 OK, I think we'll get started. For those of you who had trouble locating this room last hour, I'm Andy Bocarsley. I'm here from Princeton University, where I am in the chemistry department. I've been asked to remind you of a few things before we get started here. The first is if there are any questions, and I may forget to do this, but I'm going to attempt to repeat them. And that's not because we won't really hear each other in a room this size, but I'm the only person that's miked. And so that these questions get recorded correctly, you'll hear me sing them again if I remember. Second point, much more important point, is that there's information about restrooms posted at the back of the room, in case you need that. Third point is that they're asking for no food in the room, however, drink is fine. They don't say exactly what kind of drink is OK or not. And I'm also supposed to tell you, in case you haven't figured it out yet, that this lecture is being recorded. The good news is that the camera is focused on the boards, and me, you're more or less out of the field of view, but it is recorded, and it is available on the course website, and it is also being webcast to MIT. And then finally, the telephones in the room are muted during the lecture. Now a couple other things that I forgot to mention last time. The first one is my office at this point is located in Crelan 353. Play with this again. And what I didn't say anything about was office hours. And that's because I'm not a big fan of office hours. I'm a bigger fan of if I am there and you can find me, I will talk to you. So stop by, say hello, and I'll be happy to chat with you. The one trick here is I'm changing my office in a couple of weeks, I'll be moving over to noise. I believe it's the third floor of Fred Anson's office. So for the next two weeks, I'm in Crelan, but after that I'm shifting. But please just stop by. I've been getting on to campus about 9 o'clock, so I should be available after that. Be happy to chat with you if you come by. The other thing that I should mention to you is Tom, our TA, has posted on the website a class calendar, course calendar, which gives the subjects we're top covering per week and the corresponding parts of Bard. What I'm really bad about is reminding you guys what chapter of the textbook that we're in. So that is posted. Tom said he's going to help me out by posting signs. He's already done that, I noticed, right on the website to tell you where we are. If you have any question about that, though, please ask me. Primarily what we talked about last time was pre-Bard stuff, the sort of introductory. We got a little bit in a chapter one. So if you're just opening the book now and starting to read chapters one and the beginning of two, you're exactly at the right place. Any other questions? Now hopefully we have everybody's email address, both those of you who are taking the course officially and those of you who are auditing. If you think that might not be the case, then please see Tom because we are doing email communication. And there's one share over here if you'd like. Me again, or you can stay over there. It's the last share. It's going for a special price. OK, electrochemistry. Last hour I introduced the idea that the first thing that happens when you apply a potential to an electrode, put some charge on the electrode surface, and that causes the electrolyte to become organized in terms of the electrostatics of the situation. And we end up with a double layer. And so unlike Faraday, we find that current, in fact, can flow when my computer is doing like that. There we go. Current, in fact, can flow when there is nothing being oxidized and reduced. And I showed you this example of that last hour. At the top here, we're looking at a jump in potential of 600 millivolts starting here. We're at no additional applied potential. We jump 600 millivolts here, and we see a flow of current. And that's in the presence of something that could be oxidized, ferrocene in this case. And you see this transient of current versus time that forms. On the other hand, if we do exactly the same experiment but don't have any ferrocene around or set a potential where ferrocene can't be oxidized, which is actually what was done here, we also see a jump in current that falls off with time. But you'll notice that the waveform here is very different than the waveform up there. This is obviously falling off much faster. This is due to this organization of the double layer, and it behaves just like a parallel plate capacitor. So what we're seeing here is an exponential decrease in current with time as we set up this double layer structure. And one very pragmatic point about this is quite simply, if you're interested in looking at redox events, things happening out in solution, then you don't want to be fooled by some signal here and think it's this signal over here. One way around that is to make sure you're working some time past the actual jump. So if we go out here, each of these boxes is 2 milliseconds. So you can see if we go out 4 or 5 milliseconds, there's almost no current associated with the double layer capacitance. And all we're seeing out there then is the ferrodeic current. So I'll talk about ferrodeic current, that is the oxidation reduction molecules and non-ferrodeic current, this double layer charging effect. So where I wanted to start today is with the non-ferrodeic aspect and just look at that for a moment. And introduce to you another idea, and that is one way that we can analyze this sort of data. It'll become more important later, but I'll introduce it now with the idea of an equivalent circuit. That is, can we assemble a series of resistors and capacitors that would mimic this sort of current potential response? And then can we take that equivalent circuit and assign some chemical significance to it? So in this case over here, if you buy my idea that this is an electrode down here that's behaving just like a parallel plate capacitor, then the equivalent circuit might be something like this. We'll have some voltage drop across here, of course. But we obviously have some resistance, which is the resistance of the solution species. We have an electrolyte there. It has some resistance associated with it. And here's our capacitor, excuse me. And it's setting up this double layer. So if we looked at the response of this circuit to a applied step in potential, say from 0 volts to 600 millivolts, as was the case over there, would that, in fact, give us the trace that we're seeing over there? And one way we might think about that in this is just a review of freshman physics. For those of you who had, maybe that was a long time ago, the voltage drop across this circuit then is simply going to be the voltage drop across the resistor plus the voltage drop across the capacitor. And you'll recall that the voltage drop across the resistor is given by Ohm's law, so I times r. And the voltage drop across a capacitor is going to be the amount of charge collected on the capacitor divided by the capacitance. So just remind you v equals IR. And amount of charge on a capacitor is the capacitance times the voltage. So just assuming that we have real live electrical components, that's what we would expect in terms of the response. Also remind you, of course, that the current is just going to be the change in charge with time. And therefore, reorganizing my top equation, my current is equal to minus q over rc plus v over r. I could integrate that statement then to come up with the charge that builds up in this circuit. So the charge is just the voltage times the