Question
Answers

A solid compound $XY$ has a structure like $NaCl$. If the radius of the cation $({{X}^{+}})$ is 100pm, the radius (in pm) of the anion $({{Y}^{-}})$ will be:
A. 275.1pm
B. 322.5pm
C. 241.5pm
D. 165.7pm

Answer
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Hint: Think about the crystal structure of $NaCl$ and calculate its coordination number. This will give you the shape of the voids and the radius ratio of the cation and anion.

Complete answer:
The compound $XY$ has a crystal structure like $NaCl$. Which implies that ${{X}^{+}}$ is equivalent to the ion $N{{a}^{+}}$, and ${{Y}^{-}}$ is equivalent to the ion $C{{l}^{-}}$. We have to find the radius for the anion that is equivalent to the $C{{l}^{-}}$ ion.
We know that $NaCl$, has a face centered cubic structure. This means the coordination number of the anions and cations is 6. The crystals that have a face centered structure with coordination number 6 have cations that fit into the octahedral voids of the anions.
We know that the radius ratio of cations to anions in structures with octahedral voids is 0.414. Thus, we can say that:
\[\dfrac{{{X}^{+}}_{r}}{{{Y}^{-}}_{r}}=0.414\]
Where,
\[{{X}^{+}}_{r}\]= radius of cation
\[{{Y}^{-}}_{r}\]= radius of anion
We know the value of the radius of the anion and we have to find the radius of the cation.
\[\dfrac{100pm}{{{Y}^{-}}_{r}}=0.414\]
\[{{Y}^{-}}_{r}=\dfrac{100pm}{0.414}\]
\[{{Y}^{-}}_{r}=241.54pm\]

So, the correct answer is “Option C”.

Note: Please do not get confused between the shape of one unit of the crystal which is face-centered cubic and the shape of the voids present between two layers of the anions, which is octahedral. The formation of the voids depends on and includes only the anions (since they are bigger, we can verify the answer using this), and the unit of the crystal structure includes the cations as well as anions.