The total number of cyclic isomers of formula `C_(6)H_(12)` is/are

Answer

To determine the total number of cyclic isomers for the molecular formula \( C_6H_{12} \), we can follow these steps:

### Step 1: Calculate the Degree of Unsaturation (DU)
The degree of unsaturation (DU) can be calculated using the formula:
\[ \text{DU} = \frac{(2C + 2 - H)}{2} \]
For \( C_6H_{12} \):
- \( C = 6 \)
- \( H = 12 \)

Substituting these values into the formula:
\[ \text{DU} = \frac{(2 \times 6 + 2 - 12)}{2} = \frac{(12 + 2 - 12)}{2} = \frac{2}{2} = 1 \]
This indicates that the compound can either have one double bond or be a cyclic compound.

### Step 2: Identify Possible Cyclic Structures
Since we are looking for cyclic isomers, we can consider different cyclic structures that can be formed with 6 carbon atoms.

1. **Cyclohexane**: This is the simplest cyclic structure.
2. **Cyclopentane with a methyl group**: This can have multiple isomers based on the position of the methyl group.
3. **Cyclobutane with ethyl or methyl groups**: We can have cyclobutane with various substituents.
4. **Cyclopropane with different substituents**: This can include various arrangements of methyl and ethyl groups.

### Step 3: Count the Isomers
Now we will count the distinct cyclic isomers:

1. **Cyclohexane**: 1 isomer.
2. **Cyclopentane with 1 methyl group**: 1 isomer (methyl at any position is the same due to symmetry).
3. **Cyclobutane with 1 ethyl group**: 1 isomer.
4. **Cyclobutane with 2 methyl groups**: 2 isomers (1,1-dimethylcyclobutane and 1,2-dimethylcyclobutane).
5. **Cyclobutane with 1 methyl and 1 ethyl group**: 1 isomer.
6. **Cyclopropane with 1 ethyl group and 1 methyl group**: 3 isomers (1-ethyl-2-methylcyclopropane, 1,1-dimethylcyclopropane, etc.).
7. **Cyclopropane with 3 methyl groups**: 1 isomer (1,1,2-trimethylcyclopropane).

Adding these together gives:
- Cyclohexane: 1
- Cyclopentane: 1
- Cyclobutane: 4 (1 ethyl, 2 dimethyl, 1 methyl-ethyl)
- Cyclopropane: 5 (3 methyl, 1 ethyl, 1 dimethyl)

Total = \( 1 + 1 + 4 + 5 = 11 \)

### Final Answer
The total number of cyclic isomers of \( C_6H_{12} \) is **11**.

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