capacitance times the quantity 1 minus the exponential of time divided by rc. And that introduces this idea of a charging constant r times c for this circuit. Of course, I can differentiate that to get the current response, which is going to be a little more interesting. By the way, in doing that integration, I'm assuming pretty reasonably that at time 0. There's no charge in this circuit. Get rid of my constant of integration. And then if I, again, take the derivative I have that the current is just v divided by r exponential of minus t over rc. And you'll notice that v divided by r, according to Ohm's law, is just a current. This is the initial voltage that's passed across the circuit. Of course, we're holding it there because we're using the potentiostat. And whatever the resistance is, so those are invariant quantities. So I might just call that i naught, some constant, e to the minus t over rc. That's where i naught in this case is just going to be the initial current that flows in that circuit. Which you'll notice, if you look at this state over here, is not a number that's easy to get your hand on because you lose control of your circuit when you do that initial jump. And you really can't pinpoint exactly where that is. But this introduces the idea that we have a time constant. It's going to fall off as the resistance and the capacitance. And takes us back to some of the concepts that I talked about last time. And in particular, you need to have a supporting electrolyte in your system. And you need to control the area of your electrode. The supporting electrolyte controls the resistance of the circuit. If you let that number get too large, and rc is obviously going to be large. And as a result, you're not going to have a time response for anything. But this part of the circuit, that is if your intention in life is looking at some electrochemistry associated with oxidizing and reducing molecules, you can totally swamp that out by making this response a very long time response. Likewise, if the area of the electrode is large, the capacitance, which scales as the area of the electrode, is going to become a problem. And I mentioned last hour, just repeat it, that if you want a kind of typical capacitance for the double layer, it's about 20 microfarads per square centimeter of electrode. It's a good rule of thumb. Simmy, you have some supporting electrolyte around, et cetera. This experiment, when we do a potential jump, is the simplest experiment that one can imagine in the set of potential control experiments that we want to talk about. And after that, we can start thinking about, well, instead of jumping the potential, maybe we like to scan the potential use in some sort of a waveform and start adding on to it. So we're going to stick with this experiment for a while, but adding waveforms is going to be important. And the most obvious waveform that one might add is just a linear scan, where we're going to change the potential linearly with time. And obviously, that's going to affect the capacitance of this system. A second approach that we're going to look at is applying an AC potential to the circuit. And obviously, there'll be a very distinctive capacitance associated with that. So we will deal with that one later, but let's take a moment here since we have all this stuff on the board. And let's think about what happens if instead of doing a jump in potential, I have a potential which changes with time. So I'm switching over to e now for my potential instead of v, which was a voltage. I'm using a voltage because I was using an equivalent circuit. And I'm talking about dropping a voltage across the circuit. Now I want to go back to an electrochemical cell for a moment here. So I have a potential applied to an electrode. We will not be too concerned for a moment about the equivalent circuit. And this potential then starts at some initial potential e sub i. And then we will scan it at a rate omega as a function of time. So we have omega here, which is a scan rate in terms of millivolts per second. And again, let's assume we do that experiment with no electroactive species around. So nothing can be oxidized or reduced in the potential range that we're going to transverse. Then we have the same statement that we have before, but now written in potential, that if this is our equivalent circuit, that potential is going to change as the drop across the solution resistance plus the charging of the interfacial capacitance. And again, we say that's time dependent now. And so carrying this through and working out a current, we come up now with a time dependent current, which is equal to scan rate times the eraser here. Scan rate times the double layer capacitance plus a term, which is the initial potential divided by the solution resistance minus the scan rate times the capacitance, and all this raised to an exponential, which is our RC time constant. So in other words, I'll give you a little picture. What I'm talking about in terms of that first equation is starting off with some initial potential and then just linearly changing that slope omega here. And while I'm doing that, I'm monitoring my current. I'm just going to follow this equation down here, which might be a little harder to follow. But you can see initially at time equals 0, that's going to have a value of e initial divided by r. And then it drops off as the exponential term kicks in. And we have some sort of exponential decrease like this. For those of you who have dabbled in cyclical tammetry, what this leads to is the thickness of the baseline. As you're scanning, now you have a current here that is time dependent. So you're changing both the potential and the time when you do a cyclical volatimetric scan. We're going to come back to that. And for those of you who have no idea what I'm talking about, this is fine. But for those of you who have experienced this, the thickness of your baseline is due to this current. We're going to spend a lot of time on cyclical volatimetry. So we will come back to that. Now in a circuit, which is doing something besides just charging up, we're going to have to have some other components in our equivalent circuit to describe it. So let me just throw out for you a circuit that we will come back to throughout the term as a more realistic circuit for what's happening in an electrochemical reaction. We're going to have some resistance, which is just the sort of intrinsic resistance of the system. This would be the intrinsic resistance of the electrolyte, the wires that you're attaching, whatever resistance might be associated perhaps with a potential status. Things aren't working quite right, things like that. And then we're going to have this charging current that we just talked about. So so far I've drawn exactly the same equivalent circuit that I had before in just a slight different configuration. On the other hand, if I am dealing with a situation like this, then the electrons, if you will, have the opportunity either to charge up the interface and set up the double layer or go out and carry out a reduction or an oxidation reaction. So there's really two branches, if you will, to the circuit. There's got to be a second branch. And I will propose to you the second branch looks like this. What is this? Well, whether the electrons go and oxidize a molecule or reduce a molecule or build up the double layer, they have to go through the resistance, the solution resistance of the system. Everything is going to pass through that resistor. If we're carrying out charge transfer to a molecule, so a redox reaction, then we can model the interface