CHEM 125a: Freshman Organic Chemistry I

Lecture 3

 - Double Minima, Earnshaw's Theorem and Plum-Puddings


Continuing the discussion of Lewis structures and chemical forces from the previous lecture, Professor McBride introduces the double-well potential of the ozone molecule and its structural equilibrium. The inability for inverse-square force laws to account for stable arrangements of charged particles is prescribed by Earnshaw’s Theorem, which may be visualized by means of lines of force. J.J. Thomson circumvented Earnshaw’s prohibition on structure by postulating a “plum-pudding” atom. When Rutherford showed that the nucleus was a point, Thomson had to conclude that Coulomb’s law was invalid at small distances.

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Freshman Organic Chemistry I

CHEM 125a - Lecture 3 - Double Minima, Earnshaw's Theorem and Plum-Puddings

Chapter 1. Distinguishing Equilibrium and Resonance [00:00:00]

Professor Michael McBride: This is the slide from the end of last time, we didn’t quite finish it. Equilibrium versus Resonance. Remember, equilibrium is a case where you have two different species, different structures, and you go back and forth between them, maybe fast, maybe slow. Like the hydrogen could be attached to the top or it could be attached to the bottom oxygen, and those are different, two different, what’s called isomers; we’ll get to that later on. Or you could imagine the same thing if you take the hydrogen off, you could have a short double bond to oxygen and a long single bond to oxygen, or they could exchange as to which oxygen was long and which was short. So you could have an equilibrium there. But it turns out you don’t, that in fact it’s not two different species, it’s one species. It’s a single minimum not a double minimum.

Now how do you know this? How do you know it’s just one nuclear geometry with an intermediate bond distance? The only way you know is by experiment or by some really fancy calculation that you have to believe. A lot of people would believe experiments before calculations; some are the other way around. But there’s evidence from a technique called electron paramagnetic resonance, or EPR, that shows that indeed this is one species, a single minimum. If you have an extra electron on it and you have a carboxylate anion, then again it’s just one species, a single minimum. And there’s evidence of that from infrared spectroscopy that we’ll talk about next semester. But don’t be disappointed that you’re not able to predict this. A lot of really smart, experienced people couldn’t predict it. This is lore. If you look it up in the Oxford English Dictionary it says that lore is, “That which is learned; learning, scholarship, erudition. Also, in recent use, applied to the body of traditional facts, anecdotes, or beliefs relating to some particular subject.” So a lot of things are lore. You just have to learn them. You can’t predict them ahead of time, they’re way too subtle.

So don’t be disappointed, because you haven’t had enough time yet to get the lore; you’re not supposed to know it yet. It would be nice if Lewis theory was so accurate and straightforward that if you draw two structures, there are two structures and it’s a double minimum. But that’s not true, and you have to know from something, and you don’t have time to know yet. As time goes on maybe you’ll figure it out. This is from a good textbook; it might be the one that we’ll adopt next semester. It’s a quote that says, “Empirical rules” (it’s going to give) “Empirical rules for assessing the relative importance of the resonance structures of molecules and ions.” That is, if you have two different pictures that you can draw for the thing, does it look halfway in between them, almost all like one, almost all like the other, or some fraction of the way between them. How far is it, one way or the other? These are two different pictures you draw to try to show different aspects of the same thing. But there is one real thing. The molecule doesn’t know about resonance, it just knows what it is. The problem is with our notation. Okay, but anyhow here’s what they say. So rules that will allow you to use this concept more productively by deciding which ones are better, which look more like the real thing. Okay, so it gives rules and it numbers them.

“(1) Resonance structures involve no change in the positions of nuclei; only electron distribution is involved.” That is, when you draw these two structures you don’t move the atoms, you just change where you draw a single bond or a double bond or a dotted bond or something like that. And in fact that’s not even true because it’s not — the electrons know where they want to be, it’s the way we draw them that’s uncertain. So when you draw two different resonance structures, you’re not changing where the electrons are, you’re just changing the lines you draw. Is that clear? It’s our notation that’s at fault.

“(2) Structures in which all first-row atoms have filled octets are generally important; however, resulting formal charges” (we talked last time about how you get formal charges) “and electronegativity differences” (and of course we have to know what electronegativity is, but you’ve heard about it at least) “can make appropriate nonoctet structures comparably important.” So if you have a bad charge distribution, even though you have octets, it still might not be a very good structure.

