Collision Theory for Unimolecular Decomposition via Lindemann Mechanism

Question:
Consider a unimolecular decomposition that proceeds via the Lindemann mechanism:

The reactant A may also be activated by a collision with a product molecule B:

Use collision theory to calculate the relative magnitudes of k1, and k3. You may assume that the two reactions have the same threshold energy. ( Hint: What is a reasonable assumption about the relative volumes of B and A?)

This is the problem 3.12 of the book “Chemical Kinetics and Reaction Dynamics, written by Paul L. Houston”.

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