our supporting electrolyte. And it turns out the problem was that we were so sensitive to cesium that we were using beakers, glass beakers, for electrochemical cells. And the first time you exposed it to a small amount of cesium, the cesium would can absorb ion exchange into the glass. And later on, even though we didn't put any cesium in, a little bit would come out, and we could detect it. So eventually, we switched over to polyethylene and the whole effect went away. But we're very sensitive here. So we have a little bit of cesium around. We actually know how much is there in this particular case. Then B, we have more cesium around. C, we have more. D, you can't even see the sodium wave. We have more cesium around. So in other words, we're keeping the sodium concentration constant at one molar. And we're adding in cesium as we do that. And we're looking at this iron 2-3 redox couple. And you'll notice that D, where we've converted the whole thing over to the cesium form, we only have in there 10 to the minus 2 molar cesium ions in solution. So we convert the whole thing over, even though we're in the presence of one molar sodium ions. That is, we have a much higher affinity, or partitioning coefficient, for cesium ions over sodium ions. OK. So that's all fine. Then you say, OK, either in one of those boxes there's a cesium ion or a sodium ion. And so as this goes down, this should go up and qualitatively see that. But if that's the case, if it's a simple two-state model like that, every spectroscopist in this group will tell you that there should be the equivalent of an isobestic or isopotential point in this case. That is, all those lines should overlap. You go through one point to tell you you have two states, and you don't. Or another way of looking at this is the amount of iron on the surface that's electroactive ought to remain constant. And so if we integrate all these curves, it should just go straight across. As we add cesium, in fact, the number of irons decreases. OK, so something's happening here. You can use this analytically, I will show you, even though that's happening. Here's the cesium peak current, how it changes the function of cesium. Here's the sodium peak current, how it changes the function of cesium. We ratio those two. And you can see as a function of log cesium versus that, those two peak currents, we get a beautiful analytical fit to the data. So that's how we were able to do this. You can't detect nanomolar concentrations of cesium in one molar sodium doing this. How are we going to handle this? Well, we're going to say there are nearest neighbor interactions, like Laveron said. And so we're going to use the form of the equation that has these r's in it, which are the nearest neighbor interactions. I've used an f here, which is the fraction of electroactive material in the oxidizer reduced state. Gamma as we talked about, scan right over there. And if you do that, and we're just fitting the first wave here now, you find a very good fit. That is, you can back out our parameters, nearest neighbor parameters, that change in a very consistent way as you add cesium into the system. So what's happening? I have, this worked out really well. I have a cesium ion that, well, I start off with a lattice that has totally sodium ions in it. And I put one cesium ion in, one sodium leaves. And when I do that, I distort the lattice, because first of all, its shape has to change. And then as it changes to accommodate that cesium, the boxes that are immediately adjacent to it get distorted also. So there's sodiums in those adjacent boxes, but they now are no longer in an ideal environment, because they were before that cesium came in. And now the lattice is distorted. And it turns out, just fortuitously, that the environment is such that a sodium ion would like to leave, and another cesium ion would therefore come in. So it's sort of an amplifying effect, which is why we can see nanomolar. Now when one cesium comes in, then the partition coefficient changes so that it's easier for a second cesium to come in. What this leads to now is you get regions of the lattice that have lots of cesium ions in it. It's a ghettoing effect. One cesium's there, and the others want to go around it. And other regions that are pure sodium, you don't get a lot of mixing going on. And that's why we see two peaks, right? They're irons. They're not cesium and sodium in these two peaks. These are irons that are near a sodium, and these are irons that are near a cesium. We have two environments. And it's this heterogeneity that allows us to get this effect of this cosy that goes the same direction, whether irons are coming in or not, or going out, I should say. So that all works fine. Now you say, OK, I understand this really well. Let's do it with sodium potassium ions. Sodium potassium ions are really important. You turn on your TV any night and you find one of those wonderful medical shows, emergency room shows, type things, right? And they're always checking blood electrolytes. Blood electrolytes are just your sodium potassium concentrations. And the interesting thing about blood