How many cyclic structural isomers are possible from `C_4 H_6` :-

Options

A. `3`
B. `4`
C. `5`
D. `6`

Answer

To determine the number of cyclic structural isomers possible for the molecular formula \( C_4H_6 \), we can follow these steps:

### Step 1: Calculate the Degree of Unsaturation (DBE)
The formula for calculating the Degree of Unsaturation (DBE) is:

\[ DBE = \frac{2C + 2 - H - N - X}{2} \]

Where:
- \( C \) = number of carbon atoms
- \( H \) = number of hydrogen atoms
- \( N \) = number of nitrogen atoms
- \( X \) = number of halogens

For \( C_4H_6 \):
- \( C = 4 \)
- \( H = 6 \)

Substituting the values into the DBE formula:

\[ DBE = \frac{2(4) + 2 - 6}{2} = \frac{8 + 2 - 6}{2} = \frac{4}{2} = 2 \]

### Step 2: Analyze the DBE Value
A DBE of 2 indicates that the compound can have:
- One ring and one double bond
- One triple bond
- Two double bonds (but this will lead to acyclic structures)

Since we are interested in cyclic structures, we will focus on the first possibility (one ring and one double bond).

### Step 3: Identify Possible Cyclic Structures
1. **Cyclobutane with a double bond**:
- Structure: A cyclobutane ring with a double bond attached to one of the carbons.

2. **Cyclopropane with a double bond**:
- Structure: A cyclopropane ring (3 carbon triangle) with a double bond on one of the carbons and a methyl group (CH3) on another carbon to satisfy the valency.

3. **Cyclopropane with a double bond at a branch**:
- Structure: Similar to the previous one, but the double bond is at a different position.

4. **Highly strained structure**:
- Structure: A very strained structure that connects two cyclopropane-like structures with a double bond.

### Step 4: Count the Isomers
After analyzing the structures, we can summarize the cyclic structural isomers as follows:
1. Cyclobutane with a double bond.
2. Cyclopropane with a double bond and a methyl group.
3. Cyclopropane with a double bond at a branch.
4. Highly strained structure.

Thus, the total number of cyclic structural isomers for \( C_4H_6 \) is **4**.

### Final Answer
The number of cyclic structural isomers possible from \( C_4H_6 \) is **4**.

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