To find the total number of isomers of \( C_7H_{14} \) that contain only a 5-membered ring, we will follow these steps:
### Step 1: Understanding the Molecular Formula
The molecular formula \( C_7H_{14} \) indicates that the compound has 7 carbon atoms and 14 hydrogen atoms. This formula suggests that the compound can have a degree of unsaturation, which can be calculated using the formula:
\[
\text{Degree of Unsaturation} = \frac{(2C + 2 + N - H)}{2}
\]
For \( C_7H_{14} \):
\[
\text{Degree of Unsaturation} = \frac{(2 \times 7 + 2 - 14)}{2} = \frac{(14 + 2 - 14)}{2} = \frac{2}{2} = 1
\]
This indicates that there is either one ring or one double bond in the structure.
### Step 2: Considering the 5-Membered Ring
Since we are only considering isomers with a 5-membered ring, we can visualize a cyclopentane structure with additional carbon atoms attached. The 5-membered ring will contribute to the degree of unsaturation.
### Step 3: Drawing Possible Structures
1. **Cyclopentane with 2 additional carbons as substituents**:
- One isomer can be cyclopentane with two methyl groups (dimethylcyclopentane).
- The two methyl groups can be positioned in different ways (1,1-; 1,2-; 1,3-; and 2,2-).
2. **Cyclopentene**:
- Another isomer can be cyclopentene, which has one double bond in the ring.
- We can have different positions for the double bond and substituents.
### Step 4: Identifying Stereoisomers
For structures with chiral centers, we will also consider stereoisomers:
- If a structure has two chiral centers, it can have stereoisomers. The formula for the number of stereoisomers is \( 2^n \), where \( n \) is the number of chiral centers.
### Step 5: Counting All Isomers
1. **Isomers with cyclopentane**:
- 1,1-Dimethylcyclopentane
- 1,2-Dimethylcyclopentane (two stereoisomers)
- 1,3-Dimethylcyclopentane (two stereoisomers)
- 2,2-Dimethylcyclopentane
Total from cyclopentane: 1 (base) + 1 (1,1) + 2 (1,2) + 2 (1,3) + 1 (2,2) = 7 isomers.
2. **Isomers with cyclopentene**:
- Cyclopentene with different substituents and positions of the double bond.
- Each of these can also have stereoisomers depending on the placement of substituents.
### Final Count
After considering all possible structures and their stereoisomers, we find that the total number of isomers for \( C_7H_{14} \) with a 5-membered ring is **8**.
### Final Answer
The total number of isomers of \( C_7H_{14} \) (only 5-membered ring) is **8**.
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