Question

A solid is formed by elements X, Y and Z. X atoms form an FCC lattice with Y atoms occupying all tetrahedral voids atoms occupying half the octahedral voids. The formula of the solid is

\[\begin{align}
 & (A)-{{X}_{2}}{{Y}_{4}}Z \\
 & (B)-X{{Y}_{2}}{{Z}_{4}} \\
 & (C)-{{X}_{4}}{{Y}_{2}}Z \\
 & (D)-{{X}_{4}}\text{Y}{{\text{Z}}_{2}} \\
\end{align}\]

Answer 1

Hint: If there are N number of atoms in a crystal lattice, then N number of octahedral voids are formed. If there are n number of atoms in a crystal lattice, then 2N number of tetrahedral voids are formed.

Complete step by step answer:
-In two-dimensional hexagonal close packing, a sphere of second layer is placed above the triangular void of the first layer or vice versa, A tetrahedral void is formed.
-these voids are called tetrahedral voids because when the centers of these four spheres are joined, a tetrahedron is formed.
-if the no. of closed packed spheres be N, then the number of tetrahedral voids generated will be 2N.
-In two-dimensional hexagonal close packing, if the triangular voids of the second layer are above the triangular voids of the first layer and the triangular shapes of these two voids do not overlap such voids are octahedral and are surrounded by six spheres
-these voids are called octahedral voids because when the centers of these six spheres are joined, a octahedron is formed.
-if the no. of closed packed spheres be N, then the number of octahedral voids generated will be N.
-Let us consider the given FCC structure,
We know, there are four atoms in a FCC unit cell and X occupies all of them.
Hence, Number of X atoms =4
-As mentioned above, the number of tetrahedral voids is double the number of atoms and Y occupies all of them.
Hence, Number of Y atoms will be equal to 8.
-As discussed earlier, the octahedral voids are equal to the number of atoms and Z occupies half of them hence the number of Z atoms will be .
\[X:Y:Z=4:8:2\]
\[X:Y:Z=2:4:1\]
 A solid is formed by elements X, Y and Z. X atoms form an FCC lattice with Y atoms occupying all tetrahedral voids atoms occupying half the octahedral voids. So, the correct answer is “Option A”.

Note:An atom present at the corner of a unit cell is shared by eight unit cells, hence \[\dfrac{1}{8}\]of atom is present in a single unit cell. An atom present at the face of a unit cell is shared by a two-unit cell, \[\dfrac{1}{2}\] of an atom is present in a single unit cell.