In which of the following rate of `S_(N) 1` is very fast ?

Options

A. `CH_(3)-overset(Cl)overset(|)(CH)-CH_(3)`
B. `Ph-overset(cl)overset(|)(CH)-CH_(3)`
C. `CH_(3)CH_(2)-O-CH_(2)-Cl`
D. `CH_(3)-CH=CH-Cl`

Explanation

To determine which of the following options has a very fast rate of \( S_N1 \) reaction, we need to analyze the stability of the carbocations formed in each case. The stability of the carbocation is crucial because the \( S_N1 \) mechanism involves the formation of a carbocation intermediate.

### Step-by-Step Solution:

1. **Identify the Options**:
- Option A: \( CH_3CH_2Cl \) (forms a secondary carbocation)
- Option B: \( C_6H_5CH_2Cl \) (forms a resonance-stabilized carbocation)
- Option C: \( CH_3COOCH_2Cl \) (forms a primary carbocation)
- Option D: \( CH_3CH_2Cl \) (forms a primary carbocation)

2. **Analyze Carbocation Stability**:
- **Option A**: The reaction forms a secondary carbocation. Secondary carbocations are relatively stable but not the most stable.
- **Option B**: The reaction forms a benzyl carbocation, which is resonance-stabilized. This makes it more stable than a secondary carbocation.
- **Option C**: The reaction forms a primary carbocation. Primary carbocations are the least stable.
- **Option D**: Similar to Option C, this also forms a primary carbocation, which is not stable.

3. **Compare the Stability**:
- Tertiary carbocations > Secondary carbocations > Primary carbocations.
- Additionally, resonance stabilization (as in Option B) significantly increases the stability of the carbocation.

4. **Conclusion**:
- Among the options, Option B, which forms a resonance-stabilized benzyl carbocation, is the most stable and thus will have the fastest rate of \( S_N1 \) reaction.

### Final Answer:
**Option B** has the fastest rate of \( S_N1 \) reaction due to the formation of a resonance-stabilized carbocation.