To determine which molecule has a bond order of 1.5, we will use the molecular orbital theory. Here’s a step-by-step solution:
### Step 1: Understand Bond Order
Bond order is calculated using the formula:
\[
\text{Bond Order} = \frac{(\text{Number of Bonding Electrons} - \text{Number of Antibonding Electrons})}{2}
\]
### Step 2: Identify the Molecules
We will analyze the bond order for the following molecules: \( O_2 \), \( O_2^+ \), and \( O_2^- \).
### Step 3: Determine the Electron Configuration
- **For \( O_2 \)**: Oxygen has 8 electrons. Therefore, \( O_2 \) has a total of 16 electrons.
- **For \( O_2^+ \)**: This ion has one less electron than \( O_2 \), giving it 15 electrons.
- **For \( O_2^- \)**: This ion has one more electron than \( O_2 \), giving it 17 electrons.
### Step 4: Fill the Molecular Orbitals
Using the molecular orbital diagram for \( O_2 \):
- **Bonding Orbitals**: \( \sigma_{1s}, \sigma_{1s}^*, \sigma_{2s}, \sigma_{2s}^*, \sigma_{2p_z}, \pi_{2p_x}, \pi_{2p_y} \)
- **Antibonding Orbitals**: \( \sigma_{2p_z}^*, \pi_{2p_x}^*, \pi_{2p_y}^* \)
#### For \( O_2 \):
- **Bonding Electrons**: 10 (2 in \( \sigma_{1s} \), 2 in \( \sigma_{2s} \), 2 in \( \sigma_{2p_z} \), 4 in \( \pi_{2p_x} \) and \( \pi_{2p_y} \))
- **Antibonding Electrons**: 6 (2 in \( \sigma_{1s}^* \), 2 in \( \sigma_{2s}^* \), 2 in \( \sigma_{2p_z}^* \))
Calculating bond order:
\[
\text{Bond Order} = \frac{(10 - 6)}{2} = \frac{4}{2} = 2
\]
#### For \( O_2^+ \):
- **Bonding Electrons**: 9 (removing one electron from the highest energy level)
- **Antibonding Electrons**: 5
Calculating bond order:
\[
\text{Bond Order} = \frac{(9 - 5)}{2} = \frac{4}{2} = 2
\]
#### For \( O_2^- \):
- **Bonding Electrons**: 10 (adding one electron to the highest energy level)
- **Antibonding Electrons**: 6 (increased by 1)
Calculating bond order:
\[
\text{Bond Order} = \frac{(10 - 6)}{2} = \frac{4}{2} = 2
\]
### Step 5: Conclusion
After calculating the bond orders, we find:
- \( O_2 \): Bond order = 2
- \( O_2^+ \): Bond order = 2.5
- \( O_2^- \): Bond order = 1.5
Thus, the molecule that shows a bond order of 1.5 is \( O_2^- \).
