Calculate the bond order of `O_(2)^(-)and O_(2)^(2-)` ions on the basis of M.O. theory and explain their magnetic properties.

Answer

To calculate the bond order of the ions \( O_2^- \) and \( O_2^{2-} \) using Molecular Orbital (M.O.) theory and to explain their magnetic properties, follow these steps:

### Step 1: Determine the Total Number of Electrons
- For \( O_2 \), the total number of electrons is 16 (8 from each oxygen atom).
- For \( O_2^- \), it has one extra electron, so the total is 17.
- For \( O_2^{2-} \), it has two extra electrons, so the total is 18.

### Step 2: Write the Molecular Orbital Configuration
1. **For \( O_2 \)** (16 electrons):
- \( \sigma_{1s}^2 \sigma_{1s}^*^2 \sigma_{2s}^2 \sigma_{2s}^*^2 \sigma_{2p_z}^2 \pi_{2p_x}^2 \pi_{2p_y}^2 \pi_{2p_x}^*^1 \pi_{2p_y}^*^1 \)

2. **For \( O_2^- \)** (17 electrons):
- \( \sigma_{1s}^2 \sigma_{1s}^*^2 \sigma_{2s}^2 \sigma_{2s}^*^2 \sigma_{2p_z}^2 \pi_{2p_x}^2 \pi_{2p_y}^2 \pi_{2p_x}^*^2 \pi_{2p_y}^*^1 \)

3. **For \( O_2^{2-} \)** (18 electrons):
- \( \sigma_{1s}^2 \sigma_{1s}^*^2 \sigma_{2s}^2 \sigma_{2s}^*^2 \sigma_{2p_z}^2 \pi_{2p_x}^2 \pi_{2p_y}^2 \pi_{2p_x}^*^2 \pi_{2p_y}^*^2 \)

### Step 3: Count Bonding and Antibonding Electrons
- **For \( O_2 \)**:
- Bonding electrons: \( 10 \) (from \( \sigma_{1s}, \sigma_{2s}, \sigma_{2p_z}, \pi_{2p_x}, \pi_{2p_y} \))
- Antibonding electrons: \( 6 \) (from \( \sigma_{1s}^*, \sigma_{2s}^*, \pi_{2p_x}^*, \pi_{2p_y}^* \))

- **For \( O_2^- \)**:
- Bonding electrons: \( 10 \)
- Antibonding electrons: \( 7 \)

- **For \( O_2^{2-} \)**:
- Bonding electrons: \( 10 \)
- Antibonding electrons: \( 8 \)

### Step 4: Calculate Bond Order
The bond order is calculated using the formula:
\[ \text{Bond Order} = \frac{(\text{Number of Bonding Electrons} - \text{Number of Antibonding Electrons})}{2} \]

1. **For \( O_2 \)**:
\[ \text{Bond Order} = \frac{(10 - 6)}{2} = 2 \]

2. **For \( O_2^- \)**:
\[ \text{Bond Order} = \frac{(10 - 7)}{2} = 1.5 \]

3. **For \( O_2^{2-} \)**:
\[ \text{Bond Order} = \frac{(10 - 8)}{2} = 1 \]

### Step 5: Determine Magnetic Properties
- **For \( O_2 \)**:
- There are 2 unpaired electrons in the antibonding orbitals (\( \pi_{2p_x}^* \) and \( \pi_{2p_y}^* \)).
- **Magnetic Property**: Paramagnetic (due to unpaired electrons).

- **For \( O_2^- \)**:
- There is 1 unpaired electron in the antibonding orbitals (\( \pi_{2p_y}^* \)).
- **Magnetic Property**: Paramagnetic (due to unpaired electron).

- **For \( O_2^{2-} \)**:
- All electrons are paired.
- **Magnetic Property**: Diamagnetic (no unpaired electrons).

### Conclusion
- **Bond Orders**:
- \( O_2 \): 2 (double bond)
- \( O_2^- \): 1.5 (between a double and a single bond)
- \( O_2^{2-} \): 1 (single bond)

- **Magnetic Properties**:
- \( O_2 \): Paramagnetic
- \( O_2^- \): Paramagnetic
- \( O_2^{2-} \): Diamagnetic