“(3) The more important structures are those involving a minimum of charge separation” (so you don’t want to have formal charge separation) “particularly among atoms of comparable electronegativity. Structures with negative charges assigned to electronegative atoms may also be important.” So if you’re going to get a charge, put it where it wants to be.

Now, look at this more carefully. Number two, it says ‘generally important.’ It doesn’t say “always important,” but “generally.” That’s not such a great rule because you have to know when there’s going to be an exception. Or ‘however;’ this doesn’t sound like the Ten Commandments graven in stone. Or “can make” a thing; not that “it will,” but “it can.” Or ‘ “particularly” or “may also be important.” These are all weasel words because the rules are not rules, and this is all lore again. So people write these rules but really the people who are writing the rules know the answers ahead of time and think “Ah, it generally works that way and mostly we can get away with it.” But these are not rules like the rules you want to learn in physics. You want to learn these because it’s sort of handy, but don’t believe them. And notice at the top that it’s “empirical” rules. It’s not a fundamental theory, these are just ways of sort of correlating a bunch of the lore that’s come in. And what’s important is the experiments, not theories like this. And we’ll get better theories later.

Chapter 2. The Structure and Surface Potential of Ozone [00:06:38]

Okay, so the goal of this Lewis stuff was from the number of valence electrons it would be nice to be able to predict the constitution, that is, the valence numbers for the different atoms, how many atoms of one kind or another get together to make a molecule. Reactivity, we’ve seen a little bit of that at least, that unshared pairs can get together with vacant orbitals and make a new bond. That wasn’t something that was part of the original rules of valence. And maybe something about charge distribution as well. Now let’s look at the case of O2 and O3, and apply some of this stuff. So O2, you can have — complete the octets by bringing two oxygen atoms together and forming two bonds between them. So that looks pretty good, and it’s a double bond, and we can draw it that way with just lines and forget the unshared pairs if we’re not particularly concerned with them at a given time. Okay, now suppose you make O3. You could do the same kind of trick there and make a three-membered ring, or you could make it linear, or bent instead of a symmetrical three-membered ring. So you could have an equilateral triangle like that, or you could have it like this, have O2 and bring in another oxygen on one of the lone pairs. Now that’s not an equilateral triangle; it’s a triangle, or it could conceivably be a straight line, we’re not quite sure what the geometric implications are. Okay, but we could call it an open structure as compared to the ring structure. And we can draw it like that, and the formal charges would be as shown. Why? Because that pair that originally belonged only to the original top oxygen of O2 is now shared with the other oxygen. So one of them loses half-interest in a pair, the other gains half-interest in the pair and it’s plus, minus. So the trivalent oxygen is positive, as you would expect.

Now, what is the true structure of the molecule ozone, O3? Well you have to do some experiment to find that out or some high-faluting calculation that you believe; a quantum mechanical calculation. So it could be a ring or it could be an open structure. And notice, in the open structure you should have two resonance structures, because you could draw it that way, or you could draw it that way, and it could be two minima — it could be a double minima — double minimum situation, or it could be distorted one way or the other and click back and forth. Or it could be that the true structure is in between with equal bond distances. So we have to find out something about this.

Now one way of finding it out is to do calculations and draw a graph that shows what the answer is, and that’s what we’re going to do here. It’s based on some fairly recent high-level calculations. But the problem is drawing the graph, because if you want to be able to show the structure, you have to — how many variables do you have to specify to say what the structure of O3 is? How many numbers would you have to have? It’s three particles, right? So if you gave this distance and this distance and this distance, that would fix the triangle. Or you could give this distance and this distance and the angle here. That would also specify the triangle. But any way you slice it, you have to have three numbers to tell the structure, and then you have to have another number in your graph to say what the energy is, when it has that structure. So you have to plot four variables. And that’s not trivial on a piece of paper, to plot a graph with four variables in it. So three distances plus energy, or two distances and an angle and energy you need to plot. So if you’re good at plotting things and have had a lot of experience, great, but if you need a little warm-up exercise you could look at this webpage that you get by clicking up at the top there, and it uses these graphs to give you a little exercise in plotting things that are in many dimensions. But we’re not going to do that in class, that’s just to do on your own or with a discussion section. And we’ll show the specific example of how we’re going to do it in the case of O3.

