A metal oxide crystallise in a hexagonal close packed array of oxide ions with two out of every three octahedral holes occupied by metal ions.The formula of metal oxide is

Options

A. MO
B. `MO_(2)`
C. `M_(2)O`
D. `M_(2)O_(3)`

Explanation

To determine the formula of the metal oxide that crystallizes in a hexagonal close-packed (HCP) structure with specific occupancy of octahedral holes, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the HCP Structure**:
- In a hexagonal close-packed (HCP) structure, the oxide ions (O²⁻) are arranged in a close-packed manner. The number of oxide ions per unit cell (Z) in HCP is 6.

2. **Identifying Octahedral Holes**:
- In an HCP structure, the number of octahedral holes is equal to the number of atoms (or ions) present in the structure. Therefore, there are 6 octahedral holes available for metal ions.

3. **Occupancy of Octahedral Holes**:
- According to the problem, 2 out of every 3 octahedral holes are occupied by metal ions (M). Since there are 6 octahedral holes, we calculate the number of occupied holes:
\[ \text{Occupied holes} = \frac{2}{3} \times 6 = 4 \]
- Thus, there are 4 metal ions (M) in the structure.

4. **Determining the Ratio of Metal Ions to Oxide Ions**:
- We have 4 metal ions (M) and 6 oxide ions (O²⁻). The formula can be expressed as:
\[ \text{Metal ions (M)} : \text{Oxide ions (O)} = 4 : 6 \]
- This ratio can be simplified by dividing both numbers by 2:
\[ \text{M} : \text{O} = 2 : 3 \]

5. **Writing the Formula**:
- The simplified ratio indicates that for every 2 metal ions, there are 3 oxide ions. Therefore, the chemical formula of the metal oxide can be written as:
\[ \text{M}_2\text{O}_3 \]

### Final Answer:
The formula of the metal oxide is **M₂O₃**.

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