In an ionic compound `XY_(2)O_(4)` oxide ions are arranged in ccp. Cations `X` are present in octahedral voids.Cations `Y` are equally distributed between octahedral and tetrahedral voids. The fraction of the octahedral voids occupied is

Options

A. `1/4`
B. `1/2`
C. `1/8`
D. `1/6`

Answer

To solve the problem, we need to determine the fraction of octahedral voids occupied in the ionic compound \( XY_2O_4 \) where oxide ions are arranged in a close-packed structure (CCP), and cations \( X \) and \( Y \) occupy octahedral and tetrahedral voids.

### Step-by-Step Solution:

1. **Understanding the Structure**:
- The oxide ions (\( O^{2-} \)) are arranged in a close-packed structure, which is similar to face-centered cubic (FCC).
- In a CCP structure, there are 4 oxide ions per unit cell.

2. **Identifying the Voids**:
- In a cubic close-packed structure, the number of tetrahedral voids is \( 2n \) and the number of octahedral voids is \( n \), where \( n \) is the number of atoms in the unit cell.
- Since there are 4 oxide ions, we have:
- Number of octahedral voids = \( n = 4 \)
- Number of tetrahedral voids = \( 2n = 8 \)

3. **Distribution of Cations**:
- Cations \( X \) are present in the octahedral voids.
- Cations \( Y \) are equally distributed between octahedral and tetrahedral voids.
- The stoichiometry of the compound \( XY_2O_4 \) indicates that there is 1 \( X \) cation and 2 \( Y \) cations.

4. **Calculating Occupied Voids**:
- Since \( X \) occupies 1 octahedral void, that accounts for 1 of the 4 octahedral voids.
- The 2 \( Y \) cations are distributed equally between the octahedral and tetrahedral voids. Therefore, each \( Y \) cation occupies \( \frac{1}{2} \) of an octahedral void and \( \frac{1}{2} \) of a tetrahedral void.
- Thus, the total contribution of \( Y \) to the octahedral voids is \( 1 \) (since \( 2 \times \frac{1}{2} = 1 \)).

5. **Total Occupied Octahedral Voids**:
- Total occupied octahedral voids = \( 1 \) (from \( X \)) + \( 1 \) (from \( Y \)) = \( 2 \).

6. **Calculating the Fraction**:
- The fraction of octahedral voids occupied is given by the formula:
\[ \text{Fraction occupied} = \frac{\text{Number of occupied octahedral voids}}{\text{Total number of octahedral voids}} = \frac{2}{4} = \frac{1}{2}. \]

### Final Answer:
The fraction of the octahedral voids occupied is \( \frac{1}{2} \).

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