Question

The atomic radius of metal X is 135 picometers (pm) and a crystal of metal X has a unit cell that is face-centered cubic. Calculate the density of metal X (atomic weight = 42.3 g/mol).

Answer 1

First remember how many atoms there are per unit cell of a face-centred cubic structure:

http://faculty.chem.queensu.ca/people/faculty/mombourquette/FirstYrChem/Solids/FCC.ht2.jpg

You can see there are a total of 4 atoms once you add up the 8 corners (⅛ of an atom each) and the 6 faces (½ of an atom each). The question tells us that the mass of the metal is 42.3 g/mol. First convert this value to g/atom - divide by Avogadro's number to find the mass per atom, then multiply that mass by 4 since you have 4 atoms in the unit cell.

To calculate the volume of the unit cell, you need to realize that the DIAGONAL of each face has a length of 4r - you get 1r from each corner, and 2r from the centre of the face (look at the picture above to see what I mean). Because this is a cube, you can easily find the side length through the Pythagorean Theorem, then cube it to find the volume.

Then just divide mass by volume, and you have density!

Answer 2

Thanks!