How many cyclic isomers isomers (structrural and geometrical only) exist for `C_(5)H_(10)`?

Answer

To determine the number of cyclic isomers (structural and geometrical) for the molecular formula \( C_5H_{10} \), we can follow these steps:

### Step 1: Identify Possible Cyclic Structures
First, we need to identify the possible cyclic structures that can be formed with 5 carbon atoms. The simplest cyclic structure is cyclopentane, which consists of a ring of 5 carbon atoms.

### Step 2: Consider Substituted Cyclic Structures
Next, we can consider cyclic structures with substituents:
1. **Cyclopentane**: The simplest cyclic structure.
2. **Methylcyclobutane**: A 4-membered ring with one methyl group attached.
3. **1,1-Dimethylcyclopropane**: A 3-membered ring with two methyl groups on the same carbon.
4. **1,2-Dimethylcyclopropane**: A 3-membered ring with two methyl groups on adjacent carbons.
5. **1-Ethylcyclopropane**: A 3-membered ring with one ethyl group attached.

### Step 3: Identify Geometrical Isomers
For some of the cyclic structures, we can have geometrical isomers due to the presence of double bonds or different arrangements of substituents:
- **1,2-Dimethylcyclopropane** can have two geometrical isomers (cis and trans).
- **1,1-Dimethylcyclopropane** does not have geometrical isomers since both methyl groups are on the same carbon.

### Step 4: Count All Isomers
Now, we can summarize the cyclic isomers:
1. Cyclopentane
2. Methylcyclobutane
3. 1,1-Dimethylcyclopropane
4. 1,2-Dimethylcyclopropane (cis and trans)
5. 1-Ethylcyclopropane

### Total Count of Isomers
- Cyclopentane: 1
- Methylcyclobutane: 1
- 1,1-Dimethylcyclopropane: 1
- 1,2-Dimethylcyclopropane: 2 (cis and trans)
- 1-Ethylcyclopropane: 1

Adding these together gives us a total of 7 cyclic isomers (including structural and geometrical).

### Final Answer
The total number of cyclic isomers (structural and geometrical) for \( C_5H_{10} \) is **7**.

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