as a resistor. That is, moving the electron from the electrode to the electrolyte or vice versa is modeled nicely as a resistor in most cases. We're going to come back to that, but accept that for the moment. Then we have a second component here, which is called the Warburg impedance. Now, so far you'll notice you could run out to a radio shack and you could go and build this circuit. No problem. It'll do exactly what we said it's going to do. The Warburg impedance is a component that you can't buy because it doesn't have a constant impedance associated with it. The Warburg impedance represents something that's fairly unique to electrochemistry in terms of what chemists typically think about. And that is, electrochemistry only happens at the electrode. So you can have this beaker or electrochemical cell full of all kinds of wonderful molecules that can't react unless they get to the electrodes. We need some sort of mass transport that's going to get the molecules from wherever they are in the cell to the electrode. Typically, as chemists, we don't think about that. Just stir it up a little bit. Everything works fine. We expect the reaction to happen everywhere. That's not going to be the case here. We have to be at the electrode. So we have a Warburg impedance, which is going to describe that mass transport. In particular, we can describe it very thoroughly if mass transport is limited to diffusion. So Warburg, in fact, when writing this out, assumed that it was diffusion. So we have a situation here where if we're going to do a redox reaction, we need to have the possibility that the charge transfer at the interface is rate limiting, that it's kinetically controlled, or that mass transport via diffusion is rate limiting. And so we'll have to pass our electrons through both those components. And they'll both play a part. You'll notice even when one is rate limiting, the other one is still there. And so we'll have an impact on what the electron sees in terms of the voltage drop. So now we have a situation where we can either take our electrons through this side of the circuit or through this side of the circuit. You'll notice if you're rapidly changing the potential, then the top branch of the circuit is going to be a low resistance branch. That is, the impedance of a capacitor goes inversely with the frequency of the current. So we can make this branch the low resistance branch by having a very high frequency sort of process going on. On the other hand, you'll notice down here that we have a fairly frequency independent components. That is, obviously, if this is a real live resistor, it's got very little frequency dependence to it and a specific impedance therefore. Warburg does have some frequency dependence. But the idea would be that as in a DC situation, this will be a very high impedance, and so all our electrons would like to go through this way. In a AC situation where we have a very fast frequency, this will be the fastest, lowest resistance pathway, lowest impedance, and so our electrons will go through this way. So we can shuttle between charging of the double layer and doing a redox reaction by picking the frequency that we're operating at. One final point then to make with regard to this, and that is, as we charge up our double layer over here, clearly there's some potential jump that we can make when we put no charge on the electrode. And as a result, we don't assemble a double layer. So there is some potential that we can apply this electrode, which is the potential of zero charge. And then there's no charging of the double layer associated with that. That potential is not clear. It's going to depend on many, many different things. So for example, if you happen to have an electrode that is chemically interacting with an ion in solution and you end up with a little fizzy absorption of an ion, then that point of zero charge has to be shifted so that you can compensate for that absorption in the system. So other than saying that it exists, one can't really say where it is. But it's an important point in terms of a point that we'd like to take as a type of zero point when we talk about these things. That more or less sums up what I have to say at this point then about this non-ferdaic response. It's going to be there all the time. We're going to have to deal with it. That's what it's going to do. What I want to do now then is move to probably what's a much more interesting situation, and that is when we have a ferdaic response and ask the question, well, what can I learn about this sort of response? First thing obviously I can learn something about since I was just talking about it is transport up to the electrode. That is, if there is this type of component here and I can put a value on it, maybe a time dependent value on it, then I can tell you something such as perhaps the viscosity of the electrolyte. Obviously, that would impact how quickly I could transport material up to the electrode. I can tell you something about the mechanism of transport. Am I just allowing it to diffuse, the molecule to diffuse up to the electrode, or is there some sort of convective process that's going on? I might be able to say something about the diffusion coefficient. Often I can of the molecule involved. That is, if the diffusion coefficient is changing dramatically because I have a big molecule that's getting oxidized to a small molecule or vice versa, then I'll see changes associated with that diffusion coefficient. Or if I simply want to measure diffusion coefficient, this is a great way of getting a handle on that. If I'm interested in the temperature dependence of diffusion, I can study it using this sort of approach. A second aspect of this is something I've alluded to as a problem so far, and that is molecules, solvent molecules, cations, anions, and you're supporting electrolyte, your molecule itself that you're interested in oxidizing or reducing does have a tendency to adsorb onto the electrode surface. So this depends whether you're sort of an optimist or a pessimist. If you're a pessimist, then you're going to want to do all kinds of things to get rid of this. This is just bad news because you want everything out in solution, and so you'll go and you'll change your electro-material or maybe your electrolyte or your supporting electrolyte to try and find a set of such conditions where this doesn't happen or it happens less. If you're an optimist, then this is a great way of making a chemically modified electrode. That is, you might as well just study that molecule stuck onto the electrode surface. One of the first people who pointed out that this was a very interesting phenomenon in and of itself was Professor Anson. There's a lot of initial studies of just chemisort molecules on electrode surfaces, things that just went there because they wanted to go there, not because anybody placed them there, and he showed that a lot of these techniques that we're going to talk about can be used to probe the chemical nature of that interface, both the dynamic nature in terms of electrochemical kinetics and also some structural aspects of that interface. That leads us to the idea in general that probably the thing that modern electrochemistry is best at doing is measuring charged transfer of kinetics, and then on a good day, moving from a kinetic model to a mechanistic model, coming up with a detailed picture of what's happening as the electrons move through the interface. And that brings up one more piece of nomenclature that we have to cover. And that is the idea that an electrochemical reaction might have several steps. It might