Solid state crystal structure question regarding face-centered cubic type and finding volume - It should be easier than this people but...?

Question

The volume of one atom of a metal is 3.57 x 10^5 pm^3.  The metal has a face-centered cubic crystal structure.  What is the volume of the unit cell?

(A)	3.65 x 105 pm^3
(B)	2.43 x 106 pm^3
(C)	1.93 x 106 pm^3
(D)	1.05 x 106 pm^3
(E)	6.70 x 105 pm^3

Okay, so let me start by thanking you for clicking on my question.  Thank you.

The FCC Structure has 4 atoms per 1 unit cell (See image).

So, take the volume that's given, which is the volume of 1 atom and multiply by 4 which gives you 1,428,000 or 1.428x10^6 pm^3 but none of the choices are even close.

Worse yet, I don't have the answer key so I am flying blind.

Any suggestions?

pisgahchemist · Accepted Answer

The question doesn't ask for the volume of the metal.  It asks for the volume of the unit cell, which includes the spaces between the metal atoms
Look at the diagonal of the unit cell.  The diagonal is 4r.  Use the equation for the volume of a sphere to compute r (it will be 44 pm), then find the diagonal of the unit cell (it will be 176 pm).  Then find the length of a side of the unit cell (divide the diagonal by the square root of 2).  Lastly, cube the length of the side and get 1.93x10^6 pm³.

Solid state crystal structure question regarding face-centered cubic type and finding volume - It should be easier than this people but...?

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