Arrange the following molecule/species in the order of their increasing bond orders.
(I)`O_2`, (II)`O_2^(-)`, (III)`O_2^(2-)`, (IV)`O_2^(+)`

Explanation

To arrange the given molecules/species in order of their increasing bond orders, we will calculate the bond order for each species using Molecular Orbital Theory (MOT). The species we need to evaluate are:

1. \( O_2 \)
2. \( O_2^- \)
3. \( O_2^{2-} \)
4. \( O_2^+ \)

### Step 1: Calculate the bond order of \( O_2 \)

1. **Determine the number of electrons**: \( O_2 \) has 16 electrons.
2. **Electronic configuration**:
- \( \sigma_{1s}^2 \)
- \( \sigma^*_{1s}^2 \)
- \( \sigma_{2s}^2 \)
- \( \sigma^*_{2s}^2 \)
- \( \pi_{2p_z}^2 \)
- \( \pi_{2p_x}^2 = \pi_{2p_y}^2 \)
- \( \pi^*_{2p_x}^1 \)
- \( \pi^*_{2p_y}^0 \)

The configuration can be summarized as:
\[ \sigma_{1s}^2 \sigma^*_{1s}^2 \sigma_{2s}^2 \sigma^*_{2s}^2 \pi_{2p_z}^2 \pi_{2p_x}^2 \pi_{2p_y}^2 \pi^*_{2p_x}^1 \]

3. **Count bonding and antibonding electrons**:
- Bonding electrons: \( 10 \) (from \( \sigma \) and \( \pi \) orbitals)
- Antibonding electrons: \( 6 \) (from \( \sigma^* \) and \( \pi^* \) orbitals)

4. **Bond order calculation**:
\[ \text{Bond order} = \frac{\text{Bonding electrons} - \text{Antibonding electrons}}{2} = \frac{10 - 6}{2} = 2 \]

### Step 2: Calculate the bond order of \( O_2^- \)

1. **Determine the number of electrons**: \( O_2^- \) has 17 electrons.
2. **Electronic configuration**:
- The configuration is similar to \( O_2 \) but with one additional electron in the antibonding orbital:
\[ \sigma_{1s}^2 \sigma^*_{1s}^2 \sigma_{2s}^2 \sigma^*_{2s}^2 \pi_{2p_z}^2 \pi_{2p_x}^2 \pi_{2p_y}^2 \pi^*_{2p_x}^2 \pi^*_{2p_y}^1 \]

3. **Count bonding and antibonding electrons**:
- Bonding electrons: \( 10 \)
- Antibonding electrons: \( 7 \)

4. **Bond order calculation**:
\[ \text{Bond order} = \frac{10 - 7}{2} = 1.5 \]

### Step 3: Calculate the bond order of \( O_2^{2-} \)

1. **Determine the number of electrons**: \( O_2^{2-} \) has 18 electrons.
2. **Electronic configuration**:
- The configuration is similar to \( O_2^- \) but with one more electron in the antibonding orbital:
\[ \sigma_{1s}^2 \sigma^*_{1s}^2 \sigma_{2s}^2 \sigma^*_{2s}^2 \pi_{2p_z}^2 \pi_{2p_x}^2 \pi_{2p_y}^2 \pi^*_{2p_x}^2 \pi^*_{2p_y}^2 \]

3. **Count bonding and antibonding electrons**:
- Bonding electrons: \( 10 \)
- Antibonding electrons: \( 8 \)

4. **Bond order calculation**:
\[ \text{Bond order} = \frac{10 - 8}{2} = 1 \]

### Step 4: Calculate the bond order of \( O_2^+ \)

1. **Determine the number of electrons**: \( O_2^+ \) has 15 electrons.
2. **Electronic configuration**:
- The configuration is similar to \( O_2 \) but with one less electron in the antibonding orbital:
\[ \sigma_{1s}^2 \sigma^*_{1s}^2 \sigma_{2s}^2 \sigma^*_{2s}^2 \pi_{2p_z}^2 \pi_{2p_x}^2 \pi_{2p_y}^2 \pi^*_{2p_x}^1 \pi^*_{2p_y}^0 \]

3. **Count bonding and antibonding electrons**:
- Bonding electrons: \( 10 \)
- Antibonding electrons: \( 5 \)

4. **Bond order calculation**:
\[ \text{Bond order} = \frac{10 - 5}{2} = 2.5 \]

### Summary of Bond Orders
- \( O_2^{2-} \): Bond order = 1
- \( O_2^{-} \): Bond order = 1.5
- \( O_2 \): Bond order = 2
- \( O_2^{+} \): Bond order = 2.5

### Final Arrangement in Increasing Order of Bond Order
1. \( O_2^{2-} \) (1)
2. \( O_2^{-} \) (1.5)
3. \( O_2 \) (2)
4. \( O_2^{+} \) (2.5)