What is the correct order of decreasing stability of the following cations?
`underset(I)(CH_(3)-overset(+)(C)H-CH_(3))" "underset(II)(CH_(3)-overset(+)(C)H-OCH_(3))`
`underset(III)(CH_(3)-overset(+)(C)H-CH_(2)-OCH_(3))`

Options

A. IIgtIgtIII
B. IIgtIIIgtI
C. IIIgtIgtII
D. IgtIIgtIII

Explanation

To determine the correct order of decreasing stability of the given cations, we need to analyze the structure and effects associated with each cation. The stability of carbocations is influenced by several factors, including resonance (M effect), hyperconjugation (H effect), and inductive effects (I effect).

### Step-by-Step Solution:

1. **Identify the Cations**:
- Cation I: \( \text{(CH}_3\text{-C}^+\text{H-CH}_3) \)
- Cation II: \( \text{(CH}_3\text{-C}^+\text{H-OCH}_3) \)
- Cation III: \( \text{(CH}_3\text{-C}^+\text{H-CH}_2\text{-OCH}_3) \)

2. **Analyze Cation I**:
- Cation I has no resonance stabilization as there are no lone pairs or pi bonds adjacent to the positively charged carbon.
- It can stabilize through hyperconjugation with the three hydrogen atoms on the adjacent carbon (alpha carbon).
- The stability is moderate due to hyperconjugation.

3. **Analyze Cation II**:
- Cation II has a methoxy group (\(-OCH_3\)) attached, which can donate electron density through resonance (M effect).
- This resonance stabilization significantly increases the stability of the cation.
- Additionally, it can also stabilize through hyperconjugation with the alpha hydrogens.
- Therefore, Cation II is the most stable due to both resonance and hyperconjugation.

4. **Analyze Cation III**:
- Cation III has the methoxy group further away from the positively charged carbon, which reduces the effectiveness of resonance stabilization.
- It has fewer alpha hydrogens (only 3) compared to Cation I (which has 6).
- The presence of the methylene group (\(-CH_2-\)) does not provide significant stabilization compared to the resonance effect in Cation II.
- Thus, Cation III is the least stable due to less resonance and fewer alpha hydrogens.

5. **Order of Stability**:
- Based on the analysis:
- Cation II (most stable due to resonance and hyperconjugation)
- Cation I (moderate stability due to hyperconjugation)
- Cation III (least stable due to less resonance and fewer alpha hydrogens)

### Final Order:
The correct order of decreasing stability of the cations is:
**Cation II > Cation I > Cation III**