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Question #44466

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The Question

Consider the thermal gas-phase bromination of $(S)$ -1-chloro-2-methylbutane ( $\ce{RH}$ ) at $400\text{ K}$ to yield 2-bromo-1-chloro-2-methylbutane ( $\ce{RBr}$ ). The reaction proceeds via a radical chain mechanism with the following elementary steps:

Initiation: $\ce{Br2 ->[k_i] 2Br^\bullet}$ (Rate $= k_i[\ce{Br2}]$ ) Propagation 1: $\ce{RH + Br^\bullet ->[k_1] R^\bullet + HBr}$ Propagation 2: $\ce{R^\bullet + Br2 ->[k_2] RBr + Br^\bullet}$ Termination: $\ce{2Br^\bullet ->[k_t] Br2}$

Assuming the Steady State Approximation (SSA) applies to all radical intermediates, let $k_{inv}$ represent the first-order rate constant for the conformational inversion of the intermediate carbon radical $\ce{R^\bullet}$ . Which of the following correctly expresses the overall rate of formation of $\ce{RBr}$ and the kinetic condition required to obtain a predominantly racemic product mixture?

Options

A.
$\frac{d[\ce{RBr}]}{dt} = k_1 \left(\frac{k_i}{k_t}\right)^{1/2} [\ce{RH}][\ce{Br2}]^{1/2} \text{ ; } k_{inv} \gg k_2[\ce{Br2}]$
B.
$\frac{d[\ce{RBr}]}{dt} = k_1 \left(\frac{k_i}{k_t}\right)^{1/2} [\ce{RH}][\ce{Br2}]^{1/2} \text{ ; } k_{inv} \ll k_2[\ce{Br2}]$
C.
$\frac{d[\ce{RBr}]}{dt} = k_2 \left(\frac{k_i}{k_t}\right)^{1/2} [\ce{RH}][\ce{Br2}] \text{ ; } k_{inv} \gg k_1[\ce{RH}]$
D.
$\frac{d[\ce{RBr}]}{dt} = k_1 \left(\frac{k_i}{k_t}\right) [\ce{RH}][\ce{Br2}] \text{ ; } k_{inv} \ll k_1[\ce{RH}]$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Mechanism_Trap

Don't get tricked by the process! Learn the mechanical traps used in competitive exam questions.

Concepts Behind This Question

Stereochemical Outcome of Halogenation|Rate Law Derivation from Mechanism|Steady State Approximation

Why This Question Is Tricky

Common Mistakes

This question tests two advanced concepts simultaneously: deriving a fractional-order rate law using the Steady State Approximation (SSA) for a radical chain mechanism, and understanding the kinetic competition that dictates stereochemistry.

Students often memorize that radical halogenation gives racemic mixtures without realizing it depends on the lifetime of the radical intermediate ( $\tau = 1/k_2[\ce{Br2}]$ ).

If the trapping step is faster than the inversion step ( $k_2[\ce{Br2}] \gg k_{inv}$ ), the reaction would proceed with retention of configuration.

The hidden variable is the concentration of the halogen, which directly controls the stereochemical outcome.

Quick Facts About This Question

Question Information

Subject	Chemistry
Chapter	Hydrocarbons - Class 11 Part 2
Concept	Stereochemical Outcome of Halogenation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free