 Well, I think we should get started. We started looking at interspere charge transfer yesterday, and I gave you some solution examples of how an interspere charge transfer works, including this famous experiment by Taube. I should say, before I go any further, it would be exceedingly easy for me to give a whole course just on charge transfer mechanisms. And I'm trying to do this in about one and a half lectures. So we are just skimming the top here, and I'm trying to hit some high points. And I hope you understand we're not getting into a lot of detail here. That's point number one. Point number two is we're going to look a little bit more at interspere charge transfer now, because I thought what I would do is try and get us back into electrochemistry. That is, I was doing charge transfer chemistry yesterday. With molecules in solution, but there weren't too many electrodes on those interspere things. And much like one can have a process where the electron is mediated between two molecules by some molecular bridge, one can have a process where the electron is pass through a wire, if you will, a molecular wire between an electrode and a molecule out in solution, probably the case that for the longest time, people have been pretty certain that's been happening, although it's really hard to see it directly, is that if you take a simple iron, a three salt, which is nonhalite, ferric nitrate is a good example, which I will show you either today or tomorrow, and you try and reduce it at an electrode, it's close to impossible to do. That is, the rate constant for the reduction is really, really slow. So you have to go to excessive over-potentials before you can even see a current associated with that. But if you go and throw a little chloride in or take some ferric chloride as your starting material or start with the nitrate and throw in some sodium chloride, it goes like a shot. And the argument has been, it's indirect, but that there is a chloro bridge in between the electrode and the molecule hooks up, and the electron goes through that chloro bridge. It seems reasonable. It's totally consistent with the data that's available, although certainly the level of data is not as sophisticated or compelling as for the TABI case that I told you about. But it certainly fits in. By the way, I should point out, there is an obvious requirement for these bridge molecules that I didn't mention, but obviously they have to be able to connect to two metals or a metal and an electrode, and therefore they need at least two lone pairs. We need to make two data bonds. So there's a minimum requirement. If I pick a molecule like ammonia as a ligand, which only has one lone pair, hence one data bond, we're not gonna expect that to be a bridge, but chloro with a ton of lone pairs will make a nice bridge. Even water, TABI showed, can be a bridging ligand. It has two lone pairs that can interact in the right circumstances. So I wanna tell you a little bit about a set of experiments that I did in collaboration with Steve Brunassik at Princeton. The student that was, actually did the project was Sandra Norton, graduated many years ago and went off and became a lawyer. But while still a chemist, well, that's the way it goes sometimes, intellectual property, I believe. While still a chemist, she did a very interesting set of experiments looking at a system that is near and dear to some of the people here, which is the photo-electrochemical oxidation of water at TIO2. So some of you know a lot about this. And the question we were asking in the late 80s was, is this a process that goes by an inner-sphere mechanism in which the electron doesn't transfer until something like a hydroxide or a water? I'm showing this hydroxide there, but it could be a water, it depends on the pH, obviously. It attaches to the electrode through a series of steps is oxygen, or is this an outer-sphere charge transfer where all that has to happen is the hydroxide or the water hits the electrode and out comes oxygen. Now, you'll notice there's multiple arrows here no matter what, because this is not a one electron charge transfer to get from water to oxygen as a four electron type of process. So obviously it's kind of complicated, but we were wondering if we could distinguish between these two, in particular at TIO2 when we were doing that because there was a bit of a controversy about that. There's still a bit of a controversy about that about whether this goes essentially through a free radical process with hydroxyl free radicals floating around in solution and eventually ending up as oxygen, or whether that one electron intermediates are all confined to the surface because it's one of these inner-sphere mechanisms and so you don't have that possibility. And the reason this confusion arose is that there was a set of experiments by Al Bard that showed that you could do things like throw benzene into these solutions and end up with phenol and that suggests maybe that there were free radical OH groups running around in these systems, but then there were other experiments that argued that the free radical was never very free, if you will, that it was stuck on the surface and the benzene was interacting potentially there. So our experiment was very simple. Actually, we said we'll do flash photolysis. That's a nice photochemical experiment, but instead of looking at a spectroscopic signal, we'll look at an electrochemical signal in this system. So we have a TiO2 electrode now, for those of you who don't do photoelectrochemistry, it's light sensitive is all you need to know. And the only time an oxidative current is gonna flow is when the light's turned on, okay? And that's all we really need to know. It turns out that it's a sort of non-trivial experiment in that you have two things to be concerned with. So what you're gonna do is you're gonna take a little TiO2 single crystal electrode that's right here. You wanna hit it with a beam of light, a pulse of light from a laser. This was in the 80s, so a fast pulse was sort of several nanoseconds, not femtoseconds, but moderately fast pulse. And even if you had a faster pulse, it wouldn't really matter, because you have an RC time constant, which you don't have in normal photochemistry, and you're not gonna go any faster in the cell's RC time constant, and that's gonna be more on the microsecond time scale, no matter what you do. So that's the first complication. We have to do with the RC time constant. The second complication is you have essentially a big time noise problem. That is you're looking for a small signal that's time varying over a fairly short time, and you don't have a clean way of getting at that signal. There's lots of opportunities at a semiconductor electrolyte interface to generate noise, and so we made everything coaxial. That is we have a coaxial wire going in through a BNC connector, and we have a counter electrode here, which is a platinum gauze in a cylinder, and we just cut a little hole in that gauze, and we put our TiO2 electrode right down the center. So the whole electrochemical cell is a coax cell, and that really cuts down on the noise. We carry the coax all the way into the cell. And when we did that, we could get pretty reasonable signals. So you see over here that we do our laser flash. That's this pulse here thing. And again, you can see that's on the nanoseconds, tens of nanoseconds time scale, and then we see the photocurrent. You notice that photocurrent is a much larger time scale, so we have to go to a microsecond time scale here to see the decay of the photocurrent. And that's because of the RST time constant of this whole system, which is relatively large. I'll point out, when you're looking at the fast time scale here, even though we've done a lot to minimize noise, you have, if we were doing NMR spectroscopy, we would call that ringing going on there. That is, we have a bit of an impedance mismatch between our oscilloscope that we're putting this out to and our wires and our cell here. And so we have some signal bouncing back and forth, and all this ringing here is due to that. So the real signal is on this longer time scale here. So that's the experiment. And one might do that experiment, say, as a function of solution pH, because I mentioned you could oxidize water to oxygen at the surface or hydroxide to oxygen at the surface. And so we have this fancy little plot here where we're looking at these decay traces as a function of time, and we do it at different pHs, and we have a photocurrent here. And what you notice here is that in the acidic region, you get a set of curves that do not very much, pretty invariant. And then you hit some pH right around 9 or 10, and there's some dramatic change that takes place here. That is, at the point in concentration where I have a reasonable amount of hydroxide around, pH 10, kind of millimolar hydroxide, I start to see some change in these time profiles, suggesting a change in the kinetics at that point. So if we want to examine this system, we have to examine it on the one hand somewhere out here where we're pH-independent and we're oxidizing water, and on the other hand over here where we are pH-dependent and we are oxidizing hydroxide. Now for those of you that actually have worked in this area, you know that, especially at a T2 electrode, but in general, it is much, much easier to oxidize hydroxide than it is to oxidize water. You always run these experiments in base if you want to be efficient. And so it's not too surprising that there's a change in mechanism. What was a little surprising was how rapidly the switchover occurred here. So it's a little bit hard to look at that three-dimensional surface, but you could take that and we could take those curves, those decays, and we could characterize them in terms of a lifetime tau. Now it turns out they don't exactly fit an exponential decay. It's