Question

The radius of a single atom of a generic element X is 127 picometers (pm) and a crystal of X has a unit cell that is face-centered cubic. Calculate the volume of the unit cell.

Answer 1

Let us take a look at the side of a FCC crystal lattice ( In the diagram, I have drawn only one side)

From the above diagram we can see that, a face of FCC crystal lattice has diagonal = r + 2r + r = 4r

Now we can apply Pythagoras Theorem to find side of the square.

We have d^2 = s^2 + s^2 ( according to Pythagoras Theorem )

But we know d = 4r , so d^2 = (4r)^2

(4r)^2 = 2s^2

16r^2 = 2s^2

s^2 = 8r^2

s = sq rt ( 8r^2)

s = 2 r * sq rt (2)

Now they have given us value of r which is 127 pm

Let us convert this to meter

127 pm * 1 m/ 10^12 pm = 1.27 x 10^-10 m

Therefore we have r = 1.27 x 10-10 m.

Using this value of r, let us find s, side of the cube

s = 2 * (1.27 x 10^-10 m) * (1.414)

s = 3.59 x 10^-10 m

Therefore side of the unit cell is 3.59 x 10^-10 m

Volume of unit cell = s^3

Volume = (3.59 x 10^-10 m)^3

Volume = 4.63 x 10^-29 m^3

Volume of the unit cell is 4.63 x 10^-29 m^3