So to specify O3 we need four dimensions. Now here we have a plot that shows one distance, a second distance and the angle. So that’ll specify where the three oxygens are. But Rutenberg and his coworkers here, in 1997, when they did the fancy calculation, did it a little differently. They constrained it. They didn’t allow everything to vary. They said we’re going to require the two distances to be the same. So now instead of three variables, you have only two, and you can plot those two on a two-dimensional paper; you can plot two things. So here we’re going to plot the position. If the two distances are equal, then if you know the position of the top right you know the position of the top left, because the original oxygen is at the origin 0,0. Everybody with me on that? So we’re going to look only at where the top right oxygen is, and the other one will be someplace symmetrically related to it. So we’re going to blow that up, and here’s the plot they give then. So now any point on this will specify a precise geometry of the three oxygens. Everybody with me? We choose a point that tells where the top right one is. The bottom one is at 0,0, and the top left is just on the other side, in the corresponding mirror-image position. So any point on that thing will specify the structure. But now we still — but that’s used up our two dimensions of the paper. How do we show the energy? Can you think of a three-dimensional graph you’ve ever used?

Student: Color codes.

Professor Michael McBride: Pardon me?

Student: Color codes.

Professor Michael McBride: You could color code it. You could make red really high and blue really low. That would be one way to do it. There are earlier ways, before printing made it — or computers made it easy to make colors.

Student: Contour lines.

Professor Michael McBride: Did you ever go hiking? Yes what?

Student: Contour lines.

Professor Michael McBride: Contour lines, like a geological map, right? Okay, so we can draw contour lines that show what the energy is for every one of these geometries. And so if that — notice that at that position, that X, it turns out to be what the red structure would be here, which is a ring; equilateral, okay? And this structure is open. And I chose those particular ones because when you do the fancy calculation, those turn out to be the ones that are at the bottom of valleys, low energy. And if you distort away from those, the energy goes up. Now you can go up — any direction you go from one of the bottoms of the valleys you go up in energy, but there’s some particularly interesting positions. A particularly interesting position is that one, because that’s the pass, that’s the lowest you can go, the lowest energy that’s required to go from one valley to the other. Everybody, I think, has probably done enough hiking to realize that that’s the way you’d like to hike, or that if you spilled a cup of water up at the pass it would run down both ways.

Now in fact we can use that concept to make it even a simpler graph, where we don’t need to draw contour lines, because we could take the steepest descent path — if you pour the water out at the pass and follow how it’ll trickle down, it’ll take the steepest way to go down. It turns out it crosses every contour perpendicularly. So you follow that and it would go down to the bottom, and then if you kept going, like if you dropped a marble or something, and it rolled down to the bottom it would keep going. So there is a particularly interesting path. Not that the true molecule would necessarily have to follow that path, but it’s a well-defined path that gets from one valley to the other. So now what you could do is take a knife and slice this thing along that green line and unfold it so it’s flat. Does everybody see how that would be?

Like I remember once when I was young we took a family vacation and the AAA sent you a map that showed where you would go, but it also showed — along a particular highway — and it also showed another map that showed how high you were all the time as you went along the roads. Got the idea? So this is exactly that kind of thing, where as you go along that green road there are different altitudes. So we could just — we could draw it this way. Does everyone see how that works? So it’s not quite as specific about geometry anymore, the way the previous one was, but it shows how much energy you need as you go along. And a particularly important one — well there are two things that are really important. One is how high one valley is compared to the other, how much more stable one is, and the other is how much higher is the pass, how hard is it to get from one to the other?

Okay, so there now is, in a two-dimensional graph that you can draw on a piece of paper, is something that gives you this information. Okay, but this required that we choose R12 equals R23, in order to make this simplification. The guys who did the calculations said if we don’t make R12 equal to R23, then it turns out that all the structures are still higher in energy. So these are the lowest energy structures, although they can’t plot them on a piece of paper. So the lowest energy structure is this open form and it does have R12 equal to R23. It’s a resonance structure, halfway between; not one double bond and one single bond but a symmetrical one. Okay, so it’s a symmetrical single minimum, found by calculation. And there’s experimental evidence that supports that too, but it’s more complicated.

