Acidic Hydrolysis Rate of Substituted Trityl Ether Derivatives

The rate of acid-catalyzed hydrolysis of an ether depends strongly on the stability of the intermediate formed during the reaction.

1. Reaction Mechanism:

The acid-catalyzed hydrolysis of the trityl ether derivative ($\mathbf{X}$) proceeds via an $\text{S}_\text{N}1$ mechanism due to the steric hindrance and the high stability of the triarylmethyl cation formed as an intermediate:

1. Protonation of Ether: $\text{Ar}_3\text{C-O-R} + \text{H}^+ \rightleftharpoons \text{Ar}_3\text{C}-\overset{+}{\text{O}}\text{H}-\text{R}$

2. Heterolytic Cleavage (Rate-Determining Step, RDS): $\text{Ar}_3\text{C}-\overset{+}{\text{O}}\text{H}-\text{R} \xrightarrow{\text{slow}} \text{Ar}_3\text{C}^+ + \text{ROH}$

3. Nucleophilic Attack by Water: $\text{Ar}_3\text{C}^+ + \text{H}_2\text{O} \xrightarrow{\text{fast}} \text{Ar}_3\text{C-OH} + \text{H}^+$

2. Factors Affecting the Reaction Rate:

Since step 2 is the rate-determining step, any factor that increases the stability of the carbocation intermediate ($\text{Ar}_3\text{C}^+$) lowers the activation energy ($\Delta G^\ddagger$) of the transition state, thereby accelerating the rate of hydrolysis.

3. Evaluation of Options:

• Replacing a phenyl ring with a methyl group (Option A): A methyl group stabilizes a carbocation via hyperconjugation and inductive effects ($+I$), whereas a phenyl ring stabilizes it via extensive $\pi$-delocalization (resonance). Replacing a phenyl ring with a methyl group reduces resonance stabilization, making the carbocation less stable and decreasing the reaction rate.

• Original Molecule $\mathbf{X}$ (Option D): The triphenylmethyl (trityl) carbocation, $\text{Ph}_3\text{C}^+$, is stabilized by resonance across three phenyl rings.

• Replacing one phenyl group with a $para$-methoxyphenyl group (Option B): The $para$-methoxy ($p\text{-OCH}_3$) substituent is a strong electron-donating group through its mesomeric effect ($+M$). The lone pair of electrons on the oxygen atom delocalizes effectively into the ring and onto the central carbocation: $\text{H}_3\text{C}-\ddot{\text{O}}-\text{C}_6\text{H}_4-\text{C}^+ \longleftrightarrow \text{H}_3\text{C}-\overset{+}{\text{O}}=\text{C}_6\text{H}_4=\text{C}$ This significant $+M$ stabilization increases the carbocation stability relative to $\mathbf{X}$.

• Replacing two phenyl groups with two $para$-methoxyphenyl groups (Option C): Having two $para$-methoxyphenyl groups introduces two strong $+M$ electron-donating groups. The additive resonance contribution of both methoxy groups stabilizes the positive charge on the central carbon atom to an even greater extent than a single $para$-methoxyphenyl group.

Conclusion:

The carbocation formed when two phenyl groups are replaced by two $para$-methoxyphenyl groups is the most stable among all the choices, leading to the lowest activation energy and the fastest rate of hydrolysis.

Thus, the correct option is C.