 So what we're going to do today is do a physical treatment of the ignored part of electrochemistry and modern electrochemistry, which is the double layer part. When we first talked about this in the very beginning of the class, it was just asserted there is such a thing. There was charge on the electrode, and therefore there must be some opposite charge to balance that in the electrolyte. And it's pretty close to the electrode, so we're not going to worry about it too much. But since the capacitance of the electrode is always bigger than that of the electrolyte, all the potential drops across the electrolyte, and then you plotted Butler-Volmer plots, and all of that happened. And it was assumed that all of that potential dropped in terms of the driving force with the TOEFL coefficient, and all that kind of happened. And there was no molecular picture of what the structure of that double layer in the electrolyte next to the electrode was. You should understand, actually, that measurements of the double layer structure were the large part of what electrochemists did intellectually in the 40s, 50s, and 60s. And in fact, modern electrochemistry, as it's called, that deals with charge transfer to electroactive species, that's the basis of cyclic voltammetry, and all the other things that we've talked about is in a much later stage in the development of electroanalytical chemistry, then is the understanding of the structure of the double layer, which itself was the focus of most of the intellectual advances in fights in the literature and conceptual of how electrochemistry worked. None of that has to do with ferritate, charge transfer flow across the interface. This is all the capacitive stuff that happens when you apply a potential to a solid in contact with an electrolyte that has salt in it, ionically conducting species. No ferritate charge flow is during that process, but without this happening, ferritate charge wouldn't flow anyway because you need to drop that potential across the liquid to get the driving force to have that charge flow. So to this point, we've just assumed that all happened, but it's definitely worth understanding how you picture what that structure is, what the charges are, and how we deal with the structure of that double layer, both for metals and as the basis for semiconductors and other electrodes. Now, you heard a little bit about semiconductors in the space charge region last time, and that is just the complementary problem to the one we're dealing with today. In that case, again, the double layer of the electrolyte was assumed not to be important. And the reason is that its differential capacitance was big compared to that of the electrode. For two capacitors in series, the smallest one will be where all the voltage drops. If the electrode has the smallest capacitance, then it doesn't really matter what the structure of the capacitance of the liquid side is. All the action is in the electrode, and that's why it developed the space charge region because you move charge in and out of the majority carriers of the electrode, and the electrolyte is mostly along for the ride. And in fact, the charge in the double layer at a semiconductor electrode is relatively small because its capacitance is big compared to that of the electrode. So most of the potential and most of the charge drops across the semiconductor. In the case of a metal, it's the opposite. The metal has the large capacitance, and so all the voltage, to first order, drops across the electrolyte and all the ionic motion to counteract that applied potential, and excesses of one charge or another occur on the electrolyte. And all of that stuff was just assumed to happen before you flipped on the switch of the potential state or faster than any of the faradaic processes that happened because this has to come in equilibrium. So is to allow you to set an electric potential relative to the solution. So whenever you plugged into the Nernst equation, whenever you plugged into the Butler-Volmer equation, whenever you did any of these potential step or psychic voltammetry experiments, the underlying assumption is that when you charge up the electrode, the electrolyte responds in some way more rapidly than the faradaic process does. If that were not the case, then you could step the electrode potential, say, to minus 1 volt versus SCE, and an ion in solution would see a time-dependent driving force as the double layer started to relax to counteract that charge. And so even though you did a potential step experiment, in addition to the cattrel part, you would have a time-dependent potential experienced by that ion in solution as the double layer started to respond to that potential step. But in fact, all of that in electrolytes of any reasonable concentration happens much more quickly, the RC time constant to charge the double layer is much more rapid than any of the faradaic current response to that, and actually has to be because that's the thing that establishes the potential drop through which the faradaic current occurs. So we're going to learn about that double layer today. And this is probably, if not the most mathematical, one of the most mathematical parts of any electrochemistry class. What I probably should have done if I had more time at hand is I would have prepared nice, crisp PowerPoint slides with just the key equations and some bullet points. But I didn't have enough time to do that. And everything is in Bard and Faulkner. And so what I'm going to do is, since I scanned in those relevant pages, is to lead you through the derivations just as if you were reading it in the book, but you can go back and read it in the books. And some of it does get pretty complicated. But I think it'll be conceptually the same as if I wrote it on the board anyway and why bother me write it in a way you can't read my writing. So I've just expanded the pages from Bard and Faulkner so we can see the equations. So now let's zoom in. And we're going to think about what happens between any two phases. And in general, there's a pure phase, which we'll call alpha, and another pure phase, which we'll call beta. Those two phases could be two different liquids with two different solid concentrations. One might be water and one might be hexane. It could be immiscible. One could be a solid and a liquid. That's conventionally what electrochemists deal with. Although there can be liquid-liquid interfaces, there are also double layers at cell membrane surfaces compared to the electrolytes either inside or outside it. Any time there are two immiscible interfaces, two immiscible phases that have an interface, you will have to deal with potential ion excesses or deficiencies at the interface compared to in the bulk solution. So we can be general about it. Although we'll keep in mind that, in general, one of the surfaces is our electrode and the other is a fluid electrolyte. And there will also be a dividing zone. Now, this dividing zone could be conceptual in the interfacial zone, or it could be real. In a case of a solid, it's more conceptual that the dividing zone from the pure solid to the liquid really goes up to the boundary of the pure solid. But in the case of a liquid-liquid interface, it might not be that way. There might be some layer where you have water on one side and hexane on the other where it has some gradient from water to hexane, because even the molecules interpenetrate each other a little bit. And so in general, we would draw an interfacial zone where it's not purely phase alpha and purely phase beta, and there's stuff happening in here. And in the case of something like a platinum electrode, this interfacial zone of where the platinum ends and the solution starts might be pretty close to the middle, but we're keeping it general. This is really a reference dividing surface that we're free to move more to the left and to the right until we end up in pure phase alpha and pure phase beta. And it's going to be our thermodynamic reference point. What's going to happen at that boundary is in general, at that boundary, relative to if the boundary did not exist, on that boundary plane, there is going to be an excess of some concentration of species that would have been in one solution or the other on that boundary plane. We're free to pick that boundary plane anywhere we want to because the thermodynamics of the situation, if we assess them consistently, are going to all work out in the end anyway. But there'll be some special boundary planes. Nevertheless, if the fluids were identical on the left and the right, then the boundary plane is purely conceptual. If I have water on the left with sodium chloride and water on the right with sodium chloride, I can put boundary planes anywhere I want. There's no excess of sodium relative to chlorine over this boundary plane in this