Using `MO` theory predict which of the following species has the shortest bond length ?

Options

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

Answer

To determine which species has the shortest bond length using Molecular Orbital (MO) theory, we will follow these steps:

### Step 1: Determine the number of electrons in each species
- **O2+**: Oxygen (O) has 8 electrons. O2 has 16 electrons. O2+ has 16 - 1 = 15 electrons.
- **O2-**: O2- has 16 + 1 = 17 electrons.
- **O2 2-**: O2 2- has 16 + 2 = 18 electrons.
- **O2 2+**: O2 2+ has 16 - 2 = 14 electrons.

### Step 2: Write the molecular orbital electronic configuration for each species
1. **O2+ (15 electrons)**:
- Configuration: σ1s² σ1s*² σ2s² σ2s*² σ2p_z² π2p_x² = π2p_y²
- Total: 2 (σ1s) + 2 (σ1s*) + 2 (σ2s) + 2 (σ2s*) + 2 (σ2p_z) + 4 (π2p_x and π2p_y) = 15 electrons.

2. **O2- (17 electrons)**:
- Configuration: σ1s² σ1s*² σ2s² σ2s*² σ2p_z² π2p_x² = π2p_y² π2p_x*¹
- Total: 2 (σ1s) + 2 (σ1s*) + 2 (σ2s) + 2 (σ2s*) + 2 (σ2p_z) + 4 (π2p_x and π2p_y) + 1 (π2p_x*) = 17 electrons.

3. **O2 2- (18 electrons)**:
- Configuration: σ1s² σ1s*² σ2s² σ2s*² σ2p_z² π2p_x² = π2p_y² π2p_x*²
- Total: 2 (σ1s) + 2 (σ1s*) + 2 (σ2s) + 2 (σ2s*) + 2 (σ2p_z) + 4 (π2p_x and π2p_y) + 2 (π2p_x*) = 18 electrons.

4. **O2 2+ (14 electrons)**:
- Configuration: σ1s² σ1s*² σ2s² σ2s*² σ2p_z² π2p_x² = π2p_y² π2p_x*²
- Total: 2 (σ1s) + 2 (σ1s*) + 2 (σ2s) + 2 (σ2s*) + 2 (σ2p_z) + 4 (π2p_x and π2p_y) = 14 electrons.

### Step 3: Calculate the bond order for each species
- **Bond Order Formula**: Bond Order = (Number of bonding electrons - Number of antibonding electrons) / 2

1. **O2+**:
- Bonding electrons = 10 (σ1s, σ2s, σ2p_z, π2p_x, π2p_y)
- Antibonding electrons = 5 (σ1s*, σ2s*, π2p_x*)
- Bond Order = (10 - 5) / 2 = 2.5

2. **O2-**:
- Bonding electrons = 10
- Antibonding electrons = 7 (σ1s*, σ2s*, π2p_x*, π2p_y*)
- Bond Order = (10 - 7) / 2 = 1.5

3. **O2 2-**:
- Bonding electrons = 10
- Antibonding electrons = 8 (σ1s*, σ2s*, π2p_x*, π2p_y*)
- Bond Order = (10 - 8) / 2 = 1

4. **O2 2+**:
- Bonding electrons = 10
- Antibonding electrons = 4 (σ1s*, σ2s*)
- Bond Order = (10 - 4) / 2 = 3

### Step 4: Compare bond orders to determine bond length
- Higher bond order indicates shorter bond length.
- Bond order ranking:
- O2 2+ (3) > O2+ (2.5) > O2- (1.5) > O2 2- (1)

### Conclusion
The species with the shortest bond length is **O2 2+**.