Question and Given Options

For an autocatalytic second order reaction R→P, the rate law is (where v is rate and k is rate constant):

• (A) v = k[R]
• (B) v = 1/[R] (or similar inverse form)
• (C) v = k[R]²
• (D) v = k[R][P]

Concept: Autocatalytic Second Order Reaction

In an autocatalytic reaction, one of the products acts as a catalyst and accelerates its own formation.
A common elementary scheme is R + P → 2P. The rate depends on the concentrations of both the reactant R and the product P,
so the overall order is two, giving the characteristic autocatalytic second order reaction rate law v = k[R][P].

Deriving the Correct Rate Law

For the elementary step R + P →_k 2P, one molecule of R reacts with one molecule of P.
By the law of mass action, the differential rate is:

v = −d[R]/dt = k[R][P]

This rate expression is first order in R and first order in P, giving an overall reaction order of two.
Therefore, the only correct form of the autocatalytic second order reaction rate law is (D) v = k[R][P].

Detailed Analysis of Each Option

Option (A): v = k[R]

This represents a first‑order rate law because the rate depends linearly on R only.
It cannot describe an autocatalytic second order reaction, since the product P—which catalyzes the reaction—is absent.

Option (B): v = 1/[R] (or similar inverse form)

An expression inversely proportional to [R] does not correspond to a valid elementary rate law.
Such terms typically appear in integrated rate equations, not in differential ones, so this form is incorrect.

Option (C): v = k[R]²

This describes a simple second‑order reaction involving two R molecules (2R → products), not an autocatalytic one.
The absence of P means there’s no self‑acceleration, so it’s not applicable here.

Option (D): v = k[R][P] – Correct

This rate law is first order in R and first order in P, for a total order of two.
As the product P appears in the expression, the reaction is autocatalytic—the rate increases as P accumulates,
confirming the autocatalytic second order reaction rate law v = k[R][P].

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