![Q.21 For product formation from only one type of reactant (e.g. A → product), the CORRECT match for the order of the reaction (given in Column I) with the half-life expression (given in Column II) is ([A]0 is the initial concentration and kr is the rate constant) Column I Order Column II Half-life expression i. Zero P. ln 2 / kr ii. First Q. [A]0 / 2kr iii. Second R. 1 / (kr[A]0) S. 2kr / [A]0 (A) i–R, ii–P, iii–S (B) i–Q, ii–P, iii–R (C) i–S, ii–R, iii–Q (D) i−Q, ii−P, iii−S](https://www.letstalkacademy.com/wp-content/uploads/2025/12/slide_21-4.png)
Q.21 For product formation from only one type of reactant (e.g. A → product), the CORRECT match for the order of the reaction (given in Column I) with the half-life expression (given in Column II) is
([A]0 is the initial concentration and kr is the rate constant)
| Column I Order |
Column II Half-life expression |
|---|---|
| i. Zero | P. ln 2 / kr |
| ii. First | Q. [A]0 / 2kr |
| iii. Second | R. 1 / (kr[A]0) |
| S. 2kr / [A]0 |
(A) i–R, ii–P, iii–S
(B) i–Q, ii–P, iii–R
(C) i–S, ii–R, iii–Q
(D) i−Q, ii−P, iii−S
For this GATE 2025 question, the correct matching is: i – Q, ii – P, iii – R (Option B).
Half‑life expressions: core concepts
For a reaction A→product, half‑life t1/2 is the time required for [A] to fall from [A]0 to [A]0/2.
Using integrated rate laws, standard half‑life expressions are:
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Zero order: t1/2=[A]0/(2kr).
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First order: t1/2=ln2/kr.
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Second order (2A → products): t1/2=1/(kr[A]0).
These three expressions are exactly what Column II options Q, P and R represent in the question.
Matching Column I and Column II
Zero order reaction (i)
For a zero order reaction, the integrated law is [A]t=[A]0−krt.
Setting [A]t=[A]0/2 gives t1/2=[A]0/(2kr), which corresponds to Q in Column II.
So, i (Zero order) → Q [A]0/2kr.
First order reaction (ii)
For a first order reaction, the integrated law is ln([A]0/[A]t)=krt.
Putting [A]t=[A]0/2 yields t1/2=ln2/kr, which is option P.
So, ii (First order) → P ln2/kr.
Second order reaction (iii)
For a second order reaction 2A→ products, the integrated rate law is 1/[A]t−1/[A]0=krt.
At half‑life, [A]t=[A]0/2, so t1/2=1/(kr[A]0), which matches R.
So, iii (Second order) → R 1/(kr[A]0).
Therefore, the correct option is (B) i–Q, ii–P, iii–R.
Why the other options are wrong
The Column II option S (2k_r/[A]_0) is the reciprocal of R multiplied by 2 and does not represent any standard half‑life expression.
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Option (A): i–R, ii–P, iii–S
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Assigns second‑order expression (R) to zero order and the non‑standard S to second order, so both i and iii become incorrect.
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Option (C): i–S, ii–R, iii–Q
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Uses S (incorrect form) for zero order; also swaps first‑ and second‑order expressions, giving the dependence on [A]0 to first order, which contradicts the fact that first‑order half‑life is independent of initial concentration.
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Option (D): i–R, ii–Q, iii–P
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Again gives R (second order) to zero order and makes first‑order half‑life proportional to [A]0 (Q), both of which contradict standard kinetics.
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Thus, only Option (B) correctly matches each reaction order with its half‑life expression.