one. It's just one solution of sodium chloride. But if it's sodium chloride and water here and if it's lithium perchlorate in acetonitrile there, then the boundary plane in this interphase region is going to have an excess in general of sodiums relative to lithiums and perchlorates relative to chloride. So that boundary plane would have a defined physical position. And we can count the excess of any species on the boundary plane by taking the difference between what is there in the real system and what is there in the reference system. The reference system is the gedonkin experiment where that boundary plane is divided so that instead of having this phase different than that phase, that phase is the same as this phase. So think about the experiment where I magically as a devious experimenter take a solution of sodium chloride and water and draw a boundary plane and then I suck out all the sodium chloride and water from one phase and put in something else on the left side of my fish tank. And I compare the surface excess of sodium chloride on a boundary plane in the real system to the one in the reference system where it was the same on both sides. So the reference system is one where the phase boundary is purely conceptual in our mind because it's the same fluid on both sides. That's going to be our reference system. There can be known at excesses there because the left is physically identical to the right. And then we're going to make our real test system when the left is different than the right and we'll refer because there can be no thermodynamic difference between the left and the right when the left is the right. So the way we're going to keep track of the thermodynamic difference is by using that as our reference system and then replacing it by the real system on the left being different than the right and subtracting what happens in the real system from what happens in the reference system. And that's called the surface excesses. And so these surface excesses are the excess of any species on that boundary plane that would accumulate in a real system when the left phase is not equal to the right phase relative to what's there. Remember, it's not that there's zero sodium or zero chloride on that boundary plane in the reference phase. It just means that in that reference phase, whatever its concentration in the solution is whatever determines the amount that's there on the boundary plane. But now if I pull out sodium chloride on the left and put in something new, there will be a different amount of sodium and chlorine on the boundary plane than there would have been if it was sodium chloride on both sides. And you subtract one from the other in order to get the surface excesses. And that's what we're doing there. Yeah, that's right. The reference plane in the reference is completely arbitrary because no matter where you put it, it just reflects the concentration of the species in that phase. So now we have to evaluate the free energies. The free energies in the reference system are a function of three things, the temperature, the pressure, and the concentration of all the species in the reference phase. The free energy in the real system is a function of four things, the same three, the temperature, the pressure, and the concentration of the species on the boundary plane, but also the area of the boundary plane. So that's new. It's clear that in the reference system, the boundary plane is just imaginary. And it's just like conceptual of slicing through a solution of sodium chloride. It doesn't matter what the area of that plane is. It's sodium chloride on one side and sodium chloride on the other. Nothing matters. But if on the other hand, there is in a fixed kind of fish tank a boundary plane that is real between one species and the next, then if that constricts down to a small contact area, the free energy difference will be different than if it's an entire large area between two things of the same volumes on the left and right. So I need to know the contact area as well as the excess of species on the real system in the boundary plane and its temperature and pressure. So there are four variables that count in the real system. This is in general a set of variables. Temperature pressure and the set of excess concentrations of all the species, the solvents, the sodium, the chlorine, whatever else is in there, because this really is an index over all the species ni, the ith species in the solution. So we need to keep track of that. Well, if those are the total free energies, then you have to remember from thermodynamics how to do the differentials. And of course the differential, the total differential, is the differential with respect to the variable that you're interested in times the delta of that variable. And so if the total free energy of the reference phase is a function of temperature pressure and the concentration in excess on the surface plane of every species, then the differential on the right is dg of the right dt dt plus the derivative with respect to pressure times dp plus the sum over all the concentration excesses times the differentials of their concentrations of all the species we care about. And in those real system, there's one more term. They're all those same terms, but one more. And that one more term is the differential of the free energy in the real system with respect to area times the area that is making contact between one phase and the other. Now we only, in general, work at constant temperature and pressure. And so if the system is set up at constant temperature and pressure, then it's clear that the first two terms in each expression are 0. Because no matter what their values are, they're multiplied by dt and dp. Or the other way to think about it is if it's constant, then yeah, there's 0. And these partial derivatives are the electrochemical potentials. Now if our two phases in the real system are at equilibrium, then the electrochemical potential of any species is the same no matter where it is in the system. So what does that mean? That means that the electrochemical potential of the ith species in the reference system is the same as it is in the real system, everywhere. Because after all, the reference system is the same in both cases. The 10th Mohr-Sodium chloride is still 10th Mohr-Sodium chloride. If the electrochemical potential of sodium is the same no matter what is on the other side of the phase, as it was when it was in pure sodium chloride, then the electrochemical potential of that species, because it's in equilibrium with the same reference phase, is the same as it was in the reference phase. So these are equal to each other. So now we can take the excess. We can take the differential free energy as the difference in free energy between the differential of our system and the differential of our reference. And I'll go back one slide to remind you what they were. These two terms were 0 in both cases. This term is the electrochemical potential of each species. And so is this. They're equal. So there's one term left, except that if there's a surface excess of this species in the real system relative to that in the reference system, then these are not the same. Those are the same, but the actual concentrations on our surface dividing line are different in the real system than in the reference system, because in the reference system, that was our imaginary boundary plane. So we end up with only two terms. The differential difference between the system and the reference, our excess, is that first term plus the electrochemical potentials of each species times the difference between the surface excess in the real system minus the reference of any species. Now we've made one more thing. Where you remember there was a differential there of the derivative of g with respect to area, that partial derivative, and we'll define that partial derivative to be the quantity gamma, which we will call the surface tension. So the derivative of the free energy with respect to area is a defined quantity called the surface tension. If there were no derivative of the free energy with respect to area, then the surface tension is 0. And in our reference system, the surface tension across the boundary plane is 0, because it's an imaginary boundary plane, because if it's the left of sodium chloride and the right of sodium chloride in water, there is no surface tension between water and water with sodium chloride on the left and sodium chloride on the right. On the other hand, when there's a real system of a difference in the left and the right, there is a surface tension. And so there is a change in the free energy, its derivative with respect to that area of that boundary plane. And that quantity is defined as the surface tension. And so you can see that because the first two terms were 0 in each case, that the