If d represents the bond length, then select the correct relation.

Options

A. `d_(N_(2))=d_(N_(2)^(+))andd_(O_(2))=d_(O_(2)^(+))`
B. `d_(N_(2))ltd_(N_(2)^(+))andd_(O_(2))gtd_(O_(2)^(+))`
C. `d_(N_(2))ltd_(N_(2)^(+))andd_(O_(2))ltd_(O_(2)^(+))`
D. `d_(N_(2))gtd_(N_(2)^(+))andd_(O_(2))=d_(O_(2)^(+))`

Answer

To determine the correct relation of bond lengths for the molecules N2, N2+, O2, and O2+, we will analyze the bond order for each species using molecular orbital theory.

### Step-by-Step Solution:

1. **Determine the number of electrons in each molecule:**
- **N2**: Contains 14 electrons (7 from each nitrogen atom).
- **N2+**: Contains 13 electrons (one electron is removed from N2).
- **O2**: Contains 16 electrons (8 from each oxygen atom).
- **O2+**: Contains 15 electrons (one electron is removed from O2).

2. **Fill the molecular orbitals for N2:**
- The molecular orbital configuration for N2 is:
\[ \sigma_{1s}^2 \sigma_{1s}^*^2 \sigma_{2s}^2 \sigma_{2s}^*^2 \sigma_{2p_z}^2 \pi_{2p_x}^2 \pi_{2p_y}^2 \pi_{2p_x}^*^0 \pi_{2p_y}^*^0 \]
- Bonding electrons: 10 (from σ and π orbitals)
- Antibonding electrons: 4 (from σ* and π* orbitals)
- **Bond Order (N2)**:
\[ \text{Bond Order} = \frac{(10 - 4)}{2} = 3 \]

3. **Fill the molecular orbitals for N2+:**
- The molecular orbital configuration for N2+ is similar to N2 but with one less electron:
\[ \sigma_{1s}^2 \sigma_{1s}^*^2 \sigma_{2s}^2 \sigma_{2s}^*^2 \sigma_{2p_z}^2 \pi_{2p_x}^2 \pi_{2p_y}^1 \pi_{2p_x}^*^0 \pi_{2p_y}^*^0 \]
- Bonding electrons: 9
- Antibonding electrons: 4
- **Bond Order (N2+)**:
\[ \text{Bond Order} = \frac{(9 - 4)}{2} = 2.5 \]

4. **Fill the molecular orbitals for O2:**
- The molecular orbital configuration for O2 is:
\[ \sigma_{1s}^2 \sigma_{1s}^*^2 \sigma_{2s}^2 \sigma_{2s}^*^2 \sigma_{2p_z}^2 \pi_{2p_x}^2 \pi_{2p_y}^2 \pi_{2p_x}^*^1 \pi_{2p_y}^*^1 \]
- Bonding electrons: 10
- Antibonding electrons: 6
- **Bond Order (O2)**:
\[ \text{Bond Order} = \frac{(10 - 6)}{2} = 2 \]

5. **Fill the molecular orbitals for O2+:**
- The molecular orbital configuration for O2+ is:
\[ \sigma_{1s}^2 \sigma_{1s}^*^2 \sigma_{2s}^2 \sigma_{2s}^*^2 \sigma_{2p_z}^2 \pi_{2p_x}^2 \pi_{2p_y}^2 \pi_{2p_x}^*^1 \pi_{2p_y}^*^0 \]
- Bonding electrons: 10
- Antibonding electrons: 5
- **Bond Order (O2+)**:
\[ \text{Bond Order} = \frac{(10 - 5)}{2} = 2.5 \]

6. **Compare bond orders to determine bond lengths:**
- Higher bond order indicates shorter bond length. Thus, we have:
- N2 (Bond Order = 3) > N2+ (Bond Order = 2.5)
- O2 (Bond Order = 2) < O2+ (Bond Order = 2.5)

7. **Final Relation of Bond Lengths:**
- The order of bond lengths from shortest to longest is:
\[ \text{N2} < \text{N2+} < \text{O2+} < \text{O2} \]