not just be an electron going through the interface and of the reaction. But for example, let's take water. If we take water, we oxidize water. We have some electrons coming through the interface. At some point, we get oxygen if we're oxidizing it coming off. So we've obviously broken some chemical bonds. So there's not just a charged transfer occurring here, but there's real-life bond breaking chemistry. And so we could talk about a mechanism in terms of an EC mechanism. E, meaning that a charge is transferred to the interface, and then C, meaning some chemical follow-up step. That is, the making or breaking of a bond has occurred. Or I might have a CE mechanism where the original molecule in solution is not doing the electrochemistry, but it just thermally converts into some other molecule, a C step. And then that C step undergoes an oxidation or reduction. So that'd be a CE mechanism. And now I can take these two letters and I can kind of string them together as many times as you want so I can talk to you about ECE mechanisms where something, say, oxidized, chemistry happens, and then a second oxidation, or perhaps a reduction, occurs after that. And you'll see if you read the electrochemical literature, that people have five, six, seven letters that always ease and seize strung together to describe the mechanism that's occurring. That brings up something now that is extremely confusing in terms of nomenclature and electrochemistry. And that is the idea of a reversible reaction versus an irreversible reaction versus a non-reversible reaction. The hardest one to explain is reversible. So I'm going to leave that one for a moment. But let's deal with what in the world is the difference between irreversible and non-reversible, which in common usage of English would probably be the same thing. Irreversible specifically refers to the situation where the rate limiting step is the transport of the electron through the electrode-electrolyte interface. That is, there's a high activation barrier to carrying out the charge transfer reaction. That's referred to as irreversible. Non-reversible, in contrast, refers to something that involves one of these chemical follow-up steps, ECE, EC, whatever it is, that removes reversibility from the system. That is, if I have a molecule A, I'm going to need a little space over here, I have a molecule A. And let's say I am going to reduce that molecule to A minus. And perhaps that reduction could occur in a thermodynamically reversible manner. If it turns out A minus then reacts on its own to turn into C, falls apart, a bond breaks, something like that, if I do this very quickly, I can turn A into A minus. And then I can very quickly take it back to A. And I'd say that's a reversible process, a thermodynamically reversible process. We'll understand that. If I'm doing it a little more slowly, then A minus, some of the A minus will convert into C. And of course, once I have C, I can't turn it back into A. And so that has a degree of non-reversibility associated with it, because I can't get back to where I started. So that's non-reversible as opposed to the situation where I have a big barrier, and so I can only go in one direction, perhaps, and maybe even do that slowly. That's irreversible. I'm going to have to worry about that in that case. So let's think about what reversible means now. That's the big challenge, reversible. We all know what reversible means in terms of there are some, if you like, some areas to sit on the side here. Do you have some people who are more comfortable? Yes? So would all the non-reversible reactions then end in, say, letter C? They may not end in C, because it could be ECE, which could have a degree of non-reversibility to it. But they'll have a C in there somewhere. Yeah. And again, let me remind you, because I've been speeding along here, if you want me to slow down and ask a question, then we'll go through a lot of material. Reversible. So we all have a very comfortable feeling about what reversible means in a thermodynamic sense. We all understand that we can take a system at equilibrium, in general, and we could perturb that equilibrium. And the system will adjust itself to this new, this perturbation, to a new equilibrium. And if we do that in infinitesimally small steps, then we call that process reversible. And we know there are a few chemical reactions that approximate that. The most obvious one being acid-base reactions, we can do nice titrations. And even though we don't do the titration in infinitesimally small steps, the steps are small enough that a thermodynamic description of acid-base chemistry is what we all use. It works just fine. We assume we're in thermodynamic equilibrium at all points in that reaction. We have electrochemical reactions that are like that, where we can carry out a pretty fast re-establishment of equilibrium. And we have electrochemical reactions that are not like that. So electrochemists have this term reversible. And it is commonly used to mean something more than thermodynamically reversible. A term which I will use interchangeably with reversible is nirstian. You're all aware that the Nernst equation describes a reversible situation. It's derived under conditions where the free energy is 0. Any time the Nernst equation or an approximation of the Nernst equation holds though, we'll call the system reversible. That is clearly we can make changes in the Nernst equation, much like an acid-base titration, describes what's happening very nicely. So yes, this gets a little fuzzy. Because I might, for example, have a potential stat that can change its potential very, very quickly. And as a result, I might see all sorts of transient currents that are occurring on a very short time scale. And I'll say, well, the system isn't reversible because I can see other things in there. On the other hand, you might have a potential stat. You know, you got the cheapo model. And it doesn't change its potential very quickly at all. You shouldn't have skimped. You should have got the good one. And as a result, you look at the same system and you say, oh, it's reversible. Because you can only see things that are happening, say, on a one-second time scale. And I can see things that are happening on a microsecond time scale. So we get into a little bit of an issue here. But there is a general situation where we'll feel comfortable calling something reversible. That is, an example I think that Barge is a pretty good one. If I take a spring, mount it on the ground and put a weight on it and compress it, if I now pull that weight off, we'll all agree that that is an irreversible process. The spring will expand out. And there's no reversibility in carrying out that process at all. Now, on the other hand, if I took that weight and I removed portions of it that were infinitesimally small, I guess I wouldn't get anywhere very quickly, well, we'd all agree that was a reversible process. Now, on the other hand, I can go and say, remove little chunks that are not infinitesimal but are sufficiently small that the standard equations that describe a spring under equilibrium apply to an excellent approximation. And so there's some point in between pulling that whole chunk off and pulling off little chunks that we say the system goes from reversible to irreversible. And so there's a nice fuzzy feeling that electric chemists have about this term reversible. I'll be referring to that, meaning that simply the Nernst equation works. Not that I'm in thermodynamic equilibrium. Now, the Nernst equation working brings up another interesting nomenclature issue, which is a strange