How many cyclic and acyclic isomers (including tautomers) can be made by the formula `C_(3)H_(6)O` ?

Options

A. 4
B. 5
C. 9
D. 10

Explanation

To determine the number of cyclic and acyclic isomers (including tautomers) that can be formed from the molecular formula \(C_3H_6O\), we will follow these steps:

### Step 1: Calculate the Degree of Unsaturation
The degree of unsaturation can be calculated using the formula:

\[ \text{Degree of Unsaturation} = \frac{(2C + 2 + N - H - X)}{2} \]

Where:
- \(C\) = number of carbon atoms
- \(N\) = number of nitrogen atoms (0 in this case)
- \(H\) = number of hydrogen atoms
- \(X\) = number of halogens (0 in this case)

For \(C_3H_6O\):
- \(C = 3\)
- \(H = 6\)
- \(N = 0\)
- \(X = 0\)

Substituting the values:
\[ \text{Degree of Unsaturation} = \frac{(2 \times 3 + 2 + 0 - 6 - 0)}{2} = \frac{(6 + 2 - 6)}{2} = \frac{2}{2} = 1 \]

### Step 2: Interpret the Degree of Unsaturation
A degree of unsaturation of 1 indicates that there is either a double bond or a ring in the structure.

### Step 3: Identify Acyclic Isomers
For acyclic isomers, we can have:
1. **Aldehyde**:
- \(CH_3CHO\) (Propanal)
2. **Ketone**:
- \(CH_3C(=O)CH_3\) (Acetone)
3. **Alcohols**:
- \(CH_2=CHOH\) (Vinyl alcohol)
- \(CH_3CH_2CH_2OH\) (Propan-1-ol)
- \(CH_3CH(OH)CH_2\) (Propan-2-ol)
4. **Ethers**:
- \(CH_3OCH_2CH_3\) (Ethyl methyl ether)

### Step 4: Identify Cyclic Isomers
For cyclic isomers, we can have:
1. **Cyclic ethers**:
- **1,3-Dioxolane** (a five-membered ring with two oxygen atoms)
- **Tetrahydrofuran** (a five-membered ring with one oxygen atom)
2. **Cyclopropanol**:
- A three-membered ring with an alcohol group.

### Step 5: Identify Tautomers
Tautomerism can occur in structures like:
- **Enol-Ketone Tautomerism**:
- From \(CH_3C(=O)CH_3\) (acetone) to \(CH_3C(OH)=CH_2\) (vinyl alcohol).

### Step 6: Count Total Isomers
Now, we can summarize the isomers:
- **Acyclic Isomers**: 5 (Aldehyde, Ketone, 3 Alcohols)
- **Cyclic Isomers**: 3 (Cyclic ethers and Cyclopropanol)
- **Tautomers**: 1 (Enol form of the ketone)

Adding these gives us a total of:
- **Total Isomers** = 5 (acyclic) + 3 (cyclic) + 1 (tautomer) = 9

### Final Answer
Thus, the total number of cyclic and acyclic isomers (including tautomers) that can be formed from the molecular formula \(C_3H_6O\) is **9**.

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