more of a stretched exponential that they fit, but there's an RC component convoluted in there, so that's understandable. So we can have the time there and the pH there. And again, you can see that there's this rather linear change as we change the pH out and then somewhere around 10 or so, things get serious and the lifetime plummets on us that is everything's going a lot faster once we get some hydroxide around. Okay, so back to the question, inner sphere or outer sphere? So we look at pH one in the acidic region and we do an experiment where we try and oxidize water that is H2O versus water that is D2O to look for an isotope effect to see if protons are important. And they're not too important. It looks like based on this experiment, we see within the experimental error virtually no change whether or not we have H2O, there are D2O, but then we go into base pH 14 and do the same experiment and we see this huge primary isotope effect. You can see that the decay in the case of the protons is much faster than in the case of the deuterons in the system. So that argues that there must be a proton transfer in a critical step in this mechanism when we are in base, when we're oxidizing hydroxide, but not when we're in water. I was gonna guess actually, by the way, that it was gonna go the other way around because of course in the water, this is an acid where putting an extra proton or deuteron onto that water is really H3O plus that we're dealing with. So I thought if there was gonna be a proton effect to be here, not over here, but clearly there's a proton transfer in the hydroxide case that is the key thing. So to us, that argued that probably what was happening is a hydroxide comes and binds to the titanium surface and there's all kinds of spectroscopy that tells you that on a titanium surface sitting in water, there's lots of these OH groups stuck on the titanium. Whether or not they are important to the oxidation kinetics, you can't tell, but they're there. And that it would be a charge transfer, essentially that would give you a tight nil species. That is a titanium double bonded to an oxygen that might be the important step. And if that was the case, you can see that proton has to get lost and all kinds of exciting things would have to happen. So that would argue for an inner sphere charge transfer mechanism. Now that particular paper that I'm taking this out of did not prove, it suggested that it was a tight nil species that was the critical intermediate. It turns out in the intervening years, that's correct, people have seen that species and it is a key intermediate. But we were addressing a more preliminary question simply, is it this species that is the charge transfer species, that is, is it an inner sphere charge transfer? So the argument that we used was, well, if we throw in a material that could bind to the titanium in place of the hydroxide, if the binding of the hydroxide was critical, if it was inner sphere, if the electron didn't transfer until the hydroxide was bound to the electrode, then we should see an impact. So we could take a material like acetonitrile. That's a weak ligand. It will bind weakly to the titanium. And you might guess, but hydroxide is a weak ligand also, you might guess that there would be some kind of an equilibrium here, or a steady state perhaps, and that you would see some effect from adding acetonitrile. But you still could bind hydroxide there. You would have some equilibrium that might be set up. On the other hand, you could take another material like this phosphite, and we happen to pick this because it's water-soluble. And when that binds to a titanium site, it would make this species, and that species would not compete with hydroxide. That is, that binding is much stronger than hydroxide binding. So one would think if you had an inner sphere mechanism and you threw your electrode into this stuff, that you would kill your electrode. You couldn't oxidize water or hydroxide, rather, anymore with that. And then finally, obviously, in the absence of those materials, hydroxide could bind. And you would go to oxygen. Well, we obviously know that when you electrolyze hydroxide, it does generate oxygen. So that path is correct. I will tell you that if you add a few millimolar of this phosphite into your solution, it is dead. You oxidize nothing at that point. And you can't just go and wash your electrode off with a little bit of water and restore it. The only way to get your nice single crystal TI2 back as a photo electrode is to polish it down. So that is binding, just like I said. And in the case of the acetonitrile, this is what you see. If you go to pH 7, remember where we're just seeing no kinetic isotope effect and where we're oxidizing water. It doesn't matter how much acetonitrile you add. You don't affect the lifetime of these transients at all. So water appears to go by an outer sphere type of mechanism. It's oxidation. But if you go to pH 14, where you have one more sodium hydroxide around, you get a huge effect on the acetonitrile. It's a reversible effect. You can go backwards and forwards on this curve. And acetonitrile does compete for surface sites and stop the oxidation of water. So there's some fairly direct evidence that there's an interspere charged transfer mechanism happening when you try and oxidize hydroxide at an electrode surface. So this not only happens at a molecular level, but it happens at an electrochemical level. Let's switch over to something else. And then exciting here, which I'm just going to prep for the moment. Get that going and hide it from you, because we're not ready to talk about that yet. That is the key points that I have to make about an interspere charged transfer. Now what I really haven't told you about at all is the outer sphere charged transfer. We need to talk about that a little bit. And that's a little bit embarrassing, actually, for me to do here in that quite obviously Professor Marcus is here and knows a lot more about this than I do. But it's worse than that. Professor Gray is here, and he's made some important contributions to outer sphere charged transfer. And Professor Lewis is here, and he's made some important contributions. But actually, the most embarrassing thing is that Bruce is sitting in the back of the room here. And I think he is infinitely more qualified to be giving this lecture than I am. So I'd rather sit down and let you come up here and do it. Go ahead. I was afraid. Since I have the mic, I guess I continue talking. So I'm going to do again. I'm going to do a skimming the top sort of Marcus theory for you very, very little math. And we could do it, but it's a whole terms worth of math. And then I'm going to switch over to some electrochemical systems that hopefully will prove the point and let us do some cyclic voltammetry and chrono-amperometry and all that stuff that is what I really want to get at here. OK, so how do we want to start this system? We would like to start it by some of the assumptions that Professor Marcus laid out in his original model. It is a model that has no right to work, really, if you think about it. He made clearly unphysical assumptions in terms of real world when he laid out this model. And he understood that. It wasn't that he was trying to mimic the real world. He just was making some simplifying assumptions. So he's talking initially about two molecules in solution. One that's going to oxidize the other. So you have an oxidant and a reductant. And the first thing he tells us is these molecules are spherical, no problem, and that they can be approximated by harmonic oscillators. Sounds great for some kind of chemical physics model, but how many people have ever seen a spherical molecule that just has one vibrational mode? That's a harmonic oscillator. We don't think about it that way, but that's what he said was the way he put his model together and asked what happens if that occurs. So the argument is a semi-classical argument. It's basically, I'm going to have these two molecules in solution. Let's call them A plus and B minus. And B would obviously be giving an electron to A in this particular scheme of things. So we have A plus and B minus going to A plus B. And since A is a cation, it'd be all about that size, let's say. And B is an anion. So in this spherical context, it's about that size to start with. And the first thing that happens, according to Marcus, is that these two species have to get close to each other. They have to collide somehow. So they're far apart in solution to start with. And you can see in this particular case, they're going to come together because one's a cation and one's an anion. So there'll be an electrostatic attraction towards these species. But Marcus isn't going to limit things to a cation and an anion interacting. It could be two neutrals interacting, in which case they'll still collide. But we won't have this electrostatic term. Or it might be two species that are both positively charged or two species that are both negatively charged colliding together to react, in which case there'd be an electrostatic term again, but it'd be a repulsive term. So there's some energy term here. And I can't tell you, actually, if it's a positive going energy or a negative going energy because it depends on the nature of the charges here. And I don't know exactly that. But typically, it's going to actually require some energy. Because typically, the species will be either neutral or one will be neutral and one will be charged or both will be charged the same way. There's only the one special case that I happen to draw here where you get a positive electrostatic interaction. So typically, we're going to have to go up in energy to get these two guys next to each other. So we will go and use some energy here. And we'll bring these two molecules close together. So we do some work to get them together. And