Discussion
A compound consists of three ions X, Y and Z. The Z ions are arranged in an FCC arrangement. The X ions occupy \(\frac{1}{6}\) of the tetrahedral voids and the Y ions occupy \(\frac{1}{3}\) of the octahedral voids. Which one of the following is the CORRECT chemical formula of the compound?
Answer: B
The formula is \(\mathrm{XYZ_{3}}\).
An FCC unit cell holds 4 lattice points, here 4 Z ions. It contains 8 tetrahedral voids and 4 octahedral voids.
\[ \begin{aligned} X &= \frac{1}{6} \times 8 = \frac{4}{3}, \quad Y = \frac{1}{3} \times 4 = \frac{4}{3} \\ X : Y : Z &= \frac{4}{3} : \frac{4}{3} : 4 = 1 : 1 : 3 \end{aligned} \]
The other formulas do not follow from these counts; they come from miscounting the ions or voids per cell, for example forgetting that an FCC cell has twice as many tetrahedral as octahedral voids.
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