Which one of the following is most reactive towards substitution nucleophilic unimolecular \((S_N1)\) reactions?

\(C_6H_5C(CH_3)(C_6H_5)Br\)

The correct answer is option 3. \(C_6H_5C(CH_3)(C_6H_5)Br\).

The \(S_N1\) reaction, meaning Substitution Nucleophilic Unimolecular, proceeds through a two-step mechanism:

Step 1 (Slow): Formation of a carbocation - The leaving group \((LG)\) departs, forming a positively charged carbocation intermediate.

Step 2 (Fast): Nucleophilic attack - A nucleophile attacks the positively charged carbon atom of the carbocation, forming a new bond and displacing a proton \((H^+)\) to regenerate the leaving group.

The key factor determining the rate of an \(S_N1\) reaction is the stability of the carbocation intermediate. More stable carbocations form faster and lead to faster overall reactions. Here's how the given molecules fare in terms of carbocation stability:

1. \(C_6H_5CH(C_6H_5)Br\): This molecule has two potential contributions to carbocation stability:

Allylic stabilization: The double bond adjacent to the carbon bearing the leaving group helps to delocalize the positive charge, making the carbocation more stable.

Benzylic stabilization: The presence of the benzene ring next to the charged carbon also helps to disperse the positive charge, further increasing its stability.

However, the leaving group, \(Br^-\), is also good at leaving, which can contribute to a faster \(S_N1\) reaction.

2. \(C_6H_3CH(Br)CH_3\): This molecule lacks any significant stabilizing effects for the carbocation. While the methyl group can donate some electron density slightly, it's not enough to offer substantial stabilization. However, \(Br^-\) remains a good leaving group, potentially leading to a moderate reaction rate.

3. \(C_6H_5C(CH_3)(C_6H_5)Br\): This molecule possesses the most stable carbocation among the options due to tertiary carbocation stabilization. Three alkyl groups (methyl groups and the phenyl group) are attached to the carbon bearing the positive charge, effectively donating electron density and delocalizing the charge over a larger area. This significantly enhances the stability of the carbocation. Additionally, \(Br^-\) remains a good leaving group.

4. \(C_6H_5CH_2Br\): This molecule forms a primary carbocation, which is the least stable among the options. Primary carbocations have a positive charge localized on a single carbon atom with only one alkyl group (the methyl group) offering minimal stabilization. While \(Br^-\) remains a good leaving group, the formation of a highly unstable carbocation significantly hinders the \(S_N1\) reaction rate.

Therefore, based on the stability of the carbocation formed:

Option 3 \(C_6H_5C(CH_3)(C_6H_5)Br\) is the most reactive towards \(S_N1\) reactions due to its combination of a highly stable tertiary carbocation and a good leaving group \((Br^-)\).

Option 1 \(C_6H_5CH(C_6H_5)Br\)comes in second due to a combination of allylic and benzylic stabilization, despite forming less stable than a tertiary carbocation.

Option 2 \(C_6H_3CH(Br)CH_3\) has minimal carbocation stabilization and might exhibit a moderate reaction rate due to the good leaving group.

Option 4 \(C_6H_5CH_2Br\) forms the least stable carbocation and likely experiences the slowest \(S_N1\) reaction rate.

In summary, the reactivity of these molecules towards \(S_N1\) reactions is primarily governed by the stability of the carbocation intermediate. The combination of a highly stable carbocation and a good leaving group leads to the fastest \(S_N1\) reactions.