of course, even the case where they're both neutral, there is some work that has to be done because we need some thermal energy to get them to diffuse together. Now once we've done that, Marcus tells us that they don't just transfer charges. Now we get a little quantum mechanical. And we say that we understand that that transfer of an electron from, in this case, B2A is a tunneling process. And that has some requirements for it. And it's going to be most facile and most reasonable to consider a condition where the orbital, the energy of the orbital that the electron starts in and the energy of the orbital that it ends up in are the same. It's an iso-energetic process. Now of course, for this arbitrary AB system, chances are that's not going to be the situation. That is, in fact, using this harmonic oscillator picture, we would say that the size of the sphere that I have here is directly related to the energy of the highest occupied and lowest unoccupied molecular orbitals. So given the fact that the spheres are two different sizes, they aren't at the same energy. And yet we need that highest occupied molecular orbital and lowest unoccupied molecular orbital to become the same energy. So what we realize now is that these molecules aren't of fixed size, but they have vibrations against a harmonic oscillator in our model here. And so this B minus molecule will vibrate so that sometimes, if I could do a time snapshot on it, it'll vibrate in and it'll be a little smaller. And at other times, it would go out to some other amplitude and get a little bigger. And of course, the A is doing the same thing. It's doing their vibrating. Oh, we need a prettier color for vibrations. Let's see. Here we go. OK, and you can imagine that while these two guys are next to each other in this encounter complex, it might occur that, in this case, B vibrates in and shrinks at the same time that A is undergoing an expansion in its size. And the two of them happen to become the same size. So we have this vibrational process. And we may end up for a transient period of time in with A and B the same size. I'll put that in quotes because it's not going to stay around for very long. This is a fleeting interaction. And of course, when I meet that requirement under the conditions set up by Marcus, that means that the highest occupied molecular orbital in B and the empty orbital, the lowest unoccupied molecular orbital in A, are at the same energy. And now, of course, the electron transfer can occur at a relatively reasonable rate. And so we are going to, at that point, get our electron tunneling from B to A. And at that instant in time then, we end up with an A molecule that is now neutral. And it's the same size because this happens pretty quickly. And a B molecule that is now neutral. And now, the vibrations continue to happen. Of course, we have different vibrations and different electrons, and everything has to readjust to this electron distribution. So there is a period of time, a short period of time, for reorganization. And that reorganization is going to lower the energy of the system. And we'll end up now with a A that was bigger than the original A and a B that was smaller than the original B. But they're not exactly the same size. A bigger than A plus and B smaller than B minus. And then these species can diffuse apart, and we have finished our reaction. So that's a little scheme. It's a pretty simple scheme. Sounds like it should work just fine. Now, if you're going to go and represent this in some sort of graphical context, what you might do, perhaps you label this whole board, Marcus's theory, Marcus points out is we can look at a plot of energy versus distance. I'm going to call that distance r, and I'm going to put it in quotes, because that's a very controversial r that we have there. We'll get back to that in a minute. But before it gets controversial, we would start off, and we could symbolize then, since everything is a nice harmonic oscillator, as a parabola. That is, this would be the A plus B minus system before reaction has taken place. And of course, there are vibrational energies associated with that, and we will go semi-classical there and point out they're quantized, et cetera. And there's some average distance between these two redox centers over there. And then the charge transfer is going to take place. And we're going to have some other average distance. We'll put it further out here. That's a parabola, believe it or not. That is one of the world's sickest parabolas I've ever drawn, but there it is. And there's vibrations, and we now have this in our product state, A and B, where the charge transfer is taking place. We have two potential energy surfaces here, and they cross each other. And we're going to follow a rule about crossing that potential energy surface of that cross. Or they can't cross. I guess it's the rule. So we're going to have to open that up a little bit on crossing surfaces. So now we have a ground state surface, this W-shaped thing, and an excited state surface up here. And Marcus is telling us that we can get ourselves into some vibrational level that allows the electron to go across here. And it may not have to go exactly through this little opening. It may be possible that it could be a tunneling process and go through the barrier a little bit over there. But it's going to go from one vibrational level over here to another vibrational level. And again, the requirement is that these two vibrational levels are at the same energy. This happens. Now, you'll notice these two parabolas are not at the same zero points, and that difference in energy between the zero points, these parabolas, is just the free energy for the reaction, which of course, within a few constants, is the change in potential for the reaction. Now, once you have that picture, you can play all kinds of interesting games with it. I mean, it's geometry at this point. And we can talk about the energies here, and we see we have an activation energy to get over the barrier. And we can talk about tunneling. And we have a ground state energy change. It's all laid out there. And all we have to do is some geometry to get from one point to the other. And I'm just going to skip all that because you're going to believe it works. And if you do all that geometry, one of the things that pops out of this is what's called the Marcus cross relationship, which is what I was trying to coax out of you yesterday. And that is there is a rate constant for this charge transfer, because there's a rate for the charge transfer, and that rate constant will call k12. That is the rate at which an electron moves from molecule one to molecule two. We call it that because Marcus called it that. Now, there's a very similar scheme to this that one could draw, and that is it's a kind of boring reaction. But what happens if instead of having a b over here, we have an a over here? So it's a plus a going to a plus a plus, the self exchange reaction. No net chemical change, kind of boring from the point of view of a synthetic chemist, but a perfectly legitimate reaction. And if we did that, of course, there'd be a delta g equal to 0, because we haven't changed the chemical composition of this system. And so all that's going to do is that's going to drop these parabolas down so they're at the same energy. Otherwise, everything stays the same here. And Marcus uses that state as, if you will, the defining state for the system. That is, you have to pick some, we'll call it a zero. It's not a zero, but some point that we're going to know. And that's what Marcus picks. And so he says k12 is equal to k11, the self-exchange rate constant for a, times k22, the self-exchange rate constant for the b minus b interaction here, times the equilibrium constant for this process, capital K12, times a factor he calls f, which some people say is a fudge factor, but it was actually a frequency factor. And we have to take the square root. That doesn't seem to surprise you. We're dealing with parabolas here, so we need an equation for a parabola. And there it is. That's this equation that sort of shocked everybody. And the first reason it shocked people was they all remembered what they learned in freshman chemistry. You remember when you were in freshman chemistry and your professor told you, there's this thing called free energy. And if it's negative, the reaction goes. But you can't tell how fast the reaction is going to go by, how negative it is. And if you ever tell me that a reaction goes faster, because it has a more negative free energy, then I'll do something like scoff at you in front of the whole class or throw an eraser at you or something like that. Some horrible fate will refall you if you make a correlation between kinetics and thermodynamics like that. And here comes Marcus. And this is exactly, of course, what he does. Because that equilibrium constant is a thermochemical parameter. We remember that the equilibrium constant is just e to the minus delta g0 divided by RT. And so there it is, bigger the free energy, more negative the free energy, the faster the rate constant. So people have now found a lot of linear free energy relationships in the world. But that bothered people at the time. It turns out, by the way, most people just leave that F as an F, because they can't deal with it. So Marcus actually told us that it is a frequency factor and that F is equal to, let me make sure I get this correct, the logarithm of that k12, that equilibrium constant, the quantity squared divided by 4 times log of k11 times k22, the self-exchange constants. The whole mess divided by z squared, where z is a collision frequency. And obviously, we should have the collisions and more collisions the fastest things should go. And this, again, just all comes out of the geometry of this problem. Now typically, people somewhat ignore this