Early crystallographers had trouble solving the structures of inorganic solids using X-ray diffraction because some of the mathematical tools for analyzing the data had not yet been developed. Once a trial structure was proposed, it was relatively easy to calculate the diffraction pattern, but it was difficult to go the other way (from the diffraction pattern to the structure) if nothing was known a priori about the arrangement of atoms in the unit cell. It was important to develop some guidelines for guessing the coordination numbers and bonding geometries of atoms in crystals. The first such rules were proposed by Linus Pauling, who considered how one might pack together oppositely charged spheres of different radii.
Pauling proposed from geometric considerations that the quality of the "fit" depended on the radius ratio of the anion and the cation.
If the anion is considered as the packing atom in the crystal, then the smaller catin fills interstitial sites ("holes"). Cations will find arrangements in which they can contact the largest number of anions. If the cation can touch all of its nearest neighbour anions then the fit is good. If the cation is too small for a given site, that coordination number will be unstable and it will prefer a lower coordination structure. The table below gives the ranges of cation/anion radius ratios that give the best fit for a given coordination geometry.
`{:("Coordiantion number","Geometry",rho =(r_("cation"))/(r_("amion"))),(2,"linear",0-0.155),(3,"triangular",0.155 - 0.225),(4,"tetrahedral",0.225 - 0.414),(4,"square planar",0.414 - 0.732),(6,"octahedral",0.414 - 0.732),(8,"cubic",0.732 - 1.0),(12,"cuboctahedral",1.0):}`
(Source : Ionic Radii and Radius Ratios. (2021, June 8). Retrieved June 29, 2021, from https://chem.ibretexts.org/@go/page/183346)
A solid AB has square planar structure. If the radius of cation `A^(+)` is 120 pm, calculate the maximum possible value of anion `B^(-)`.

Options

A. 240 pm
B. 270 pm
C. 280 pm
D. 290 pm

Explanation

To solve the problem, we need to determine the maximum possible value of the anion radius \( B^- \) for a solid \( AB \) that has a square planar structure. The radius of the cation \( A^+ \) is given as 120 pm.

### Step-by-Step Solution:

1. **Identify the Coordination Geometry and Radius Ratio:**
- The problem states that the solid \( AB \) has a square planar structure. According to the provided table, the radius ratio \( \rho \) for square planar coordination is between 0.414 and 0.732.

2. **Set Up the Equations:**
- Let the radius of the anion \( B^- \) be \( x \).
- The radius ratio \( \rho \) is defined as:
\[ \rho = \frac{r_{\text{cation}}}{r_{\text{anion}}} = \frac{120 \, \text{pm}}{x} \]

3. **Calculate the Maximum Anion Radius Using the Lower Limit:**
- Using the lower limit of the radius ratio (0.414):
\[ 0.414 = \frac{120 \, \text{pm}}{x} \]
- Rearranging gives:
\[ x = \frac{120 \, \text{pm}}{0.414} \approx 289.8 \, \text{pm} \]

4. **Calculate the Minimum Anion Radius Using the Upper Limit:**
- Using the upper limit of the radius ratio (0.732):
\[ 0.732 = \frac{120 \, \text{pm}}{x} \]
- Rearranging gives:
\[ x = \frac{120 \, \text{pm}}{0.732} \approx 164.0 \, \text{pm} \]

5. **Determine the Maximum Possible Value of the Anion Radius:**
- The maximum possible value of the anion radius \( B^- \) corresponds to the lower limit of the radius ratio, which we calculated as approximately 289.8 pm.

6. **Final Answer:**
- The maximum possible value of the anion \( B^- \) is approximately **290 pm**.