Now how about the charge distribution? What would the Lewis structures that we’ve drawn here predict about charge separation? Would there be charge anyplace, do you think, on the basis of this, in the real molecule, which is symmetrical? What do you think? Anybody got a suggestion? Yes?

Student: Well, it would be positive on the central oxygen because even converted, regardless of which of the two structures are on that side, that…

Professor Michael McBride: Both structures have it positive in the middle, so that looks like a good prediction.

Student: And the ends would be negative.

Professor Michael McBride: Right, and then partially negative but equally negative on the two ends. Now is that true? Well here’s a picture that, based again on — so the prediction is positive in the middle, negative on the ends. Is it true? Well there’s the structure, the minimum energy according to calculation, quantum-mechanical calculation. And we can draw what’s called the surface potential of that structure, which again comes from calculation. So you put a proton — you define the molecular surface — and that is a little bit of a problem but we’ll talk about that later, where the molecular surface is — but then you put a proton at a point on it; and we talked about this before, for ammonium chloride. You put a proton on and you find — or no, it was the BNH6 we talked about the surface potential. Here’s the same thing for O3. And you’ll notice the surface potential is high in the middle, a bad place to put a proton because there’s positive charge there, and it’s low on the two ends, at least some place on the two ends. So more or less the Lewis structure predicted this right. So that’s great. We’re not confident that it will always work but maybe the lore will build up that way; and it does. Okay, so charge distribution also.

So reactivity we saw was a special attribute of this nice Lewis theory, and charge distribution, at least qualitatively if not in detail, for O3 and also for the BNH6. Now how about specific distances and specific angles? We’ll get into that later on as to whether you can do that from something like a Lewis structure. And how about the energy content? Well that’s not so good actually, with just Lewis structures, but we’ll test it later. Okay, so the Lewis-dot structures. It attempts to provide a physical basis for the valence rules, based on completing octets and sharing electrons for bonds. What it gives that’s new is reactivity due to unshared pairs where both of the “hooks” in the bond come from the same atom, you might say. And it’s convenient for electron bookkeeping. It certainly tells you what the molecular charge is. The formal atomic charges are qualitatively realistic, as we’ve just seen, at least in the case of O3. And there’s this question of stability and resonance. But resonance actually is not something nature knows anything about. It’s just a correction we try to apply to having made drawings based on Lewis theory. It’s not something serious.

Chapter 3. Visualizing Electrostatic Force: Earnshaw’s Theorem [00:20:58]

Now, but this leaves us with some serious questions. Why does Lewis theory work? What’s so great about octets; why not have a different number? Or if you have a sestet, instead of an octet, how bad is it? Or how bad are structures that have charge separation? It said in those rules that you don’t want to have charge separation. Well how bad is it? Suppose you had a choice, you either had to have a sextet instead of an octet, or you had to have charge separation. Which one would win? How bad is “bad” charge separation? Remember it said that if you put the negative charge on an electronegative atom, that’s not so very bad, but how bad?

Now last year in the Wiki there came an interesting comment. Somebody said, “I have a question when drawing these structures. Is it more important to try to fill the octet or to have the lowest formal charge on as many atoms, especially carbon, as possible? And why?” That’s a good question, because Lewis doesn’t tell you that at all. And there’s further the question, “Is this at all true?” Are there electrons — this is a really important question, we’re going to spend some time on it — are there electrons in between the nuclei that hold them together, pairs of electrons? It’ll take us several lectures to get to answer that question. Are there electron pairs between nuclei and are there unshared ones on some atoms? What’s the nature of the force laws? That’s what we’re after, remember, here altogether.

Now, as to whether Lewis theory is right, there’s a very fundamental theorem in physics that was developed in 1839 by Samuel Earnshaw, who was a tutor at Cambridge. And the statement of the theorem is that in systems governed by inverse-square force laws, things like gravity, magnetism, electrostatic interaction, there can be no local minimum or maximum of potential energy. He proved it mathematically. It’s not our business to repeat his proof, but that’s the statement of it. But we want to understand what that means. One thing that means is that if you have Coulombic interaction, positive/negative, you can’t have a minimum energy structure that has a nucleus here and eight electrons at the corner of a cube, because there are inverse-square force laws and that can’t be a minimum energy; if it distorted, it would keep going.

