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Who Wants to Answer a Crystal 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 10^5 pm^3
(B) 2.43 x 10^6 pm^3
(C) 1.93 x 10^6 pm^3
(D) 1.05 x 10^6 pm^3
(E) 6.70 x 10^5 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.
And what's worse, I don't have the answer key so I am flying blind.
Any suggestions? If you do happen to make a calculation that is one of the multiple choice answers, please walk me through on what you did to arrive at that solution.
Thanks guys :-D
2 Answers
- pisgahchemistLv 77 years agoFavorite 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.
- Steve4PhysicsLv 77 years ago
I suspect that you are confusing the volume of the atoms with the volume of the cell.
Volume of cell = volume of atoms + volume of space between atoms.
The packing-efficiency for FCC = 74% (=0.74). This means 74% of a cell is occupied by atoms, the remaining 26% is space..
Volume of 4 atoms = 4 x 3.57x10⁵ pm³
Volume of cell = 4 x 3.57x10⁵ / 0.74 = 1.93x10⁶ pm³ (Answer c).
______________________
If you haven't done packing-efficiency, you will have to use a 'long' method . Here is one way...
For 1 atom:
V = (4/3)Ïr³
3.57x10⁵ = (4/3)Ïr³
r = ³â[3 x 3.57x10⁵ / (4Ï)] = 44.0pm
The diagram in the 1st link below shows the diagonal of a face of the cell to be 4r which is 4 x 44.0 = 176.0pm
If the unit cell has sides length d, then using Pythogoras:
d² + d² = 176²
d = 124.45pm
Cell volume = d³ = 124.45³ = 1.93x10⁶ pm³ (Answer c).
Look at the 2nd link too.
Source(s): http://www.chemteam.info/Liquids&Solids/fcc-face-v... http://www.chem.wisc.edu/areas/clc/general/103/cry...