electrolytes I've learned is a doctor will use blood electrolytes to figure out one of the primary things, whether or not you're going into shock. That is, you carefully regulate the amount of potassium in your bloodstream. It's right around 5 to 6 millimolar. Only changes by a few tenths of a millimolar. You're very concerned about that. You want to kill a person, how do we execute people these days, lethal injection? What's the lethal thing we inject? Potassium ions. It disrupts your heart function. So you've got to hold that. If you want your nerves to work right, you've got to hold your potassium right around 5 to 6 millimolar. Sodium we have to regulate also, but not quite as tightly. It's about 135 millimolar. And it can fluctuate about 5 millimolar. Well, if that ratio of sodium to potassium gets out of whack, that's the sign you're going into shock and bad things are going to happen. So right now, how does the doctor figure this out? Well, he pulls some blood out of you. You're in a hospital, right? He sends it down to the lab. Some technician takes it, does flame photometry or ion selective electrodes, figures out what it is, writes a report, sends it back to the doctor. And you've been in shock already. Time, it gets back. There is no fast way. And so we said, we have these electrodes. You throw them into the sodium and potassium. We'll see it. We said, in fact, we know. This is just a simulation over here, the top half of the signal of titanium. We know that in pure sodium, a CV will look like that, the top half of it. And in pure potassium, we know it'll look like that. I showed you that. And so there's 150 millivolts in between those two. And we expect a wave in between. This is the simulation, that gray wave. And we'll pick it out. There's no way you can miss that. There it is. Turns out that you don't get two waves, like I'm showing you there. When you have mixtures of sodium and potassium, you get one wave. It's not like the sodium cesium. You get one wave that marches all the way from the pure sodium case to the pure cesium case, as you change that ratio. So you can use it analytically. But it doesn't do the same thing. It doesn't give you the two waves. Even though there's enough potential spread there, that you should be able to see both the waves. It turns out it's complicated. This student, David Kuhn, worked all this out. But you get this complicated dependence of the peak potential based on the sodium-deseasium ratio and the ionic strength. It turns out, fortunately, for us, in terms of doing blood electrolytes, your ionic strength is very constant, so that's OK. So we can go and look at a region where we have the sodium mole fraction here and the peak potential. And we can get a nice analytical curve. And then we can actually just take this lower part of it and get a nice linear curve there, because this is the only region where people's sodium-potassium ratios exist. That is, if it's outside that region, you're not alive. And so we're not interested. So now here's the problem. We have a cyclical tamagram that allows us, from a working curve like this, to get a sodium-to-potassium ratio. But what we actually need are sodium concentrations and potassium concentrations. So we have one equation and two unknowns is the problem. So how do we handle that? So we need to get a second equation, obviously. So we found the second equation. It turns out it's very simple. The second equation is the person you take the blood out of is alive. That's the case. Then it turns out that there is a unique, it's a very limited range of concentration, and there's a unique sodium concentration that goes with that. That is, I can find other solutions to this ratio, but not in the regime where you're alive. So I can get the sodium-potassium concentrations out of it. Here's my favorite cyclical tamagram. It looks an awful lot like the ones I've shown you so far, right? That thing too unusual there. We can pull out the sodium-potassium ratio from that. The reason I like this cyclical tamagram is given in the figure down here, cyclical tamagram and heparinized human whole blood. David Kuhn was a very brave person. He allowed my wife to stick a needle into him. We pull out his blood. It was legal. We just took our electrode. We took our chemically modified electrode. We threw it into his blood. And we ran a cyclical tamagram. Plenty of supporting electrolyte, it turns out. And there it is. It looks as good as water. What? Did he get his name on the page? Oh, he was the primary. I mean, it turns out there's all kinds of rules that affect this. And you can take your own blood. You can't go to Tom here and say, I want to write a paper, give me your blood, really, without all kinds of permissions. That's the Korean thing, really, since you're in that way. That's the start of it. Yeah, there's all kinds of rules that protect Tom from you sneaking up at night with a needle. So to do the simple experiment, you use your own blood. Actually, the poor graduate student got his PhD and he went off to the University of Hawaii, where he's a postdoc with Gary Reknitz, who's an analytical