`(I) H_(2)S_(2)O_(5) (II) H_(2)SO_(5) (III) H_(2)S_(2)O_(6)`
Increasing order of number of `pi`-bonds in above compounds

Options

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

Explanation

To determine the increasing order of the number of pi bonds in the compounds \( H_2S_2O_5 \), \( H_2SO_5 \), and \( H_2S_2O_6 \), we will analyze the structure of each compound and count the pi bonds present.

### Step-by-Step Solution:

1. **Identify the Compounds:**
- The compounds we need to analyze are:
- (I) \( H_2S_2O_5 \)
- (II) \( H_2SO_5 \)
- (III) \( H_2S_2O_6 \)

2. **Draw the Structure of \( H_2S_2O_5 \):**
- The structure can be represented as follows:
- \( S \) is bonded to another \( S \) and has three double bonds with \( O \) atoms and two \( OH \) groups.
- The structure looks like:
```
O O
\\ //
S
/ \
O OH
|
S
|
OH
```
- **Count the Pi Bonds:**
- There are 3 double bonds (each double bond contributes 1 pi bond).
- Total pi bonds = 3.

3. **Draw the Structure of \( H_2SO_5 \):**
- The structure can be represented as follows:
- \( S \) has two double bonds with \( O \) atoms and two \( OH \) groups.
- The structure looks like:
```
O O
\\ //
S
/ \
OH O
```
- **Count the Pi Bonds:**
- There are 2 double bonds.
- Total pi bonds = 2.

4. **Draw the Structure of \( H_2S_2O_6 \):**
- The structure can be represented as follows:
- \( S \) is bonded to another \( S \) and has four double bonds with \( O \) atoms and two \( OH \) groups.
- The structure looks like:
```
O O
\\ //
S
/ \
O OH
|
S
|
O
|
OH
```
- **Count the Pi Bonds:**
- There are 4 double bonds.
- Total pi bonds = 4.

5. **Summarize the Number of Pi Bonds:**
- \( H_2S_2O_5 \): 3 pi bonds
- \( H_2SO_5 \): 2 pi bonds
- \( H_2S_2O_6 \): 4 pi bonds

6. **Determine the Increasing Order:**
- The increasing order of the number of pi bonds is:
- \( H_2SO_5 \) (2 pi bonds) < \( H_2S_2O_5 \) (3 pi bonds) < \( H_2S_2O_6 \) (4 pi bonds).

### Final Answer:
The increasing order of the number of pi bonds in the compounds is:
\[ H_2SO_5 < H_2S_2O_5 < H_2S_2O_6 \]