Question

The cubic unit cell structure of a compound containing cation $M$ and anion $X$ is shown below. When compared to the anion, the cation has a smaller ionic radius. Choose the correct statement(s).

• A. The ratio of the ionic radii of cation $M$ to anion $X$ is $0.414$.
• B. The empirical formula of the compound is $MX$.
• C. The ratio of $M-X$ bond length to the cubic unit cell edge length is $0.866$.
• D. The cation $M$ and anion $X$ have different coordination geometries.

Answer 1

Answer: (B), (C)

The ratio of $M-X$ bond length to the cubic unit cell edge length is $0.866$.

According to the given diagram, the structure seems to be BCC.

(1) Empirical formula $\to MX$

(2) Coordination number of $M$ ion $=8$ [same co-ordination geometry]
Coordination geometry of $X$ ion = $8$ [same co-ordination Geometry]

(3) $M-X$ Bond Length = $\sqrt{{\left(\frac{a}{2}\right)}^2+{\left(\frac{a}{\sqrt{2}}\right)}^2}$

$M-X\text{Bond Length}=a=\sqrt{\frac{1}{4}+\frac{1}{2}}$
$\frac{M-X\text{bond length}}{a\left(\text{edge length}\right)}=\sqrt{\frac{3}{2}a}=0.866$

(4) As it is BCC.

$r_{x-}+r_{m-}=\sqrt{\frac{3}{2}a}$

$0.732\le \frac{r_{m+}}{r_{x-}}<1$