Four isomeric optically active compounds `overset (NaHCO_3) rarr CO_2 (g) (A, B , C, D) (C_4 H_8 O_3)`
Compound `(A) overset (LAH) rarr ( E)` (Archiral compound)
Compound `(B) overset (KMnO_4) underset (or CrO_3) rarr` Inert to oxidation.
Compound `( C) underset (NaOI//H_3 O^(oplus)) rarr CHI_3 (Yellow ppt.) + Compound (F)`.
Compound `(B)` is :

Options

A. `(I)`
B. `(II)`
C. `(III)`
D. `(IV)`

Explanation

To solve the problem, we need to identify compound B from the given information about the four isomeric optically active compounds with the formula C4H8O3. We will analyze the information step by step.

### Step 1: Determine the Degree of Unsaturation (DU)
The formula for calculating the degree of unsaturation (DU) is:

\[ DU = \frac{(2 \times \text{number of Carbons}) + 2 - \text{number of Hydrogens}}{2} \]

For the compound C4H8O3:
- Number of Carbons (C) = 4
- Number of Hydrogens (H) = 8

Substituting these values into the formula:

\[ DU = \frac{(2 \times 4) + 2 - 8}{2} = \frac{8 + 2 - 8}{2} = \frac{2}{2} = 1 \]

### Step 2: Analyze the Implications of DU
A degree of unsaturation of 1 indicates that there is either one double bond or one ring present in the structure. Given that we are dealing with carboxylic acids and their derivatives, it is likely that one of the functional groups is a carboxylic acid (-COOH).

### Step 3: Identify Compound A
From the information given, compound A reacts with LAH (Lithium Aluminium Hydride) to form an achiral compound. This suggests that compound A contains a carbonyl group (C=O) that can be reduced to an alcohol.

Assuming compound A has a structure like:
- CH3-CH(OH)-COOH (a chiral center)

When reduced with LAH, it could yield:
- CH3-CH2-CHOH (an achiral compound)

### Step 4: Identify Compound B
Compound B is described as being inert to oxidation. This typically indicates that compound B is likely a primary alcohol or a ketone that does not have any hydrogen atoms on the carbon adjacent to the carbonyl group that could be oxidized.

Possible structures for compound B could include:
- A structure such as CH3-CH(OH)-OCH3 (methoxy group) or CH3-CH2-COOH (where the alcohol is not oxidizable).

### Step 5: Conclusion for Compound B
Given that compound B is inert to oxidation, it is likely a structure that includes a methoxy group and a carboxylic acid group. Therefore, compound B can be represented as:

\[ \text{Compound B: } CH3-CH(OH)-COOH \]

### Final Answer
Compound B is likely to be a structure that is a methoxy derivative of a carboxylic acid, specifically:

\[ \text{Compound B: } \text{Methoxyacetic acid} \, (CH3-COOH) \]