Question

If atoms along an axis connecting the opposite edge centres on a face are removed from NaCl type solid AB then new empirical formula of the remaining solid would be:
A.${{\text{A}}_{\text{8}}}{{\text{B}}_{\text{5}}}$
B.AB
C.${{\text{A}}_{\text{3}}}{{\text{B}}_{\text{4}}}$
D.${{\text{A}}_3}{{\text{B}}_8}$

Answer 1

For a simple cubic unit cell, the number of atoms is 8 and their contribution per unit cell at corners is $\dfrac{1}{8}$. Thus, the total number of atoms per unit cell is 1. In a face centred cubic cell (one constituent particle present at the centre of each face, at corners), the number of atoms at the corners are 8 and their contribution per unit cell is $8 \times \dfrac{1}{8} = 1$ and that at the centre of each face, so $6 \times \dfrac{1}{2} = 3$. Therefore, for Fcc, a total number of atoms per unit cell is 4.

Complete step by step answer:
As mentioned above, A occupies 8 corners of the cube and thus contribution per unit cell is $\dfrac{1}{8}$.
So, $8 \times \dfrac{1}{8} = 1$. There are 6 atoms at each face. So its contribution is $\dfrac{1}{2}$.
$6 \times \dfrac{1}{2} = 3$. Thus, the total of the A atom per unit cell is 4.
Now let us consider B. As discussed above, the number of atoms per unit cell is 4 and B occupies voids which are $\dfrac{1}{4}$.
The coordination number of fcc is 12( number of nearest neighbours).
Therefore, its contribution to the unit cell is $1 + 12\dfrac{1}{4} = 4$.
So, we get the formula as ${{\text{A}}_{\text{4}}}{{\text{B}}_{\text{4}}}$ or it is same as AB.
Now, in the question, it is told that atoms along one axis connecting opposite edge centres on the face are removed. This means that 2 atoms are removed.
Therefore $8 \times \dfrac{1}{8} = 1$ and $4 \times \dfrac{1}{2} = 2$. Thus the total of B atoms per unit cell is 3.
Thus, the correct empirical formula is ${{\text{A}}_{\text{3}}}{{\text{B}}_{\text{4}}}$.
Therefore, the correct option is (C).

Note: The coordination number of a simple cubic cell is 6, fcc is 12 and the body-centred cubic cell is 8. Fcc is also called cubic close packing. The total number of atoms per unit cell for bcc is 2 ( 8 at the corners with contributing $\dfrac{1}{8}$ and 1 at the centre). The highly efficient close packing is fcc.