Question

Atoms of an element A occupy $\frac{2}{3}$ of tetrahedral voids in the hexagonal close-packed (hcp) unit cell formed by element ‘B’. The formula of the compound formed by the elements A and B is:
a) $A{{B}_{2}}$
b) ${{A}_{4}}{{B}_{3}}$
c) ${{A}_{2}}{{B}_{3}}$
d) ${{A}_{2}}B$

Answer 1

To answer this question, we should first know about hexagonal close packing. And we should know about numbers of atoms that are shared in a unit cell.

Step by step answer:
So, first of all we should know about hexagonal close packing. We should know that in a hexagonal closest packed structure, we will observe that a third layer has the same arrangement of spheres as the first layer and covers all the tetrahedral holes. We should know that the structure repeats itself after every two layers. And we should know that, the stacking for HCP may be described as "a-b-a-b-a-b." The atoms in a hexagonal closest packed structure efficiently occupy 74% of space while 26% is empty space.
Now, we will come back to our question. We have to find the formula for A and B. $\frac{2}{3}$of tetrahedral voids is occupied by atoms of an element A.

Now, in the question it is given that the atom of B forms HCP lattice. We should know that the atoms at corners are shared by 6 unit cells. Now, we should know that its contribution is$\frac{1}{6}$​. We know that face centered atoms contribute $\frac{1}{2}$​ and middle layer atoms contribute 1 each.
So, from this we can say that, the effective number of atoms in unit cell will be equal to:
Effective number of atoms in unit cell =$\frac{1}{6}\times 12+\frac{1}{2}\times 2+1\times 3$= 2+1+3=6

We know that:
Number of tetrahedral voids =2$\times$Number of atoms forming the lattice.
So, it will be equal to:
Number of tetrahedral voids=2 $\times$ 6=12
Now, we will find number of atoms of A= $\frac{2}{3}\times 12=8$
Number of B atoms= 6
So, now we will find the number of atoms A and B.
So, A:B= 8:6
A:B=4:3

So, the formula between A and B is${{A}_{4}}{{B}_{3}}$. From this we can say that option B is correct.

Note :
We should learn the following table:

Closed pack structure	Coordination number	Number of atoms in unit cel
Hexagonal closest packed (HCP)	12	6
Face-centered cubic (FCC)	12	4
Body-centered cubic (BCC)	8	2
Simple cubic	6	1