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And so last time we talked about the different types of solids you can have. And so we specified that for a crystalline solid, you have a defined flautist. And so we'll talk about one specific Gladys, which is the cubic structure, Um, but essentially, there are seven types of unit cells that we can generally organize all of the different types of solid as, um but that it's more out of the scope of, of course, and so, if you're interested in it, you can take a course in material science or solid state chemistry. But for thes lecture topics, I'll be mainly talking about the cubic structure because there are different types of import structures that would be useful for you to know anyway. And so we'll actually talk about three different types of the cubic, and so the first one that will talk about is called Simple Cubic. And so, for a simple cubic structure, it is essentially a cube, and so we have an Adam at each of the four corners of a cube, And so again, um, a lot of points are points that tell us where each of the atoms or molecules or ions are in some kind of material, and the unit cell is essentially the smallest building block of the material. And so we can imagine that these cubes, specifically the simple cubic will repeat itself in three dimensional space in all different directions and so on. The simple cubic is made up of atoms on each of the four quarters. And so if we think about how the atoms are arranged in space, we can see that each atom is surrounded by six neighbors. And we know this because if we draw an extended cube like so we know that we have another neighbor over here and on the back and it down here. And so we have one, 23 45 and six. And so we know that in three dimensional space, each atom is surrounded by six other Adam and soon, Um, and another important thing to note is that for all the unit cells, we can define a unit cell length, which we often designate as a not. And so this is simply just the length of the whole, um, you know, itself. And so in this case, we can see that the, um, everything about how this is structured without the spaces. This would simply just be, um he points connected to each other. And so let's actually make this a bit more clear. Um and so we know that if we take out the spaces, the packing looks like this. So if this is one face, we have all of the atoms. If we pretend that there is an atom on each corner, we can see that the unit cell length would essentially be from here to here. And so if we zoom in a little closer, we see that it is essentially, um, the center point of the atom to the other. CenterPoint. And so if we define us as the radius as, um, Adam, the pain, it would be to our And so we could say that in this case, you know, it is equal to do have. And so when we think about the unit length, we're talking about the specific corners, and so it wouldn't stretch out for the whole atom because these parts of the atoms would be a part of another units on. And so we only count from the points directly in the center of these spirits and So in this case, we can just say that again. The unit selling is simply just to our if r is the radius of these spheres. Um, and so again, this is the genetic structure of simple cubic. And so the second type of, um, cubic structure is the bodied center. Cubic. And we've actually seen the body centered cubic before from one of our examples. But essentially, this is very similar to the simple cubic in the sense that we have thes spheres which again can represent an atom or molecule on DSO. These are located at the corners. Well, thank you. But what's special about this is that it has an Adam right in the center as well. And so we imagine that we have one den center. And so if we draw like little guidelines, we know that this is from the center of the cute and so for the body center. Cubic. This one has eight neighbors, which makes sense because all of these points are actually identical in space. And so if we look at he so no one, we know that we have eight because there are the quarters of acute and so that is pretty clear. And in it is actually, um, different. Because if we think about this in terms of how this is packed in space, we know that these items are not touching each other, like in the simple cubic, because we have this extra atom in the center. And so, um, the points at which the spirits connect is actually right through the body diagonal. And so, um, it would be this line over here connecting thes, um, three years, and so a not is actually equal to four are over route three. And we can actually calculate this based on the geometry of the triangle, because we know that unit Cell Cube is equivalent here, and so you can find the length of this. And we know the length of this and this, and then you just need to find the body Dagnall. And so again, the body centered cubic example has the close packing across the body diagonal, which is different from simple cubic because all the atoms are touching each other, and the last type of cubic shocker that will learn about is called the face centered. Cubic. And so they face centered. Cubic, Juan. Um is also very similar. Thio the body centered cubic in the sense that we have additional items on three units. L accept. These are located on the faces. And so for this one, we can still have some proponents of a simple cubic in the sense that we still have serious located at the different corners Cube. But what's different is that now we have Sears on the faces of the Cube and so we can have one here on one base as well as thes faces as well as the front and Libya. And so the face under cubic thes have 12 neighbors. And so we can see this. If we draw out and extended, you didn't sell. And so let's say that we have this quarter and we have this neighbor and this