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Question #44810

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The Question

The decomposition of a substrate $S$ in the presence of a homogeneous catalyst $C$ follows the mechanism:

$\ce{S + C <=>[k_1][k_{-1}] I ->[k_2] P + C}$

The rate constants at $298\text{ K}$ are given as $k_1 = 2 \times 10^7 \, \text{M}^{-1}\text{s}^{-1}$ , $k_{-1} = 10^5 \, \text{s}^{-1}$ , and $k_2 = 10^5 \, \text{s}^{-1}$ . The initial concentration of the catalyst is $[C]_0 = 10^{-6} \, \text{M}$ and the initial concentration of the substrate is $[S]_0 = 0.1 \, \text{M}$ . Applying the steady-state approximation for the intermediate $I$ , what is the time required for the concentration of $S$ to reduce to $0.05 \, \text{M}$ ?

(Assume the reaction volume remains constant and no products are initially present.)

Options

A.
$(0.5 + 0.1 \ln 2) \, \text{s}$
B.
$(0.1 \ln 2) \, \text{s}$
C.
$0.5 \, \text{s}$
D.
$(0.5 - 0.1 \ln 2) \, \text{s}$

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

First Order Half-Life Independence|Rate Law Derivation from Mechanism

Why This Question Is Tricky

Common Mistakes

This question tests the transition regime of saturation kinetics (Michaelis-Menten type).

Students often blindly apply the first-order half-life formula ( $t_{1/2} = \ln 2 / k_{eff}$ ) or the zero-order half-life formula ( $t_{1/2} = [S]_0 / 2V_{max}$ ).

The trap is that at $[S]_0 = 0.1 \, \text{M}$ and $K_M = 0.01 \, \text{M}$ , the reaction is neither purely zero-order nor purely first-order throughout the half-life period.

It starts predominantly zero-order, but the first-order term ( $K_M \ln 2$ ) is highly significant.

Failing to integrate the full rate law derived from the Steady-State Approximation leads to selecting the pure zero-order (0.5 s) or pure first-order ( $0.1 \ln 2$ s) distractors.

Quick Facts About This Question

Question Information

Subject	Chemistry
Chapter	Chemical Kinetics - Class 12
Concept	First Order Half-Life Independence
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free