Now we can visualize Earnshaw’s theorem here in terms of for electrostatics by the analogy that you’ve all seen of magnetic lines of force; everybody’s seen something like this I think, right? And the idea, if it’s electrostatic, rather than magnetic, the idea is that lines of force emerge from a positive charge and converge on a negative charge, and then you’d make them continuous by drawing these lines of force; which iron filings of course did in that case. And this was the idea of Michael Faraday, whom we met a little while back, and he thought these lines of force were real physical things. Most people don’t think that now, they think they’re just graphs that involve inverse-square force laws, but he thought they were real. And the neat thing about them is they not only show the direction of the force that a charged body would feel, because of the other charged body, the one that created the lines of force, they not only show the force, the direction of the force, they also show how strong the force is. And the strength of the force is by the density of lines. The denser the lines, the stronger the force.

Everybody’s familiar with this idea? Speak up if things are not familiar because I’m assuming they are, if I say so. Okay? So you’ve seen that. Now let’s just think about that a little bit. Suppose you look at the line density here. Through that little line, there are three lines of force that pass. So let’s say that means there are three, the force is three at that — and it’s obviously pointing to the right. Okay? Now suppose you checked it out at that distance. Stronger force or weaker force?

Students: Weaker force.

Professor Michael McBride: Weaker. In fact only one or one and a half, or something like that, lines of force are going through that. Now, how does it depend on distance, the number of lines going through this standard linked line, the blue one? Well you see that in flatland — we’re just doing this in two dimensions, we’ll get to three in just a second — in flatland, in two dimensions, the circumference through which all these lines of force pass, the length of the circumference is proportional to the radius. Right? So if you go out twice as far there’s twice the circumference. So the density of lines passing through it is half as big. Everybody got that? Okay, so that means the force, which is proportional to the line density, must be proportional to 1/r. Okay? So then as you move out twice, there’ll be half as many lines; move out three times, a third as many lines. Okay? But this is just in two dimensions. Now let’s think about, how’s it going to be different in three dimensions? Katelyn? Speak up a little bit, my hearing isn’t so great. I said I went to my 50th high school reunion, right?

Student: Instead of a line it would be more like a square, some sort of an area.

Professor Michael McBride: Yes, it would be a two-dimensional area that this stuff’s going to be. Now if you have a certain number of lines passing through at a certain distance, and you go out twice as far, what’s the density of the lines going to be? Can you think? Anybody help? Yes?

Student: It’ll be inverse-square.

Professor Michael McBride: Inverse-square, because now we’re not talking about the circumference of a circle, we’re talking about the area of a sphere as we go out. Right? So if we go to three dimensions here and take your area and move it out, the surface is going to be proportional to r2, as you go out. So the line density is going to be 1/r2. So if you have an inverse-square force law, then you can draw lines of force. The lines of force won’t work if you don’t have an inverse-square force law, because they have to drop off this way in order to give lines of force.

So in 3D such diagrams work only for inverse-square forces. You can’t draw a thing like that for Hooke’s law. Right? Now, so here’s a whole bunch of charges, positive and negative, and the lines of force between them. And notice something interesting. The lines of force start on positive charges and end on negative charges. There’s no other place in space where all the lines go away from it, or go toward it. Everybody see that? It’s only at the charges that they all go away or all go toward. Right? And that has a very important meaning, that you can’t have someplace off in free space where all the lines of force would converge. Notice if you were to have something that was positively charged, and it’s at a minimum of energy, then any place you displace it, it’ll get pushed back, if it’s at a lowest point in energy. You push it any direction, it’ll come back. That is, all the lines of force must converge on that point. Okay, everybody with me on that? But it can’t be. The only place that all the lines of force converge is on a negative charge. It can’t be any place — if you have inverse-square laws, you can’t have someplace in space which is the lowest energy place for the charge to be, except on another charge. The same is true, you can’t have a maximum either. That’s a visualization of Earnshaw’s theorem.

So if you have inverse-square force laws, or any combination of inverse-square force laws, like a combination of gravity and electrostatics and magnetics, you can’t get a minimum energy structure, unless everything just falls together or blows apart. Okay? So Earnshaw’s Theorem: “In systems governed by inverse-square force laws there can be no local maximum (or minimum) of potential energy in free space.” Okay? And that’s why I don’t just float here, I have to be on the floor. Right? Did you ever see anything truly levitating, something that just sits in space and doesn’t move, not touching anything else? That’s been levitating for ten or fifteen years. It’s not plugged into anything, it’s just this thing right here. What do you conclude from seeing that, about the force laws that are involved?

