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A quantitative measure of how efficiently spheres pack into unit cells is called packing efficiency, which is the percentage of the cell space occupied by the spheres. Calculate the packing efficiencies of a simple cubic cell, a body-centered cubic cell, and a face-centered cubic cell. (Hint: Refer to Figure 11.22 and use the relationship that the volume of a sphere is $\frac{4}{3} \pi r^{3},$ where $r$ is the radius of the sphere.)

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scc: $52.4 \% ;$ bcc: $68.0 \% ;$ fcc: $74.0 \%$

Chemistry 102

Chapter 11

Intermolecular Forces and Liquids and Solids

Liquids

Solids

Rice University

University of Kentucky

Brown University

University of Toronto

Lectures

04:08

In physics, a solid is a state of matter characterized by rigidity and resistance to changes of shape or volume. Solid objects have a definite volume, they resist forces (such as pressure, tension and shear) in all directions, and they have a shape that does not change smoothly with time. The branch of physics that studies solids is called solid-state physics. The physical properties of solids are highly related to their chemical composition and structure. For example, the melting point of ice is significantly lowered if its crystal structure is disrupted.

03:07

A liquid is a nearly incompressible fluid that conforms to the shape of its container but retains a (nearly) constant volume independent of pressure. As such, a liquid is one of the four fundamental states of matter (the others being solid, gas and plasma). A liquid is made up of tiny vibrating particles of matter, such as atoms, held together by intermolecular bonds. Water is, by far, the most common liquid on Earth. Like a gas, a liquid is able to flow and take the shape of a container. Most liquids resist compression, although others can be compressed. Unlike a gas, a liquid does not disperse to fill every space of a container, and maintains a fairly constant density. A distinctive property of the liquid state is surface tension, leading to wetting phenomena.

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A quantitative measure of …

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Packing efficiency is defi…

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Calculate the percent of v…

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A simple cubic unit cell i…

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Assuming that in a simple …

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to start with approaching this question, I've gone ahead and drawn out each of the different unit cells and to calculate their various packing efficiencies, we've assumed that each cell has each add impact as closely as possible to determine the maximum possible efficiency that we can get. And this efficiency is gonna be defined as the volume of our spheres divided by the total volume of the self. And we can calculate the volume of the cell just by cubing the edge because it is in fact, Cuba working with and the volume of the spheres is just going to be 4/3 pyre cute, which is the volume of each individual sphere times the total number spheres that we're working with. So to get started with simple case of a simple cubic cell, let's get started. We know right away that V cell is gonna be equal to our edge. Cute and our edge is exactly two times are radius when they're smushed as closely as possible together. And that is just eight are cute. And we know that there are 18 of a sphere at each corner which add up to a total of one sphere in our simple cubic case. So therefore, V Sphere is just one times for third pie are cute. Now let's go ahead and calculate our efficiency to V sphere over the cell which equals 4/3 pi r cubed over eight r cubed, we go ahead and simplify this fraction down. We see that we are going to get pi over six, which is roughly equal to 0.5 to 4. Or in the simple cubic case, we're gonna have 52.4 percent. All right on to the more interesting case of a body centred Cubic, the body centred Cubic. We don't have everything lining up right along the edge, which is problematic because this is something that we need to find. However, we do know that everything lines up across the middle like so. Okay, that should be a straight line. But I'm not the greatest at drawing this so all draw it as though we can envision it right across a center cross section. It's gonna look like this right across the middle, where we have the body centred cubic, great in the middle and each other sphere pushed up as close as possible against that one right in the middle. And this line right here across the center is what we know the distance off. That is gonna be 123 for our for our distance right across the center of this. But we need to see what that is in terms of the entire cube of the whole. And we can do that using Pythagorean theorem. So if we consider just the bottom of this cube will have something that looks a little bit like this where we have some spheres here, some spheres here, sphere here and is fear here. And if we calculate this high part news, that is gonna be the edge that eventually weaken. Use Pythagorean theorem again to determine what this is in terms of the edge. So this distance here is one edge which we can denote you. And this year is also eat. So this high part news some value Age Square is going to equal to e square just by basic Pythagorean theorem that it's gonna be e squared plus b squared equal or I pontin e squared. Now let's go ahead and look at this case here where we know that this edge here on this bottom side is, in fact, that high part news that we just computed that high partners is gonna help us here because we know that this edge here as Ling H and we know that this edge here is like e because it's a right triangle by Pythagorean theorem, we can relate them so we can say that four are all squared is going to equal this e squared plus h squared, which simply equals three e square. So as a result of this, we can say that our edge is going to equal. Well, that's going to be four are over. Square it of three. Okay, so now we have the cell, which is just gonna be this cute. So let's go ahead and figure out V Sphere. The volume of our sphere is gonna be just 4/3 pi r cubed times how many spheres we have. We have an eighth of the Cuban age corner and we have a sphere in the middle, which gives us a total of two spheres here. So that's two times for third pi r cubed. And now we can go ahead and figure out this ratio, so v sphere over V so equals 8/3 pie are cute over four huge r cubed over three to the 3/2. Okay, let's simplify this down a little bit and we see that the ark you will drop from each still have pile on top. We're going to have 32 power of 1/2 one talk, and that's or well, rather we can just say that that's the square root of three. Because the three on the bottom will cancel history, which will be on top. And we have an eight on top and we have four cube on the bottom. So 8/4 to the power of three is going to get us value off. I see two little mouth, and that's just gonna be won over eight. So we have pie route three covert eat. And if we comfortably get a BCC value of approximately 0.680 so BCC equals 68% efficiency. Now, lastly, we have our FCC cell, and this case is gonna be much more simple because our edge where everything is lined up is right here in front of us. So using python were in there and once again, we know that this edge is gonna be a length of four are. So if we go ahead and square that that is gonna be equal to feeling of our edge squared times two because it's gonna be the edge squared plus the Edgware. I assume we know that edge is going to equal just for our over square it of to or if we want to clear the fraction that would be route to are times two. Okay, so now let's calculate v Sphere. How many spheres do we have? Well, in this case, we have 1/2 sitting on each edge, and each of these are gonna add up to a total of three spheres, plus the 8 1/8 in the corner for an extra one sphere gets us a total of four spheres, so that's four times for thirds Pi r cubed. So are packing efficiency here. These fear over visa is going to equal 16/3. Pie are cute, divided by eight. Route eight are cute. Let's go ahead and simplify this down and we're going to have two time over three Route eight, which is gonna be equal to square root of eight over three time all over. Believe believe that will be times for. So we end up with Route eight times pi, and that will be just over 12. And if we go ahead and calculate that out, that's going to give us a final value of 0.74 which means that in this case, our FCC equals 74% efficiency. So finally weaken some arrests, all by saying we have 52% and 68% and 74 percent.

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