neighbor as well as thes neighbors. And so we have three here and we also know that this is connected. Thio the faces So we have right now 123 45 and we know that we have another one right across from it. And we also know that if we extend this in three dimensional space, there's also one that is right in the center of those things. So we have seven. You also have one right on the bottom, which is eight. And we also know that we have some more on the middle sections of each. You know, it's all and so that would be nine and 10. And we also know that we have one on the bottom faces as well. And so that would be the remaining serious. And so we have thes how 12 neighbors, if you think about how they're oriented in three dimensional space, um and specifically in this case we have do not is equal to four are over route to because again, this is calculated based on the geometry. And so we know that for the face centered cubic, the atoms that are touching, um, each other would be, um, here. And so we can use this to make a relationship between the radius of the sphere and the unit cell, Actually, and so again, we can calculate this because we know that across here we have four are and we know that these edges are the same like and so we can calculate this, um, using some geometry. And so now that we've learned about the different types of you know itself. We can also talk about how these fears can be packed together. And so there are three different types of spheres or packing that we can have. And so one is called the H C P Packing or ABC packing, and one is called the CCP, which is a B A B pack. And so HDP represents hexagonal close packing and CCP represents closed cubic packing. And so let's actually compare the two. And so let's say that you have some generic layer of spheres lined out, and so the spheres are spears and they're all touching each other. And so in the unit. So we added the spaces just because it's a lot easier to see how it's structured. But in real life, the materials, um, can be thought of as thes spheres and do not have spaces in between them. And so again, let's actually compare the two different types of structures, and so on the left will actually be showing, um, a B A packing and on the right will show BBC. And so right now we just have Larry A and so again, this is simply just the serious being connected to each other in space. And if we add a second layer, we can see that in both cases we have layer A and we also have a Larrabee. And so Larrabee is placed on top, and we basically added another layer of these fears on a very specific point on A and so we can see that we have placed a sphere on each of these particular points, and it's important to note that all of the points with an upward facing triangle are filled. But if we look, these spaces are unfilled, and so let's actually highlight that's it's easier to see. We can see that the down triangles have still have an open space, but on the second layer we see these upward triangles so being occupied. And so this is really important to note, because this is how we're going to distinguish between the A B A packing as well as the ABC packing. And so we see that for both of these, these air occupying these red triangle spaces and the green triangle spaces are still open. Um, and so we essentially put all of the Sears onto these holes and it's important to note that you can't occupy the red region and the green region at the same time because there is simply no space between these years. And so this would be considered close packing just because this minimizes thes space in between, you do like it. And now what? Actually look at the last layer, which will, um, tell us more information about the differences between the two types of clothes backing. And so the bottom layer is actually be, but is in a different color. But this darker layer is again, and so when you have a B A B packing, we see that we are putting the A again where it was located originally, and so you're only occupying the read spaces for the layers, and so the green space will never be filled. And so this is called a B A B packing, because DBS will always be located in the same spot, and the A's will always be located in the same spot as well. But what's different is that for ABC packing, we see a difference in the sense that we're occupying different spaces. And so let's recall, um, that for Thebe packing we occupy thes red triangles and we leave the green spaces open has seen here. And so I'll just fill these in. But when you look at ABC backing, we see that in the So it's actually was, This is a This is B and this is C And so we see that for the bi layer were occupying the rent spaces, but in the sealer were actually occupying the green spaces that we couldn't fill before. And so this is important to note because this is the biggest difference between A B, A, B Packing and ABC ABC packing because we're adding another layer on a different type of empty space. And so because of that, we can see that the layers become staggered because of the fact that we are putting this extra layer on a different type of space. And so, um, be face centered cubic structure actually follows the ABC ABC packing, Um, and so that's really cool to think about in terms of seeing how this is structured in three dimensional space versus a different kind of structure, like the A B A B packing. And so, for our first example, let's actually determine the number of atoms in a unit self for different types of Cuba instructions. And so we know that if we have some kind of structure, we know that the atoms or the spheres actually do not take up all the space that shows because it is being shared by different unit cells. So let's actually calculate this for a simple cubic structure first. And so let's say that we have some kind of generic