This question involves analyzing the stability of carbocations to determine which compound would undergo the fastest SN1 reaction. SN1 reactions proceed through a carbocation intermediate, and the rate is determined by the stability of this carbocation.

Carbocation Stability Order:

Tertiary carbocation (3°) > Secondary carbocation (2°) > Primary carbocation (1°) > Methyl carbocation

Additionally, carbocations stabilized by resonance are more stable than those without resonance.

Analysis of each option:

1. ${Ph}_{3} C - I$ : The leaving group is iodide (I⁻). The carbocation formed is (C₆H₅)₃C⁺ (trityl cation). This is a tertiary carbocation with extensive resonance stabilization from three phenyl rings, making it very stable.

2. $({Me}_{3} C) 3 C - Cl$ : The leaving group is chloride (Cl⁻). The carbocation formed is ((CH₃)₃C)₃C⁺. This is a tertiary carbocation, but it is highly crowded (neopentyl type) and lacks resonance stabilization. It is less stable than the trityl cation.

3. ${(CH 3)}_{2} CH - Br$ : The leaving group is bromide (Br⁻). The carbocation formed is (CH₃)₂CH⁺ (isopropyl cation). This is a secondary carbocation with no resonance, so it is less stable than tertiary carbocations.

4. ${(CH 3)}_{3} C - Cl$ : The leaving group is chloride (Cl⁻). The carbocation formed is (CH₃)₃C⁺ (tert-butyl cation). This is a tertiary carbocation with no resonance, so it is stable but less stable than the resonance-stabilized trityl cation.

Conclusion: The trityl carbocation from option 1 is the most stable due to resonance, so it forms the fastest and leads to the fastest SN1 reaction.

Related Topics:

Carbocation Stability: Carbocations are electron-deficient species. Stability increases with:

Increasing substitution (3° > 2° > 1° > methyl)
Resonance (e.g., allylic, benzylic)
Hyperconjugation
Neighboring atom effects (e.g., lone pairs)

SN1 Reaction Mechanism: This is a two-step process where the rate-determining step is the formation of the carbocation. The rate law is first order: rate = k[substrate].

Formulae:

Rate of SN1 reaction: $r a t e = k [R - X]$

Where k depends on the stability of the carbocation intermediate.