differential excess free energy of our real system relative to the reference one is the surface tension times da plus the sum of the electrochemical potentials of all the species times their differential excess concentrations on the boundary plane. Yeah. Yes. But I'm going to have different concentrations. That's not going to be at equilibrium. I have systems at equilibrium. They would have equilibrated until they're the same concentration. There could be a surface excess as a result of that. This real system is still going to be at equilibrium. Otherwise, the electrochemical potentials of the left and right of all species can't be the same. Right? OK? It still is equilibrium. There's a left phase that's different than a right phase, but it's still at equilibrium. Whatever has happened could be a different solvent. Could be lots of different things. So now we need to stop here for a bit because this we've just derived as a statement of truth almost by definition of subtraction of what the reference phase is relative to the system. And we can go further into a parallel argument of a different statement of truth and make a relationship between them. Because we've got a differential as a function of two variables, one area and the other electrochemical potential, or in this case, the dn, the excess. And those are extrinsic variables. Then you can go back to invoke Euler's theorem, which says that for any function, if you can write the function in terms of extrinsic variables, then by definition, that function can also be described as the derivative of that function with respect to that extrinsic variable times that extrinsic variable. And since these are the extrinsic variables, it must be the case that independent of how we got to this point that g as a function of the excess can be written just purely on a mathematical basis as dgda times a plus the sum of dg dn sigma i times n sigma i. So that's just a statement of fact. Since this is the surface tension, this is equivalent to saying that this is dg sigma of the surface tension times the area plus the sums of the electrochemical potentials times the surface excesses of all the species. We can take this and find the differential from this just statement of truth, because this must be the case for any such function g. Then this means that delta g sigma must be that plus that and then take the derivative of the other term a delta gamma and the derivative of the other term here, the change in electrochemical potential times the n i's. This is just chain rule. If this is the function and there are four terms, then it's the first d the second and the second d the first and then the same thing there. So that's also just a statement of truth. On the other hand, if this is the differential as a statement of truth for a function g, but we know that in our system, that same differential must be this, then clearly this must equal that, right? Since these two are equal, that term is common. That term is common. That means the other two terms must be 0. And so that means that a delta gamma plus these derivatives of electrochemical potential with respect to the surface excesses must be equal to 0. And we're close to an important result. It's convenient to write these surface excesses not just in terms of the number of excess of sodiums in the boundary plane. There are 10 to the 12th excesses in the boundary plane. But we really want to know the surface excess concentration. How many are there per unit area? So we'll divide this whole thing by area and define the quantity gamma i to be the actual not just number of excess of that species i, but the excess per unit area. So for those of you thinking about a surface, it might be the number of sodium atoms per unit area on that surface plane, or sodium ions adsorbed on a surface plane, right? And therefore, just bringing this to the left and dividing through by a and moving that to the right, negative d gamma is just the surface excess times the derivative of the electrochemical potential of the ith species. And that's actually the Gibbs adsorption isotherm. It relates the surface tension to a number density of adsorbed species times their change in electrochemical potential. Now, this is general. We haven't yet invoked anything to do with one phase as a solid and another phase as a liquid. This was general for any two phases. It's purely thermodynamics, and it comes about from one phase and the other phase not being identical and therefore having an excess of species at the boundary plane and our definition of the surface tension there. So now we need to go further and go into a real electrochemical situation to evaluate what all the contributions of the various different species are in that master system. Now, if you do that, I'm not going to go through that derivation for you now because it's in the book and it's very long and convoluted, I think. But you could consider an electrochemical cell like something having copper and silver and silver chloride and then KCL and solution and the other electrode having some metal and mercury and then nickel and then copper wire and then connected. This is just a phase diagram of a full electrochemical cell where there is a mercury electrode on one side and a silver electrode on the other and they both are connected via a copper wire and they're dunked in a solution of potassium chloride and there's a silver chloride reference potential in there. This is just a phase diagram you read of left to right of how you stack up all the components of this big sandwich. You can show that because there are certain things that are related to other things that if, say, there was a surface excess of chloride due to some electrochemical potential on one part, then if there's excess of chloride on one side, there must be a deficiency of chloride on the other because it's conserved. And if because the solution is always electro neutral, if there's an excess of chloride, it tells you what the potassium ions have to do too. And by going through all these relationships, you can reduce them down to a few key ones that you substitute for the other ones to eliminate them. This is a system-specific derivation, but the end result is general no matter what. The end result that you get is that that surface tension is, of course, related to the change in those surface excesses of key species times the derivative of their electrochemical potentials. In this case, in the case of that cell, there are only two surface excesses that you need. You can write it as a surface excess of potassium ions times the change in electrochemical potential of KCL and the surface excess of metal ions times the change in the chemical potentials of those metals. It doesn't always have to be this way. This is just because of that cancellation. We could have written it some other way. If we had chosen not to write potassium as a surface excess, we could have written chlorides as negative of the potassiums or some other relationship like that. So in general, if you took that general Gibbs absorption isotherm equation and applied it to the system, there would be less terms here than all the species because some define what the others must be. And the exact numbers I don't need to get into because it was system-specific anyway. But there are these surface excesses of certain key species. And the other point is that there's a charge density times the derivative of the potential. The reason there's a charge density times the derivative potential is that some of the other key species that we left out that were buried in that derivation are that in the case where one of our phases is an electrode, if it has an excess of negative charge, then that must be compensated by an excess of a cation on the other side. So one of the variables that is a species of concern in the case of one of the materials being an electrode as it was there is actually the charge on the surface of that electrode. But it's just another species, right? It's just another surface excess. Instead of potassium, mine's a surface excess of electrons. It's just another of these ite species that we're keeping track of. And we're taking it and multiplying it by the derivative of the thing that's related to it. It's dE, the electrode potential, just like we're taking an excess species of potassium and multiplying it by derivative. It's chemical potential. These should have been electrochemical potentials, but they're simplified because in the solution, as you could see, some of the electrochemical potentials are the chemical potentials. Now one other point about what these surface excesses are, just so we don't confuse them, these surface excesses are not just saying that there is an excess of potassium, for instance, as an ion on that surface boundary plane. But it's saying that there is an excess of potassium