Correct order for N-O bond length in the given species is
`underset(I)(NO) underset(II) (NO^(+)) underset(III)(NO_(2)^(-)) underset(IV)(NO_(3)^(-))`

Options

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

Answer

To determine the correct order of N-O bond lengths in the given species (NO, NO⁺, NO₂⁻, NO₃⁻), we need to analyze the bond order of each species. The bond length is inversely related to the bond order; that is, a higher bond order corresponds to a shorter bond length.

### Step-by-Step Solution:

1. **Identify the Valence Electrons**:
- For NO (Nitric Oxide):
- Nitrogen (N) has 5 valence electrons.
- Oxygen (O) has 6 valence electrons.
- Total = 5 + 6 = 11 valence electrons.

- For NO⁺ (Nitric Oxide Cation):
- NO has 11 electrons, and removing one electron gives us 10 electrons.
- Total = 10 valence electrons.

- For NO₂⁻ (Nitrite Ion):
- NO has 11 electrons, and adding one electron gives us 12 electrons.
- Total = 12 valence electrons.

- For NO₃⁻ (Nitrate Ion):
- NO has 11 electrons, and adding two electrons gives us 13 electrons.
- Total = 13 valence electrons.

2. **Calculate the Bond Order**:
- The bond order can be estimated using the formula:
\[ \text{Bond Order} = \frac{\text{Number of bonding electrons} - \text{Number of antibonding electrons}}{2} \]
- For NO:
- Bond order = 2.5 (as derived from molecular orbital theory).

- For NO⁺:
- Bond order = 3 (since it has one less electron than NO).

- For NO₂⁻:
- Bond order = 1.5 (with the addition of one electron).

- For NO₃⁻:
- Bond order = 1.33 (with the addition of two electrons).

3. **Arrange the Bond Orders**:
- Now we can summarize the bond orders:
- NO⁺: Bond order = 3
- NO: Bond order = 2.5
- NO₂⁻: Bond order = 1.5
- NO₃⁻: Bond order = 1.33

4. **Determine the Bond Length Order**:
- Since bond length is inversely related to bond order, we can arrange the species in increasing order of bond length:
- NO⁺ (shortest bond length)
- NO
- NO₂⁻
- NO₃⁻ (longest bond length)

5. **Final Order of Bond Length**:
- The correct order for N-O bond lengths is:
\[ \text{NO₃⁻} > \text{NO₂⁻} > \text{NO} > \text{NO⁺} \]

### Conclusion:
The correct order for N-O bond lengths in the given species is:
\[ \text{NO₃⁻} > \text{NO₂⁻} > \text{NO} > \text{NO⁺} \]