In sapphire, oxide ions are arranged in hexagonal close packing and aluminium ions occupy two-thirds of the octahedral voids. What is the formula of sapphire?

Explanation

To determine the formula of sapphire, we will follow these steps:

### Step 1: Determine the number of oxide ions (O²⁻) in hexagonal close packing (HCP)

In hexagonal close packing, the number of atoms (ions) can be calculated as follows:
- There are 12 corner atoms, each contributing \( \frac{1}{6} \) (since each corner atom is shared by 6 unit cells).
- There are 2 face-centered atoms, each contributing \( \frac{1}{2} \) (since each face atom is shared by 2 unit cells).
- There are 3 body-centered atoms, each contributing 1 (since body-centered atoms belong entirely to one unit cell).

Calculating the contributions:
- Contribution from corner atoms: \( 12 \times \frac{1}{6} = 2 \)
- Contribution from face-centered atoms: \( 2 \times \frac{1}{2} = 1 \)
- Contribution from body-centered atoms: \( 3 \times 1 = 3 \)

Adding these contributions together gives:
\[ \text{Total O²⁻ ions} = 2 + 1 + 3 = 6 \]

### Step 2: Determine the number of octahedral voids

In a hexagonal close-packed structure, the number of octahedral voids is equal to the number of atoms in the lattice. Since we have 6 oxide ions (O²⁻), the number of octahedral voids is also 6.

### Step 3: Determine the number of aluminum ions (Al³⁺)

It is given that aluminum ions occupy two-thirds of the octahedral voids. Therefore, the number of Al³⁺ ions can be calculated as:
\[ \text{Number of Al³⁺ ions} = \frac{2}{3} \times \text{Number of octahedral voids} = \frac{2}{3} \times 6 = 4 \]

### Step 4: Write the empirical formula

Now we have:
- 4 aluminum ions (Al³⁺)
- 6 oxide ions (O²⁻)

The empirical formula can be written as:
\[ \text{Formula} = \text{Al}_4\text{O}_6 \]

### Step 5: Simplify the formula

To simplify the formula, we can divide both the subscripts by their greatest common divisor, which is 2:
\[ \text{Simplified Formula} = \text{Al}_2\text{O}_3 \]

### Final Answer

Thus, the formula of sapphire is:
\[ \text{Al}_2\text{O}_3 \]

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