In a crystalline solid, having formula `AB_(2)O_(4)` oxide ions are arranged in cubic close packed lattice while cations A are present in tetrahedral voids and cations B are present in octahedral voids.
(a) What percentage of the tetrahedral voids is occupied by A?
(b) What percentage of the octahedral voids is occupied by B ?

Explanation

To solve the problem, we need to analyze the arrangement of ions in the crystalline solid with the formula \( AB_2O_4 \). The oxide ions are arranged in a cubic close-packed (CCP) lattice, and we need to determine the occupancy of tetrahedral and octahedral voids by cations A and B, respectively.

### Step-by-Step Solution:

#### Part (a): Percentage of Tetrahedral Voids Occupied by A

1. **Determine the number of oxide ions (O²⁻)**:
- The formula \( AB_2O_4 \) indicates there are 4 oxide ions.

2. **Calculate the number of tetrahedral and octahedral voids**:
- In a cubic close-packed structure, the number of tetrahedral voids (T) is given by \( 2 \times Z \) and the number of octahedral voids (O) is given by \( Z \), where \( Z \) is the number of formula units per unit cell.
- For CCP, \( Z = 4 \).
- Therefore, the number of tetrahedral voids = \( 2 \times 4 = 8 \).
- The number of octahedral voids = \( 4 \).

3. **Determine the occupancy of tetrahedral voids by cation A**:
- According to the problem, cation A occupies 1 tetrahedral void.

4. **Calculate the percentage of tetrahedral voids occupied by A**:
\[ \text{Percentage of tetrahedral voids occupied by A} = \left( \frac{\text{Number of tetrahedral voids occupied by A}}{\text{Total tetrahedral voids}} \right) \times 100 \]
\[ = \left( \frac{1}{8} \right) \times 100 = 12.5\% \]

#### Part (b): Percentage of Octahedral Voids Occupied by B

1. **Determine the occupancy of octahedral voids by cation B**:
- According to the problem, cation B occupies 2 octahedral voids.

2. **Calculate the percentage of octahedral voids occupied by B**:
\[ \text{Percentage of octahedral voids occupied by B} = \left( \frac{\text{Number of octahedral voids occupied by B}}{\text{Total octahedral voids}} \right) \times 100 \]
\[ = \left( \frac{2}{4} \right) \times 100 = 50\% \]

### Final Answers:
- (a) The percentage of tetrahedral voids occupied by A is **12.5%**.
- (b) The percentage of octahedral voids occupied by B is **50%**.