Question

Atoms of element $A$ forms HCP and atoms of element $C$ occupy only two-third of octahedral voids in it, then the general formula of compound is:
A) $CA$
B) $C{A_2}$
C) ${C_2}{A_3}$
D) ${C_3}{A_2}$

Answer 1

Hint: The formula for any compound can be determined using the voids that the atoms are forming in crystal lattice. We know that there are three main types of unit cell which when rotated in three dimensions forms the crystal lattice. The position of atoms in HCP is such that the coordination number is $12$ . Find out the octahedral and tetrahedral void to proceed on the question.

Complete step-by-step answer:
Here, the arrangement is HCP where one spherical atom gets surrounded with $12$ other atoms hence we can call that the coordination number is $12$ . Now the formula determination is carried out with the help of voids that the atoms usually formed when they place in the cell.
For hexagonal close packing (HCP) the octahedral voids are $12$ in number same as of the coordination number and the tetrahedral voids are always double to the octahedral voids thus, the number of tetrahedral voids are $24$ .
$\text{Tetrahedral voids} = 12$
$\text{Octahedral voids} = 24$
Now let’s see that which atom is forming the lattice and which atom is placed in the voids. According to the question, atom $A$ is forming hexagonal close packing; it means that it will be $12$ in number. While the atoms $C$ occupy two-third of octahedral voids, it means two third of $12$ .
$atom\,C = \,\dfrac{2}{3}\, \times 12$ = $8$
This overall calculation gives us the ratio of atoms, which we can write as-
$A:\,C$
$12:\,8$ On solving further we get the ratio as $3:2$
It means the formula is ${C_3}{A_2}$

Therefore the correct answer is option ‘D’.

Note: The determination of formula also depends upon the type of packing that the crystal was doing. If there is formation of CCP which is also called as face centered crystal which is also having coordination number $12$ thus the formula remains the same, but there are cases where the coordination number differs. Coordination number is the nearest particle which is surrounding one atom.