To determine which of the given ions has the minimum bond length, we need to analyze the bond order of each ion using molecular orbital theory. The bond length is inversely proportional to the bond order; therefore, the ion with the highest bond order will have the shortest bond length.
### Step-by-Step Solution:
1. **Identify the Ions and Their Electrons**:
- The ions to consider are O2^2+, O2, O2^-, and O2^2-.
- Oxygen has an atomic number of 8, so O2 has a total of 16 electrons.
- The number of electrons for each ion is:
- O2^2+: 15 electrons (16 - 1)
- O2: 16 electrons
- O2^-: 17 electrons (16 + 1)
- O2^2-: 18 electrons (16 + 2)
2. **Construct Molecular Orbital Diagrams**:
- For each ion, we will fill the molecular orbitals based on the number of electrons.
- The molecular orbital filling order for O2 is:
- σ(1s)², σ*(1s)², σ(2s)², σ*(2s)², σ(2p_z)², π(2p_x)², π(2p_y)², π*(2p_x)¹, π*(2p_y)¹
3. **Calculate Bond Order**:
- The bond order is calculated using the formula:
\[
\text{Bond Order} = \frac{\text{Number of bonding electrons} - \text{Number of anti-bonding electrons}}{2}
\]
- **For O2^2+ (15 electrons)**:
- Bonding: 6 (σ(2s)², σ(2p_z)², π(2p_x)², π(2p_y)²)
- Anti-bonding: 1 (π*(2p_x)¹)
- Bond Order = (6 - 1) / 2 = 2.5
- **For O2 (16 electrons)**:
- Bonding: 8 (same as above)
- Anti-bonding: 2 (π*(2p_x)¹, π*(2p_y)¹)
- Bond Order = (8 - 2) / 2 = 3
- **For O2^- (17 electrons)**:
- Bonding: 6 (σ(2s)², σ(2p_z)², π(2p_x)², π(2p_y)²)
- Anti-bonding: 3 (π*(2p_x)¹, π*(2p_y)¹, one more added)
- Bond Order = (6 - 3) / 2 = 1.5
- **For O2^2- (18 electrons)**:
- Bonding: 6 (same as above)
- Anti-bonding: 4 (π*(2p_x)², π*(2p_y)²)
- Bond Order = (6 - 4) / 2 = 1
4. **Compare Bond Orders**:
- O2^2+: Bond Order = 2.5
- O2: Bond Order = 3
- O2^-: Bond Order = 1.5
- O2^2-: Bond Order = 1
5. **Determine Minimum Bond Length**:
- The ion with the highest bond order is O2, which has a bond order of 3. Therefore, it has the minimum bond length.
### Conclusion:
The ion with the minimum bond length is **O2**.
