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About this Lesson
 Type: Video Tutorial
 Length: 14:59
 Media: Video/mp4
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 Posted: 07/14/2009
This lesson is part of the following series:
Chemistry: Full Course (303 lessons, $198.00)
Chemistry: Condensed Phases: Liquids and Solids (15 lessons, $25.74)
Chemistry: Solid State: Structure and Bonding (5 lessons, $7.92)
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, oxidationreduction 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 electronrich 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 moleculebased 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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Solids can be divided into crystalline solids and amorphous solids. And crystalline solids have the property that there's a periodic lattice that makes up the bulk. Now, to understand what a periodic lattice is in three dimensions, it's a lot easier to think about it in terms of two dimensions and talk about the terms used in two dimensions, and then extend it into three dimensions.
So, consider this periodic array of dots in two dimensions. They have an x direction and a y direction. And when we say "periodic", what we can think of is we have this rubber stamp and the rubber stamp just comes along and thumps down on the paper, and it creates a dot. And it creates a dot at regular intervals and, in this case, it creates it in regular intervals where the distance between the dots going in the up and down direction is exactly the same as the length between the dots going in the crossways direction. But that's not necessarily the case. For instance, we could have dots that are much further apart in one direction than in the other direction, and then, instead of having square lattice, we'd have a rectangular lattice. And what we'll see is that's a perfectly reasonable thing.
Now, we can define something called a unit cell. And what the unit cell is, is it's a repeat unit, which, when you extend it in the two directions of the lattice, can create the whole lattice. So, for instance, suppose our unit cell is this square. Then you can see that, if we take the square and we stick a next square on top of it, we'd get the dot. And we do another square and we'd get the dot. So, in other words, the stamp doesn't have to include the red, but the point is, if we had a stamp that was as big as the red, then, by translating this red square, we can create the entire lattice. And this is what is known as a unit cell. Now, you'll note that we could have made it twice as big. We could have had the unit cell include 2 dots, so it would be twice as long as it was wide, and we would still be able to translate and create the whole lattice. And, in fact, it could have been 4. It could have included 4 dots. The point is that what we've drawn here is just a repeat unit, but that's not unique. Now, what is somewhat unique is that you can draw the smallest unit cell, and this is clearly the smallest unit cell. The smallest unit cell has to contain 1 dot. If it contained only part of a dot, then it wouldn't be able to recreate the entire lattice. Now, another thing about the smallest unit cell is it isn't unique so long as it has 1 complete dot in it. So, instead of drawing the unit cell like this, we could have drawn the unit cell like this green spot, where, instead of having a dot in the middle of the unit cell to translate, we could have created the square that has a of a dot in each of the 4 corners. And you can see that, if you take this green box and translate it sideways and up and down, we can still recreate the entire lattice. So this unit cell is exactly the same size, so the green is exactly the same size as the red, except the red has the dot in the middle, whereas the green has of each of the dots in each of the corners.
Let's look at something a little more complicated. Here's a periodic lattice, but the periodic lattice is not square, not rectangular. In fact, the unit cell is a parallelogram. So we take this parallelogram of these fuchsia flowers and, if we take this one and we translate it one to the right, we would make the next orange one. We'd go one to the left, we can create this one. And, if we go up diagonally, we would create this flower. So we can reproduce the entire twodimensional lattice, but, in this case, the unit cell is not square, it's not rectangular, it's a parallelogram. Once again, you can see that the unit cell we draw doesn't have to entirely contain one of our repeat units. It could contain little bits of each repeat unit, like the unit cell that's drawn here. But you can see that, if we take this piece, and that little bit, and this little bit and this little bit and add them all up, what we would get is the whole repeat unit. So we would get one whole fuchsia flower, except that, the way we've drawn the unit cell here, we'd get a little bit of each one. There's not really any great advantage in two dimensions of drawing it. My point is that we can draw it more than one way, but the volume of the unit cell or, in this case, the area of the unit cell, is exactly the same for both.
