Before You Begin

Question #44819

Difficulty: 8/10 | 🟢 Free

The Question

The mutarotation of D-glucose in aqueous solution at 298 K proceeds via an acyclic aldehydic intermediate ( $I$ ). The mechanism is represented as:

$\alpha\text{-D-glucopyranose} \ce{<=>[k_1][k_{-1}]} I \ce{<=>[k_2][k_{-2}]} \beta\text{-D-glucopyranose}$

The observed mutarotation rate constant is defined as $k_{mut} = k_{f,eff} + k_{b,eff}$ , where $k_{f,eff}$ and $k_{b,eff}$ are the effective forward and backward rate constants for the direct $\alpha \rightleftharpoons \beta$ interconversion.

A student applies the Steady-State Approximation (SSA) to the intermediate $I$ to derive $k_{mut}^{SSA}$ . However, under specific acid-catalyzed conditions, the ring-opening of the $\alpha$ -anomer is rapid and reversible compared to the ring-closure to the $\beta$ -anomer ( $k_{-1} \gg k_2$ ), making the Pre-Equilibrium (PE) approximation more appropriate for finding the forward rate.

Which of the following correctly represents the expression for $k_{mut}^{SSA}$ and the ratio of the effective forward rate constants $\left( \frac{k_{f,eff}^{SSA}}{k_{f,eff}^{PE}} \right)$ ?

Options

A.
$k_{mut}^{SSA} = \frac{k_1 k_2 + k_{-1} k_{-2}}{k_{-1} + k_2}$ ; Ratio = $\frac{k_{-1}}{k_{-1} + k_2}$
B.
$k_{mut}^{SSA} = \frac{k_1 k_2 - k_{-1} k_{-2}}{k_{-1} + k_2}$ ; Ratio = $\frac{k_2}{k_{-1} + k_2}$
C.
$k_{mut}^{SSA} = \frac{k_1 k_{-1} + k_2 k_{-2}}{k_{-1} + k_2}$ ; Ratio = $\frac{k_{-1}}{k_{-1} + k_2}$
D.
$k_{mut}^{SSA} = \frac{k_1 k_2 + k_{-1} k_{-2}}{k_{-1} + k_2}$ ; Ratio = $\frac{k_{-1} + k_2}{k_{-1}}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Exception_Exposure

Learn to identify edge cases and boundary exceptions that JEE examiners hide in math problems.

Concepts Behind This Question

Steady State Approximation|D-Glucose Anomeric Equilibrium|Chemical Kinetics

Why This Question Is Tricky

Common Mistakes

This question tests the critical distinction between the Steady-State Approximation (SSA) and the Pre-Equilibrium (PE) approximation in complex reaction mechanisms.

Students often blindly apply SSA to any low-concentration intermediate.

However, in mutarotation under specific conditions, the acyclic form is in rapid equilibrium with the anomers.

The trap lies in failing to recognize the exact kinetic form of reversible first-order reactions ( $k_{obs} = k_f + k_b$ ) and incorrectly assuming SSA and PE yield identical rate laws without evaluating the limit where $k_{-1} \gg k_2$ .

Quick Facts About This Question

Question Information

Subject	Chemistry
Chapter	Chemical Kinetics - Class 12
Concept	Steady State Approximation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free