Question

A mineral has molecular formula $ \text{X}{{\text{Y}}_{\text{2}}}{{\text{O}}_{\text{4}}} $ . $ {{\text{O}}^{\text{2-}}} $ Ions are arranged in ccp and cations of X are present in octahedral voids. Cations of Y are equally distributed among octahedral and tetrahedral voids. The fraction of the octahedral voids occupied will be
(A) $ \dfrac{1}{6} $
(B) $ \dfrac{1}{8} $
(C) $ \dfrac{1}{4} $
(D) $ \dfrac{1}{2} $

Answer 1

Hint: Arrangement of atoms in the solids are in different forms and patterns. In one pattern the layer of the sphere in three-dimensional space is arranged such that one layer covers the gaps in the previous layer and the third layer will be in alignment with the first layer and is known as hexagonal close packing (hcp). In the second pattern, the gaps in the first layer are closed by the second but the third layer is not in alignment with the first and is known as cubic close packing (ccp). In the ccp lattice of one oxide ion, there will be one octahedral void and two tetrahedral voids.

Complete step-by-step solution
In the given question, a mineral with molecular formula $ \text{X}{{\text{Y}}_{\text{2}}}{{\text{O}}_{\text{4}}} $ is arranged in a three dimensional space where, the oxide ions that is four oxide ions are arranged in ccp. In the ccp lattice of one oxide ion, there will be one octahedral void and two tetrahedral voids. Hence, for four oxides we will have,
octahedral voids=4 and tetrahedral voids = 8.
As we have two cations X and Y, where X is present in the octahedral void while Y is equally distributed among octahedral and tetrahedral voids. Cation X = 1 and cation Y = 2.
Out of 4 octahedral voids, 2 voids are occupied, one by X and another by Y; out of 8 tetrahedral voids 1 is occupied by Y.
The fraction of octahedral void occupied will be $ \dfrac{\text{2}}{\text{4}} $ which will be $ \dfrac{1}{2} $ .
Hence, the correct answer will be option D- $ \dfrac{1}{2} $ .

Note
In ccp, the total number of atoms will be $ \left( \text{8 }\!\!\times\!\!\text{ }\dfrac{\text{1}}{\text{8}} \right) $ atoms, these are corner atoms contributing $ \dfrac{\text{1}}{\text{8}} $ and $ \left( 6\times \dfrac{1}{2} \right) $ atoms, these are atoms from face contributing $ \dfrac{1}{2} $ which is equal to 4 atoms. In ccp, the tetrahedral voids generated are two times the octahedral void. Tetrahedral void is a triangular void and is surrounded tetrahedrally by four spheres around it whereas an octahedral void is a void with one triangle vertex upwards and the other triangle vertex downwards with six surrounding spheres.