Question

In a face centred cubic unit cell, what is the volume occupied?
$A.\dfrac{4}{3}\prod {r^3}$
$B.\dfrac{8}{3}\prod {r^3}$
$C.\dfrac{{16}}{3}\prod {r^3}$
\[D.\dfrac{{64{r^3}}}{{3\sqrt 3 }}\]

Answer 1

Hint: To solve this question use the concept that in a face centred cubic unit cell there are 6 faces, 12 edges and 8 corners. Atoms are arranged in such a manner that atoms are present in the centre of the face and at the corners of the cube. This will help to solve the problem.

Complete step by step solution:

In face centred cubic unit cell, number of faces in a cube are 6 and number of corners are 8
So, on each face there will be half atoms. And the other half of the atom will be in the adjacent cell.
Then we can observe that in each of the 8 corners there will be $\dfrac{1}{8}{\text{th}}$ of the atom. And the remaining will be in the adjacent cells.
Now when we calculate or 1 cell, then we get
For faces, since there are 6 faces, and on each face, there is half atom.
Total number of atoms becomes
$ \to \left[ {\dfrac{1}{2} \times 6} \right] = 3$
Now as we have mentioned earlier in this numerical that there are 8 corners and each corner has $\dfrac{1}{8}{\text{th}}$ of the atom,
So, our total number of atoms in 1 cell becomes
$ \to \left[ {\dfrac{1}{8} \times 8} \right] = 1$
There total number of atoms in 1 cell becomes
$ \to 3 + 1 = 4$ atoms
We know that volume of a sphere is given by $\dfrac{4}{3}\prod {r^3}$
The atoms present in the cell is in the form of sphere and hence volume of all the 4 spheres of the atoms in a cell is given by
${\text{Total number of atoms in cell}} \times {\text{Volume of sphere}}\left( {\dfrac{4}{3}\prod {r^3}} \right)$
$ \to \left( {4 \times \dfrac{4}{3}\prod {r^3}} \right)$
$ \to \left( {\dfrac{{16}}{3}\prod {r^3}} \right)$
Hence, we can see that the total number of atoms in a face centred unit cell is given by $\left( {\dfrac{{16}}{3}\prod {r^3}} \right)$. Here we must remember that $r$is the radius of the atom which is the same for all the atoms present in the cell.
Therefore, option $C$ is the correct answer and as per this in a face centred cubic unit cell, the volume occupied by the atoms will be $\left( {\dfrac{{16}}{3}\prod {r^3}} \right)$

Note: For solving these types of questions we must remember the properties of the distribution of atoms in a face centred cubic unit cell and also, we must remember the volume of the sphere because we saw in the numerical that volume of sphere was required. Other than face centered there can be other configurations even like body centered or even simple cubic, the arrangement of the atoms differ in all three of these.