Now, let's consider what this looks like when we extend it to three dimensions. And, in three dimensions, it turns out that there are 7 primitive crystal systems. And the 7 primitive crystal systems are cubic  so, obviously, if you take a cube and you can take a bunch of cubes and you can stack them together to form a periodic array, except that now we're going to go in three dimensions. And so, if we were going to stack, for instance, Styrofoam balls, like this, we could build a lattice of Styrofoam balls and it would look something like this. And this is a cubic lattice of Styrofoam balls. What's the repeat unit? Well, the repeat unit could be a cube that just surrounds this sphere. That's one way to think about the repeat unit, or the unit cell. And another way to think about it, and you'll see a drawing in the graphic right over here is that, if you take this, which looks like it contains 8, but, if you shave off all of the pieces that aren't within 1 unit cell, what you're going to see is that it contains 1/8 of each one of these balls. In exactly the same way this picture here contained of each one of the round dots, the unit cell of a simple cubic, and this is called cubic or simple cubic, is going to contain 1/8 of each of the corner atoms. Well, 8 times 1/8 is equal to 1, so there's 1 atom in the unit sphere, or 1 sphere in the unit cell, just like, if we make the unit cell surrounded on a dot instead or on a sphere instead, we'd have 1 atom in the unit cell. Well, as you progressively distort the cube more and more, you get the various other ones. So tetragonal, for instance, comes from  and you can see in the graphic what the prescription is for making the unit cell. But, if you take a cube and you stretch it in one direction, you get a tetragonal cell. For instance, instead of packing baseballs, if we were packing footballs, we would have something that would pack in a tetragonal cell. If you stretch in the other direction, but still keeping all the angles 90 degrees  so now we have a not equal to b not equal to c, but we still have 90 degree angles, that's known as an orthorhombic unit cell. If we start with a cube and pull on diagonal corners, so, for instance, we take this cube and we pull on corners on opposite ends, we would get a trigonal unit cell. And then, if we take this cube and we tip it over, tip it sideways, so move the top pack sideways with respect to the bottom one, we'd have 2 rectangular faces and 1 face that was a parallelogram. And, if all the lengths of the sides are not the same, that would be monoclinic. If we tip it so that all the bound angles are no longer 90 degrees, that's triclinic. And then a somewhat special case is hexagonal, where and are equal to 90, so we have rectangular faces on this parallelepiped, but the angle that isn't 90 degrees is 120. And you can see that it is 1/3 of a hexagonal pyramid. And so, that's why it's called hexagonal. And we'll talk more about this later on.
These are the 7 primitive unit cells. And one question you might ask is, "Well, how do we know that this is really what's going on? How do we know that this is how atoms and molecules pack in crystalline lattices?" And the answer is a technique called xray diffraction. And xray diffraction was first suggested by Max von Laue in 1912. He suggested that, since the wavelength of an xray is on the order of the spacing between atoms, that atoms in a crystalline lattice should be able to serve as a diffraction grading for xrays. And this was shown by the father and son team of William Bragg and Lawrence Bragg just a little bit later than that. And all three of them won the Nobel Prize separately  well, von Laue won the Nobel Prize separately, and then the two Braggs won the Nobel Prize. And the idea of xray diffraction is this: suppose we have an incident beam of xrays  and remember, xrays are electromagnetic radiation, and so the electric field looks like a wiggle. So this wiggle is coming in and we get reflection, and reflection is just a way to think about it  it's really diffraction, but reflection works as a way to think about it. We get reflection of these waves coming out. And it turns out that this reflected beam gives rise to a spot or no spot, depending on how the reflection from, say, the top layer here and the next layer down interact with each other. So the idea is that sometimes the wave coming off the top layer is going to reinforce the wave coming off the second layer, and sometimes the wave coming off from the top layer is going to cancel the second layer. And why is that? The reason is, because these are waves, waves have the property that they can either interfere constructively or destructively. And so, for instance, if the wave coming off the top layer interacts constructively with the wave coming off the bottom layer, then these two guys will add to each other, their intensity will reinforce, and what happens is the diffracted or reflected xray is going to be very intense and you'll see a spot when you put a fluorescent screen and collect the light that's coming off of your lattice. But, in contrast, if you don't satisfy the criterion, where you have the wave coming off the first plane and the wave coming off the second plane, being exactly lined up like this, you can get what is known as destructive interference. And again, this is because the xrays or waves  it could be that the phases are exactly 180 degrees out of phase. And so, where this one is small, this one is large, and where this one is large, this one is small. And what happens when you add waves that are 180 degrees out of phase is you get nothing. You get complete cancellation.