electrochemist for a while. And then he started teaching there. And then he said, you know what I really want to do is go to medical school. And he went to medical school, and he got his MD. And then he went and was a resident at Mass General. And he's doing really well today. The poor graduate student is doing just fine. And it's all because of the blood. He's a pathologist. He does analytical chemistry, essentially. OK. Why only one peak when there's sodium potassium, but two peaks when there's sodium cesium? Well, sodium and potassium are very similar in size. And so when a potassium goes into the lattice, it does not distort the lattice significantly. And the boxes immediately adjacent don't want to have potassiums in them now. So now we have more or less a homogeneous distribution of sodium and potassium ions that just depends on the fixed partition coefficient between ions out in solution and ions in the lattice. And so it's more like an alloy or a mixture, if you will. And it just moves slowly from one point to the other. We don't have those two different environments. And so here you see, if you will, a lattice viscosity change that's rather constant with the changes, rather not constant, but changes homogeneously as you change the sodium potassium ratio. But it doesn't matter whether ions are coming in or going out. At that ratio, sodium and potassium, you have that lattice parameter, and hence that viscosity. It's a distortion, yes. But it's not such a big distortion that the adjacent box has a different affinity for the ion that's in it than it started with. So we can see it at the electronic level of the redox chemistry, but not at the structural level. OK. So let's try and summarize this a little bit. And see what else we can do here. So here's my solid electrode. Here's my chemically modifying surface of A molecules that are stuck to the electrode. And I'm just listing here all the different processes that might happen, and things that might be rate limiting. So we saw already that there is a effective diffusion coefficient, a charge transfer diffusion coefficient, for moving the charge through these As by self-exchange. So that can be a rate-limiting process. A second possibility is the heterogeneous charge transfer electrode out to the layer could be rate-limiting. We've seen that in other cases. Another possibility we've already talked about is the ions have to come in, these counter ions, the supporting counter ions, and that, in theory, could be rate-limiting. Now let me add another complication to this system. So far, all we've had is redox features on the surface. But what if we put a redox molecule out in solution here, a B molecule, and ask how it interacts? Well, one possibility is B would just diffuse out to the surface. That's probably going to happen pretty fast. It's out in the solution, but that could be a limiting process. A second possibility is B might start diffusing through this layer, and that would have a different diffusion coefficient could be limiting. A third possibility is that this whole charge transfer thing is very fast, and we push A-minuses out to the outer surface here very quickly. And so as soon as a B gets the outer surface, it reacts with A-minus in a charge transfer event to make an A and a B-minus, like a rate constant associated with that by molecular now. Or these B's that diffuse in, if the A-minus coming out isn't so fast, might meet somewhere in the middle here, and we might get an A plus B going to B-minus, and then the B, of course, has to diffuse, B-minus now has to diffuse back out. So all of those things could be happening if I have, in particular, a B molecule out in solution. And any one of those could be rate-limiting. So I can come up with all kinds of wonderful kinetic schemes. We're doing this. Savion looked at this problem very carefully and looked, as I said yesterday, did the Nicholson and Shane sort of treatment for all these different kinetic schemes, both by CV and rotating disk, and tells you what the diagnostics are and how to treat it and figure out what are those as rate-limiting, what's the right mechanism. Whoops. Let me give you just one example here how this might work out. Here is Bruce's friend. Here's my nickel-ferric-sine-nide-derivatized nickel electrode, nothing in solution, but salt, water. Here is a platinum electrode, and I have it sitting in what electrochemically would be a ton of ferric ions, ferric nitrate. And I attempt to reduce them in ferric nitrate, and nothing happens. I mean, teeny little current way out here redox potential is way back here. You just can't do it. I put it in my chemically modified electrode in, and I see this wave. So this wave is not, they're on the same current scale, these two. So this wave down here is obviously bigger than this wave, so it's the reduction of this plus the reduction of iron in the system. This is just iron-3 going to iron-2. That's the process I want to look at, the electrocatalyzed reduction then of iron-3 to iron-2. Why? Because Bruce and the rest of the world says that there's one redox reaction you're going to study. That's it, Bill