term because experimentally it's very hard to get a handle on this term. And it tends to be a number somewhere around 1. So let's just throw 1 in there and make our life simple, because then we can calculate k12 just based on some parameters that we can easily get our hands on, if it's 1, if you don't know that. So the first point here is this famous relationship. The second point is, which is not going to be as obvious from this discussion, but as I change the delta G here, then I move these parabolas with respect to each other on this energy axis. And I can actually get to a point where initially, well, initially obviously what happens here is that as delta G gets more negative, that as I start to move these apart, k12 gets bigger. The thing goes faster. But I get to a point where moving a part no longer does that. And then eventually, when I make the delta G even bigger, this relationship falls apart. And I find out that actually the reaction slows down according to Marcus's theory, when delta G gets too large. Because these parabolas don't overlap the right way anymore. And why is that? It's easy to see chemically what's happening. And that is as the free energy gets different here, then obviously the energy of the orbitals has to change. And you don't have to overlap between your HOMO and your LUMO. And if things get so bad that there's very little overlap, then the electron doesn't matter what the vibration is, isn't going to move from A to B or B to A. So you're going to have things slow down. So Marcus predicted based on this simple model, what's called the Marcus inverted region. And actually, I think a lot of people laughed at this. It just doesn't make any physical sense that as you increase delta G, things should go slower. But he was in pretty safe ground because experimental chemists couldn't figure out how to get delta G big enough to figure this out. That is, you cannot observe any rate constant, electrochemically, spectroscopically, whatnot, that is faster than the diffusion limit. Things could only diffuse together so fast. And the kinds of rate constants that Marcus was saying you would need to have were going to be extremely fast. And so if nobody ever saw the inverted region, he was still safe because he would simply say, you didn't go to a big enough delta G. There wasn't enough driving force in there. So for a long time, people were skeptical that it was going to occur. But then people came up with some really clever experiments, flash electrolysis experiments, where they would photochemically do this reaction, which I'll get to in a moment. When you do that, you generate this redox pair, these two molecules, right next to each other. Because you photochemically pump the electron from one molecule to the other. And now you have a thorough back reaction. And it can be faster than the diffusion limit because the molecules are already there. And in fact, when you do this, when you look at the back reaction, you find out Marcus theory is correct. It does predict this inverted region, and it's there. Now this isn't to say, by the way, that every single molecule does what Marcus says it's going to do. But it turns out that a remarkable number of molecules do what Marcus says they're going to do. And they don't necessarily have to look like spheres. And they still seem to work. But before we get to that, a couple more points. This is an electrochemistry course. Marcus handled that also. He pointed out that you could also think about this in terms of an electrode in solution and a molecule hitting the electrode. And if the molecule was in the right vibrational state when it hit the electrode, so you can get an iso-energetic charge transfer, your gold. And so he tells us for the self-exchange rate constant, case of S, no potential in this term, divided by this collision frequency with a little e down there. This is a collision between two molecules. This is a collision between a molecule and the electrode surface. You'll notice they'll be different in the same circumstances. It's a little different geometry there. That that will equal the self-exchange rate constant for the molecule. The ups was, that goes down here, divided by the solution collision frequency and all that to the one-half power. So we have a little different geometry here. We don't have two species that are spheres that are colliding together and oscillating like harmonic oscillators. We have one species that's doing that and hitting a flat surface. We're assuming the normal semi-infinite linear diffusion and all that wonderful stuff that you know about. And so changes things a little bit but not too spectacularly. Now, there were other things in this model. There were things in this model that Marcus didn't even see originally. And a gentleman by the name of Hush came along and saw some things in this model that Marcus didn't see. Marcus was thinking about ground state electron transfer. And one of the things that Hush did was he took the same model and said, well, you know, there's another way of getting the electron from parabola 1 to parabola 2. And that is, we could put a photon in. And I've already strived that experiment to you. And that's a vertical charge transfer, believe it or not. That is a straight line going up. Let me try that again. The problem is when I put a photon in, promise I can't draw straight, but when I put a photon in, that happens very quickly. And there's no chance that the molecule reorganizes its geometry on the time scale of the absorption of the photon. And so I can't have an angle there. It's got to stay at the same arc. And so we can go over there. We could then end up on this excited state surface. You'll see. I'm going to do that. I could have my molecule now rattle down this excited state surface into the parabola 2 and into the ground state there. And so I can do a photo induced charge transfer. And Marcus and Hush played around with that. And you can work out some nice relationships there. And that works fine. I already told you about the flash photolysis experiment that's based on this that gives you the back electron transfer data that one is interested in. So in other words, this is a process that could occur the way I showed you, but it could also occur from a system that has a positive delta G under thermal conditions. That is, I can go up this way, but I certainly could also do this. Experiment right here, go up, extend my parabola so I can hit it. And rattle down this way. So this allows me to take a system that thermally has a positive delta G and move the electron in the wrong direction, if you will. And then I can get the back reaction. That was the experiment I was just explaining to you. And you can, again, work out all kinds of wonderful geometric relationships there that one might want to write down. For example, let's see, do I want to do a for example yet? I guess we could do a simple for example. Yes? So right there, there's still some probability that it rattles down one way or the other. It could. There is a possibility that it will fall down, yeah, on this side also. Yeah, you're going up to the top of a activation barrier. And depending on the symmetry of the barrier, you might fall back down and do a nonproductive reaction. But of course, a certain percentage of the time, you will go over the barrier. Absolutely. And hopefully, probably not today, but probably tomorrow. I'll show you an example where you can control that actually on a good day. Anyway, so we have, in the case for an optical charge transfer, we have that the free energy of activation for that process, according to this, is equal to the square of the optical energy, that is the size of the photon we're putting in there, absorbing, divided by four times the optical energy minus the ground state standard free energy. And in the case where we don't have this delta G difference here, we have a symmetric system or a self-exchanging system that reduces down to the delta G double daggers equal to the optical energy divided by four. Now we have to take into account one more important concept. And that is this idea that once the electron has transferred over here, that the system has to reorganize. And that's going to take some energy as well as some time. That is, I need to think about this in a lot of detail and Marcus did. And there are two kinds of reorganization, at least, that I have to be concerned about. The first is, if this is not just a sphere but actually a real-life molecule with bonds in it, there may be bonds in the molecule that have to readjust their bond lengths. For example, perhaps I'm moving this electron into an anti-bonding orbital, and that would make bonds longer, or potentially I'm moving it into a bonding orbital, and that'll make bonds shorter. So I expect there to be some bonds that are changing. And that would show up as a vibration if I change the life of a bond. So this thing doesn't just change, but it does that and settles down to this new shape. And then I have all the solvent molecules that are immediately adjacent to this. And I've changed the charge there, and the solvent probably has a dipole moment associated with it. And so the number of solvent molecules and the direction that they're facing and whatnot, all those dipoles, it has to relax also. So I have a reorganizational process. That reorganizational energy is called lambda. And I can divide that into an inner sphere process and an outer sphere process, outer sphere being solvent and the inner sphere being bonds. I can get a handle. I think I'm not going to go into the details on this. But I can get a handle on this spectroscopically by looking at vibrational spectroscopy. In fact, not only do I think I'm not going to go into the details of this, I know I'm not going to go into the details on this because we could write a book on this. And it probably wouldn't be finished