[Students speak over one another]

Professor Michael McBride: There must be some force involved in that that’s not an inverse-square force law; otherwise it wouldn’t work. You can read about it on the Web, if you want to, it’s not the business of this class. There is a non-inverse-square force law involved in that little magnet sitting there. The only stationary points allowed by Earnshaw’s theorem are saddle points where it’s flat in energy, for all directions, but you go one direction and then you go down, go the other direction and you go up. That’s like a potato chip or a saddle. So you can have saddle points, but you can’t have absolute minima or maxima of energy. There’s the picture of that thing. Let me do the — this reminds me to get back to seeing who’s here.

[Professor McBride takes attendance ]

Chapter 4. J. J. Thomson’s Plum Pudding Model [00:35:08]

Professor Michael McBride: Now we’ll continue, got a few more minutes here. And yet it stands still. So there must be something that’s not an inverse-square force law there. Okay, so J.J. Thomson, in 1897, discovered the electron. So the idea is maybe electrons have something to do with bonds; that’s what brought Lewis into the game. But Thomson himself came up with what came to be called — and I don’t know by who first, I’ve tried to find out and can’t — the “plum-pudding” atom. Has anyone ever heard of that? Yes, okay. And I suspect that when people told you about the plum-pudding atom they sort of snickered. Am I right? This was sort of a naïve idea. It’s not that naïve at all. Let me show you why.

Here’s the book by J.J. Thomson called The Corpuscular Theory of Matter, and if you look in there you’ll find out what he’s talking about. He says — he has a model of electron configuration — he says: “Consider the problem of how to arrange 1, 2, 3, up to n corpuscles” (that’s what he called electrons, he called them corpuscles); “Consider the problem as to how they would arrange themselves if placed in a sphere filled with positive electricity of uniform density.” So that’s the idea of the plum-pudding. You have this sphere of uniform positive density — why it should be like that nobody knows — but suppose you have a sphere, and you put the electrons in it, like plums in a plum-pudding, which is like a fruitcake in England. Okay, now notice that he said, “placed in a sphere filled with positive electricity.” Why did he do that? Why didn’t he just have — like Rutherford ultimately did it — a nucleus with positive charge and electrons around and about? Why did he put the electrons inside the positive charge? Zach?

Student: Was it lowest potential energy?

Professor Michael McBride: I can’t hear very well.

Student: Lowest potential energy?

Professor Michael McBride: No. Yes?

Student: Well they were thinking [inaudible] maybe that they were going in.

Professor Michael McBride: No, no, they weren’t thinking about things moving, they wanted them just sitting there.

Student: Maybe because on a macroscopic scale opposite charges attract. So maybe he might not…

Professor Michael McBride: Yes, but that’s well known, Coulomb’s law. Yes Keith — or Kevin, right?

Student: If you have them all, you have a positive sphere and if you have the negative corpuscle inside, then they’ll cancel each other out and be a neutral body.

Professor Michael McBride: Yes, but that could be — it wouldn’t have to be a big sphere of uniform density, you could have a little particle of positive charge, of the same charge, right? It’d be the same deal.

Student: They didn’t know that the inverse existed, so they thought that…

Professor Michael McBride: No, none of this is it. It’s what we’ve just been talking about. Yes?

Student: Isn’t it about accounting for the bonding between the electrons and the protons and containing the electrons?

Professor Michael McBride: No, bonding hasn’t come up yet. This is something very fundamental. Yes?

Student: He was a big fan of plum-pudding.


Professor Michael McBride: He probably liked plum-pudding at Christmas.

Student: Did he have experimental evidence?

Professor Michael McBride: I can’t hear.

Student: Did he have experimental evidence?

Professor Michael McBride: No. Yes?

Student: Because Earnshaw said that there should be no minimum or maximum…

Professor Michael McBride: Ah ha! Earnshaw said you can’t have an energy minimum for separate particles, but if you put the negative ones inside the positive ones, then you could get a stable structure. It’s Earnshaw’s theorem that required it to be a plum-pudding. Okay? Now why a sphere, why not a doughnut or some other shape, a barbell or something, right? And he said that the positive charge is distributed in a way most amenable to mathematical calculation. He chose a sphere so it would be simple. You’ve heard about spherical cows and so on like that, that physicists like to calculate. Let’s suppose there’s a cow, let it be a sphere, right? So that’s what he did. That’s why it’s a sphere.