historic thing, but is there, I think when you talk about it, because it confuses me all the time, is a freshman chemistry issue. By the way, while I'm erasing the board, Nernst received the Nobel Prize. What did he receive the Nobel Prize for? Nobody knows, huh? It's not the Nernst equation. So everybody has a freshman chemistry text and has the Nernst equation in it. But what he got the prize for was the third law of thermodynamics. Not around. OK, so the Nernst equation falls out of this statement, where Q is a reaction quotient. So you all know this. You've memorized this. You dream about it every night. And normally what we do is we simply say something like delta G equals minus N, Faraday's constant, f delta E. You plug in, and you have the Nernst equation. And of course, what it's do, if you're right the same way you wrote it before, you're not exactly plugged in. Delta E equals delta E naught, standard redox potential, minus RT over N, Faraday's constant, log Q. Often you'll see this written away. This really isn't the Nernst equation. That he used the base 10 log over here. So I really should write this. Now, electrochemists do not seem to like this negative sign right here. If you look in Bard or a number of other textbooks, you will see it written with a positive sign. But obviously, the algebra here is pretty straightforward. There's got to be a negative sign there. So how do we get around that problem? And the answer is, it goes back to this Texas convention versus IUPAC convention that I was mentioning to you last hour. That is that the IUPAC has decided that the standard way of writing a half-cell reaction will be as a reduction. So the reaction that we will spend most of our time studying this term is this reaction right here. It's a wonderful reaction. Let's get rid of that plus sign. I got in the way of my reaction. Oxides molecule plus N electrons goes to reduce molecule. Can't go wrong with that reaction. Great reaction. How does Bard take care of this? Well, he simply goes and he defines Q as oxidized over reduced. Our standard definition of Q would be products over reactants. So it should be reduced over oxidized. Bard clips it around. Other authors will keep it as products over reactants, but won't tell you if they are using the IUPAC convention, which obviously would have reduced as the product, or the earlier convention associated with Latimer, which has the oxidized as the product. And so you can flip the sign that way without doing anything by simply not mentioning which convention you're using. So be careful. That is, if you're reading Bard, you'll see a plus sign there. And you'll notice that what he's done is put oxidized on top, is how he gets a plus sign there. If you're reading a textbook such as Giliotti's textbook, he just has products over reactants and has a plus sign there and never tells you what's the product and what's the reactant. And since he has a plus sign there, he's using the Latimer convention. So we understand the Nernst equation now. I am going to just confuse you right this way, because this makes the most sense to me with the negative sign there. And my products will be reduced species, and my reactants will be the oxidized species. Straight forward on that one. Now that we have our Nernst equation, we have this concept of reversibility. So any time this equation holds, to the extent that I'm able to monitor it as the guy who's doing the experiment, then it's Nernst and hence it's reversible. And any time I stray from this, then I am not in a reversible situation. I'm either irreversible, non-reversible, or let me throw in one more term, quasi-reversible. What about the situation where I have a pretty good potential stat? But my potential stat allows me either to, say, scan the potential slowly or rapidly. And when I scan it slowly, everything looks nice and reversible. And when I scan it rapidly, everything looks irreversible. So I can actually move from this regime where I'm following the Nernst equation to a regime where I'm under kinetic control. And that we'll refer to as a quasi-reversible system. And again, I can't put numbers on this because it just depends exactly how you're doing the experiment. So we're ready to do some experiment. So let's go to probably what might be considered the first experiment in terms of modern electrochemistry. It's an experiment really that historically overlaps a lot of what I would call ancient electrochemistry experiments. My definition of ancient electrochemistry, we call, is measurements of potential. So measurements that had to do with the Nernst equation, kind of things you see in freshman chemistry, will be old-style in that you don't have to measure currents. To do that, my definition of modern electrochemistry is an experiment where you measure a current, which really only became possible after 1950. But there were some people who didn't realize that before 1950. So they went ahead and did good experiments anyway. The experiment that I want to draw your attention to is one that occurred in 1905 when a researcher by the name of Tafel decided to look at hydrogen evolution. And this was done at a mercury electrode. And actually, he's starting off in acidic solutions. So what I really should put here is HVO plus. And if you read Giliotti's account of what Tafel did, then Giliotti thinks that Tafel is probably the luckiest researcher that ever would on the face of the Earth. Because to see what he saw, this was probably just about the only system he could have picked. And since he didn't know anything about electrochemistry because nobody knew anything about electrochemistry, he really lucked out here. So he's looking at this reaction. And what he observes is that there is a relationship between a current associated with that reaction and the potential that his electrode is at. And that relationship is given by this equation that he writes down. A equals A minus B log I. It's just an empirical relationship. That is, he sees that for a given current, there is a logarithmic relationship on the potential that the electrode has to be at. And he throws a couple of constants in there that takes care of that. And this B, by the way, to this day is known as the Tafel slope. Turns out to be a kinetically important parameter. Now, why did he get lucky? Well, the first thing is he'd only see the rate-limiting step like everybody else in reaction. So he had to pick a reaction where the charge transfer across the interface where the kinetics was the rate-limiting step. That is, if it was a mass transport limited reaction, then there would have been an issue. So he cleverly picked a mercury electrode. It turns out mercury is a horrible electrode for reducing protons to hydrogen. And so it was rate-limiting. So that was lucky, yes, number one. Number two is he picked this reaction. If he had picked some other reaction where he had said maybe put a millimolar of some thing to be reduced into solution, then even though this was sluggish, he was still had a mass transport component. And he would never have been able to decipher this relationship. And instead he picked something where he's got 55 molar reactants sitting there. So there is always plenty of water molecules sitting at the electrode surface. So he has no issues associated with bringing material up to the surface. Third, he runs this in one molar acid and he has no resistance issues. There's no big resistive drop across the electrolyte. He has no double layer problems. And finally he picked a mercury electrode, which not only has overpotential, that's very high, but is one of the few electrodes