The most stable free radical is

Options

A. `overset(*)CH_3`
B. `CH_3 overset(*)CH_2`
C. `(CH_3)_2 overset(* )CH`
D. `(CH_3)_3overset(*)C`

Explanation

To determine the most stable free radical among the given options, we need to analyze the stability of each free radical based on the concept of hyperconjugation and the number of alpha hydrogens.

### Step-by-Step Solution:

1. **Identify the Free Radicals**: The given options are:
- A. CH3• (Methyl radical)
- B. CH3CH2• (Ethyl radical)
- C. (CH3)2CH• (Isopropyl radical)
- D. (CH3)3C• (Tert-butyl radical)
- E. C• (Carbon radical)

2. **Understand Hyperconjugation**: Hyperconjugation is the interaction of the electrons in a sigma bond (C-H or C-C) with an adjacent empty p-orbital or a free radical. The more hyperconjugative structures a free radical can have, the more stable it is.

3. **Count Alpha Hydrogens**:
- **For CH3• (Methyl radical)**: There are no adjacent carbons, hence 0 alpha hydrogens.
- **For CH3CH2• (Ethyl radical)**: There is 1 adjacent carbon (CH3), contributing 3 alpha hydrogens.
- **For (CH3)2CH• (Isopropyl radical)**: There are 2 adjacent CH3 groups, contributing 6 alpha hydrogens.
- **For (CH3)3C• (Tert-butyl radical)**: There are 3 adjacent CH3 groups, contributing 9 alpha hydrogens.
- **For C• (Carbon radical)**: There are no adjacent carbons, hence 0 alpha hydrogens.

4. **Stability Ranking**: Based on the number of alpha hydrogens:
- CH3•: 0 alpha hydrogens (least stable)
- C•: 0 alpha hydrogens (least stable)
- CH3CH2•: 3 alpha hydrogens
- (CH3)2CH•: 6 alpha hydrogens
- (CH3)3C•: 9 alpha hydrogens (most stable)

5. **Conclusion**: The most stable free radical is (CH3)3C• (Tert-butyl radical) because it has the highest number of alpha hydrogens contributing to hyperconjugation.

### Final Answer:
The most stable free radical is (CH3)3C• (Tert-butyl radical).