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Chemistry: Band Theory of Conductivity

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  • Type: Video Tutorial
  • Length: 11:21
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 121 MB
  • Posted: 07/14/2009

This lesson is part of the following series:

Chemistry: Full Course (303 lessons, $198.00)
Chemistry: Chemistry of Metals (8 lessons, $13.86)
Chemistry: An Introduction to Metals (4 lessons, $6.93)

This lesson was selected from a broader, comprehensive course, Chemistry, taught by Professor Harman, Professor Yee, and Professor Sammakia. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/chemistry. The full course covers atoms, molecules and ions, stoichiometry, reactions in aqueous solutions, gases, thermochemistry, Modern Atomic Theory, electron configurations, periodicity, chemical bonding, molecular geometry, bonding theory, oxidation-reduction reactions, condensed phases, solution properties, kinetics, acids and bases, organic reactions, thermodynamics, nuclear chemistry, metals, nonmetals, biochemistry, organic chemistry, and more.

Dean Harman is a professor of chemistry at the University of Virginia, where he has been honored with several teaching awards. He heads Harman Research Group, which specializes in the novel organic transformations made possible by electron-rich metal centers such as Os(II), RE(I), AND W(0). He holds a Ph.D. from Stanford University.

Gordon Yee is an associate professor of chemistry at Virginia Tech in Blacksburg, VA. He received his Ph.D. from Stanford University and completed postdoctoral work at DuPont. A widely published author, Professor Yee studies molecule-based magnetism.

Tarek Sammakia is a Professor of Chemistry at the University of Colorado at Boulder where he teaches organic chemistry to undergraduate and graduate students. He received his Ph.D. from Yale University and carried out postdoctoral research at Harvard University. He has received several national awards for his work in synthetic and mechanistic organic chemistry.