So how can we know when we're going to see a spot and when we're not going to see a spot or, alternatively, when we see a spot, what does that tell us about how far apart the planes are? And to understand that, we need to essentially derive what is know as Bragg's law, or the Bragg equation, and the idea is this: so here's that first light beam coming in, so it actually represents a wiggle, but I'm just drawing it as a straight back line. And then here's the second one coming in that goes off the second plane. And the idea is that, if the light coming off on the reflection side is in phase, what must necessarily be the case is the additional distance that it travels in reflection off the second plane  so, in other words, this guy coming off, and then reflecting off the first, the one that reflects off the second travels a little bit further. And, in order for it to have reinforced amplitude  so, in other words, to add constructively  what has to happen is the distance from the black green line where my index finger is here to this point to this point  so the sum of this line here and this line here  has to be an integral multiple of the wavelength. In other words, we're basically saying that you can shift these with respect to each other and, if you shift them such that one of them is high while the other one is low, it's going to completely cancel, whereas, if they're shifted such that when one is low the other one is low and, when one is high the other one is high, you're going to get constructive interference. And so, what we're looking at is we're hoping to have the path length difference between the black line that bounces off the first plane of atoms and the black line that bounces off the second plane of atoms reinforce constructively. What it means is that the path distance, which is this distance and this distance added together, has to be an integral multiple of the wavelength of the light we're using. And we know what the wavelength of the light we're using is, because it's an xray and we can essentially filter the xrays so it's monochromatic, meaning that it's only of 1 wavelength.
We can measure the angle that the light comes in and the light goes out. And we'll measure it and we'll call that , and that is the same for both. So, in other words, it comes in at one angle and it goes out at exactly the same angle. So that's . What we're trying to determine is we're trying to determine d. We're trying to determine how far apart the atoms are between two different layers. And the path difference, again, is this line plus this line. Well, if we do a little bit of trigonometry, we can see that this black line here is just d times sin of , where , again, is this angle. Trigonometrically, it's also the interior angle there. So it's between this dotted blue line, and the green line is also by geometry. And so, the path length's difference is d sin , and we have to take 2 of them, because this far and that far are both exactly the same distance. And again, what we say is that has to be an integral multiple of the wavelength, or the difference in the distance traveled has to be n times , where is the wavelength. And it depends on how far apart these are, and it depends on the angle that we're measuring, and it depends on the wavelength of light we're using.
So, we can then set the difference in the distance traveled, which is n times , equal to 2d sin , where we had to take 2, because there was 1 piece for the incoming and 1 piece for the outgoing, where n can be any number, but, if we let n be 1  what we say is that, if n is 1, it's first order diffraction. When n equals 2, it's second order, when n equals 3, it's third order, but don't worry about that. The point is let's just let n equal 1. If we know the wavelength of the xray, that's . And we know what is, because that's just the geometrical angle that the light comes in and the light goes out. Then we have a relationship, where we can solve for d. d is equal to over 2 sin . So we know , we know , that means we can know d, that means we can know how far apart the atoms are. Well, if we know how far apart the atoms are, that gets us well on our way to understanding how the atoms are put together in the lattice.
So the bottom line is that, in a crystalline lattice, there is a repeat unit in three dimensions. It isn't necessarily exactly the same distance in each of the 3 directions. It isn't necessarily the case that the angles are all 90 degrees, but it is a regular lattice. And we can learn something about the regular lattice by a technique called xray diffraction that tells us, in the most rudimentary sense, how far apart the atoms are. It turns out that, if we know more, for instance, we know the intensity of the xrays that are scattered, we can actually learn the actual positions of the atoms within the unit cell. And we're not going to talk about that, but just sort of take it on faith that that can be true. And if you accept everything that I've told you, we can understand a lot about crystalline lattices, well beyond the sort of dots and fuchsia flowers to actually talk about molecules in crystalline lattices.
Condensed Phases: Liquids and Solids
Solid State: Structure and Bonding
Crystal Structure Page [3 of 3]
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