A solid `A^(+)B^(-)` has `NaCl`-type close-packed structure. If the anion has a radius of `250` pm, what should be the ideal radius for the cation? Can a cation `C^(+)` having radius of `180` pm be slipped into the tetrahedral site of the crystal `A^(+)B^(-)`? Give reason for your answer.

Explanation

To solve the problem step by step, we will first determine the ideal radius for the cation in a NaCl-type structure and then check if the cation with a radius of 180 pm can fit into the tetrahedral site.

### Step 1: Determine the ideal radius for the cation

1. **Understanding the NaCl-type structure**: In a NaCl-type structure, the cation (A^+) occupies the tetrahedral voids formed by the anions (B^-). The relationship between the radius of the cation (r+) and the radius of the anion (r-) can be expressed as:
\[ \frac{r^+}{r^-} = \sqrt{2} - 1 \]

2. **Given data**: The radius of the anion (r-) is given as 250 pm.

3. **Calculate the ideal radius for the cation (r+)**:
\[ r^+ = (\sqrt{2} - 1) \times r^- \]
Substituting the value of r-:
\[ r^+ = (0.414) \times 250 \text{ pm} \]
\[ r^+ = 103.5 \text{ pm} \]

### Step 2: Check if the cation C^(+) with radius 180 pm can fit into the tetrahedral site

1. **Given data**: The radius of the cation C^(+) is 180 pm.

2. **Calculate the ratio of the cation radius to the anion radius**:
\[ \frac{r^+}{r^-} = \frac{180 \text{ pm}}{250 \text{ pm}} = 0.72 \]

3. **Determine the acceptable range for the radius ratio for tetrahedral sites**:
- For a cation to fit into a tetrahedral void, the ratio \( \frac{r^+}{r^-} \) should be less than 0.414 and greater than 0.225.

4. **Conclusion**: Since 0.72 is greater than 0.414, the cation with a radius of 180 pm cannot fit into the tetrahedral site.

### Final Answers:
- The ideal radius for the cation (A^+) is **103.5 pm**.
- The cation C^(+) with a radius of **180 pm cannot be slipped into the tetrahedral site**.