Molecular orbital theory as developed by Hund and Mulliken concerns with the formation of molecular orbitals formed by linear combination of atomic orbitals. The electrons are present in these molecular orbitals. The molecular orbitals are filled. The molecular orbital configuration helps us to calculate bond order which gives information about the number of bonds present between atoms. The bond order is related to bond length and bond strength.
Which of the following is expected to have largest bond length?

Options

A. `O_(2)`
B. `O_(2)^(+)`
C. `O_(2)^(-)`
D. `O_(2)^(2-)`

Explanation

To determine which molecule has the largest bond length, we need to analyze the bond orders of the given molecules using molecular orbital theory. The bond order is inversely related to bond length: a lower bond order corresponds to a longer bond length.

### Step-by-Step Solution:

1. **Identify the Molecules**: The question likely refers to O2, O2+, O2-, and O2^2-. We will calculate the bond order for each of these species.

2. **Determine the Electron Configuration**:
- **O2**: Oxygen has 8 electrons, so O2 has a total of 16 electrons.
- Molecular orbital configuration: σ1s² σ*1s² σ2s² σ*2s² σ2p_z² π2p_x² π2p_y² π*2p_x¹ π*2p_y¹
- **O2+**: O2+ has 15 electrons (16 - 1).
- Molecular orbital configuration: σ1s² σ*1s² σ2s² σ*2s² σ2p_z² π2p_x² π2p_y² π*2p_x¹ π*2p_y⁰
- **O2-**: O2- has 17 electrons (16 + 1).
- Molecular orbital configuration: σ1s² σ*1s² σ2s² σ*2s² σ2p_z² π2p_x² π2p_y² π*2p_x² π*2p_y¹
- **O2^2-**: O2^2- has 18 electrons (16 + 2).
- Molecular orbital configuration: σ1s² σ*1s² σ2s² σ*2s² σ2p_z² π2p_x² π2p_y² π*2p_x² π*2p_y²

3. **Calculate the Bond Order**:
- **Bond Order Formula**:
\[ \text{Bond Order} = \frac{1}{2} \times (\text{Number of bonding electrons} - \text{Number of antibonding electrons}) \]
- **O2**:
- Bonding electrons = 10, Antibonding electrons = 6
- Bond Order = \(\frac{1}{2} \times (10 - 6) = 2\)
- **O2+**:
- Bonding electrons = 10, Antibonding electrons = 5
- Bond Order = \(\frac{1}{2} \times (10 - 5) = 2.5\)
- **O2-**:
- Bonding electrons = 10, Antibonding electrons = 7
- Bond Order = \(\frac{1}{2} \times (10 - 7) = 1.5\)
- **O2^2-**:
- Bonding electrons = 10, Antibonding electrons = 8
- Bond Order = \(\frac{1}{2} \times (10 - 8) = 1\)

4. **Compare the Bond Orders**:
- O2: Bond Order = 2
- O2+: Bond Order = 2.5
- O2-: Bond Order = 1.5
- O2^2-: Bond Order = 1

5. **Determine the Largest Bond Length**:
The molecule with the lowest bond order will have the largest bond length. From our calculations:
- O2^2- has the lowest bond order of 1.

### Conclusion:
The molecule expected to have the largest bond length is **O2^2-**.