yet, either, by the way. Close to, though. But that would be vibrational spectroscopy because that's bonds reorganizing. And so by knowing something about the vibrational frequencies of the molecule, I can figure this out. And there's this wonderful theory developed by Eric Heller that uses Raman spectroscopy, typically, and as a function of the absorption wavelength allows you to come up with some information about this. There's other ways of doing it. But like I said, I've stopped talking about that now because we don't want to hear about that. And then we have this solvent term. Now, that's harder to get a handle on. From a vibrational point of view, if you want to think of this in terms of vibrations as the solvent relaxes, those are very low frequency vibrations. And so you experimentally have difficulty looking at that. In fact, not only difficulty, it really hasn't been done. There are a way that Marcus initially handled this and Hush, is simply saying that this has to do with dielectric relaxation of the solvent. And so we can use the dielectric constants, that's the static and dynamic dielectric constant of the solvent, to approximate this term. And that is something that is being re-looked at right now, and that dielectric constants are a bulk phenomena. And we're trying to get to a molecular model here. And so that's made some people a little bit unhappy, and they would like to get a more detailed, molecular picture of that. And there are several theories out there now that aim to give you a more molecular image, if you will, of this term, exactly what the molecules are doing. The problem experimentally is the kind of data you get more or less is well-fit by the dielectric relaxation, that the simple Marcus relationship there works great from an experimental point of view. Now, probably if we could do better experiments, we would find out that this bulk phenomena falls apart, and we need a better molecular picture. But we're not good enough to do that yet. So of course, as the theory comes out, it gives you an idea of the kinds of experiments you might do to look for a molecular phenomena here. But everybody knows that's wrong, but it sure works well for something wrong. OK, one last important point here now. That R. What is that R? We have all the spectroscopy. We can measure electrochemistry, so we can get free energies, and all this wonderful stuff. We can get rate constants. All that has to do with stuff happening on the vertical axis here, the energy axis. We don't have good measurements. We can't take out a tape measure and say, oh, look, A is 2 and 1 half centimeters away from B. Never be that far away anyway. When the electron transfers, we have very little information about R. Now, if our molecules were really spheres, then we could just take, say, the center to center distance, and that would be the end of the story. That's what Marcus was using. No problem. But our molecules aren't spheres, and so the question is, what is the length parameter we should be using? And we could have the simple one. Well, if they're not spheres, should we still take a center to center distance and just say, that's OK? Or should we take the distance of closest approach, or the edge of one molecule to the edge of another molecule? And people have done experimental studies, and sometimes the center to center seems to fit best, and sometimes in a different class of molecules, the closest approach seems to fit best. More importantly, though, what about a molecule that has several degrees of freedom and several different vibrations in it? And what have you? I obviously cannot characterize that with one distance parameter. So really, if I want to do this right, I need to be in n space, where n takes into account all the different vibrations in the molecule that might be occurring and allows everything to adjust and relax just the right way. Nonetheless, if I do that, I obviously can't draw a simple picture like this. And I lose a lot of my intuitive feel for this. But if you want to see how that's handled, you could look, for example, at the work of Jeff Zink at UCLA, who has analyzed vibrational spectroscopy from excited states in terms of reorganization and has done a full vibrational treatment, and shows us how that can be done, looking at wave packets traveling on these upper excited state surfaces. Another topic we won't get close to right now. That is a quickie outer-sphere charge transfer. Questions? I probably have said things so quickly there can't be any questions, I don't know. Bruce, do you have any comments you would like to add? It's we're glossing over the surface, but OK. Now, actually there is a comment Bruce would like to add because he made it yesterday, and I have to repeat it because it's a very important comment. And I'm going to show this in a minute, actually. And that is this theory, again, is supposed to work for outer-sphere reactions. And hence you would think it may not hold for inner-sphere reactions, but in fact it often does hold. We can use the same formalism to describe an inner-sphere charge transfer. So this is not something where rigorously, again, we need two spheres banging up against each other and then floating away, doing their thing. So I'm going to push the envelope now on Marcus' theory a bit. I'm going to do it for electrodes and I'm going to do it for chemically modified surfaces, which is in fact even another topic that I want to cover here. So I'm throwing a lot of stuff at you right now. And I'm going to use my favorite system, the nickel ferrocyneide nickel modified electrode. Back in 1980, I came to Princeton and I was going to see some really fancy chemically modified surfaces that had nothing to do with electrochemistry, because I had done that for my dissertation and I thought, ah, let's apply chemical modifications to non-electrochemical systems in particular. How about heterogeneous catalysis? And I immediately ran into a problem. I'm going to go and I'm going to attempt to stick a molecule on a catalyst surface as a modifying agent. How will I know if it's there? When I do the electrochemistry, I know it's there because I can see the redox processes, but I'm not doing electrochemistry anymore. So I decided I needed to put a molecule that was redox active on the electrode surface as a tag so I could figure out if it was there. So I said, OK, what is the best understood molecule, both in terms of coordination chemistry and charge transfer chemistry, because I don't really care about the electrochemistry of this molecule. It's just there as a reporter. And so I don't want to be confused by this molecule. So let me take something that everybody understands. So I go and I take ferrocyanide couple, which has been studied for 300 years in terms of this coordination chemistry, not quite as long in terms of electrochemistry, but close. So I can't go wrong, because here is the one molecule that coordination chemists understand. I have been studying that molecule for the past 25 years since I came to that conclusion because I don't understand it. And this was the first thing that happened that I didn't understand. It reacts with nickel electrodes. Now, that gets us into this idea of chemically modified electrodes, CMEs. So I have an electrode, and it does certain electrochemical processes. It can oxidize certain things. It can reduce certain things. And if I want to change the rate constants for that process, typically what I do is I change the electrode. I change the solvent. Sometimes I have a problem with molecules in solution chemizorbing onto the electrode that is typically considered a problem. And I might live with it, but I might go and try different solvent systems and different electro systems to change that. And the idea is, well, instead of being beat up by this game, why don't we just join this game? If I have an electrode and I want to change its properties, why don't I attach some molecules to its surface that will endow it with the properties, the new properties that I want? And that's the idea of a chemically modified electrode. The concept is really first enunciated by Tecawanna and Royce Murray. And most of it really Royce Murray's work, but Ted said it first, I think, actually. And Royce, in particular, has also developed a fair amount of the theory that goes with a chemically modified electrode. However, it turns out that the person who, in this case, sort of inadvertently worked out some of the key theory before anybody even thought about doing a chemically modified electrode is Fred Anson. Because in the later 60s, early 70s, Fred was interested in what he called, and others have called a thin layer electrochemical cell, TLC. Why initially was that of interest? Well, initially it was of interest because you could use it with hybrid techniques. That is, one thing that one would like to be able to do is do, say, spectroscopy and electrochemistry at the same time, spectral electrochemistry, or perhaps do x-ray diffraction and electrochemistry at the same time, or ESR and electrochemistry at the same time. All these things have been done. The first was optical spectroscopy. And so Fred was interested, the way you do this is use an optically transparent electrode so you can get light through it if you're going to do optical spectroscopy, and you make a really thin cell. Well, I gave you a problem in the homework the other day asking when you would run into trouble if you put your electrode too close to the edge of the cell. Well, if your cell was really thin, your electrode is too close to the edge of the cell. You can make a cell that is sufficiently thin that you don't have the length you need to generate the full diffusion layer in there. You hit the other side of the cell before you develop your full gradient. And if you make it sufficiently thin, then essentially nothing diffuses. That is, you're