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I told you that metals are characterized as having high electrical and thermal conductivity as well as being valuable and ductile. What that means is that metals can carry current when you apply a voltage. And they also, if you heat up one end, the other end gets hot. So if you take a bar and you heat up one end, the other end gets hot. And then you can pound them into flat sheets or draw them into wires.
Let's look at how scientists think about how we can model a metal and explain these properties that descriptive of what a metal is. And the first model I want to look at is the oldest one. It's called the free electron model. And the idea is that let's the take the nuclei of each individual atom that make up our metal and its core electrons and just symbolize it by a plus. And then all the valence electrons are going to be poured into this solid. But they're basically free to move about the entire solid. The nuclei are fixed by their lattice sites, but the electrons are delocalized over the whole solid. And this is something called the free electron model.
Now, it sort of makes sense, because we know that metals are relatively easy to ionize, so they're relatively easy to take an electron away from the neutral atom so that this electron isn't particularly tightly bound to this positive nuclear site. And so, the idea that the electrons might be delocalized makes a lot of sense.
This model allows us also to understand why a metal might be malleable or ductile, because the idea is that the electrons are sort of this delocalized glue that holds the solid together. But if we wanted to take this nucleus and move it over a little bit to the side or something like that, the electrons are just going to go along for the ride and provide the bonding that's going to allow this solid to stay in one piece, as opposed to sodium chloride, fracturing and cleaving along crystal plains.
Now, the idea that it might be thermally conductive fits with this picture as well, because the electron - if we heat up this end for instance - the electron is going to have more kinetic energy. And so having more kinetic energy, it's going to move faster and the electrons, then, can carry that kinetic energy over to the other side. Having more kinetic energy, it will heat up the other side.
And as a little story, if you take a pure metal, it typically has both high electrical conductivity and high thermal conductivity. And what people historically have done with things like silver is make utensils and champagne buckets out of these metals and it costs a lot of money to make a sterling silver spoon or a sterling silver champagne bucket. Well, if you take such a spoon and you stir your hot coffee, what you soon find is that the handle is too hot to touch. And similarly, with the champagne bucket, when you put the ice into the champagne bucket, it actually melts reasonably fast compared to if you put it in a Styrofoam cooler. Why is this? Well, it's the thermal conductivity of the pure metal - it's really high. And why is this a problem? Who wants to stir coffee and burn your hand? Wouldn't it be better if you used a plastic spoon, which clearly doesn't get hot? Or who wants to ice to melt in your ice bucket? What you want is you want to keep the ice around and keep your champagne cold. The point is that people who spend a whole lot of money to buy these really expensive items, like sterling silver spoons and sterling silver champagne buckets, are wasting their money if their goal is to have something that's functional, because you burn your hand if you stir your coffee with a silver spoon.
Well that whole idea is explained very nicely with the free electron model. It also explains why alkali metals, having only one valence electron per atom, are typically not as conductive as some of the other metals, because they just don't have as many electrons in this electron sea to carry the electrical current. Now this picture works, as far as it goes, but it turns out it doesn't explain some things. For instance, it doesn't explain what a semiconductor is. And so we go to a more evolved model that's more difficult to get your mind around, but it does explain, ultimately what semiconductors are all about, but first let's look at what they have to do with things that are metals - again, things that conduct electricity.
And where we're going to start is to consider an extension of molecular orbital theory. From what we've looked at previously, if we take a carbon that has two hydrogens on it, it's going to have a 2p orbital left over, and that 2p orbital will have 1 electron. And if we take 2 of those fragments and then stick them together, we make ethylene. And ethylene is going to form both a bonding and an anti-bonding molecular orbital. And because there's one electron in each of the 2p orbitals, those electrons are going to go to fill the sigma bonding level. And this is molecular orbital theory picture for ethylene. Now if we go to butadiene that has 4 carbons and then we develop them like an orbital theory picture, what happens is we have 4 molecular orbitals that arise from the overlap of the 4p orbitals. Again, it's the 4 2p orbitals that are perpendicular to the piece of paper. And then, because we form 4 orbitals but we only have 4 electrons. We pour the electrons in and only the 2 sigma bonding levels are filled.
Now let's extend this out to say Avogadro's number of CH groups. And so I misspoke earlier. I didn't mean to say a carbon with 2 hydrogens. I mean a carbon with 1 hydrogen. So these points are carbon with 1 hydrogen. If we go to Avogadro's number, we can imagine a chain that has Avogadro's number of CH groups that are in a chain like this. And this compound is known as polyacetylene and you can actually make it. It turns out if you take acetylene, which is HC, triple bond C, single bond H, you can polymerize it where you trade one of the pi bonds for a new sigma bond and form an infinite chain, or at least a chain of macroscopic dimensions that has alternating double bonds and single bonds and double bonds and single bonds - exactly like butadiene - except now we've got Avogadro's number of CH groups in this thing. And so what happens is we end up with Avogadro's number of molecular orbitals. And I'm just using Avogadro's as a big number to represent the number of atoms in this chain.
So given that we have Avogadro's number of molecular orbitals and that each orbital holds 2 electrons, what we do is we say each carbon comes in with 1 p orbital. So what we're going to do is we're going to end up with something that puts electrons into each one of those levels going all the way up until we've filled up this level that's exactly half-way up. And then we run out of electrons.
In much the same way, we filled the bonding orbital and not the anti-bonding orbital. If you count the number of orbitals that we've got, we've got Avogadro's number, going all the way up to the top. But each one holds 2, so we only fill it up halfway, and this is known as a partially filled band and the notion of a partially filled band is what gives rise to electrical conductivity. And the analogy - this is a Chinese checkerboard. We're not playing the game. What we're doing is we've got these indentations and the indentations represent molecular orbitals that the electrons can move into with a relatively small input of energy. So in other words, there are orbitals here that you can put in just the tiniest bit of energy and the electron can move into that orbital. And the idea is that having available orbitals into which the electrons can move is what gives rise to electrical conductivity. In other words, the electron can move along the chain, meaning that it has a new orbital that it can move into.
So clearly, these electrons each have open orbitals that they can move into. But if you think about what would happen if the board were filled, the electrons couldn't move around - the marbles couldn't move around - because if all the slots on the board were filled, then there would be no open space for the electron to go to. And we'll talk about that later on.
But let's now think about how this idea of polyacetylene extends to a real metal. Let's consider lithium. Lithium has 1 electron and a 2s atomic orbital. And if we take 2 lithiums and we put them together, we form a dilithium. Just like Star Trek - they're always talking about dilithium. Turns out, dilithium is a reasonable molecule in the gas phase. It consists of 2 lithium atoms together. The 2 electrons go into the bonding orbital. This is a sigma bonding orbital. If we go to lithium-4, then we'd have a picture that looks like this. And we'd go to lithium with an Avogadro's number worth of lithiums. We're again going to have a band and once again, It's going to be half-filled. The energy of the very top of the band is something called the Fermi energy. It's not really important to understand what that is. The point is, with lithium we get a partially filled band. And the partially filled band is what gives rise to electrical conductivity.
Now, if you're thinking about what's going on, you'd say the next element over beryllium, which has 2 electrons in the 2s, ought to give rise to a filled band. In other words, with 1 electron in this orbital, it fills up halfway. So if we had 2 electrons in the 2s orbital, it would fill all the way up. And that's equivalent to - on our Chinese checkerboard - putting electrons into all of these orbitals. And then, we have no free orbital. So what do I do with my marbles? I'm not going to do it, but the idea is that if we had beryllium, that would mean putting electrons into all of these open orbitals as well. And so you can see that as we fill it up - as we put electrons into all those orbitals - there are going to be no open spaces. And remember, open spaces are the key to electrical connectivity. So the idea is I could fill this up and, in fact, if you counted the electrons for beryllium, it would go all the way up to the top.
Now, we know that beryllium is a metal. So how do we explain why beryllium is a metal, even though by this picture, beryllium shouldn't be a metal? And the answer is that we have other orbitals that we can consider. For instance, in beryllium, we can talk about the 2p orbital. The reason we didn't talk about the 2p orbitals in lithium is we didn't need to. We could adequately explain the metallic properties of lithium without considering the 2p. The 2p are there as well and they probably do play a role. But in the beryllium, we need them in order to explain the metallic properties. The 2p gives rise to a band as well and that band overlaps with the 2s. Could we have predicted this without any other information? Can we use this to explain why it's electrically conductive? Absolutely. What happens is we're basically bringing in another Chinese checkerboard. And that other Chinese checkerboard will take some of the electrons that would have been over here to fill up this 2s band, and they'll go into the 2p band. Why should it do that? Because electrons are like water. They seek their lowest level. And so if the energy of the lowest part of this 2p band is sufficiently low, the electrons will go into this band as well. So now we have a partially filled band that's rising from the 2s and a partially filled band arising from the 2p. Again, it's like 2 separate Chinese checkerboards, and each one of those is going to give rise to electrical conductivity. Okay, so this band structure model, which is really molecular orbital theory but applied to metals, gives us a way to explain the electrical conductivity. Is it absolutely, necessarily much better than the free electron model? The free electron model explains a lot as well. But what we're going to see is the band structure model allows us to explain things like insulators - electrical insulators - things that don't conduct electricity - and things that are semiconductors. They conduct electricity, but not really well - not as well as metals do, and we'll see that in the future.
Chemistry of Metals
An Introduction to Metals
Band Theory of Conductivity Page [3 of 3]

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