Karsley. So I say I have this powerful technique, though. I have these ions, and I can throw these different ions in, and I can shift the potential of this wave and see how shifting that potential affects the rate, that is the current, of the iron redox reaction. And so I see over here, if I put its cesium ions in and shift away positive, I'll actually, the current is associated with just reducing the surface. I get none of the iron in solution reduced. And then rubidium ions, I start to reduce the surface. Potassium ions, even a bigger current. Sodium ions really get going there. Lithium ions, I get my largest current. Well, qualitatively, this is what Marcus is telling us is supposed to happen. As the redox potential between the iron and the ferricine I get further apart, the rate should go up. That's this cross relationship that I've been talking about. You can't easily get the data out of those cyclic voltamograms or linear sweet voltamograms. So we went to the rotating disk experiment. So here's rotating disk voltametry. And what we're doing is we have a nickel ferricinide rivetized electrode. We're in lithium ions. Our thing goes the fastest. In this particular case, we do it for all the ions. We have our ferricinide, and now we've backed off to a more reasonable concentration there of half a millimolar for an electrochemistry experiment. And we are doing these voltamograms as a function of rotation rating. You see, we come out here to these plateaus, and these plateaus get larger, more current, the faster we rotate our electrode. And you'll recall, if you can think back to what we were talking about, the initial purely diffusion or mass transport limited actual case. This is the voltamogram that I said you would generate based on the equations that we saw and a nurse D in response, and that it should depend on the scan rate. And you're seeing it here. This is sort of beyond electrochemistry in a sense. There had to be a really good chemical engineer that came along and showed electrochemists how to solve hydrodynamic equations. And when you do that, you can find out the actual relationship between the rotation rate. So this omega now is not a scan rate. We're scanning really slow, like 10 millivolts a second. So that doesn't come into play. But this omega now is the rotation rate of the electrode, the rpms of the electrode. And the current. And you notice the current there has a little L, capital L next to it, which stands for Levich, because Professor Levich in the Chemical Engineering Department, I believe at NYU, is a person who worked this out. Why did it when he was in Russia? Well, he was later, yes. He did it when he was in Russia, but he was later at NYU. No, he was in the city, I think. Oh, was he at city? OK, I may have that wrong. OK. He was in New York somewhere. He was not in California. OK. Anyhow, you have an equation here that depends on what? There's some constants. There's your n, number of electrons that transfers in the Nernst equation. Your f, your a, you know that. You notice the diffusion coefficient now goes as to the 2 thirds when you work this thing out. A little different dependence there. Rotation rate again to the 1 half. This new here to the minus 1 sixth power is the kinematic viscosity. And that's the number you look up. And it's the viscosity of the medium. And it turns out it goes that way. And chemical engineers understand this. And then the c is the bulk concentration there. Right? Right? Yeah. OK. And so what this predicts is that they're in a purely diffusion, this is for the reversible diffusion limit case, in a purely diffusion limited case, that there should be a linear dependence between the plateau current out here and the square root of the scan rate. So you take our data and you plot it versus square root of the scan rate. And there's the plateau current. And there's the data right there. And it's not very linear. And you put in the right constants. That leverage tells you to put in right there. And you should get this dashed line. We fail miserably at a diffusion limited process, a simple diffusion limit process. So then you go to the next most complicated equation, the simplest one didn't work. This is how you play the game, guys. And that's the Kutaki-Levich equation, which says that if you look at this leverage current, you might also have a kinetic limitation. And so you may have a current associated with the charge transfer kinetics. And in the rotating disk case, the total 1 over the total current will equal 1 over the kinetic current plus 1 over the leverage current. And of course, for the leverage current, we have this equation up here. And for the kinetic current, this is all very interesting. This is going to relate to what's happening in the layer, the nickel-prysinite layer. Now, we have a molecule. The reason to rotating disk here is we have a molecule out in solution. It is affected by the rotation, obviously. We have a molecule glued to the electrode surface. It doesn't care if it's rotating or not. It does the same thing. So in this particular case, the kinetics has no rotational dependence to it. That is, it