Question

A crystal is made of particles X, Y, and Z. X forms FCC packing. Y occupies all the octahedral voids of X and Z occupies all the tetrahedral voids of X. If all the particles along one body diagonal are removed, then the formula of the crystal would be:
A.$XY{Z_2}$
B.${X_2}Y{Z_2}$
C.${X_8}{Y_4}{Z_5}$
D.${X_5}{Y_4}{Z_8}$

Answer 1

Hint:In a face centered cubic cell, the octahedral void is present in the body centre and all edge centre. In one unit cell eight tetrahedral voids are present. The atom in the face centered cubic cell is shared by two unit cells adjacent to each other and only 1/2 belongs to each cell.

Complete step by step answer:
In a face centered cubic unit cell, X represents the FCC packing, Y represents the octahedral void and Z represents the tetrahedral voids.
Given, all the particles along one body diagonal are removed.
Then, 2X, 1Y and 2Z atoms are removed.
The effective number of X atoms in 1 unit cell $= 6 \times \dfrac{1}{8} + 6 \times \dfrac{1}{2}$
$\Rightarrow X = \dfrac{3}{4} + 3$
$\Rightarrow X = \dfrac{{15}}{4}$
The effective number of Y atoms in 1 unit cell $= 12 \times \dfrac{1}{4}$
$\Rightarrow Y = 3$
The effective number of Z atoms in 1 unit cell$= 6 \times 1$
$\Rightarrow Z = 6$
The ratio of X:Y:Z $= \dfrac{{15}}{4}:3:6$
To determine the simplest ratio of X, Y and Z divide with 3 and multiply with 4.
The ratio of X:Y:Z$= \dfrac{5}{4}:1:2$(division by 3)
The ratio of X:Y:Z$= 5:4:8$ (multiplication with 4)
Thus, the formula of the crystal will be ${X_5}{Y_4}{Z_8}$.
Therefore, the correct option is D.

Note:
Make sure to derive the simplest ratio of X, Y and Z. The total number of atoms present Face centered cubic unit cell is 4.The face-centered cubic unit cell is the simplest repeating unit in a cubic closest-packed structure.