Question

Atoms of element B from the HCP lattice and those of element A occupy $\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$ rd of tetrahedral voids. What is the formula of the compound formed by the elements A and B?

• A. ${\mathrm{A}}_1{\mathrm{B}}_1$
• B. ${\mathrm{A}}_2{\mathrm{B}}_1$
• C. ${\mathrm{A}}_3{\mathrm{B}}_2$
• D. ${\mathrm{A}}_4{\mathrm{B}}_3$

Answer 1

The correct option is D

${\mathrm{A}}_4{\mathrm{B}}_3$

Explanation for correct option:

(D) ${\mathrm{A}}_4{\mathrm{B}}_3$

Step 1: Calculation of number of atoms in unit clell

• Given that, atoms of B form an hcp lattice. So, the atoms at corners are shared by $6$ unit cells. So, its contribution is$\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.$; Face centred atoms contribute$\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$​ and middle layer atoms contribute $1$ each.
• So, the effective number of atoms in a unit cell $=\frac{1}{6}\times 12+\frac{1}{2}\times 2+1\times 3$

$$\begin{gathered} =2+1+3 \\ =6 \end{gathered}$$

Step 2: Calculation of the number of tetrahedral voids:

$$\begin{gathered} \mathrm{Number}\mathrm{of}\mathrm{tetrahedral}\mathrm{voids}=2\times \mathrm{Number}\mathrm{of}\mathrm{atoms}\mathrm{forming}\mathrm{the}\mathrm{lattice}. \\ \mathrm{Number}\mathrm{of}\mathrm{tetrahedral}\mathrm{voids}=2\times 6=12 \end{gathered}$$

Step 3: Finding the formula:

• So, the number of atoms of A$=\frac{2}{3}\mathrm{x}12$

$=8$

• Number of atoms of B$=6$

$$\begin{gathered} \mathrm{So},\mathrm{A}:\mathrm{B}\Rightarrow \mathrm{A}:\mathrm{B} \\ 8:64:3 \end{gathered}$$

• So, the formula is ${\mathrm{A}}_4{\mathrm{B}}_3$. This is the correct option.

Explanation for incorrect option:

(A) ${\mathrm{A}}_1{\mathrm{B}}_1$

• The obtained formula does not match the above option. So, this is an incorrect option.

(B) ${\mathrm{A}}_2{\mathrm{B}}_1$

• The obtained formula does not match the above option. So, this is an incorrect option.

(C) ${\mathrm{A}}_3{\mathrm{B}}_2$

• The obtained formula does not match the above option. So, this is an incorrect option.

Hence, the correct option is (D) ${\mathrm{A}}_4{\mathrm{B}}_3$.