is the intercept. So you go and re-plot your data, now not square root of scan rate versus current, but 1 over rotation rate, because that is to the 1 half power, because that is the leverage functionality, versus 1 over the total current. And there is your linear plot, just like these guys say you're supposed to get, with a non-zero intercept down there. And from that non-zero intercept, you back out this. And you know what your coverage is, gamma. And you know your solution concentration right there. And you know the area of your electrode. And you know n. And you know f. And therefore, you can get the rate constant, k, for the oxidation of the solution irons by the nickel-ferri cyanide. And so you do that for all the different cations, shifting around the redox potential for the irons. And you make a plot of log of that rate constant, log k12 versus delta E, the difference between the known iron 2, 3 redox couple and the nickel-ferri cyanide couples of function of iron. And you say, wow, Marcus theory really works. Because there it is. That's what Marcus theory predicts, right? A linear, there's the equation I gave you the other day. k12 equals the self-exchange rate constant, signs the equilibrium constant, which is really that delta E right there, times this f, square root, right? So log of k12 should be equal to this thing right here, where if you take all these constants out here and throw them together, it's 8 and 1 half per volt. That slope is 8.49. We've got 8.47 here. That's 8.49 is the slope on that line. So you say, Marcus theory has no right to work here. Two featureless spheres colliding in solution to get the rate. And here, once pasted on the surface, the other featureless sphere out in solution. And it all comes together, and it still works. Now to get to that, by the way, you have to be a little bit clever. You have to think about what is actually the rate-limiting process. Because remember on that other cartoon I showed you, there were lots of possible rate-limiting processes. And we were able to rule out that this right here was the rate-limiting process, which is the charge transfer flux. That is, this is the charge transfer diffusion coefficient. If that had been rate-limiting, the rate that we could push electrons into the nickel-ferry cyanide and through it wouldn't have worked. And we went through and we showed that the only thing that had to be a slow step had to be this Kentucky Leavich term, which has in it the rate constant for the bimolecular charge transfer. Works perfectly. That's happening on the surface, though. That is happening right at the surface, that which helps a lot also. The charge transfer is so fast that the iron never has to diffuse in. It goes right to the surface. OK. We just covered a ton of material. What about questions? Yes? In this case, it is the outer layer. An iron comes up, it hits it. As soon as it hits it, it's getting oxidized or reduced in this case. But it could have turned out that any of these processes were rate-limiting. And then that's what we would have learned about. But it turns out, in that particular case, it is this process right here that is the rate-limiting process. And so that's what led us to relate it back to the Marcus theory. But you're allocating the product L. Yes. Yeah. Well, it turns out there's always going to be an equation that holds that's 1 over the total current is equal to 1 over a sum of other currents. But they're different currents. In other words, there's an intercept. You can always extract an intercept. But what does that intercept physically respond to? Depending on what's rate-limiting here, it responds to different things. And of course, if it had turned out it was something like the diffusion, B through the lattice or something like that, or even this process, we would have not got the Marcus dependence that we were looking for. Why doesn't the platinum electrode reduce ion? Because it turns out that the reduction of ferric to ferrous is an inner sphere mechanism. It goes very, very slowly by outer sphere. So I could have thrown in chloride ions, actually, and it would have gone. And what happens though is the electrode, the chloride, would attach the electrode and the ion, and we get a charge transfer through the chloride. We're using the ferric cyanide to play that role. It goes by inner sphere. In other words, that iron must be coming up on a cyanide out here attached to an iron, and there's a bridge form. Most of why some electrodes will do that kinetic on reducing some species and others, though? Well, you could always just implicate a difference in rate constant, so it depends how big a difference. In other words, you could have two electrodes, one that appears to do the process, one that doesn't. They both go by outer sphere, but one, the difference is there's a different activation barrier, and hence a different rate constant. So that's a possibility. So not always, but certainly there is the explanation in many cases that the mechanism changes when you change the nature of the metal and the electrode. And one's inner sphere, one's outer sphere. It can be more