A crystalline solid AB has NaCl type structure with radius of `B^(-)` ion is 250 pm. Which of the following cation can be made to slip into tetrahedral site of crystals of `A^(+)B^(-)` ?

Options

A. `P^(+)` (radius = 180 pm)
B. `Q^(+)=` (radius = 56 pm)
C. `R^(+)=` (radius = 200 pm)
D. `S^(+)=` (radius = 150 pm)

Answer

To solve the problem, we need to determine which cation can fit into the tetrahedral site of a crystalline solid with the NaCl structure, given the radius of the anion \( B^{-} \) is 250 pm. We will use the Radius Ratio Rule to find the appropriate cation.

### Step-by-Step Solution:

1. **Identify the Radius of the Anion**:
- We are given that the radius of the \( B^{-} \) ion is 250 pm.

2. **Understanding the Radius Ratio Rule**:
- The Radius Ratio Rule helps us determine the type of cation that can fit into a specific site in a crystal structure based on the ratio of the cation radius (\( R^{+} \)) to the anion radius (\( R^{-} \)).
- For tetrahedral sites, the radius ratio (\( R^{+}/R^{-} \)) should be in the range of 0.225 to 0.414 for stability, but for just slipping in, we will consider the upper limit of 0.225.

3. **Calculate the Maximum Radius of the Cation**:
- Using the formula for the radius ratio:
\[ \frac{R^{+}}{R^{-}} = 0.225 \]
- Rearranging gives:
\[ R^{+} = 0.225 \times R^{-} \]
- Substituting \( R^{-} = 250 \) pm:
\[ R^{+} = 0.225 \times 250 \text{ pm} = 56.25 \text{ pm} \]

4. **Determine the Suitable Cation**:
- Now, we need to check which cation has a radius close to 56.25 pm. Common cations that might fit this criterion include \( Li^{+} \), \( Na^{+} \), \( K^{+} \), etc.
- From known ionic radii, \( Li^{+} \) has a radius of approximately 76 pm, \( Na^{+} \) is about 102 pm, and \( K^{+} \) is around 138 pm.
- However, cations like \( Be^{2+} \) (approximately 31 pm) or \( Mg^{2+} \) (approximately 72 pm) might be considered, but we need to find one that is closest to 56.25 pm.

5. **Conclusion**:
- Based on the calculated radius of 56.25 pm, the cation that can fit into the tetrahedral site of the crystal would be one with a radius close to this value.
- If the options provided include cations with radii around this value, we can select the appropriate one.