working on a time scale where you do not have time to set up the diffusion gradient. And so you're totally limited by charge transfer, even though you're not a charge transfer limited process in a normal sized cell. And so that was actually the problem, the thin layer problem that Anson worked out. And in doing that, he came up with this relationship right here for the cyclic voltamogram relating current to a peak current and the potential versus the standard redox potential for the system, where that peak current has some numbers in it to get the right units and has a few things that are a little different here. First of all, it has a scan rate, you'll notice down there again, but now it's linear in scan rate, not it's a square root because there is no diffusion. And we pointed out the other day that that square root dependence comes from diffusion. So that goes away. The concentration terms are the same. We now have a volume term. The volume of the cell becomes important. You'll notice, by the way, concentration times volume is just the number of moles of molecules that's in the cell. All of the electrochemistry, I didn't make a big deal of this, that I've shown you so far, is concentration dependent. And concentration right next to the electrode. So you could have a trivial number of molecules in your cell, if they all happen to be next to the electrode, the concentration is high, and you see a large current for a very small number of molecules. In this system, we're actually looking at the number of molecules, but we still have a concentration term so we can still play that game. In the normal cyclic voltamogram, the number of electrons and that flows was raised to the three house power I showed you the other day. Here it's a squared term, so there's another little change. So we have a slight difference in shape of the cyclic voltamogram in that this thing becomes symmetric when I do this. We have a difference in the scan rate dependence. That is, the cyclic voltamogram is no longer going to be a system where I see a displacement between my oxidized and reduced redox processes. Remember that displacement has to do with diffusion, that's this 60 millivolts business. I don't have that anymore, reversible case. And so I expect redox potential right there, a symmetric structure now centered on the redox potential. And you can actually see, just look at that equation, that that's going to happen. We're going to hit B maxima, you notice, when E equals E0. That term drops out to zero. So we have Royce, Fred worked this all out for us. And then when people started pasting molecules onto electrode surfaces, and actually, for instance, one of the people that started pasting molecules on the electrode surfaces in the 70s and 80s, they initially just started off using his theory for the thin layer cell. Lavron came along and modified that theory and made it more sophisticated. She developed it specifically for a chemically modified electrode and a cyclical tammetry of a chemically modified electrode. Take a few more things in effect. And then more recently, but still in the 80s, late 80s, now Savion comes along and works out a full-blown theory for cyclical tammograms and protein disc electrodes of a chemically modified electrode where you have all kinds of possible kinetic schemes going on. Savion has done for the chemically modified electrode what Nicholson and Shane did for solution cyclical tammetry. Thought about all the kinetic schemes. Thought about, well, what if there's some molecules on the surface and some out in solution, and how do they interact the whole nine yards? One of the things that Lavron does is it gives us the current potential relationship and points out now that once you paste these molecules onto the surface that you have the possibility that two molecules next to each other will ignore each other, will interact attractively with each other, or will interact repulsively with each other. That is, nearest neighbor interactions now become important. And so she throws in a nearest neighbor term in here. That is these r's. The theta is your normal sort of Nernstien thing. But she throws in these r's, which are a nearest neighbor interaction for the oxidized species, and the rr nearest neighbor interaction for the reduced species. Gamma now is the t, is the total concentration of molecules on the electrode surface. And so this is a nearest neighbor term down here that's modifying the overall Anson type result. And it also turns out it affects the potential of the redox process, and it shows up here. And I actually meant to take this out of this slide. So we'll just ignore that. That's something we did once for another thing that you can see later on. Now, out of those equations, a few things fall out. Some I've mentioned already. First obvious one is a linear scan rate dependence. That is, I peak is directly proportional to scan rate. We do not have the square root dependence. Yeah, I dropped gamma out of this relationship. There should be a, not gamma, it's omega, right? Whatever they agree. OK, gamma is the surface concentration. This is the concentration of oxidized molecules on the surface. This is the concentration of reduced molecules on the surface. And gamma sub t is the total concentration on the surface. And there should be an omega up here that I dropped out. Thank you. Add that omega in. So we have a linear dependence there. We have now a peak to peak, peak anodic to peak cathodic dependence that is equal to 0 for a reversible process. I should point out that although you can think of lots of different kinetics schemes, E, C, C, E irreversible and whatnot on an electrode surface, when you have a chemically modified electrode, it's the reversible one that is the one that is the interesting one. Because with anything else, you only have a chemically modified surface for one cyclic volatometric scan. And after that, you've converted it into something else. So we are specifically interested in the reversible case here. So not working out the other cases is OK. So we have 0 peak to peak separation. And now we have a situation where the peak is equal to the half wave potential. And that is equal to the standard redox potential in the case where we can ignore these nearest neighbor terms. There's another parameter to think about here. And that is, we now have some interest in the full width at half max of that peak. It has a well-defined full width at half max. It's given by the Anson equation. And it turns out that for the reversible system, that will be 90 millivolts divided by m, the number of electrons in the Nurentzian equilibrium. So delta E, let's see. I have to continue my list down here. Delta E, full width at half max, full width at half max, equals 90 millivolts divided by n. And finally, I can go and integrate this area under the curve, under one-half of the curve. And that's the number of electrons that takes the total oxidize the surface. And if I know whether n is 1 or 2 or 29, then I can calculate the number of molecules that are on the surface and are redox active. So I can tell you the surface coverage. I can tell you my gamma total by doing this. And of course, gamma r and gamma o change as I scan through this thing. But always equal gamma total, hopefully, doesn't fall off. So I get a lot of information from doing this. And I have just, by the way, given you, again, the ideal case, it also turns out that this number will change according to Lavron if these r's come into play. So again, this is r0 equal, well, not, yeah, equal r, r equal 0. I'm doing ideal, the ideal reversible case. So the first thing that this lets me do is figure out whether or not the molecule is on the surface. I attempt to put it on the surface. Did I succeed? And it's the linear scan rate dependence. That's probably the best indicator of that. But of course, the peaks lining up with each other is another good indicator. Like solution cyclovoltametry, though, the peaks may not exactly line up. And that doesn't mean it's not on the surface and it's not reversible, because once more you have that IR loss that can come into play in these systems. I also can run into an issue if my activity coefficients are non-zero and not equal to each other. That is my nearest neighbor interactions. I can use the 90 millivolts to figure out if it's on the surface also. But again, I might find a system. In fact, 9 out of 10 systems are broader than 90 millivolts because there aren't nearest neighbor interactions. So actually, this term is better to use to work on the nearest neighbor interactions. That is, figure out it's on the surface and use this to get the nearest neighbor interaction parameters out of it. And finally, I can figure out how much material is there that is electroactive. I can't say anything about stuff that might be there that is not redox active. So here's my favorite system. This is the nickel-nickel ferricyanide electrode. So here's a chunk of nickel. That's the electrode. And sitting on its surface is a very thin layer of nickel ferricyanide. So we have these green balls right here that are nickel ions in the 2 plus oxidation state. They are locked into that oxidation state for kinetic reasons. They're not going to change. Over here we have these gold balls. Every other ball is a gold ball. That is an iron. The iron can be in the 2 oxidation state or the 3 oxidation state depending on the electrode potential. So I can cycle those. Each of these nickels and irons is hooked together by a cyanide ligand. Carbon is always on the iron side. Nitrogen is always on the nickel side. You end up with this cubic lattice on the surface. How do we make this? We simply take ferricyanide, dump it into solution, take our electrode and set it out of positive potential, generate nickel ions by anodizing the electrode. The nickel ions shoot out aqueous solution. You