To determine which molecule has the smallest bond length among the given options, we will calculate the bond order for each molecule. The bond length is inversely proportional to the bond order; therefore, the molecule with the highest bond order will have the smallest bond length.
### Step-by-Step Solution:
1. **Identify the Molecules**: We have four molecules to analyze: O2, O2+, O2-, and O2^2-.
2. **Calculate the Total Number of Electrons**:
- For O2: Each oxygen atom has 8 electrons, so O2 has a total of 16 electrons.
- For O2+: This molecule has lost one electron, so it has 15 electrons.
- For O2-: This molecule has gained one electron, so it has 17 electrons.
- For O2^2-: This molecule has gained two electrons, so it has 18 electrons.
3. **Write the Molecular Electronic Configuration**:
- **O2**:
- Configuration: σ1s², σ*1s², σ2s², σ*2s², σ2pz², π2px², π2py², π*2px¹, π*2py¹
- **O2+**:
- Configuration: σ1s², σ*1s², σ2s², σ*2s², σ2pz², π2px², π2py², π*2px¹ (one less electron in the π* orbitals)
- **O2-**:
- Configuration: σ1s², σ*1s², σ2s², σ*2s², σ2pz², π2px², π2py², π*2px², π*2py¹ (one more electron in the π* orbitals)
- **O2^2-**:
- Configuration: σ1s², σ*1s², σ2s², σ*2s², σ2pz², π2px², π2py², π*2px², π*2py² (two more electrons in the π* orbitals)
4. **Calculate the Bond Order**:
- **Bond Order Formula**:
\[
\text{Bond Order} = \frac{(\text{Number of electrons in bonding MOs}) - (\text{Number of electrons in antibonding MOs})}{2}
\]
- **O2**:
- Bonding MOs: 10 (σ1s², σ2s², σ2pz², π2px², π2py²)
- Antibonding MOs: 6 (σ*1s², σ*2s², π*2px¹, π*2py¹)
- Bond Order = (10 - 6) / 2 = 2
- **O2+**:
- Bonding MOs: 10
- Antibonding MOs: 5 (one less in π* orbitals)
- Bond Order = (10 - 5) / 2 = 2.5
- **O2-**:
- Bonding MOs: 10
- Antibonding MOs: 7 (one more in π* orbitals)
- Bond Order = (10 - 7) / 2 = 1.5
- **O2^2-**:
- Bonding MOs: 10
- Antibonding MOs: 8 (two more in π* orbitals)
- Bond Order = (10 - 8) / 2 = 1
5. **Compare Bond Orders**:
- O2: Bond Order = 2
- O2+: Bond Order = 2.5
- O2-: Bond Order = 1.5
- O2^2-: Bond Order = 1
6. **Determine the Smallest Bond Length**:
- Since bond length is inversely proportional to bond order, the molecule with the highest bond order is O2+ (2.5), which means it has the smallest bond length.
### Final Answer:
The molecule with the smallest bond length is **O2+** (Option B).
