Question: Atoms of an element Z form a hexagonal close packed (hcp) lattice and atoms of element X occupy all ...

Question

Atoms of an element Z form a hexagonal close packed (hcp) lattice and atoms of element X occupy all the tetrahedral voids. The formula of the compound is:
(A) $\text{XZ}$
(B) $\text{X}{{\text{Z}}_{\text{2}}}$
(C) ${{\text{X}}_{\text{2}}}\text{Z}$
(D) ${{\text{X}}_{4}}{{\text{Z}}_{\text{3}}}$

Answer 1

Solution

Hint : In the hexagonal close packed (hcp) lattice arrangement, the atoms are arranged in the shape of a hexagon and the gaps are left uncovered. Once the first layer is laid down, the second is placed on top of the hexagon. The vacant spaces that are left behind due to the arrangement are called “holes”. The number of tetrahedral voids are double the octahedral voids.

Complete step by step solution
Since the structure repeats itself after every two layers, the stacking in hcp can be described as “a-b-a-b-a-b”. The atoms in a hexagonal close packed structure efficiently occupy 74% space while 26% space is left empty. From the above discussion it is clear that the number of atoms of element Z = 6,
And the number of tetrahedral voids = 2 times the number of atoms of Z = 12.
The number of atoms of Z in one unit cell = 6, so the ratio of the tetrahedral voids to the number of atoms = 12 : 6 = 2 : 1.
Hence, the formula of the compound will be ${{\text{X}}_{\text{2}}}\text{Z}$ , the correct option will be C.

Note
The term “close packed” refers to tightly packed or space efficient composition of the crystal structures or lattice. In the hcp arrangement, the constituent particles are arranged in the hexagonal shape at first which are left with trigonal holes in between the centre of the atoms in the hexagon. The second layer of atoms covers up these trigonal holes that exist between the second and the first layer. This arrangement is again left with some holes which are called the tetrahedral holes. The holes from the first layer that fall directly on the hole form the second layer are called the octahedral holes. The third layer of atoms again has the same arrangement of atoms as the first layer and covers up all the tetrahedral holes.