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Question #44703

Difficulty: 8/10 | 🟢 Free

The Question

The gas-phase photochemical decomposition of ozone is represented by the overall reaction: $\ce{2O3(g) -> 3O2(g)}$ .

The reaction proceeds via the following mechanism: Step 1: $\ce{O3(g) ->[h\nu] O2(g) + O^*(g)}$ (Fast initiation) Step 2: $\ce{O^*(g) + O3(g) -> 2O2(g)}$ (Slow, Rate Determining Step)

A photon of energy $E$ strikes a stationary $\ce{O3}$ molecule, causing the dissociation in Step 1. The bond dissociation energy for $\ce{O3 -> O2 + O}$ is $E_0$ .

Assuming the intermediate $\ce{O^*}$ generated in Step 1 directly participates in the RDS without any collisional energy loss, what is the De Broglie wavelength of the $\ce{O^*}$ particle?

(Let $m$ be the mass of a single oxygen atom. Assume non-relativistic mechanics and that the excess energy is entirely converted into the translational kinetic energy of the products in Step 1).

Options

A.
$\frac{h}{2} \sqrt{\frac{3}{m(E - E_0)}}$
B.
$\frac{h}{\sqrt{m(E - E_0)}}$
C.
$\frac{h}{\sqrt{2m(E - E_0)}}$
D.
$h \sqrt{\frac{3}{2m(E - E_0)}}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Numerical_Precision

Improve calculation stamina and avoid rounding errors in high-stakes competitive examinations.

Concepts Behind This Question

De Broglie Wavelength Logic|Rate Law Derivation from Mechanism|Stoichiometric Coefficient Division

Why This Question Is Tricky

Common Mistakes

This problem tests the intersection of Quantum Mechanics, Conservation Laws, and Chemical Kinetics.

The 'Hidden Variable' is the conservation of momentum during the elementary photochemical initiation step.

Students often fall into the 'Stoichiometric Trap' by looking at the overall reaction ( $\ce{2O3 -> 3O2}$ ) and erroneously dividing the excess energy by 2 or 3 to find the kinetic energy of the intermediate.

Alternatively, they may assume all excess energy goes to the lighter $\ce{O^*}$ atom.

By forcing the student to isolate the elementary Step 1 and apply physics principles (equal momentum, unequal kinetic energy partition based on mass ratio $m$ vs $2m$ ), the question bypasses rote memorization of kinetic formulas and tests true conceptual depth.

Quick Facts About This Question

Question Information

Subject	Chemistry
Chapter	Structure of Atom - Class 11 Part 1
Concept	De Broglie Wavelength Logic
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free