Okay, we can solve the special case where the corpuscles are confined to a plane, if you do it in two dimensions — it’s difficult mathematically in three dimensions — but you can do it in two dimensions. And he gives a picture in this book, like this, which is a solenoid magnet that attracts little needles that are magnetized, and those needles are stuck into corks, which float in the water, right? So they have to, the corks have to stay at the level of the surface of the water. So it’s a two-dimensional problem. The needles are parallel, the magnets, so they repel one another. But the big magnet attracts them to the center. Okay? So what he does is toss a certain number of needles and corks in there and see what pattern they form. Okay? So here are some patterns. You can get these on the Web, at Greg Blonder’s website here. So if you have just one, it goes to the center; no big deal. If you have two, you get a line; there’s no big deal about that. Three make an equilateral triangle. Put in four, they make a square. Put in five, you make a pentagon. No one’s surprised so far I suspect. Except that sometimes when you put in five and shake it up, you get a square with one in the middle. Can you see where this might be going?

Let’s keep going. Okay, if you put in six, you get one inside a pentagon; seven, one inside a hexagon; eight, one inside a heptagon; nine, two inside; ten, two inside. And sometimes it’s like that, and sometimes it’s like that. These are experimental, right? And if you put in eleven, it’s three inside, and then three inside nine; although sometimes you get two — pardon me, I’m screwing up here — sometimes you get — there are two different patterns — sometimes you get three inside eight, sometimes you get two inside nine, for that number. Okay, and then you can get four inside nine, four inside ten, five inside ten, and then after five, if you put in more, if you put in sixteen, you get one inside five inside ten. What is this reminding you of? Yes?

Student: Of orbitals.

Professor Michael McBride: The shell structure of atoms. Right? As you go down the periodic table you complete a shell. So it’s a model of shells, and then you can get more and more and more. So this is what Thomson was thinking about. But that’s just two dimensions. Right? Three dimensions is a bigger problem. But he could say, he was able to say something mathematically about eight. “The equilibrium of eight corpuscles at the corners of a cube is unstable.” Even if you have the spherical charge — so it’s not exactly Earnshaw — still you can’t get eight at the corners of a cube. Now Lewis comes along and in 1923, as I’ve told you before, he writes: “I have ever since regarded the cubic octet as representing essentially the arrangement of electrons in the atom.” This is long after Thomson had written this about not being able to do that. So was Lewis ignorant of Earnshaw’s theorem? Because by 1923 they know that the nucleus is not a plum-pudding — that is the positive sphere — that it’s a point. So was Lewis just naïve?

No, look what he wrote, in 1916. “The electric forces between particles which are very close together do not obey the simple laws of inverse-squares which holds at greater distances.” So Coulomb’s law breaks down. You don’t have inverse-square. So then you don’t have Earnshaw’s theorem, and you don’t have to worry, and you can get a structure if it’s not an inverse-square force law. Okay? No trouble. But what is the force law? Well Thomson thought the same thing in 1923 in his book The Electron in Chemistry. He wrote: “If electron nuclear attraction were to vary strictly as the inverse-square, we know by Earnshaw’s theorem that no stable configuration is possible with the electrons at rest or oscillating about positions of equilibrium… I shall assume that the law of force between a positive charge and an electron is expressed by this equation… Then a number of electrons can be in equilibrium about a positive charge without necessarily describing orbits around it.” And look at the — we’re going to end right now by looking at this equation. What’s that bit of it, the first bit, the “one” part?

Students: Coulomb’s law.

Professor Michael McBride: That’s Coulomb’s law. But he’s got a correction to Coulomb’s law, here at the end, c/r, where c is a distance — you divide it by r and you get a number — and c is a distance that’s on the scale of atomic lengths. Right? And that means that as long as c is very small — pardon me, as long as — okay so have I got this right? Okay, so when distance r gets smaller than c, then the force changes sign. Okay? So what was attractive becomes repulsive, and then you can have the electrons sitting around the nucleus. Okay? So we’ll see what happened three years later next time.

[end of transcript]

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