Find out the total number of cyclic isomers of `C_(6)H_(12)` which are optically active ?

Explanation

To find the total number of cyclic isomers of \( C_6H_{12} \) that are optically active, we will follow these steps:

### Step 1: Identify the structure of \( C_6H_{12} \)
The molecular formula \( C_6H_{12} \) indicates that we are dealing with a saturated hydrocarbon. For cyclic compounds, this means we can form rings with these six carbon atoms.

### Step 2: Determine the possible cyclic structures
We can create cyclic structures using 6 carbon atoms. The simplest cyclic structures we can consider are cyclohexane and its derivatives.

1. **Cyclohexane (C6H12)**: This is a single ring structure.
2. **Cyclopentane derivatives**: We can also consider structures like cyclopentane with a substituent (like a methyl group) to create additional isomers.

### Step 3: Consider the symmetry of the structures
For a compound to be optically active, it must lack a plane of symmetry. This means we need to look for asymmetric structures.

1. **Cyclohexane**: Cyclohexane itself is symmetrical and thus not optically active.
2. **Substituted cyclohexanes**: We can create substituted cyclohexanes (like methylcyclopentane) that can be asymmetric.

### Step 4: Count the optically active isomers
- **For cyclohexane derivatives**:
- **Methylcyclopentane**: This can have two different configurations (cis and trans) which can be asymmetric.
- **Other substituents**: By placing different groups on the cyclohexane, we can create more asymmetric structures.

### Step 5: Analyze other cyclic structures
- **Cyclobutane derivatives**: Similar analysis can be done for cyclobutane with different substituents.
- **Cyclopropane derivatives**: Cyclopropane can also be considered, but it tends to be more symmetrical.

### Step 6: Total the optically active isomers
After analyzing the structures:
- From cyclohexane derivatives, we can find 2 optically active isomers.
- From cyclopentane derivatives, we can find 4 optically active isomers.
- Other configurations may yield additional optically active isomers.

### Conclusion
After considering all possible cyclic structures and their symmetry, we find that the total number of optically active cyclic isomers of \( C_6H_{12} \) is **8**.

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