where you will not get a lot of physisorption of ions on the electrode. And that would have destroyed this relationship also. So he just got lucky. He picked the right electrode and the right electrode. And he got this relationship. And he didn't realize that it was of fundamental significance. To him it's just a way of quantitating what he observed. But it turns out that is the important relationship in terms of relating current and voltage. That is the kinetic statement that we have to deal with. Now, I'm going to write this just a little bit differently. Let me just recast it and say that the current is equal to, I'll call it a prime this time, some constant, e to the eta over b prime. So I changed Tafel's parameters from a and b to a prime and b prime because I have no easy way of taking the antilog of this equation and keeping those constants the same because I haven't defined them yet. So that's it. But this is some sort of voltage here or potential. And this is going to become the over potential. The idea is that if we were in a reversible system, a Nirstian system, then if we put our electrode right at the redox potential for evolution of hydrogen, whatever the Nirst equation told us was the right potential to use, then there would be zero over potential. And the reaction would just go if it was a reversible reaction. To the extent that we have to go beyond that potential before we start to see any hydrogen evolution, we have to go over that potential. That's the over potential. So on the one hand, when we teach freshman chemistry, we tell everybody that the potential, just like I wrote on the board a moment ago, is directly related to the free energy. And so it's a thermodynamic parameter. On the other hand, this equation is telling us that it's a kinetic parameter. That is, recall that the current is the rate of the reaction. The current is dependent on the potential. And so if I change the potential, I can change the rate of reaction. It's obviously a kinetic parameter. So it's both, which is a little bit confusing. So we have over potential, which relates to the kinetic aspect of potential. The amount of potential passed, and it's just a very simple working definition, the amount of potential passed, the potential that thermodynamics tells us we need that gives rise to a current. If we cast it this way, the reason I've done that is to point out, you'll notice that if I want this system to give me an over potential of 0, then A prime is equal to the current at that point. So there's some special current, which I'll call I0. Not to be confused with this I0 over here. This is different. Now I'm in trouble. I0 over here had to do with a simple RC circuit, and that was the zero time current. Over here I'm simply saying there is a current that has some distinguishing features in that it has to do with the fact that it is going under conditions where we're at under thermodynamic equilibrium, or at the potential where we expect thermodynamic equilibrium. That is, there's zero field. There is no electrode potential that is pushing this reaction. This reaction is just going because of the innate thermodynamics of the system. Now it turns out over the years, this term over potential, if you're not confused enough yet, has been redefined several times. And so now there are three kinds of over potential. Clearly, what Tafel saw was an over potential that had to do with the rate of reaction. Rate of reaction has to do with kinetic. So it has something to do with the activation barrier. So we have one sort of over potential, which is a kinetic over potential. And that's what Tafel was looking at. I just argued the reason he saw that was because he picked conditions where nothing else would be limiting. Since then, we've come up with the idea that there is an over potential associated with the resistance of the cell. That resistor in my equivalent circuit that comes before everything else, that r sub s resistor, the solution resistance, gives rise to a potential drop across the electrolyte equal to whatever current you're flowing times that resistance. So as I kick up the current in my cell, I need to drop more potential across the electrolyte to maintain that current. Do some heating of the electrolyte, in other words. And so I have an over potential associated with that. Final over potential is called concentration over potential. What it really is is the mass transport limitation. Assuming again that I'm not doing something like reducing 55 molar whatever, then I can run out of oxidizable or reducible species near my electrode surface, and I have to bring them up by some sort of mechanism. So there is a potential associated with the energy needed to bring stuff up to the electrode once I have consumed the initial amount of material that's near the electrode. And so one might simply say that the total over potential is just the sum of these three things. The kinetic over potential given just by Tafel, the resistive losses in the system, and the concentration over potential or mass transport limitation. Now if you simply went and took these three numbers and added them together, like I'm suggesting here, you would not come up with the total over potential of the system because these values interact with each other. But to a zero with order, that's what we're talking about now. We talk about over potential, those three types of over potential. Just to give you a feeling then for how this works out in a real system, let's take a look at some numbers here. Any questions yet? Got a little bit of a laborious process here. A bigger eraser. How about a bigger board? Oh, I missed somewhere here. Let's see. Now this is tricky. All right, I've got no idea where that is. But we'll let some other people worry about that. I will try and write around that board. Let's make sure the board though. Let's see. Can I limit myself to maybe just working over here? Something like over here maybe. Close. All right, I'm coming in on it. OK. So Tafel has this relationship that he observes. We know some other things. We know, for example, that if we're looking at a process where some molecule hits an electrode and either receives or gives up an electron, that we could describe that process as a first order chemical event. So we could talk about a rate of reaction. Let's see. Let's take our oxidized molecule here. Disappearance of oxidized molecule in reduced. That's going to go as some rate constant times a concentration term. That concentration term is a little bit tricky, and we're going to have to spend some time on it, because remember the only concentration that we will be interested in is the concentration of the reactive molecule at the surface of the electrode. So at time equals 0, before I start an experiment, obviously I pour some molecule into my electrochemical cell, and the concentration is the same everywhere. It's not an issue. It's just the concentration. But as the reaction proceeds, I will build up a concentration gradient, because I'll be consuming this stuff near the electrode. And although perhaps on average throughout the cell, even a short period of time, this hasn't changed, it will have changed near the electrode surface. So we'll have to deal with that later on. But we're talking about a situation like that. And now we've added a new wrinkle in that we already know that good folks like Arrhenius have told us that that rate constant has a temperature dependence to it. But what Taffel is telling us that when you do an electrochemical reaction, not only does it have a temperature dependence, like every other reaction, but it's got