The correct decreasing order of `SN^(1)` reactivity of the folliwng is:
I. `PhCH_(2)X`. II. `CH_(2) = CHCH_(2)X`
III. `Me_(2)CHX`, IV. `C_(2)H_(5)X`

Options

A. `(I) gt (II) gt (III) gt (IV)`
B. `(IV) gt (III) gt (II) gt (I)`
C. `(II) gt (I) gt (III) gt (IV)`
D. `(IV) gt (III) gt (I) gt (II)`

Explanation

To determine the correct decreasing order of SN1 reactivity for the given compounds, we need to analyze the stability of the carbocations that would form during the SN1 reaction mechanism. The stability of carbocations is crucial because the rate of an SN1 reaction is dependent on the formation of a stable carbocation intermediate.

### Step-by-Step Solution:

1. **Identify the Compounds**:
- I. `PhCH₂X` (Benzyl halide)
- II. `CH₂=CHCH₂X` (Allylic halide)
- III. `Me₂CHX` (Tertiary halide)
- IV. `C₂H₅X` (Ethyl halide)

2. **Determine the Type of Carbocation Formed**:
- I. `PhCH₂X` will form a **benzylic carbocation** (`PhCH₂⁺`), which is highly stable due to resonance with the benzene ring.
- II. `CH₂=CHCH₂X` will form an **allylic carbocation** (`CH₂=CH-CH₂⁺`), which is also stable due to resonance with the double bond.
- III. `Me₂CHX` will form a **tertiary carbocation** (`Me₂C⁺`), which is stable due to hyperconjugation and inductive effects.
- IV. `C₂H₅X` will form a **primary carbocation** (`C₂H₅⁺`), which is the least stable among the four.

3. **Rank the Stability of Carbocations**:
- The stability order of carbocations is as follows:
- Benzylic > Allylic > Tertiary > Secondary > Primary
- Based on the carbocations formed:
- I (benzylic) > II (allylic) > III (tertiary) > IV (primary)

4. **Establish the Order of SN1 Reactivity**:
- Since the rate of SN1 reactions is directly related to the stability of the carbocation:
- `PhCH₂X` (I) > `CH₂=CHCH₂X` (II) > `Me₂CHX` (III) > `C₂H₅X` (IV)

5. **Final Order**:
- The correct decreasing order of SN1 reactivity is:
- I > II > III > IV

### Conclusion:
The correct decreasing order of SN1 reactivity for the given compounds is:
**I. `PhCH₂X` > II. `CH₂=CHCH₂X` > III. `Me₂CHX` > IV. `C₂H₅X`**