make nickel ferricyanide. And either you would make so much nickel ferricyanide that you consume your electrode and you would not have a future career as an assistant professor. Or you could be fortunate like Professor Bocazli. And you would find out that it's a self-limiting process. And you can make about 1,000 angstroms at the most of nickel ferricyanide. And at that point, you generate no more nickel ions. That is, it shuts down and you have a stable surface with a nickel that no longer decomposes, which is actually what I was trying to do, and a nickel ferricyanide lattice that I can put electrons into and out of via these irons here. Now, when I do that, I obviously change the charge on this lattice. And I have to keep electroneutrality. In the body-centered cubic position of these lattices, we'll have whatever cations I use for my supporting electrolyte and have around. And so when I pump an electron in here, I will cause a cation to move into counterbalance of charge. And when I pull an electron out through the electrode, I'll push a cation out into solution. And so I have an electrical current, if you will, a redox current on this side of the interface and an ion current on that side of the interface, moving in and out. There goes my ion. OK. I can take cyclical tanograms of the system. Nickel ferricyanide on nickel. First, we just did a whole bunch of them. There's scan 1425 versus scan 50. And to show at 100 millivolts a second, that it's a pretty stable system. A nickel electrode itself lasts for about 10 scans. By that point, it's got so much oxide on the surface that it's passivated. No current can flow. We have a little drop off after 1,400 scans, but it's not bad. We have stabilized the system, practically speaking. Now, what have I done? Is the molecule on the surface? What is the molecule and whatnot? So we do a scan rate dependence. And the first thing you'll notice is that it does have a very different shape from the solution cyclical tanograms I showed you the other day. Everything is more or less lining up here. There are peak-to-peak separation. Actually, it's about 10 millivolts. A little bit of IR drop. And we have a linear scan rate dependence. So you'll notice here's 200 millivolts, for example. And if I go up twice the distance, there's 400 millivolts. 200 millivolts, cut that in half. Go half down, there's 100 millivolts. So a very linear scan rate dependence over this whole range from 50 millivolts per second to 500 millivolts per second. Only one order of magnitude, but that's good enough. And I have a redox potential in this. This is done in a supporting electrolyte, which is aqueous sodium nitrate, one molar sodium nitrate. And you can see about a little shy of 0.4 as 0.38 or something like that. That was about what you expected for your nickel ferrocyanide. So everything looked good. And you publish a paper saying you've developed a new chemically modified surface in the germ of electron-analytic chemistry. And it is nickel ferricyanide on nickel. And it's wonderful because it stabilizes everything. And you know it's nickel ferricyanide for a couple reasons. The first reason you know it's nickel ferricyanide is it's got the right redox potential. And that's obviously a good way of characterizing a molecule. What do you expect for nickel ferricyanide? The second reason you know it's nickel ferricyanide is you take some IR spectra of the nickel layer with the nickel ferricyanide on it. And you see a cyanide peak that you associate with the nickel ferricyanide and another one that you associate with nickel ferrocyanide. That is this is an electrode that we held at the redox potential. So it's a 50-50 mixture of both of them. And so we see both peaks. But if we had the electrode held just at a very oxidizing potential, we'd just see this peak. And at a reducing potential, we'd just see this peak. And you could also do fancier things like some spectral electrochemistry in the optical region in the cell. You can't do this experiment easily in the cell because it's IR and the cell has water in it and the two things don't get along. So we did an optical experiment where we looked at the diffuse-reflectin spectrum of an authentic sample of nickel ferricyanide. And this is a different spectrum, actually, between the ferri and the ferro, excuse me. And you see this subtraction peak. And then we go in the cell and we do a grazing reflectance experiment. And you can see on the low energy side it matches very well. And we have a little bit of discrepancy on the high energy side, but we understand why that is. So we have lots of spectroscopic data, and a lot of chemical data saying it's nickel ferrocyanide. So we publish our paper and we're really happy. This just reviews my little drawing over here. Shows you that it works. That is, there's the nickel ferricyanide. A few more details here. So this is worth going over. We get the redox potential by the halfway potential. If these two ways were split, they really are not in the nickel ferricyanide case. We would take halfway in between them and that would still be a good measure of the redox potential. By the way, I got to get Tom out of a little bit of trouble. I was sitting there hammering with my fist on the podium yesterday, saying that if you didn't do a scan rate dependence, you didn't know anything. And Tom said, well, what if you just want to know? You're pretty sure you have a reversible couple. And you just want to know the redox potential for a whole bunch of things. So I really have to sit there. I have 20 molecules. I need to measure the redox potential. Go have to do a scan rate dependence for each one. And the answer is no. You probably don't have to, as long as you're not too, too particular about the redox potential to 29 decimal points. Because if you see something that looks more or less reversible at a single scan and you measure the halfway potential and you get the redox potential, then the most you could be off is by the amount greater than 60 millivolts that your peak-to-peak separation is. So assuming your peaks aren't 200 millivolts apart, maybe they're 80 or 100 millivolts apart, the biggest error you're going to make is on the order of 20 or 30 millivolts. And as long as you're happy knowing your redox potential to within 20 or 30 millivolts, you're in good shape. So we get our redox potential by taking the halfway potential, halfway in between these two peaks, which is 0 in this particular case, but not always on the surface because of these parameters, and IR losses on the surface. We measure the 90 millivolts peak-to-peak separation there, which turns out to be 90, which suggests there are no nearest neighbor interactions in this particular case. It actually turns out there are either no nearest neighbor interactions, or there are exactly as many attractive interactions as there are repulsive interactions. Now, if you had to guess, which one would you guess it is? You know, you go with the simple one. There are no nearest neighbor interactions. So of course, that's what we did. That's, of course, wrong. It turns out that the attractive interactions and the repulsive interactions cancel each other out in this particular case. So we get 90. So like the lattice isn't it just a- Yeah, that we weren't. Yes, you're right. Yes, it is. It's crystalline, and it's a big glob of interactions. And then it's, yeah. But you would think of big glob of interactions. There might be some that are more repulsive, or some that are more attractive, not a 50-50 mix. That's the way it goes. We have to make a bit of a baseline correction here. You'll notice if we want to do the integration under there, because we don't have a flat baseline. Another important point. And the next point to make is now this striped area, of course, is current versus potential. But again, that potential is really a time axis. So I could take a little box under here of known dimensions. And on the vertical axis, I have current. And on the horizontal axis, I have time, given by the organization of that equation. And therefore, current times time is the number of coulombs. So I can take and integrate this area under here and figure out how many coulombs there are. And if every iron on the surface, since I know the iron 2, 3, it's a 1-electron charge transfer, it's oxidized and reduced. And that scan, I can tell you how many irons are on that surface. I was talking to a couple of you this morning, and you were amazed to hear that the way one might do this is cut and weigh. As I can simply cut out that shape, cut out that, and know how many coulombs that mass is, it works perfectly. And it doesn't use a computer or anything. It works very well. So that's how we used to do all this, just cut and weigh it. You could go and get a fancy computer and integrate the area under the curve, but you won't do any better. Yeah, you could do it that way also, but cut and weigh is very easy. Everybody understands cut and weigh, right? What do I mean so does that you can't charge? Absolutely. It's great. I mean, think about it. Using a precision balance so you're going to get, as long as you're a good cutter, as long as when you were back in kindergarten, you really practiced with those scissors, you're in good shape. It's a highly accurate way of determining number of coulombs. This, we can also do the same experiment, but we have ruthenium cyanide out in solution, and we get nickel-ruthenium cyanide on the surface. And it has a different redox potential. And so the whole peak shape is shifted more positive. And you can see this isn't quite as clean out here. There's something else happening out there that's the decompositor of the surface, the oxidation of the surface. But you see a little bit now, you see our peak-to-peak separation is not quite zero, but it's a chemically modified surface. You have to have a little flexibility