In a cubic closed packed structure of mixed oxides the lattice is made up of oxide ions, one eighth of tetrahedral voids are occupied by divalent ions `(A^(2+))` while one half of the octahedral voids occupied trivalent ions `(B^(3+))`. What is the formula of the oxide ?

Options

A. `A_(2)BO_(4)`
B. `AB_(2)O_(3)`
C. `A_(2)BO_(3)`
D. `AB_(2)O_(4)`

Answer

To determine the formula of the oxide in a cubic closed-packed structure with the given conditions, we will follow these steps:

### Step 1: Determine the contribution of oxide ions (O²⁻)
In a cubic closed-packed structure, oxide ions occupy the lattice points. The contribution of oxide ions can be calculated as follows:

- **Corners Contribution**: There are 8 corners in a cube, and each corner contributes \( \frac{1}{8} \) of an ion. Therefore, the total contribution from corners is:
\[ 8 \times \frac{1}{8} = 1 \]

- **Faces Contribution**: There are 6 faces in a cube, and each face contributes \( \frac{1}{2} \) of an ion. Therefore, the total contribution from faces is:
\[ 6 \times \frac{1}{2} = 3 \]

- **Total Contribution of O²⁻**: Adding both contributions together:
\[ 1 + 3 = 4 \]

### Step 2: Determine the contribution of trivalent ions (B³⁺)
The problem states that half of the octahedral voids are occupied by trivalent ions (B³⁺).

- **Total Octahedral Voids**: In a cubic closed-packed structure, there are 4 octahedral voids.
- **Occupied Octahedral Voids**: Half of these are occupied by B³⁺ ions:
\[ \frac{1}{2} \times 4 = 2 \]

### Step 3: Determine the contribution of divalent ions (A²⁺)
The problem states that one-eighth of the tetrahedral voids are occupied by divalent ions (A²⁺).

- **Total Tetrahedral Voids**: In a cubic closed-packed structure, there are 8 tetrahedral voids.
- **Occupied Tetrahedral Voids**: One-eighth of these are occupied by A²⁺ ions:
\[ \frac{1}{8} \times 8 = 1 \]

### Step 4: Write the formula based on the contributions
Now we can summarize the contributions:
- Contribution of A²⁺ = 1
- Contribution of B³⁺ = 2
- Contribution of O²⁻ = 4

Thus, the formula can be represented as:
\[ \text{A}_1\text{B}_2\text{O}_4 \quad \text{or} \quad \text{AB}_2\text{O}_4 \]

### Final Answer
The formula of the oxide is \( \text{AB}_2\text{O}_4 \).

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