The correct order of S-S bond length in following oxyanions is:
`S_(2)O_(4)^(2-)`
(II) `S_(2)O_(5)^(2-)`
(III) `S_(2)O_(6)^(2-)`

Options

A. `I gt II gt III`
B. `I gt III gt II`
C. `III gt II gt I`
D. `III gt I gt II`

Explanation

To determine the correct order of S-S bond lengths in the oxyanions \( S_2O_4^{2-} \), \( S_2O_5^{2-} \), and \( S_2O_6^{2-} \), we will analyze the structures of these compounds and apply concepts related to hybridization and bond character.

### Step 1: Draw the Lewis Structures
1. **For \( S_2O_4^{2-} \)**:
- Sulfur atoms are bonded to each other and to four oxygen atoms.
- The structure will have two sulfur atoms connected by a single bond, and the remaining valences are satisfied by double bonds to oxygen atoms.
- Each sulfur atom will have two lone pairs.

2. **For \( S_2O_5^{2-} \)**:
- This structure will have two sulfur atoms connected by a single bond, with five oxygen atoms attached.
- One of the sulfur atoms will have a double bond with one oxygen and single bonds with the others, resulting in one sulfur having one lone pair.

3. **For \( S_2O_6^{2-} \)**:
- In this case, both sulfur atoms are connected by a single bond, and there are six oxygen atoms.
- Each sulfur atom will have no lone pairs, and they will be bonded to oxygen through double bonds.

### Step 2: Analyze Hybridization and Lone Pairs
- All three compounds have sulfur atoms that are \( sp^3 \) hybridized.
- The presence of lone pairs affects the s-character of the bonds:
- **\( S_2O_4^{2-} \)** has 2 lone pairs on each sulfur, leading to a higher s-character in the S-S bond.
- **\( S_2O_5^{2-} \)** has 1 lone pair on one sulfur, reducing the s-character compared to \( S_2O_4^{2-} \).
- **\( S_2O_6^{2-} \)** has no lone pairs, resulting in the least s-character in the S-S bond.

### Step 3: Determine Bond Length Order
- According to the concept of s-character:
- More s-character in a bond leads to a shorter bond length.
- Therefore, the order of s-character and hence bond lengths will be:
- \( S_2O_6^{2-} \) (least s-character, longest bond)
- \( S_2O_5^{2-} \) (intermediate s-character)
- \( S_2O_4^{2-} \) (most s-character, shortest bond)

### Conclusion
Thus, the correct order of S-S bond lengths in the given oxyanions is:
\[ S_2O_6^{2-} > S_2O_5^{2-} > S_2O_4^{2-} \]