In this lesson, we're going to be talking about how we can apply some of these abstract concepts of packing factors in face-centered cubic and body-centered cubic materials. We'll begin with our calculation in the following way. First we know that the density is mass per unit volume. And in this particular case, when we think about mass per unit volume, it defines a quantity that we've described previously as an intensive quantity. That is, it doesn't depend upon the amount of material that we have. So, in order for us to begin the calculation, we have to come up with the description of the mass. So, for example, let's consider the possibility of calculating a material that is in the hearts sphere packing arrangement, either BCC or FCC. So in this case we need to know, how many atoms do we have in the unit cell? We need to know what the mass of each of those atoms are in that unit. For example, if we want to calculate a body-centered hard sphere cubic packing. What we know is that we have a total of two atoms in the unit cell, we know the relationship between the volume and the edge of the unit cell, and we also know what the relationship between the edge of the unit cell is and the radius in the packing of that hard sphere BCC structure. Alternatively if we look at the FCC material, here's the picture of our FCC unit again. And we have a total of four atoms in the unit cell, and we know what the relationship between the volume of the unit cell is, and the dimensions of the edge of the unit cell. And we know then the relationship between the edge of the unit cell and the radius of the hard sphere. So this provides us all the information that we need to know to do a calculation of density. So let's take a look at a couple of problems. First of all, we'll take a look at calculating the density of a material like iron. Iron can be, can exist in the form of material which is body-centered cubic. So we need to return to the body center cubic lattice that we've been describing. We know that there are, in the case of the BCC hard sphere, there are two atoms and those two atoms lie in the BCC unit cell. We know that the atomic mass of iron is going to be 55.85 grams per mole. And now if we continue, we need to know what the relationship between the number of atoms and the number of moles we have and we do that by incorporating the constant 6.023 x 10 to the 23rd. Atoms in a mol of material. So given the radius of the iron atom, so now we're providing the radius of the iron. With that we can calculate the volume of the unit cell and by completing the calculations we can come up with the density of 7.87 grams per centimeter cubed. So that's the density that we can calculate by knowing only the radius of the iron atom and knowing that it forms a crystalline body centered cubic array. Alternatively we could calculate the same information for a material that crystallizes in an FCC structure. An example of that is aluminum. Aluminum is face center cubic, and that means that we have four atoms in the unit cell. And the mass is 26.98 grams per mole. And now we're looking for the relationship between mols and the number of atoms in the mole for aluminum. And this will again use Avogadro's number, 6.023 x 10 to the 23rd. We know the relationship between the volume and we can calculate then the density. So we have a density of 2.7 grams per centimeter cubed. So now we've used two different types of materials, face centered cubic, body centered cubic and knowing the characteristics of the cells, that is. What direction do these atoms touch when they form these densely packed arrays? So we can then calculate theoretical density assuming that every atom is occupying a position and that we have hard spheres. Now it's going to be important for us to be able to calculate and make some simple calculations of a variety of different materials. We haven't gotten to some of the materials in terms of their structures. But we do know that there are some important characteristics that are associated with the density and a normalized factor that we refer to as the specific strength. So if we take a material, and we know its strength, and we divide it by its density, we can come up with the specific strength. Here I have indicated some very important materials that we use in the metals area. Magnesium, aluminum, titanium, iron, and nickel. It turns out that as we look at these different structures, we find out that the densities of these increase from magnesium to aluminum to titanium to iron, and then ultimately to nickel. Now what the idea specific strength provides us is the opportunity to make some comparisons between materials like aluminum, which is a very low density material and a material like iron which is a high density. And if may very well be the case that we can substitute the one structure for another, by compensating with a reduced density, for example, that we have in aluminum. And we'll touch on this more and more as we go through the remaining of this course. Thank you.