Question

In a face-centred lattice of $X$ and $Y,X$ atoms are present at the corners while $Y$ atoms are at face-centres. Then the formula of the compound if one of the $X$ atoms is missing from a corner in each unit cell would be:

• A. $X{Y}_{24}$
• B. $X_7{Y}_{24}$
• C. $X_{24}Y$
• D. $X_{24}{Y}_7$

Answer 1

The correct option is B $X_7{Y}_{24}$

$\text{Calculating the contribution}\text{of X atoms:}$

Each atom $\left(X\right)$ at the corners of the unit cell contributes $\frac{1}{8}$ to the unit cell.

There are $8$ corners in a unit cell. It is given that one corner atom $\left(X\right)$ is missing from the unit cell. So, there are a total of 7 atoms $\left(X\right)$ are present at the corners of the unit cell.
Therefore, the contribution of the corner atoms $\left(X\right)$ to the unit cell $=7\times \frac{1}{8}=\frac{7}{8}$


$\text{Calculating the contribution}\text{of Y atoms:}$

Each atom $\left(Y\right)$ in the face-centre of the unit cell contributes $\frac{1}{2}$ to the unit cell.
There are 6 face-centres in a unit cell.
Therefore, the contribution of the face-centred atoms
$\left(Y\right)$ to the unit cell $=6\times \frac{1}{2}=3$

$\text{Determining the formula}\text{of the compound:}$

$X:Y=\frac{7}{8}:3$

$=7:24$

Therefore, the formula of the compound is $X_7{Y}_{24}.$

So, option (a) is the correct answer.
The ratio of number of atoms in the unit cell