on that boundary plane relative to the bulk species or other species that are important. It's the surface excess of potassium corrected for the mole fraction of KCl in water that you're using on that side times the surface excess of water. So if there were no surface excess, that doesn't mean that there wouldn't be in this fraction there couldn't be excess of sodium or potassium, in this case, relative to its bulk. It just means that if you drew this boundary plane in this system relative to the reference, that at this given potential and where you're operating, that there's a different fraction of a surface excess of potassium relative to the surface excess of water than there would have been if you didn't apply this potential. This is the surface excess of potassium relative to what it would have been weighted by the mole fractions of the stuff it displaced. So it means a little different thing than the pure surface excess we saw before. And this, again, just works out of how the arithmetic works. And that bottom equation is taken into account that one of our phases, that the phase alpha in our real system relative to our reference system is an electrode. And so it doesn't have any ionic species in it that we're worrying about for its electrochemical potentials. It only has charge, but the other side has all the ionic species in the solution. This is perfectly general. It's derived exactly from that Gibbs isotherm we had before. This is a specific implementation for KCl on this side in that cell. But no matter what the specific ions are here, there's always going to be a surface charge and a derivative of the potential when the electrode is one of the phases. And then there's all the ion stuff on the other side. And the exact details of the ion stuff we don't need to worry about because you can derive them in each case individually. The other important point is that actually everything in this equation can either be controlled or measured. We can measure experimentally the derivative of the surface tension. We'll be able to measure the charge density. We'll be able to measure the surface excess of potassium or of the other species or control that by changing the concentrations of those things to determine the dependence of these things. And so if we know some and hold the others constant, we could solve for the other ones. So this is the link between the theoretical description and what experimentalists can do to understand double layer structure because this allows you to probe variables under certain conditions and actually determine values of these things, that the other things we couldn't determine. The other things were before this were theoretical, right? They're these free energies and we're subtracting them to get these quantities. But these are under experimental measurement or control, every one of them. Let's keep going. How do we measure the surface tension? Let's see if we have a picture. I don't think we have a picture. Okay, we're going to have a hanging mercury drop. We're going to have a mercury drop go through a capillary of a constant radius and we're going to keep feeding mercury into that capillary. So it's got an infinite big jug of mercury that can drop down a radius of that capillary and gravity will just keep it dropping down. And why does the ball of mercury get bigger and bigger before it finally drops off? Because there's surface tension acting on the mercury opposing the gravitational force from the head pressure. And there's a certain mass and a certain gravitational force and there'll be a certain time at which that drop is going to release. And the drop releases just at the instant after when the gravitational pressure exceeds the surface tension that's holding it together in the first place. And so just at that critical time those forces are equal to each other. That surface tension acts on that capillary of radius r across the circumference because that's what holds that drop in that capillary. So there's a force acting over two pi r times the surface tension. And that's equal and opposite to the mass of the mercury drop at that instant times the gravitational force. You can measure the mass by looking at the density of mercury and you know how big that drop is because you can look at it under a microscope. And so you can determine, it's radius, you can determine its mass, you know g, and you measure the time and therefore you can measure the surface tension. So the surface tension is an experimentally measurable quantity and furthermore because this is a mercury conducting material, you can do that as a function of the potential that you would apply to that mercury as it's dropping and expanding in a solution of an electrolyte. So that means you can measure the surface tension of that mercury under different electrode potentials because you just wait until the drop is just ready to fall off and you measure the time at which it falls off for a fixed capillary diameter. Okay, so this is an experimentally measurable quantity and you can change the concentration of sodium and the concentration of chloride and the potential at which you measure it and furthermore the reason people love mercury and the reason all classical electrochemistry was done in mercury is twofold. Number one, surface contamination is relatively minimal as an artifact problem because you get a fresh drop on surface every time. Two, it's relatively easy to purify mercury because it's so sensitive in its surface tension to the presence of impurities that when you have a big pool of mercury that you're using or planning on using for your drop, you can tell whether or not it's pure by measuring its surface tension and watching whether or not there's junk absorbed on the top of that pool. So it's easy to know when it's pure and the third is that mercury happens to have a very large over potential for hydrogen evolution and so you have a very large window of accessible electrode potentials where just in the presence of water, no ferritate current is going on and so you can probe what's happening in these non ferritate double layers over a very large potential range like two and a half or three volts. So it's those three things that made conveniently mercury by far the electrode of choice for 40 years for electrochemical experiments and in fact, there was a disdain of any other electrode for those 40 years because solid electrodes could not be purified, could not be sworn to be clean and all the classical electrochemists would never believe any measurements at any electrode other than a mercury drop and you couldn't measure the surface tension. So you might measure something like this. The time that it takes for that drop to release as a function of the electrode potential and it might peak somewhere and then get smaller either side. Now that peak, let's see. I shouldn't be skipping this, but I have no way. Good, we'll go next to it. Great, tells us something important. So the charge density we know is the derivative of the surface tension with respect to the electrode potential if we've held all the other species in the solution constant. Because just look at that last equation I showed you. It had all those other terms as derivatives. If I held those things constant, those derivatives go to zero and therefore since the surface tension was the charge density times delta E, if these things are constant, the only differential that matters is that one. So now under constant everything else except electrode potential, that electric capillary equation simplifies to the charge density is the derivative of the surface tension which I can measure with respect to the electrode potential which I control. And so that curve I showed you measured the surface tension as a function of electrode potential. If I differentiate that by taking its slope then at every point I can experimentally determine the excess charge per unit area on that electrode. Now that must tell me something about the double-arrowing solution but not very much. I just know that when I do this, there's an excess of charge relative to the solution and there's a special point. That special point of course is when the charge equals zero. And that's the peak. Because this is zero when the derivative of the surface tension with respect to potential is zero and that occurs either at a maximum or a minimum because the curve looks like that, it peaks, then it's the maximum value of that that is the point of zero charge of that electrode. So measuring the surface potential, the measuring the surface tension as a function of electrode potential is the classic method to determine the point of zero charge of mercury of an electrode in contact with a given electrolyte. Okay, because you measure that surface tension by measuring a drop time, you can from that directly calculate from the