cube, and we know that for a simple cubic structure we have Sears on each of the four corners and so let's just rallies in blue. Uh, and so we know that we have spears. And we also know that, um, the answer would not be eight Adams just because of the fact that these items are three D spheres and they're actually being shared between a couple other unit cells. And so if we actually draw this in three dimensional space, we know that there can be a cube here and in the front as well in the back. And so you know that each of these spheres are actually being shared by eight other cubes, so we stock for over here that's basically splits thesis fear into four pieces. And if we imagine this in the back as well, we can visualize that, um, each corner only hold 1/8 of an atom if an atom is located at the last point. And so we actually know that because of that, we have eight of these. We have eight of these Adams, and we know that 1/8 of the atom is actually inside of one unit cell. And so if you multiply that out, we get one. And so we know that there is one atom per unit self in the case of a simple cubic structure. And so let's actually do the same thing. But for the FCC structure or the face centered cubic. And so from before we know that they face centered. Cubic is again essentially this cube. And if we just draw one face, we know that we have Adam's on each thes four corners of the cube. But we also have one located right in the center. And so we know that from the previous problem for the simple cubic structure, all of these corners will contribute 1/8 of the volume for one unit cell and we have eight of these corners, and so we have a count of one. But now we actually need to consider the fact that we haven't Adam located on the face as well. And so if we think about this in three dimensional space, half of the atom is inside one unit cell. And if we imagine another unit cell that is connected on the back, we can imagine that the, um, Adam is being cured between doing it sells. And so half of the volume is being shared. And so we know that cubes have six faces, and each of these years will contribute one half, and so does it be three. And so we have one plus three is equal to four. And so we have four atoms bringing itself. And so this is how you would calculate the atoms per the unit self. And since we're doing these Thio, we might as well do the last one, which is B. C. C. And so BCC can be represented again by a simple cubic. Except in this case, we have and Adam right in the center of the Cube. And so we have some items at the corner. We also have one right in the center, and so for this one, we can see that we can count the corners as well. But with different is that we have a whole atom in the center. And so we have eight times 18 because we have eat atoms in total for all of the corners, which all contribute a volume of 1/8. And we also have one to the center and so every obvious, the other. We have two atoms per unit cell. And so the reason why this is useful is because we can actually use this to calculate the density. And so, if you know, uh, the number of atoms, we can actually use avocados law to find the density. Given that we know the length of the unit self and so let's recall the identity is equal to mass over volume. And let's also recall that if you have atoms, you can use avocados numbers who actually get the moles on. Then, from there we can use the molar mass of an atom to get grams. And so let's recall that avocados number is 6.0. Do you do time sentence whether atoms for one more and we can use Mitchell analysis to make those grams So we actually can multiply this by the molar mass to get units of Mass on top. And we also know that if we have an edge length, we know that if we do length cubed, this would actually give us the volume. And so we can actually calculate the United States of different types of materials and so that let's actually do a couple of examples. And so let's say that we want to calculate the identity of some kind of material given that the material has an FCC structure. Yeah, and the atom has a Mueller Mass. 52 g per mole. And we also know that the unit, cell length ISS 562 parking meters. And so for this problem, we basically have all of the information that we need to calculate the mass and the volume. And so we know that for a SEC structure from before, we know that four atoms make up one unit self and so we can actually get the mask from this based off of the Miller math. And so we know that for he done to me. This is simply mass over volume and served with the mass. We know that we have four atoms and we can divide this by Allah God rose number to convert atoms. Two moles. And we can also multiply this by any more math to get the mass on top. And so if we do, these conversions are units will cancel. And so we're just left with the grams on top and now for the volume or given that the unit cell length is 562 pack a meters and so density is usually calculated in or over Centimeters cube and so will convert us. But we know that if we have some kind of cube, if the unit cell length is 562 we know that we can keep this to get the volume and so we can do 570 piper meters and keep this together volume. But we also need to convert this so we know that there is one times 10 winning and 10 centimeters for one pack a meter. And so if we keep this, the centimeters will be cubed and the PA communities will be cubed and the acute by commuters would cancel out. And so we're lasting Centimeters Cube. And so if you play this into our tacular, we get a density of 1.95 grams per centimeters keep. And so again, it's useful to understand three number of atoms that make up the unit cell, because we can use this information to find the density of some kind of material, given the appropriate pieces of information.
Acids and Bases