The order of decreasing stability of the following cations is :
`CH_(3)underset((I))overset(+)CHCH_(3),CH_(3)underset((II))overset(+)CHOCH_(3),CH_(3)overset(+)CHunderset((III))COCH_(3)`

Options

A. `IIIgtIIgtI`
B. `IgtIIgtIII`
C. `IIgtIgtIII`
D. `IgtIIIgtII`

Explanation

To determine the order of decreasing stability of the given cations, we will analyze each cation based on the effects of substituents and hyperconjugation.

### Step 1: Analyze the first cation, \( CH_3-CH^+-CH_3 \)
- This cation has a positive charge on a secondary carbon (the carbon with the positive charge is attached to two other carbon atoms).
- It has a total of 6 alpha hydrogens (3 from each methyl group).
- The stability of this cation can be attributed to hyperconjugation, where the adjacent C-H bonds can donate electron density to the positively charged carbon, stabilizing it.

**Hint:** Look for the number of alpha hydrogens and the type of carbon (primary, secondary, tertiary) to assess stability.

### Step 2: Analyze the second cation, \( CH_3-CH^+-OCH_3 \)
- In this cation, the positive charge is also on a secondary carbon.
- The presence of the \( OCH_3 \) group introduces a mesomeric effect (resonance) where the oxygen can donate electron density through resonance, further stabilizing the positive charge on the carbon.
- This cation is more stable than the first one due to the additional stabilization from the electron-donating effect of the \( OCH_3 \) group.

**Hint:** Consider the presence of electron-donating groups and their resonance effects when evaluating stability.

### Step 3: Analyze the third cation, \( CH_3-CH^+-C(=O)-CH_3 \)
- Here, the positive charge is on a secondary carbon adjacent to a carbonyl group (C=O).
- The carbonyl group is an electron-withdrawing group due to its electronegativity and resonance effects, which destabilizes the positive charge on the adjacent carbon.
- This cation is the least stable among the three due to the destabilizing effect of the carbonyl group.

**Hint:** Identify electron-withdrawing groups and their impact on cation stability.

### Conclusion: Order of Stability
Based on the analysis:
1. The second cation \( CH_3-CH^+-OCH_3 \) is the most stable due to resonance stabilization.
2. The first cation \( CH_3-CH^+-CH_3 \) is next in stability due to hyperconjugation.
3. The third cation \( CH_3-CH^+-C(=O)-CH_3 \) is the least stable due to the electron-withdrawing effect of the carbonyl group.

### Final Order of Decreasing Stability
The order of decreasing stability of the cations is:
\[ CH_3-CH^+-OCH_3 > CH_3-CH^+-CH_3 > CH_3-CH^+-C(=O)-CH_3 \]