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# CHEM 125a: Freshman Organic Chemistry I

## Lecture 3

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

### Overview

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.

Professor McBride’s website resource for CHEM 125 (Fall 2008)

http://webspace.yale.edu/chem125_oyc/#L03

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html## 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]
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 “(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 O Now, what is the true structure of the molecule ozone, O 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 O So to specify O
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 R 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?
So reactivity we saw was a special attribute of this nice Lewis theory, and charge distribution, at least qualitatively if not in detail, for O ## 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?” 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?
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 McBride takes attendance ] ## Chapter 4. J. J. Thomson’s Plum Pudding Model [00:35:08]
Here’s the book by J.J. Thomson called
[Laughter]
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?
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
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