mass of mercury what the surface tension must be for that drop radius. So that's a arithmetic calculation. You do that as a function of electrode potential, you get then a plot of the surface tension as a function of electrode potential and its peak value is where the electrode has zero net charge by that exact electrocapillary relation. Now we know more than that because since you know the charge, not only do you know the place at which the charge is zero, but if you took this derivative, you know the value of the charge density at every electrode potential. So now I make a second column by Excel spreadsheet. The first column is what the surface tension is as a function of every electrode potential. The second is what the derivative of the surface tension is which is the charge density at that electrode potential. And then I make a third column which is the other answer, which is another derivative because the derivative of the charge density with respect to the electrode potential is the differential capacitance, right? Because that's the definition of the differential capacitance of that double layer. It's the derivative of the charge with respect to the potential because q equals cv. So this just says that c is q over v or in fact for a small change in v that the differential capacitance delta c is delta q delta v. In our case, delta v, the voltage is the electrochemical potential of the electrode difference and delta q, the charge is this charge density xs per unit area on the electrode. So the key quantity here is the drop time that gives us a surface tension whose first derivative gives us the charge density whose second derivative gives us the differential capacitance. So for mercury, all three quantities are experimentally measurable. Now for solid electrodes, you can't measure directly the surface tension. How do you do that? You have no drop. You have a platinum electrode. You can't make a surface tension measurement. You can, however, measure the differential capacitance. You need another assumption to work backwards to the surface tension. The reason is that when you take derivatives, the constants of integration drop out. But if you start out with the capacitance, the first integration of that gives you the charge density. The second integration of that gives you the surface tension, but you need to know the constants, which amounts to saying you need to know where the point of zero charge is. Because if you knew where the point of zero charge was, then that would have allowed you to determine those two constants of integration. But you often don't know that. And so you all know is the differential capacitance that you measure. And you have to independently spectroscopically or some other method infer what the point of zero charge is for a solid electrode. Another reason why people like mercury. And so, exactly what I said, let me do one more thing here, which is to remind you of the difference between the differential capacitance and the integral capacitance. So let's suppose that I was smart enough to have done this on mercury and get this curve of what the surface tension for my drop time is as a function of electrode potential. I could take the first derivative of that and I could take the second derivative of that and I could measure from that second derivative the change in charge density with respect to an applied potential but I could do that one of two ways. I could do it for a small applied potential change or I could do it for a large one. If I do it for a large potential change, the only potential that is defined uniquely is the point of zero charge. So I'd start at the point of zero charge and then step somewhere and I'd ask how the surface tension changed and I'd ask how my capacitance changed. That's called the integral capacitance. That's the total charge difference between where I start and where I end divided by the total potential between where I start and where I end. The differential capacitance is just the slope of that second derivative and the slope of the second derivative instantaneously is not the same thing as the chord defined by starting from one point and going to another point ending some kind of average of the total instantaneous small step capacitance. The differential capacitance there's just a picture of it somewhere, yeah. So if this was the charge density that you got by taking the first derivative of that curve it might look something like this. The second derivative, the derivative of this is the capacitance. If I do it for a small voltage change then of course it's the slope of the tangent to that curve around any potential that I choose to measure but if I did it for a large voltage change relative to the point of zero charge I stepped one volt that way then it'll be the slope of the chord. The only way this is a unique definition is if I know where to start. Because if I know that I'm going to start at the point of zero charge then I could have mapped out the whole curve by defining the integral capacitances what this change was with respect to zero for half a volt, for one volt, for two volts and I could have mapped all those points and regenerated the curve. Alternatively I could have measured the differential capacitances and that would have generated the curve but not located it in absolute value relative to the point of zero charge. Right? So that's where this constant of integration comes in. I can generate the curve and its functional form from measuring all the differential capacitances which you can do with a solid electrode that you can't locate it along the potential axis unless I have another variable to determine that constant of integration. On the other hand, as I said, if you work the other way and if you work from the surface tension then this uniquely defines that curve because its first derivative now this is the charge density so the surface tension was this other thing we showed you about before right there so there's the surface tensions the first derivatives of them the densities those and the first derivatives of them uniquely define the capacitance right? So we can make measurements of these the other, yeah yes well I don't want to get into it but that one's a little easier to deal with so I'll defer on that okay there's another thing we can do the other thing we can do is we can go back to that electric capillary equation and not change the electric potential we can hold it constant and we can probe the surface excesses because we made the assumption to determine sigma and the capacitance that we held the other terms constant the concentrations of species and we were going to change the electric potential and then the solution would respond but we could hold the electric potential constant and change the concentration of a species and solution and then go use the same equation to measure how the surface excess changes in response to the change in concentration of stuff because I can still measure my surface tension right? and so you can see that if we went back to the electric capillary equation for that specific case of KCL but we did not change the electric potential that term in the derivative goes away and then you would get something like if we held the metal concentration constant but varied the concentration of the salt varied the KCL concentration that the change in surface tension with respect to KCL would have given us the surface excess of potassium ions because remember there were only two terms we needed in that equation it so happened for that particular cell and if we're holding one of them constant the concentration the electric chemical potential like mercury and we're holding the electric potential constant then the only thing that affects what we measure our surface tension is the change in the stuff that we add to the solution to sodium chloride it could have been there were ten terms and we held nine constant but it's always possible to only change one of those things and then measure the change in the surface tension as a response to that one thing and that gives you the surface excess of the thing that is related to that on the electrode surface now remember this is a surface excess this doesn't mean that it's zero so if you remember those charge densities they tell you the total charge density on the electrode and of course that must be equal and opposite to the excess of the charge in the solution because the whole system is neutral over a big enough chunk of that box so it makes sense that relative to the point of zero charge that an electrode that is positive in potential is going to have more anions in the solution relative to caveins because at a positive potential of the electrode relative to its point of zero charge the electrode we would measure to have a net positive charge density therefore the solution must have a net