All right class. So this Alex topic is called finding an atomic radius from an FCC or BCC lattice constant. So FCC, that's face centered cubic and BCC, that's body centered cubic. And we need to again have a good idea picture in our minds of what those look like.
So this tells us that a certain metal am doesn't tell us what the metal is it's irrelevant. In this case, crystallizes in a lattice described by a face centered cubic unit cell. So you know if we want to draw a picture of that I've got my unit. Cell here and then face centered means that I would have the atoms in the middle of the face right. So on each edge, I would have an atom. And then in the middle I would also have an atom. And then it says, calculate the radius of atom M, and it gives us the lattice constant a.
So this distance here, the distance of one edge is our alpha our lattice constant. And that gives us our distance of 464 PICO meters. So it says, calculate the radius of an atom. And so the radius is going to be from the center out.
That's going to be the radius R, and we need to use this information basically to figure this out. So this is really a geometry problem it's, not really that related to chemistry, it's, definitely more geometry. And the first picture that I would draw is going to be just looking at one of these faces, and then I'm going to draw three atoms in that line. So these atoms they will be sort of touching each other more or less. And if I draw a diagonal line through them, I've got half of an atom.
So this would. Be one radius here, and I've got two radii here. This should be in the middle here, and then one more here. So this line here is essentially going through two of the atoms. And then I know, this distance here I know that that distance is my alpha distance so that's, 464, picometers.
And here I've got a right triangle. So this right here, you know, I'll just color this in a little more. This is what we're going to call a right triangle.
And if I want to find the distance for this hypotenuse it's going to be. A squared if I call this a plus B squared equals C squared so that's the equation to find the distance length here. And my a value is the same as my B value is 464 PICO meters. So I can go ahead. And do this calculation for 64 squared, plus 4, 64 squared equals C squared solve for C and C is going to be this distance here. And that distance is 650 six point, two PICO meters. So I'm, just staying in PICO meters since this was given in PICO meters, I might as well just stay in PICO meters.
And that tells. Me that this distance here is 650 six point, two PICO meters. So now I need to figure out well. What is the radius here? And essentially we've got four radii here 1 2, 3 4. So if I just divide this by 4 that's going to tell me the distance for one of these little pieces, so 6 56.2 divide that by 4 that's going to equal 164, PICO meters and that's going to equal my radius. So the distance between here to here that would be a radius and that's going to be 164 PICO meters.
And then the whole distance 1 2, 3 4. Equivalents, that's going to be 650, 6.2, PICO meters and I just figured that out because I drew this, you know, triangle, essentially this right, triangle and solve for that hypotenuse for that distance there, all right hope that helps.
Dated : 18-Apr-2022