Consider a process where a product is formed from a reactant A via two different mechanistic pathways, characterised by rate constants k1 and k2. Which rate law correctly describes this process?

Question

BlackTom AI · Accepted Answer

For two parallel first-order pathways converting A to product, the individual rates add, giving rate_total = (k1 + k2)[A].

1) Relevant knowledge points and concepts
- For a unimolecular (first-order) step A → product with rate constant k, the rate = k[A].
- When a single reactant can convert to the same product by two independent, parallel pathways, each pathway contributes separately to the formation of product.
- Overall formation rate is the sum of the rates of all contributing pathways (principle of superposition for reaction rates).

2) Step-by-step reasoning
- Pathway 1: A → product with rate constant k1 ⇒ rate1 = k1[A].
- Pathway 2: A → product with rate constant k2 ⇒ rate2 = k2[A].
- Because both pathways consume the same pool of A and produce the same product independently, the total rate of product formation is the sum:
  rate_total = rate1 + rate2.

3) Intermediate calculation steps (explicit)
- rate_total = k1[A] + k2[A]
- Factor out [A]: rate_total = (k1 + k2)[A]
- Example: if [A] = 1.0 M, k1 = 2.0 s^-1, k2 = 3.0 s^-1:
  rate_total = (2.0 + 3.0) × 1.0 = 5.0 M·s^-1.

4) Why the correct answer is correct
- The algebra and physical principle (each pathway contributes positively and independently) directly produce the sum (k1 + k2). There is no mechanism or reason for the rate constants to subtract or for one to be ignored.

5) Why other options are incorrect
- (a) rate = k2[A]: ignores the contribution from pathway 1 (k1).
- (b) rate = (k1 – k2)[A]: subtraction is unphysical here; rates do not cancel unless one pathway removes product via a reverse process, which is not stated.
- (c) rate = k1[A]: ignores the contribution from pathway 2 (k2).

6) Conclusion
- The correct rate law is d(rate)/dt = (k1 + k2)[A], i.e., option d, confirming the provided answer.

Consider a process where a product is formed from a reactant A via two different mechanistic pathways, characterised by rate constants k1 and k2. Which rate law correctly describes this process?Single choice

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