`NH_(4)Cl` crystallises in a body centred cubic lattice with a unit cell distance of 387 pm. Calculate
(a) the distance between the oppositely charged ions in the lattice.
(b) the radius of `NH_(4)^(+)` ion if that of `Cl^(-)` ion is 181 pm.

Explanation

To solve the problem step by step, we will break down the calculations for both parts of the question.

### Given:
- NH₄Cl crystallizes in a body-centered cubic (BCC) lattice.
- Unit cell distance (edge length, a) = 387 pm
- Radius of Cl⁻ ion = 181 pm

### (a) Calculate the distance between the oppositely charged ions in the lattice.

1. **Understanding the BCC structure**:
- In a BCC lattice, Cl⁻ ions are located at the corners of the cube, and the NH₄⁺ ion is located at the body center of the cube.

2. **Finding the body diagonal**:
- The body diagonal (d) of a cube can be calculated using the formula:
\[ d = \sqrt{3} \times a \]
- Substituting the value of a:
\[ d = \sqrt{3} \times 387 \text{ pm} \]
- Calculating this gives:
\[ d \approx 1.732 \times 387 \approx 673.56 \text{ pm} \]

3. **Distance between oppositely charged ions**:
- The distance between the Cl⁻ ion at a corner and the NH₄⁺ ion at the body center is half of the body diagonal:
\[ \text{Distance} = \frac{d}{2} = \frac{673.56 \text{ pm}}{2} \approx 336.78 \text{ pm} \]

### (b) Calculate the radius of NH₄⁺ ion.

1. **Using the relationship between the radii**:
- The distance between the Cl⁻ ion and the NH₄⁺ ion can also be expressed as the sum of their radii:
\[ R_{NH_4^+} + R_{Cl^-} = \text{Distance} \]
- We know \( R_{Cl^-} = 181 \text{ pm} \) and the distance calculated is approximately \( 336.78 \text{ pm} \).

2. **Substituting the values**:
- Rearranging the equation gives:
\[ R_{NH_4^+} = \text{Distance} - R_{Cl^-} \]
- Substituting the known values:
\[ R_{NH_4^+} = 336.78 \text{ pm} - 181 \text{ pm} \approx 155.78 \text{ pm} \]

### Final Answers:
(a) The distance between the oppositely charged ions in the lattice is approximately **336.78 pm**.
(b) The radius of the NH₄⁺ ion is approximately **155.78 pm**.