Step 1: Understanding the Question:
We need to determine the IUPAC name of the final product Y formed through a three-step reaction sequence starting with isobutane.
Step 2: Key Formula or Approach:
1. Alkanes containing a tertiary carbon are oxidized selectively by potassium permanganate ($\text{KMnO}_4$) to form tertiary alcohols.
2. Alcohols react with sodium metal (Na) to form sodium alkoxides.
3. Alkoxides react with alkyl halides via Williamson ether synthesis to yield ethers.
Step 3: Detailed Explanation:
• Step 1: Formation of X:
Isobutane ($\text{(CH}_3\text{)}_3\text{CH}$) contains a tertiary carbon-hydrogen bond ($\text{3}^\circ\text{ C-H}$).
Oxidation with $\text{KMnO}_4$ selectively converts this C-H bond into a C-OH bond, yielding tert-butyl alcohol (X):
\[ \text{(CH}_3\text{)}_3\text{CH} \xrightarrow{\text{KMnO}_4} \text{(CH}_3\text{)}_3\text{C-OH} \quad (\text{X}) \]
• Step 2: Reaction of X with Na:
Tert-butyl alcohol reacts with sodium metal to form sodium tert-butoxide:
\[ \text{(CH}_3\text{)}_3\text{C-OH} + \text{Na} \rightarrow \text{(CH}_3\text{)}_3\text{C-ONa} + \frac{1}{2}\text{H}_2 \]
• Step 3: Reaction with $\text{CH}_3-\text{Br}$ (Williamson Ether Synthesis):
The alkoxide nucleophile attacks methyl bromide in an $\text{S}_{\text{N}}2$ reaction to form product Y:
\[ \text{(CH}_3\text{)}_3\text{C-ONa} + \text{CH}_3-\text{Br} \rightarrow \text{(CH}_3\text{)}_3\text{C-O-CH}_3\text{ (Y)} + \text{NaBr} \]
• Step 4: IUPAC Naming of Y:
The structure of Y is:
\[ \text{CH}_3-\text{C}(\text{CH}_3)(\text{OCH}_3)-\text{CH}_3 \]
The longest continuous carbon chain is a 3-carbon chain (propane).
At carbon-2, there is a methyl substituent ($-\text{CH}_3$) and a methoxy substituent ($-\text{OCH}_3$).
According to IUPAC alphabetical priority, "methoxy" is listed before "methyl".
Therefore, the IUPAC name is 2-Methoxy-2-methylpropane.
Step 4: Final Answer:
The IUPAC name of the product Y is 2-Methoxy-2-methylpropane.