GATE 2023 PH – Question 18
A compound consists of three ions X, Y and Z. The Z ions are arranged in an FCC arrangement. The X ions occupy 1 6 of the tetrahedral voids and the Y ions occupy 1 3 of the octahedral voids. Which one of the following is the CORRECT chemical formula of the compound?
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Correct answer: (B) $\mathrm{XYZ_3}$
Explanation
**Counting voids in an FCC arrangement.** An FCC cell contains 4 atoms (here Z ions). There are
- $2$ tetrahedral voids per atom, so $8$ tetrahedral voids, and
- $1$ octahedral void per atom, so $4$ octahedral voids.
**Occupancy:**
- X ions occupy $\tfrac16$ of the tetrahedral voids: $\tfrac16\times8=\tfrac43$.
- Y ions occupy $\tfrac13$ of the octahedral voids: $\tfrac13\times4=\tfrac43$.
- Z ions: $4$.
**Ratio** X : Y : Z $=\tfrac43:\tfrac43:4=1:1:3$.
The formula is $\text{XYZ}_3$ (option B).