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).

The correct option is C

The ratio of $\mathrm{M}-\mathrm{X}$ bond length to the cubic unit cell edge length is $0.866$

The correct options are (A) and (C):

Explanation for the correct answers:

Option (A):The empirical formula of the compound is $\mathrm{MX}$:

1. As the structure in the figure is of Body centered cubic (B.C.C).
2. The M is present at the center.
3. So, No. of $\mathrm{M}$ atom per unit cell = $2\times \frac{1}{2}=1$.
4. No. of $\mathrm{X}$atom per unit cell = $4\times \frac{1}{4}=1$
5. Hence, the empirical formula would be = $\mathrm{MX}$

(C): The ratio of $\mathrm{M}-\mathrm{X}$ bond length to the cubic unit cell edge length is $0.866$.

1. $\mathrm{M}-\mathrm{X}$ bond length would be : $\sqrt{{\left(\frac{\mathrm{a}}{2}\right)}^2+{\left(\frac{\mathrm{a}}{\sqrt{2}}\right)}^2}$
2. So, the bond length : $\mathrm{a}=\sqrt{\frac{1}{4}+\frac{1}{2}}$
3. Hence, $\frac{\mathrm{M}-\mathrm{X}\mathrm{bond}\mathrm{length}}{\mathrm{a}(\mathrm{edge}\mathrm{length})}=\frac{\sqrt{3}}{2}=0.866$

Explanation for the incorrect answers:

(B): The cation $\mathrm{M}$ and anion $\mathrm{X}$ have different coordination geometries.

1. As the structure in the figure is of B.C.C
2. So, the$\frac{{\mathrm{r}}_{+}}{{\mathrm{r}}_1}=0.732(\mathrm{cation}\mathrm{present}\mathrm{in}\mathrm{cubical}\mathrm{void})$
3. Hence, Coordination number of cation $\mathrm{M}$= 8 and that of anion $\mathrm{X}$= 8.

Option (D): The ratio of the ionic radii of cation $\mathrm{M}$ to anion $\mathrm{X}$ is $0.414$.

1. Because the structure is of B.C.C
2. So, ${\mathrm{r}}_{\mathrm{x}-}+{\mathrm{r}}_{\mathrm{m}-}=\sqrt{\frac{3}{2}\mathrm{a}}$
3. $0.732\le \frac{{\mathrm{r}}_{\mathrm{m}+}}{{\mathrm{r}}_{\mathrm{x}-}}\le 1$

Hence it would be greater than $0.414$.

Therefore, the correct options are

(A): The empirical formula of the compound is $\mathrm{MX}$:

(C): The ratio of $\mathrm{M}-\mathrm{X}$ bond length to the cubic unit cell edge length is $0.866$.