negative charge density and so it must have an excess of adsorbed anions relative to caveins at of course the point of zero charge there is no net excess this doesn't mean that there is no adsorbed charge there's just no net excess adsorbed charge right at the point of zero charge there could be in this case there's not well by definition of this this is with respect to the excess there could be equal moles or equal amounts of potassium and fluoride both adsorbed on that mercury electrode at the point of zero charge right we don't know that that's not the case all that we know is relative to the point of zero charge when there is zero net excess of cations relative to anions because the electrode has no charge therefore the electrolyte has no excess of charge relative to its neutral state that there is an excess of negative charge at positive potentials and an excess of positive adsorbed charge at negative potentials and these can be measured independently because you can change the potassium concentration and change the fluoride concentration you can independently change one electrochemical potential and the other you can measure the response of the surface tension at fixed electrode potential to those changes and from that equation above deduce what the surface excess of each individual species are and if this all works out then it has to be the case and the check is you know independently what the charge density is because you measure the surface tension you've taken its derivative if you measure independently the amount of surface excess potassium and independently the surface excess fluoride and that's all that's in the solution then these two numbers one subtracted to the other has to be equal and opposite to the electrode charge you deduced from your electrocapillary measurements because if it's positive 5 on the electrode there have to be 5 net negatives in the solution but they're all measured independently because they're independent variables that are being changed one is changing the potential the others are changing the electrochemical potentials of the species that you're interested in determining the surface excesses of and you can see that there's no reason these are linear in this case for mercury in tenth molar potassium fluoride positive the point of zero charge the fluoride starts to shoot up and the potassium levels off and you can build molecular models of sites that might be sites of specific absorption of fluoride that would be blocking some potassium from absorbing and at negative potentials the potassium actually does increase linearly at negative potentials and the fluorine levels off at some point we would have no way of knowing what it's going to behave like all that we know is from the arithmetic is that this counterbalance by that has to be equal and opposite to the total charge density at any potential but this allows you to determine these numbers independently not just the total so it's more knowledge than just you would have gotten from the electrocapillary measurements versus potential we know what the species are and how much they're absorbed so now we're starting to learn something about the double layer structure because we know that at this potential more fluoride is absorbed than potassium and here more potassium is absorbed than fluoride but we don't know whether this is right or wrong we just make a measurement of this we would like a molecular model that tells us more about the double layer structure than just measuring these surface excesses so we need to build a model of that there are three levels of such models one is the easiest mathematically it's the most intuitive it also is the least close to reality for a double layer that's real if you have a certain charge density on the electrode we could just think about it it's a pure capacitor where the electrode is one plate and the solution that is the other plate and as I add more potential across those two plates I just bring more charge density up to the second plate as I equally and opposite put it on the metal on the first plate and there will be a certain dielectric constant and a permittivity of vacuum and a certain thickness of the medium in between that I was charging that capacitor with a given voltage that I charge it with and it produces a certain charge density right and this would imply that if I measured the charge density and I knew the voltage drop that there would be a fixed value of that capacitance independent of the voltage drop and we know that's obviously not the case let me see if I have a picture of that all you need to do is remember it looks something like that right the slopes of these things are obviously not constant as a function of electrode potential so we know a priori that the simple model that these ions are just going up to an intervening gap and snuggling up to a fixed distance and no further and no closer doesn't quantitatively describe what the real double layer is doing and in fact I do have a picture of that I just have to find it of what a real system looks like so let me go down to 100% here right there so the real capacitance might look something like this so these are real electric capillary measurements whose second derivatives are taken relative to the PZC in this case is sodium fluoride and again you can see 1947 when people were measuring all the non-ferritic properties of mercury in water in different electrolytes and you can see that this differential capacitance is hardly a constant it has structure to it and therefore our molecular model of treating this as a simple parallel plate capacitor is not adequate to explain what we observe from these second derivatives of these surface tensions with respect to electrode potential and you see two things you see one that it tends to flatten out and be more or less constant at very high ionic strength and it tends to have a dip at much lower ionic strength and the capacitance there is lower than you would predict from the pure Helmholtz model not higher these always dip, they don't peak so even if you constructed a simple Helmholtz like parallel plate capacitor model you would see that at low ionic strength the capacitance is much lower in certain regions than you would predict from using that fixed parallel plate capacitor that you got from high ionic strength even if you were bold or foolish enough to call this all a constant the key message is that it's lower not higher and it's not constant with potential so we need another model that will help us get closer to this structure and there are two components of the structure one is that the potential the capacitance will change will go up relative to some minimum value around both sides of the point of zero charge because obviously the differential capacitance should be very close to zero at the point of zero charge because that's where our slope is zero and so the change in the slope is going to go near zero and so we want to get that general thing right account for this dip which occurs at low ionic strength but seems to be washed out at high ionic strength and there are two different phenomena that are built to explain these because the first phenomenon that is intuitive as to what really happens with ions in the double layer will explain one part of this but not the second and so we're going to need to add something else to explain the second part of this look at what could be happening and then we need to build a molecular model of this double layer so in one possible molecular model the conventional molecular model that is built we break up what happens in the solution and say it's really not just a parallel plate capacitor when the solution is completely electroneutral outside of one boundary plane and all the charge just gets sucked up to the boundary plane as if I physically built a capacitor when there was two defined metal lines and if there's positive charge here there's only negative charge there and no negative charge anywhere else in the solution but that's not the way a real solution will ever behave the real solution is going to have some negative ions here and lesser ones here and a bunch of plates with decreasing concentration of negative excesses to offset the positive that I put on the electrode plate and so we at least know there has to be that much in it so let's build that in the model so we can divide the solution up into various different lamellae at different positions x with respect to our boundary plane and there'll be a certain potential that'll be dropped because we've applied the potential between this and that and we measure the differential capacitance there and we want to try to model that from the ground up and we want to know that if we purely take the ions in solution and put them into some fixed distance in a Helmholtz capacitor that we don't get the right