complicated than that. For example, remember what I talked about when we did Tafel's work? So in that case, platinum is great at reducing, say, water to hydrogen, but mercury stinks. It's been shown that you make platinum hydrides on the surface in the platinum case, and that they go off and make hydrogen, no problem. But in the mercury case, you're going through free hydrogen atoms, which are very hard to generate. And so you have this horrible overpotential problem. So it certainly can indicate a mechanistic shift, but it may not be as simple as saying, oh, one's inner sphere and one's outer sphere. It could be a different mechanistic shift. And you might be thermodynamics as an effect for the hydrogen case producing a hydrogen atom, which thermodynamically is very, very hard to produce. So you've shifted the potential by a long way. And, of course, the rate will change. But you can also, by going from the inner sphere to the outer sphere, you actually change the kinetic point of the reorganization energy. And that will also have an effect. So it's sort of a comp. It's complicated. The other point to make is, well, I guess you made the point. It's complicated. But the world, it's nice to say the world could be divided into outer sphere processes and inner sphere processes. But there are processes that are complex mixtures of those two mechanisms. And actually, ferricynide itself is a good example of that. Ferricynide is not gaining and losing a ligand in a charge transfer process because the cyanides are perfectly happy. So it's not an obvious candidate for an inner sphere charge transfer. Most people in the earlier literature did call ferricynide a pretty good outer sphere reagent. But in fact, in most cases where ferricynide goes and oxidizes or reduces something, there's an intervening cation. So there's a cyanide ligand. There's a cation on the negative side of that cyanide ligand. And that cation, it's going to hook up to the other molecule that's going to be oxidized or reduced. It's like that's an inner sphere process. But not through the first coordination sphere. It's through the second coordination sphere now. So it's got an outer sphere component to it and an inner sphere component to it. What do you call that? So it turns out, in this case, it's well-modeled by Marcus theory. So you could call it outer sphere if you wish, but there is this other dynamic going on there. And as we learn more and more in particular about this reorganization term, that lambda that Bruce was just mentioning, that's the great leveling in all of this, because that has an inner component, right, and an outer component. And you could say all reactions are outer sphere reactions. It's just that inner sphere reactions have a huge lambda inner associated with them, whereas what we call outer sphere reactions are more affected by the lambda outer. So it gets very gray, in other words. Other comments or questions? When you always talk about it, you're talking about it always as single molecules that are separate. But of course, the nickel phase, the nickel-ion-sinide crystal, why not talk about it as a semiconductor? And then you are talking about oxidizing site, you're talking about pumping electrons into a band. Into a band, yeah. So several comments. The first is these mixed metal cyanometallates have been investigated as semiconductors. Prussian blue has received the most investigation that way. And it is well established that they are lousy semiconductors. So that is they have very narrow bands. And second of all, in our case, we know from SEM work that although I draw a nice little single crystal on the electrode surface, they're micro-crystalline. We have some micron-sized crystallites. So we start off with a material that is a poor semiconductor to start with. And we chop it up into little teeny pieces, which of course makes it even worse. So we don't expect semiconducting properties, really. Number one. And number two is all the electrochemistry is easily interpreted in terms of molecular phenomena that it holds. In other words, I used all the equations that hold for molecules, and they hold here. You can't talk about a charge-transfer diffusion coefficient in a semiconductor that is of the same order of magnitude as the one I'm talking about here. It's much larger. If it's an electron diffusing through a semiconductor, that has a certain diffusion coefficient. If it is self-exchange, it's got a very different diffusion coefficient, but many, many orders of magnitude slower. And that's what we're seeing here. We're seeing diffusion coefficients on the order of 10 to the minus 9, 10 to the minus 10 centimeter squared per second. That clearly puts you in the solid-state molecular range. Now, the one other thing we have to do, but we don't get to until next week, is we still haven't totally nailed this problem about the viscosity and the cations moving around. We've got a lot of peripheral data, but the way to get a handle on the whole thing is to do AC impedance analysis. So I'm going to bring that technique in on Tuesday, and we'll look at this system from that point of view next.