The correct order of S-O bond length is :

Options

A. `SO_(3)^(2-)gtSO_(4)^(2-)gtSO_(3)gtSO_(2)`
B. `SO_(3)^(2-)gtSO_(4)^(2-)gtSO_(2)gtSO_(3)`
C. `SO_(4)^(2-)gtSO_(3)^(2-)gtSO_(2)gtSO_(3)`
D. `SO_(4)^(2-)gtSO_(3)^(2-)gtSO_(3)gtSO_(2)`

Explanation

To determine the correct order of S-O bond lengths in the given species (SO3^2-, SO4^2-, SO3, and SO2), we need to calculate the bond order for each species and then relate it to bond length. The bond length is inversely proportional to the bond order; thus, a higher bond order corresponds to a shorter bond length.

### Step-by-Step Solution:

1. **Identify the Species**: We have four species to analyze: SO3^2-, SO4^2-, SO3, and SO2.

2. **Calculate the Bond Order**:
- **SO3^2-**:
- Structure: S has one double bond with one O and two single bonds with two other O atoms.
- Total bonds = 1 (double bond) + 2 (single bonds) = 3
- Total canonical forms = 3 (due to resonance structures)
- Bond order = Total bonds / Total canonical forms = 3 / 3 = **1.33**

- **SO4^2-**:
- Structure: S has two double bonds and two single bonds with O atoms.
- Total bonds = 4 (2 double bonds + 2 single bonds) = 6
- Total canonical forms = 4 (due to resonance structures)
- Bond order = 6 / 4 = **1.5**

- **SO3**:
- Structure: S has three double bonds with three O atoms.
- Total bonds = 6 (3 double bonds) = 6
- Total canonical forms = 3 (due to resonance structures)
- Bond order = 6 / 3 = **2**

- **SO2**:
- Structure: S has two double bonds with two O atoms.
- Total bonds = 4 (2 double bonds) = 4
- Total canonical forms = 2 (due to resonance structures)
- Bond order = 4 / 2 = **2**

3. **Compare Bond Orders**:
- SO3 = 2
- SO2 = 2
- SO4^2- = 1.5
- SO3^2- = 1.33

4. **Determine Bond Length Order**:
- Since bond length is inversely related to bond order, we can establish the following order:
- Higher bond order = shorter bond length
- Therefore, the order of bond lengths (from longest to shortest) is:
- SO3^2- > SO4^2- > SO2 = SO3

5. **Final Answer**:
- The correct order of S-O bond lengths is:
- **SO3^2- > SO4^2- > SO2 = SO3**