answer that that would predict the constant capacitance with respect to applied potential and wouldn't get this dip and wouldn't get the functional form right so the first thing you immediately see is what we should be doing is describing the concentration of the ith species of an ion if this is a positive electrode surface charge then we'll have a buildup of negative ions in the solution to offset that that doesn't mean that there aren't positives here it just means that there are negatives being offset partially by positives and then there are negatives offset by less positives at somewhere else but we'll treat the ith species independently here so this could be you could think about a Godonkin experiment when we only worry about one species but it would be general for the excesses right so the concentration or the excess of the ith species at any position at any point has got to be related to what it would have been times the potential that is dropped in that point right so we have a reference lamella far from the electrode there and there's no potential drop there right and so in that reference lamella there's some surface concentration of that species say chloride anions and now if there's a potential dropped because this is positive and this potential is dropped in the solution as I get closer to this electrode there are going to be more chlorides here than here if this is positively charged and since the potential is dropping in that solution as a result of those chlorides then we can just use the Boltzmann relation at a fixed temperature if they are in equilibrium and we wait long enough to express the relationship between the number density that are in this plane the number density are in this plane because if this potential is different than this one then the relationship between those two must be in Boltzmann equilibrium with respect to that potential drop this is no different for people taking Chem 140 then establishing the concentration of the electrode at the electrons at a semiconductor electrode surface relative to the bulk is just in Boltzmann equilibrium the number at the surface is the bulk times the potential drop Boltzmann factor this is the exact same thing so this is general and z is the sine charge it's going to be distance dependent this is in the eith lamella it's an i this is the eith lamella right bulk concentration and then it's going to be it's going to be distance dependent right and if you don't believe that it's right there so the charge density is in fact and remember this is that z is the sine charge on that ion so if it's chloride that's negative one if it's sodium it's positive one we need to keep track of the signs of these ionic charges and e is the charge on the electron no matter what but this allows us to keep track of excess positive charge density consistently with respect to excess negative charge density and a barium ion is worth twice as much as a sodium ion because of 2 plus versus 1 plus okay so let's not forget about z i's being the because these are concentrations of species and we care about charges and so we need to keep track of how many charges there are per unit species of concentration and what the sign of those charges are okay so the charge density is a function of position here it is right this is the continuous integrated form for those lamella being discrete but very small in distance delta x right is exactly the sum of all the excess charges of that ion in each lamella out to that position this is in any one lamella right so this is distance dependent right right so this is this dependent is evaluated at the distance of that particular lamella right okay so what do we know from this well we know independently this is the model right this is a molecular model that says that any ionic species must be in Boltzmann equilibrium with respect to the potential that it experiences at that position and there's some self consistency between how many ions that are in front of it and the potential applied and what it experiences right we built a model now we don't build we don't need a model to derive this the charge density in any phase as a function of x is related to epsilon epsilon not times the second derivative of the potential with respect to that position because that's Poisson's equation that's true no matter what since this is true no matter what and this comes out of our model then we can use this in conjunction with that to determine what the concentration of any species must be at any given lamella position as a function of the electric potential because we'll just say this equals that so this equals that so now we can solve for what d squared psi dx squared is we'll just bring around over into the right side a one over epsilon epsilon not there's the charge here's all the other ionic species excesses weighted by their Boltzmann factors of the potential that they experience so now we have a linear second order differential equation for the potential as a function of distance except that of course the potential is a function of distance is in turn a function of a concentration of the species that build up when we applied that potential to that species so implicit in here is some function that depended on that so you can't solve this directly without using one of these evil tricks but evil tricks are not beyond us especially if they're in the book here's one of these evil tricks the second derivative of any function with respect to x squared is one half of d by d psi of the first derivative of the x just chain rule it and you'll see that that's true so we have the second derivative here we know the second derivative is one half of d by d of the first derivative squared so let's plug in for the second derivative of this thing so then this is d of the first derivative and we're going to bring the d phi over there so we can integrate so here's d of d phi d x squared and that's going to equal bring the two to the other side all this stuff is on the other side and there's the d phi and so now what we have is we can integrate because this is a derivative that's a derivative and now we can say that the integral the left side must be equal to the integral with some constant so the trick of writing this allows you to solve that even when one is implicitly a function of the second I didn't think of this trick someone thought of it in 1945 that's okay so now we're going to integrate well the integral of a derivative of something is just the something and the integral of a bunch of constants with respect to that variable is just the stuff plus a constant if this is equal if this is an equality then the integral of that must be equal to the integral of that except that there can be a constant relating one to the other because if I took the derivative of this and the derivative of this the constant would go away and I would get back that so this is a classic trick of how you solve a differential equation like this when the differential on one side related to a differential of something else on the other side you just integrate both sides at the same time so you can do that then we get to remember that it distances far from the electrode not only is the electrode potential going to be screened by all the irons in solution so far enough away the potential drop in the electrolyte goes to zero because if it's negative on the surface and positive in the solution and infinite distance away it was a double layer there so not only is the potential zero but the potential gradient with respect to position is also zero either one of those would be sufficient to tell you the constant but both are true and of course like all things in physics the constant is one and it has to be because after all if this is going to go to zero potential drop this exponential is going to go to one since this thing has to go to zero gradient at far enough away then one minus one is going to bring that to zero so the constant of integration is one and so we can just group it in with that thing and we see that this derivative of this potential with respect to position squared is that stuff so this came about from our model of having this B lamella of species responding to the potential that we applied now that's perfectly general if we make one more mathematical assumption which you don't have to make but if you make one more mathematical assumption that the electrolyte itself is a so-called Z to Z electrolyte that is one to one and what I mean by one to one is the electrolyte is something like sodium chloride or something like calcium sulfide that's a bad example cadmium sulfide would be good cadmium 2 plus and a sulfur 2 minus that's a Z to Z electrolyte where one is 2 plus and you need two one minuses if the cation has an equal and opposite charge to the anion that's a Z to Z electrolyte because one has charge plus