This question involves comparing the rate of abstraction of iodine by silver ion (Ag⁺) from different alkyl iodides. The rate depends on the stability of the carbocation formed in the SN1 reaction mechanism, as Ag⁺ aids in the dissociation of the C-I bond.

Step 1: Understand the Reaction Mechanism

The reaction is an SN1-type process facilitated by Ag⁺. Ag⁺ coordinates with the iodine atom, helping to break the C-I bond and form a carbocation intermediate. The rate of this reaction is determined by the stability of this carbocation.

General Reaction: $R-I+{\mathrm{Ag}}^{+}\to \left[R^{+}\right]+\mathrm{AgI}\downarrow$

Step 2: Identify the Carbocations Formed

We need to analyze the structure of each reactant (a, b, c) to see the carbocation that would be generated.

• Reactant (a): This is a benzyl iodide derivative. Loss of I⁻ would form a benzyl carbocation, which is highly stabilized by resonance with the phenyl ring.
• Reactant (b): This is a primary alkyl iodide. Loss of I⁻ would form a simple primary carbocation, which is very unstable.
• Reactant (c): This is an allyl iodide. Loss of I⁻ would form an allylic carbocation, which is stabilized by resonance between two carbon atoms.

Step 3: Rank the Carbocation Stabilities

The order of carbocation stability is: Tertiary > Secondary > Allylic ≈ Benzyl > Primary > Methyl.

Comparing our cases:

• Benzyl (from a) is highly stable.
• Allylic (from c) is stable.
• Primary (from b) is very unstable.

Therefore, the stability order is: a > c > b.

Step 4: Relate Stability to Reaction Rate

A more stable carbocation is formed faster because the transition state leading to it is lower in energy. Therefore, the rate of iodine abstraction follows the same order as carbocation stability.

Final Answer: a > c > b

Related Topics:

• SN1 Reaction Mechanism
• Carbocation Stability
• Effects of Resonance on Reactivity
• Nucleophilic Substitution

Key Formulae and Theory:

• The rate law for an SN1 reaction is: $Rate=k\left[RX\right]$
• The general order of carbocation stability is: $3^{\varepsilon }>2^{\varepsilon }>1^{\varepsilon }>{\mathrm{CH}}^{+}3$
• Resonance stabilization (e.g., in allylic and benzylic carbocations) significantly increases stability compared to their non-resonance counterparts.
• Ag⁺ salts are often used to promote ionization of alkyl halides by precipitating the halide ion (e.g., AgI) and shifting the equilibrium towards carbocation formation.