The fraction of octahedral voids filled by `Al^(3+)` ion in `Al_(2)O_(3)(r_(Al^(3o+))//r_(O^(2-)) = 0.43)` is

Options

A. `0.43`
B. `0.287`
C. `0.667`
D. `1`

Explanation

To find the fraction of octahedral voids filled by Al³⁺ ions in Al₂O₃, we can follow these steps:

### Step 1: Determine the number of oxygen ions (O²⁻) in the FCC lattice.
In a face-centered cubic (FCC) lattice, the number of O²⁻ ions can be calculated as follows:
- There are 8 corner atoms, each contributing \( \frac{1}{8} \) to the unit cell.
- There are 6 face-centered atoms, each contributing \( \frac{1}{2} \) to the unit cell.

Calculating this gives:
\[ \text{Number of O}^{2-} = 8 \times \frac{1}{8} + 6 \times \frac{1}{2} = 1 + 3 = 4 \]

### Step 2: Determine the number of Al³⁺ ions based on the stoichiometry of Al₂O₃.
The formula for aluminum oxide is Al₂O₃, which indicates that for every 2 aluminum ions, there are 3 oxide ions. Therefore, the ratio of Al³⁺ to O²⁻ is:
\[ \frac{Z_{\text{Al}^{3+}}}{Z_{\text{O}^{2-}}} = \frac{2}{3} \]
Given that \( Z_{\text{O}^{2-}} = 4 \), we can find \( Z_{\text{Al}^{3+}} \):
\[ Z_{\text{Al}^{3+}} = \frac{2}{3} \times 4 = \frac{8}{3} \]

### Step 3: Determine the number of octahedral voids.
In the FCC lattice, the number of octahedral voids is equal to the number of anions (O²⁻). Therefore, the number of octahedral voids is also 4.

### Step 4: Calculate the fraction of octahedral voids filled by Al³⁺ ions.
The fraction of octahedral voids filled by Al³⁺ ions can be calculated using the formula:
\[ \text{Fraction filled} = \frac{\text{Number of Al}^{3+}}{\text{Number of octahedral voids}} = \frac{\frac{8}{3}}{4} \]
Calculating this gives:
\[ \text{Fraction filled} = \frac{8}{3} \times \frac{1}{4} = \frac{8}{12} = \frac{2}{3} \approx 0.667 \]

### Conclusion
The fraction of octahedral voids filled by Al³⁺ ions in Al₂O₃ is **0.667**.

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