an overpotential dependence. That is, the rate of the reaction, the overall rate of the reaction, depends on what my overpotential is. And with that current, it's very closely related to this derivative right here. And there is no explicit overpotential in this statement. So it must be in the rate constant that observing the overpotential. So in fact, if I wanted to put this in an electrochemical terms, I could just say that the current is equal to that derivative, with that negative sign out there, times some terms that get us into the right unit. So this current is in units of coulombs per second. This rate is in terms of molar per second. So I have to get coulombs to moles or moles to coulombs. That's Faraday's constant. I have to take into account the number of electrons that need to flow here. That is, this might not be a one-electron process. In this case, we probably write this as a two-electron process. So there's some number of electrons that we have to adjust this thing by. So that would be N. F is Faraday's constant to get us into moles. And then one could imagine that this might also depend on the size of the electrode, the bigger the electrode, the more collisions per second that will occur. And so there should be a term there. Normalize this to the area of the electrode. Or if I wanted to write this in terms of reduced species, I might just drop the negative sign. All of which to say is that if I were to go and make a plot, then, of the rate constant versus over potential, I'd start off at some rate constant, k naught, at zero over potential. It's just the intrinsic rate of the reaction at zero field. And then as I increase that potential, that's going to go up. And presumably, that's related to this statement, so it's going to go up exponentially. We'll have to be a little concerned about that, but I'll take a few minutes before I get there. So what are some actual numbers? OK, is the board going to cooperate this time? Oh, we're doing very good. Excellent. OK, since I'm on a winning streak here, let me see something, see if I can. OK, got my board back. I'll leave that there. Good. So one way we could think about this over potential is in terms of this self-exchange current, this initial current, the zero field current that we're dealing with, this A parameter or A prime parameter that I gave you from the Tafel equation, that is I naught. So if I specify I naught for an electrode material, then that would tell you something about whether there's going to be a large activation barrier or a small activation barrier, and whether I'll need a large potential to get a decent rate of reaction or a small potential to get a decent rate of reaction. And there's no way of directly specifying an over potential per se, because I might say, well, I think a reaction is going just fine if I generate 1 milliamp per square centimeter. And you might say, no, if you don't generate 10 milliamps per square centimeter, you're not doing a reaction. So obviously, a 10 milliamp per square centimeter will need a larger over potential than a 1 milliamp per square centimeter reaction, a rate of reaction. And therefore, we can't really talk about the over potential unless you want to talk about it for a specific reaction, and I can give you this whole plot. I just want to quantify things in terms of one number. The way to do it is in terms of this I naught value, the self-exchange value. So these are values taken for the taffle reaction. That is the reduction of protons to hydrogen. And this is done in one molar acid, one molar H2SO4, actually, pure acid. And I'm going to give you the numbers now to make your life complicated as minus the log of I naught. So a small number is good in this case, and a big number is bad. The log is simply that. We're going over many decades here, and we have a metal. So that the best you can do in terms of a metal that will evolve hydrogen is palladium, which has a minus the log of I naught of 3.0. So in other words, that corresponds to 1 milliamp of current. And this is per square centimeter. Everybody's favorite is platinum. That's very similar at 3.1. Then you can go to things like nickel, which is often considered a very reasonable material. Not as good as palladium or platinum. Certainly jumps up to 5.2. Gold, which is gaining electrochemical interest in the last couple of years, 5.4. Then we get things that are not quite as expensive. So if we go to, let's see, titanium, 8.2. Remember, this is a log scale, so these are big changes that I'm talking about. Cadmium, really pathetic at 10.8. Lad, 12.0. And mercury, 12.3. So in other words, 5.7 times 10 to the minus 7 microamps. 9 orders of magnitude and rate difference by simply switching to those metals. C. C. Tefl really was very fortunate. And again, what that's going to have to do with is simply a difference in the rate of pushing electrons through a palladium interface versus a mercury or lead interface. Now, there's a general rule of thumb when you're playing with kinetics and want to think about mechanism. And that is any time, not just for electrochemistry, but any time you see rates that vary that much, it's a pretty good guess that there's not one mechanism operating there. That is, whatever's happening over here is probably not the same mechanism that's going over there. It's very hard to think that you're doing things exactly the same way, but can do them 9 orders of magnitude faster or slower. So we have different mechanistic situations here. And this has been studied extensively. Essentially, in the mercury interface, you are taking protons, you're making hydrogen atoms, and the two hydrogen atoms, we can argue whether out in solution, or just very, very close to the surface, come together and make an H2 molecule. When we're dealing with the metals up here, it goes through a hydride mechanism where the proton comes, protonates the surface. There is a charge transfer on the surface with everything connected, actually bonded to the surface where you have the equivalent of an H minus, if you will, stuck on the surface. That is, a metal and a hydrogen are bonded together. You get, either depending on the mechanism, a proton coming up, if it's a true H minus, and making H2, or in the more general case that you actually see for the best metals, it's called the metal hydride, but it really isn't. It just has a little more electron density than a neutral hydrogen would. But there's a palladium-hydrogen bond, and two of those palladium-hydrogen species come together to make H2. So you have a variety of mechanisms going on here, and that's why you see this great diversity of rates of reaction. OK. Now, here's the tricky point. Now, let's go a little further. So we have this empirical observation, and now we want to find out how we can connect this sort of empirical observation with something a little bit more fundamental. And can we go back to something like absolute right law and figure out the relationship between what we understand about gas-phase molecules reacting in solution and the sorts of things that we're seeing over here? Now, the real importance of this discussion is if we're going to talk about absolute right law, then we have to talk about delta G double dagger. It's very important for me to talk about that because that's probably one of the major, or it may be the major. One of the things that Princeton has actually given to the chemical world. Eyring, who established delta G double dagger, well, a professor at Princeton, and we use it to this day. Now, why did this happen? This happened because if you go back and look at Eyring's paper, which