The intermediate stages in the convertsion
`(CH_(3))_(2)-underset(OH)underset(|)C-underset(OH)underset(|)C-(CH_(3))_(2)underset(Delta)overset(dil. H_(2)SO_(4))rarrCH_(3)COC (CH_(3))_(2)` are

Options

A. `(CH_(3))_(2)-underset(OH)underset(|)C-underset(o+)underset(OH_(2)^(-))underset(|)C(CH_(3))_(2)`
B. `(CH_(3))_(2)-underset(OH)underset(|)C-overset(o+)C(CH_(3))_(2)`
C. `CH_(3)-underset(OH)underset(|)overset(o+)C-C(CH_(3))_(3)`
D. `CH_(3)-underset(o+)underset(OH)underset(||)C-C(CH_(3))_(3)`

Answer

To solve the problem regarding the intermediate stages in the conversion of the given dihydroxy compound to the final product using dilute sulfuric acid, we will follow these steps:

### Step-by-Step Solution:

1. **Protonation of the Alcohol**:
- When the dihydroxy compound is treated with dilute sulfuric acid (H₂SO₄), the first step involves the protonation of one of the hydroxyl (OH) groups. The lone pair of electrons on the oxygen atom of the OH group attacks a proton (H⁺) from the sulfuric acid, forming an oxonium ion.
- **Intermediate 1**: The structure now has a positive charge on the oxygen, forming an oxonium ion.

2. **Loss of Water**:
- The oxonium ion is unstable, and one of the hydroxyl groups (now a water molecule) will leave, resulting in the formation of a carbocation. This step involves the departure of a water molecule (H₂O).
- **Intermediate 2**: The structure now has a positively charged carbon atom (carbocation) where the water molecule was lost.

3. **Methyl Shift**:
- The carbocation formed can undergo rearrangement through a methyl shift. One of the methyl groups from the neighboring carbon atom shifts to the positively charged carbon, stabilizing the carbocation.
- **Intermediate 3**: The structure now has a new carbocation with an additional methyl group attached, resulting in a more stable tertiary carbocation.

4. **Formation of Double Bond**:
- The final step involves the deprotonation of the hydroxyl group, where the lone pair of electrons on the oxygen atom forms a double bond with the positively charged carbon. This results in the formation of a carbonyl group (C=O).
- **Intermediate 4**: The structure now has a carbonyl group, indicating the formation of a ketone.

5. **Final Product**:
- The final product is a ketone, specifically the compound CH₃COC(CH₃)₂, which is formed after the rearrangement and stabilization of the intermediates.

### Summary of Intermediates:
- **Intermediate 1**: Oxonium ion (protonated alcohol)
- **Intermediate 2**: Carbocation (after loss of water)
- **Intermediate 3**: More stable carbocation (after methyl shift)
- **Intermediate 4**: Carbonyl compound (after formation of double bond)