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: Identify the structure of cyclic compounds
Cyclic compounds are those where the carbon atoms are connected in a ring. For \( C_6H_{12} \), we can start by forming a six-membered ring.

### Step 2: Construct the six-membered ring
1. **Single six-membered ring**: The simplest cyclic structure is a cyclohexane, which consists of six carbon atoms connected in a ring. This structure has the formula \( C_6H_{12} \).

### Step 3: Explore smaller rings with substituents
2. **Five-membered ring with one substituent**: We can create a five-membered ring (cyclopentane) and add a carbon chain (methyl group) to it. The substituent can be attached to any of the five carbon atoms. This gives us several isomers based on the position of the substituent.

### Step 4: Explore four-membered rings with substituents
3. **Four-membered ring with two substituents**: A four-membered ring (cyclobutane) can also be formed, and we can add two carbon substituents. The two substituents can be positioned in different ways, leading to multiple isomers.

### Step 5: Consider three-membered rings
4. **Three-membered ring with three substituents**: A three-membered ring (cyclopropane) can be formed, and we can add three carbon substituents. This will also lead to different arrangements.

### Step 6: Count the isomers
- **Cyclic hexane**: 1 isomer
- **Cyclic pentane with one substituent**: 5 isomers (one for each carbon where the substituent can attach)
- **Cyclic butane with two substituents**: 6 isomers (considering different arrangements of the two substituents)
- **Cyclic propane with three substituents**: 1 isomer (only one way to arrange three substituents)

### Total Count
Adding all these together, we have:
- 1 (cyclohexane)
- 5 (cyclopentane with one substituent)
- 6 (cyclobutane with two substituents)
- 1 (cyclopropane with three substituents)

Thus, the total number of cyclic isomers for \( C_6H_{12} \) is \( 1 + 5 + 6 + 1 = 13 \).

### Final Answer
The total number of cyclic isomers of the formula \( C_6H_{12} \) is **13**.

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