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

Options

A. `II gt I gt III`
B. `II gt III gt I`
C. `III gt I gt II`
D. `I gt II gt III`

Explanation

To determine the correct order of decreasing stability of the given cations, we need to analyze the structures and the effects of the substituents on the carbocations. Let's break down the solution step by step:

### Step 1: Identify the Structures of the Cations
1. **Cation I**: `CH₃ - C⁺H - CH₃` (Secondary carbocation with two methyl groups)
2. **Cation II**: `CH₃ - C⁺H - OCH₃` (Secondary carbocation with one methyl and one methoxy group)
3. **Cation III**: `CH₃ - C⁺H - CH₂ - OCH₃` (Secondary carbocation with one methyl and one methoxy group, but the methoxy is further away)

### Step 2: Analyze the Electron-Donating Effects
- **Cation I** has two methyl groups. Methyl groups are electron-donating through the +I effect, which stabilizes the carbocation.
- **Cation II** has one methyl group and one methoxy group. The methoxy group can donate electrons through resonance (+R effect) and also has a +I effect, making this cation more stable than Cation I.
- **Cation III** has a methyl group and a methoxy group, but the methoxy group is further away from the positively charged carbon. The distance reduces its stabilizing effect. Additionally, the methoxy group can also act as an electron-withdrawing group due to the electronegativity of oxygen, which can destabilize the carbocation.

### Step 3: Compare the Stability
- **Cation II** is the most stable due to the strong electron-donating effects from both the methyl and the methoxy groups.
- **Cation I** is next in stability because it has two methyl groups that provide stabilization.
- **Cation III** is the least stable because the methoxy group is further away, and its electron-withdrawing nature can destabilize the carbocation.

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

### Conclusion
Thus, the final answer is:
**Cation II > Cation I > Cation III**