Question

In a cubic close packed structure of mixed oxide, the lattice is made up of oxide ions, one eighth of the tetrahedral voids are occupied by the divalent ion \[\left( {{{\text{X}}^{2 + }}} \right)\] ions, while one half of the octahedral voids are occupied by trivalent ions \[\left( {{{\text{Y}}^{3 + }}} \right)\] , then the formula of the oxide is?

Answer 1

Hint: The number of octahedral voids is half as that of the number of tetrahedral voids. The number of octahedral voids is the same as the number of lattice points. We will calculate the ratio of each of the ions present and hence will form the general formula.

Complete step-by-step answer:
Let us assume that there is N number of anion that is oxide ion is present in the lattice. We know that the number of octahedral voids is the same as that of the number of anions occupying the lattice site. According to this the number of octahedral void is also N. The \[\left( {{{\text{Y}}^{3 + }}} \right)\] ion occupies the half of the octahedral void.
Hence the number of octahedral void of \[\left( {{{\text{Y}}^{3 + }}} \right)\] ion is \[\dfrac{1}{2} \times {\text{N}} = \dfrac{{\text{N}}}{2}\]
Now if there is N number of octahedral void then there must be 2N number of tetrahedral void. Now tetrahedral voids are occupied by \[\left( {{{\text{X}}^{2 + }}} \right)\] ions. The one eighth of the tetrahedral void is occupied by the ions.
Hence number of tetrahedral void will be \[\dfrac{1}{8} \times 2{\text{N}} = \dfrac{{\text{N}}}{4}\]
Now we will calculate the ratio of the number of each ion present.
The ratio of \[\left( {{{\text{X}}^{2 + }}} \right)\] : \[\left( {{{\text{Y}}^{3 + }}} \right)\] : \[\left( {{{\text{O}}^{2 - }}} \right)\] ion is
\[\dfrac{{\text{N}}}{4}:\dfrac{{\text{N}}}{2}:{\text{N}}\]
To convert it into whole number ratio we will get the ratio as:
\[1:2:4\]
Hence, the formula of the crystal will be: \[{\text{X}}{{\text{Y}}_2}{{\text{O}}_4}\]

Note: The crystal is made out of the unit cell. The properties of a unit cell are the same as that of the crystal in case of crystalline solids. There are various types of unit cell such as face centred unit cell, body centred, end centred unit cell and edge centred unit cell corresponding to different crystal structures.