Z and the other has charge negative Z where Z is an integer in that particular case then this expression can be simplified further because if you add up these excesses of concentrations and one is an E to the plus one something and the other is an E to the minus one something then if you remember that that will give you a hyperbolic sign because that is E to the plus X plus E to the minus X over E to the plus X minus E to the minus X over 2 and the hyperbolic cosine is the other way but this only works if they're equal if they weren't equal it wouldn't simplify to the hyperbolic sign this is still perfectly general this is just a simplified expression for this in the case of a Z to Z electrolyte and then all these terms out here are just this stuff because now we're just solving for the square root of this right because this is d phi dx squared so now we've gotten d phi dx we need phi so we've got to integrate we can do that so if this is what d phi dx is what the derivative of the potential is with respect to distance so now we know its slope at any position in the solution but we want to know what the total potential drop is from the electrode to any arbitrary point now if we know the slopes we have to integrate those slopes over the total distance to get the total potential drop so we'll do our normal thing on the left so we'll bring the cinch down there and keep the phi on the left we'll bring the x over to the right and leave the constants over there so now we integrate to get the total potential dropped from any position 0 and infinity to x and that's equal to the potential phi 0 relative to the real potential phi there's a potential difference between wherever we started and whatever position we're choosing to evaluate as our end point as we go through this buildup of charge and the closer we get to the electrode the more potential is going to be dropped relative to our infinite solution case when there's no potential dropped because we're adding up all these derivatives to get the total okay so this is true and they may know what the integral of 1 over a hyperbolic sign is probably gas that's a tangent it's a hyperbolic tangent so when you solve for that the integral of that is a hyperbolic cotangent just look it up in the handbook if you don't want to derive it but we have to evaluate it we have to evaluate it because it's a definite integral it's a difference between the potential on our reference point and where the potential is dropped that we care about the position which was x equals 0 at infinity and then coming off to some place okay so here's the constant that was still there on the right here's the total distance we've traveled the definite integral has two terms one where phi is evaluated at phi 0 and the other where it's really phi because we integrated from phi 0 to phi to get the total potential drop in that definite integral if you remember right here there's a phi and there's a phi 0 and there's an x and a 0 since this term is a hyperbolic tangent of the thing which has phi in it we need to evaluate that at both phi and phi 0 and when we do that we get exactly that expression where here's our constants again here's our position x and it should go like the log of these hyperbolic tangent ratios just bring this exponent group the constants into one constant called kappa so bring this over this says that the log of this is that thing times x that the ratio of that is e to the constant x right that constant is this stuff all of which is known right because this is just the concentration of stuff at infinite in the solution that we know because we added to the solution this is the z the charge on this ion that's either the sodium or the chloride or whatever the valence is charge on an electron epsilon epsilon not is known k and t is known so what this says especially if it's dilute because if it's dilute then the potential drop is relatively small and if the potential drop is small right so this thing turns into the potential drop because it will just be linearized so the potential at any position that's dropped relative to its value in the bulk will go exponentially with some screening length from our molecular model of these lamellas that are in thermal equilibrium with respect to their initial potentials that they've responded to so the way to think about this is this predicts something like this the potential relative to the total potential that would be dropped is gonna fall off essentially exponentially with distance and eventually there's zero potential experienced by a point charge far away from the electrified interface in the solution this is a dilute ionic strength because it goes like those ratios of hyperbolic tangents that expand to be just the values of the arguments of those potentials when it's small this length is called the device screening length it's the length at which a charge applied to one of the plates of this capacitor of this system is effectively fallen off by one over e of its value in the potential dropped in the other side of the interface and you can see that at higher ionic strengths that's a shorter distance because there are more charges to screen the potential that would have come about from applying to the electrode that surface charge and so that potential in the solution falls off more quickly until it's neutralized and then a point charge in the solution doesn't see anything except some residual dipole and eventually it sees nothing so this is right it's in the right direction and at higher ionic strength the small or the screening distance needs to be before the solution is electroneutral again but the functional form of this is completely analytically predicted by that function right there and the screening length is a knowable number so for dilute aqueous solutions all you need to know is the dielectric constant of water and room temperature and for a one to one electrolyte this is what the device screening length should be and I can tell you what the numbers are good and so you see that for one molar with a one to one electrolyte like sodium chloride the device screening length is on the order of three angstroms at tenth molar it's nine angstroms this is just all going to come out of that constant where the only variable really is the concentration so these are completely knowable and to first order are close to what we can remember a zeroth order these kind of orders of magnitude and they better be close to right and it makes sense that one molar ionic strength if you're ten angstroms away from that electrode surface the ions have done all their work they've neutralized that charge to a couple values of one over e of it and so the solution is essentially electroneutral if you're down at ten to the minus third molar ionic strength you've got to go at almost a hundred angstroms before the solution is electroneutral because if this is really negative then the first one has an excess of sodium but there aren't enough sodiums built up at one millimolar to neutralize all that negative charge and you have to keep building up lamella of sodium ions until you have enough positives to neutralize all the negatives and they're in thermal equilibrium with each other with respect to the potential that's remaining as you go farther out in the solution and that gives you the functional form of their concentration with respect to distance and so we just solve for based on that molecular model of how things are working right right right we assume the linearization of that right there right if not we could have plugged in analytically and still computed what would have happened right this is a mathematical non-approximation for a one-to-one electrolyte so this is analytically true and Mathematica would spit out what should happen right okay so that's good so this predicts something like this we could go further today but I'm going to have to stop from time but I'll tell you that we will pick this up on the next class on Thursday because now if we know the potential that results from any given electrolyte in the solution we can calculate what the capacitance should be according to this molecular model and this will get one of the two key things right it will get right the gradual increase in capacitance relative to a minimum value around the point of zero charge but it won't quantitatively describe that dip that we saw and for that we're going to need something else but if we add them both together you can then see that we'll actually have a molecular model that will be able to explain the observations of what the second derivatives of these electric capillary curves are that will then allow us to say something chemically about where the ions are in this double layer that are happening all the time as we're applying a potential to pass current and so we're going to finish that on Thursday