Arrange the given species in increasing order of O-O bond length
`underset(I)(H_(2)O_(2)) , underset(II)(KO_(2)) , underset(III)(Na_(2)O_(2)), underset(IV)(O_(2))`

Options

A. `I lt III lt IV lt II`
B. `II lt I lt IV lt III`
C. `IV lt I lt III lt II`
D. `IV lt II lt I lt III`

Answer

To arrange the given species \( H_2O_2 \), \( KO_2 \), \( Na_2O_2 \), and \( O_2 \) in increasing order of O-O bond length, we need to analyze the bond order of each species. The bond order is inversely related to bond length; that is, a higher bond order corresponds to a shorter bond length.

### Step 1: Determine the bond order for each species

1. **For \( O_2 \)**:
- The molecular orbital configuration is:
\[ \sigma 1s^2 \sigma^* 1s^2 \sigma 2s^2 \sigma^* 2s^2 \sigma 2p_z^2 \pi 2p_x^2 = \pi 2p_y^2 \pi^* 2p_x^1 \pi^* 2p_y^1 \]
- Total electrons = 16 (8 from each O)
- Bonding electrons = 10 (from \(\sigma\) and \(\pi\) orbitals)
- Antibonding electrons = 6 (from \(\sigma^*\) and \(\pi^*\) orbitals)
- Bond order = \(\frac{10 - 6}{2} = 2\)

2. **For \( O_2^- \) (as in \( KO_2 \) and \( Na_2O_2 \))**:
- Adding one electron to \( O_2 \) gives:
- Bonding electrons = 10
- Antibonding electrons = 7
- Bond order = \(\frac{10 - 7}{2} = 1.5\)

3. **For \( O_2^{2-} \) (as in \( Na_2O_2 \) and \( KO_2 \))**:
- Adding another electron gives:
- Bonding electrons = 10
- Antibonding electrons = 8
- Bond order = \(\frac{10 - 8}{2} = 1\)

4. **For \( H_2O_2 \)**:
- The O-O bond in \( H_2O_2 \) is a single bond.
- Therefore, bond order = 1.

### Step 2: Compare the bond orders and arrange the species

- \( O_2 \) has the highest bond order (2), thus the shortest bond length.
- \( O_2^- \) has a bond order of 1.5, which is longer than \( O_2 \) but shorter than \( O_2^{2-} \) and \( H_2O_2 \).
- \( O_2^{2-} \) and \( H_2O_2 \) both have a bond order of 1, but \( H_2O_2 \) has a covalent bond which is generally longer than the ionic bond in \( Na_2O_2 \) and \( KO_2 \).

### Final Arrangement of Species in Increasing Order of O-O Bond Length

1. \( H_2O_2 \) (longest bond length)
2. \( Na_2O_2 \) (next longest)
3. \( KO_2 \)
4. \( O_2 \) (shortest bond length)

### Increasing Order of O-O Bond Length
\[ H_2O_2 < Na_2O_2 < KO_2 < O_2 \]