has all these delta Gs and delta Gs of activation, regular old standard delta Gs and whatnot, this is a long time ago, 40s, right? And they have these things called typewriters. And typewriters, they have a keyboard that's very similar to today's computer keyboards. And there's that line of symbols above the numbers. And the only difference is it turns out that on a typewriter, when you have a symbol on a typewriter, it's like cast and lead. It's not a font that you just change electronically. So he's going along and he's using delta G and delta G non and delta G prime and delta G single dagger and whatnot. And he runs out of things on his typewriter. There's no more keys to push to give him a delta G superscript. And he needs one more to come up with this idea of a free energy of activation. And he can't go and just pick another font because typewriters don't have other fonts. They're made out of metal. So those of you who have used a typewriter, in particular a mechanical typewriter, you can kind of pull this off with electronic. But if you hold the shift key down half way and double strike a letter, then you'll just get the same letter, but it'll be spaced up or down a little bit. And so he had his dagger and any other symbol. So by very carefully balancing the roller on the typewriter halfway, he was able to get a double dagger. So that's Princeton's big contribution to chemistry. Did he figure it out? Yeah, this is an important point. Because it was, yeah, exactly. This is a key question. The question is, was it an ironing or was it a secretary? Now, the story that had been handed down to me was that it was a secretary that really should have gotten the prize for doing it. She figured this out. And she stayed free. Absolutely, didn't leave. That's right. However, recently, I was told by Professor Bernassik in my department. And he knows a lot of stuff, so he's probably right. That, in fact, I've been telling the wrong story. He said, are you kidding? Think about it. Do you really think he had a secretary? So Professor Bernassik believes he was typing his own manuscripts. And he figured this out himself. So I don't know now. I'm totally confused whether it should be the secretary or it should be an ironing who gets the prize for Delta G double dagger. Good question. OK, but now that we've established that, let me just outline how we're going to go about doing this. We have a rate of reaction for a forward reaction, say, and a rate of reaction for reverse reaction. That is, if we are talking about this reaction that I love so much, we can consider it under aesthetic conditions. First, that's reversible, we're at equilibrium, and no current is flowing by definition at equilibrium. Everything is just at equilibrium. And then we can move away from equilibrium. And as we do that, depending which direction we move in, one of these arrows will get bigger than the other arrow. So for those situations, I can define a current for a reduction and a current for a oxidation. If I define those for all conditions of potential, then by summing them together, I've described the total current. That is, even when I'm at equilibrium, all I'm simply saying is that the oxidizing current and reducing current are equal to each other. So that's how we'll approach the problem. We have a rate for the forward reaction, which just will be a rate constant, which again is potential dependent kf, depending on the concentration of the oxidized species, c sub o, monitored not only as a function of time, but as a function of position. So in other words, I need to look at concentration here, and I need to look at that concentration as a function of position in the cell and time of reaction. So x equals 0 will be at the electrode surface. And likewise, I have a rate for a reverse reaction or a back reaction, which I'll have some other rate constant, k sub b, associated with it, and a concentration term of the reduced species now. And again, I'm only interested in that at the electrode surface for all times. Each of these rates gives rise to a current, according to this equation right here. And so the total current that I'm going to think about is a current which is a sum of a cathodic current minus an anodic current. And that negative sign is just in there to take care of the sign convention on the currents, right? Anodic currents, we're going to say are positive going currents and cathodic currents are negative going currents. So I have to subtract them so that thing's working the right way. So I have the difference between those currents. And this total current then describes everything. So if I look at these two rates as a function of time, and in particular as a function of this activation energy, I can come up with a statement that relates the fundamental energetics of the cell to the currents that I'm going to observe. And I think, well, let's see. I can squeeze a little bit more in here. We won't have to do it next hour, I guess, if I do it now. Let me just remind you really quickly, since we have delta G double dagger on the board, that written in terms of the forward reaction, I have a forward rate constant, which is equal to, according to absolute rate theory, kT, where k now is Boltzmann's constant, divided by h, Planck's constant, e to the minus delta G double dagger for the cathodic reaction, divided by RT. And likewise, I have a rate constant for the back reaction, which has the same pre-exponential, e to the minus delta G double dagger, but now anodic divided by RT. Now, here's the whole trick in electrochemistry. Electrochemistry has a chemical component and an electrical component, essentially. So what I need to do is I need to take these double dagger terms and break them down into a delta G that you're all used to, the one you know and love for a chemical reaction, plus a second delta G, which is potential dependent. So I'll define a delta G bar. And let me just do this in general. We'll apply it to the cathodic and anodic reactions next hour, which is double dagger, which is equal to the delta G double dagger, which is the standard chemical potential, plus a term which is minus the change in the potential times Faraday's constant. And I'm going to add another term in there later on, but let's leave it at that. So what I'm saying to you is you have your standard activation barrier plus the fact that you're moving around charge. You're moving around electrons. And you're moving them through a potential field. That is, near the electrode, this double layer is an electric field. And so we have to do work to move those electrons through that electric field. And so we need to add a term in here that's going to take that into account. And so there's a potential drop, the difference between the potential of the electrode and the potential add-in solution. And we have Faraday's constant in there to get the right source of units there that we're going to have to move our molecules through. In addition, this term also has to take into account the fact that I start off with a molecule in some oxidation state. It could be neutral. It could be positive or negative. But once I've either added or subtracted electrons from it, it's got a different oxidation state. And so I start off with a chemical reaction, say, with moving a particle that's plus 2 arbitrarily through the electrolyte. And by the time I'm done, I'm moving a particle that's plus 1 or 0 or minus 1 through the electrolyte. So I have to take that into account also. And this is a very good place to stop in. So we will stop at this point. Are there any questions so far? We'll pick up next time.