here. And again, gamma is just that striped area, the area under the electrode, assuming that everything is redox active. Here we took a very similar molecule on the nickel surface, but we took off one of the cyanides and put on the amino acid histidine. Why? Because we could do that, and it's a big bulky group. And we figured that would induce repulsive nearest neighbor interactions. It would distort that lattice, so that nice cubic lattice would have a mass. And again, you know it's on the surface because you have a linear scan rate dependence here. You have a slightly larger peak-to-peak separation, but still it's below 60 millivolts, so you know it's on the surface. But you notice that these waves are much broader than everything I showed you before. That is due to these repulsive interactions that Laveron tells us about. You can equally well have things that are narrower than 90 millivolts because it's attractive nearest neighbor interactions. OK, here's where the clever young Professor Bocazuli got himself into big trouble. So you do a really simple experiment. Remember, I told you that cations move in and out. That lattice, how do you know that? Well, you put in different cations and see what happens. And you put in different anions. You find out anion doesn't do a thing. Cations do quite a bit. So here's a series of cyclovolcaneograms for nickel-fairy cyanide and different nitrate salts, lithium nitrate, potassium, sodium, cesium nitrate. Probably the first thing that jumps out at you is that they have different shapes to them. But what's more important is that the peak potentials are shifting. And so we start with a very positive peak potential here. It gets not very positive here, but negative here. It gets more positive here, more positive as we drop down the periodic table and even more positive over there. These two, the potassium case and the sodium case, those are your 90 millivolt peak-to-peak separation. That's where the attractive and repulsive interactions happen to cancel out. These two, we have both some kinetic limitations and some repulsive interactions. And so we get different wave shapes there. But they're all on the same electrode. It's all nickel-fairy cyanide. OK, take those cyclovolcaneograms and take those redox potentials that you measured from the halfway potentials and plot them now versus the ionic radius of the cation. And you get an amazing linear relationship. Now this was really astounding because what are those ions? Those are the spectator ions. They're not supposed to do a thing. And here, they're changing the redox potential over a little over a half a volt in doing this. So we were all excited about this. We thought about all kinds of things that we could do with this. But the first thing we could do is just say, hey, we have this neat system where the cation changes the redox potential over a half a volt. Because one of the issues when you do all this Marcus theory stuff is that you might want to change, for example, this delta G term. But typically to do that, you have to change the nature of the molecule. And although you may see something that appears to follow in that Marcus theory, there will always be somebody sitting out in the audience who says, no, no, no. You changed the molecule. It's not that Marcus theory is working. It's that you picked a different molecule and it's different. So of course the kinetics are different. So here's a system where it's nickel free cyanide all the time, and yet we changed the redox potential. We can test Marcus theory with this. So but the first thing you do, you've written a communication to the Journal of Electroanalytical Chemistry telling the world about this great new nickel free cyanide surface. And so you write a second communication to the Journal of Electroanalytical Chemistry telling them about this great cation dependence. And the referee reports come back and they say, wait a minute, Bo Karsley. You can't have it both ways. And paper number one, you said you knew you had nickel free cyanide on the electrode surface because the redox potential is right for nickel free cyanide. And paper number two, you tell us you can make the redox potential anything you want by putting in different supporting electrolytes. So you get to publish one of these papers, not both of them. We got really lucky in that we happened to run the original experiment in sodium ions, and that gave us the redox potential we were expecting. But we also, this wasn't luck, this was thinking, did not just rely on one piece of evidence to identify that the molecule on the surface was nickel free cyanide. We had done the spectroscopy also. And so we were able to write back and say, well, we didn't expect this redox potential shift, but then you can't honestly tell me anybody else did either. And we do know that we have the right molecule because we have the spectroscopic information. And we presented that, so it got published. By the way, one of the things you can do is you can use this sort of as a ruler. That is, I got size down there, I got redox potential over there. So I can ask myself in this system, what is the size of a proton, for example? So for my supporting electrolyte, I use a little acid instead of that. And I just look on here and find out the redox potential. And I find out that a proton is about intermediate between a lithium ion and a sodium ion in the system as I have my little H3O plus. It's a cute little thing you can do. Another thing you can do is say, let me go back to that. Great. You drew a relationship here between the ionic radius and the redox potential, very linear. I like to think that linear things aren't an accident. But maybe you could have drawn a relationship between other things down here in the redox potential, and it might be linear. For example, wait a minute. Those are the ionic radii. And these guys are sitting in water to start with. Shouldn't I have used the hydrated radius of those systems? So let's try and draw that one, because people know how many waters there are. They think they know. Per lithium ion in solution, they say 12. And sodium is supposed to be 6. And cesium is 0 or 1, depending who you talk to. So people have numbers based on these transference coefficients. So you make that plot. And it turns out it's linear also, but it goes in the other direction. Because the sea, the hydrated lithium is bigger than hydrated cesium. It's not quite as linear, but it's linear with an experimental error. So now you have a big problem. So you try and figure out which one is the right one. It turns out this is the right one. But a lot of confusion, because the problem is this. Although you have a fully hydrated, say, lithium ion or whatnot out in solution, when it goes into that lattice, when you stuff it into that lattice, it can't be fully hydrated. Now, does it lose all its waters or two-thirds of its waters when it goes in? If you allow me to arbitrarily use the data to tell you how many waters you lose, I can get a wonderful linear relationship, but of course it doesn't mean anything. Because I don't really know how many waters it's lost. There were some nice experiments that Dan Butry did on this system who used a quartz crystal microbalance to actually assess the mass of these ions going in. It's very sensitive. And he can tell you something about the number of waters. And the simple answer is some of the waters fall off when it goes into the lattice. But you cannot nail the problem this way. Now, let's see what's the argument here would be that there's a delta G term for stripping the waters off of the ions. And that delta G term would end up feeding into this redox potential. And so if I'm doing a dehydration experiment, of course there should be a linear relationship between the redox potential and the hydrated radii. You know what I'm thinking? I'm thinking you're getting tired of hearing me. And there's plenty of time to Marta continue with this story, so let me just tell you where this is going to go. We have this water problem. And so to start to try and get a handle on this water problem, we switch from cyclic voltammetry to chronoamperometry, actually, cronocoolometry. And we do it two ways. We do standard cronocoolometry. That's this scale over here, or a set of data I should say. And we do spectral electrochemistry. We're using our ability to monitor the diffuse reflectance of the surface of the electrode to see if we have iron two or iron three there. And follow that as a function of time. And the point to make is for all these curves, and I'll have to explain what these curves are later, it doesn't matter what you do. The curves are identical. So in other words, when we see current flowing, Faraday is right, molecules get oxidized. So we turn for every coolant we pass over here, we see 10 to the minus 5 moles of iron two oxidized to iron three. So I can use either of these panels. So we're going to take a look at that. We're going to apply what we know about the kinetics of a chronocoolometric experiment and try and get a handle on this process this way. Both to learn something about dynamics, but also to address this problem of, are the waters coming off and going on? And what is the right radius? Because if the waters are coming off, going on, not only is there a thermodynamic component, but there should be a kinetic component. It takes some time to lose the waters, potentially. So we will look at that in our next encounter, which is tomorrow. So we have a problem set due tomorrow. And I'm also going to hand you a little take-home exercise tomorrow, which will be due the following Tuesday. We'll see you then. Yes? It's going to be open book, open notes, not open with other people. Work on it yourself, but you may use your textbook and you may use your notes. And again, it's going to be problem-based, not derivation or theoretically-based.