Which one of the following statements is correct for a first order reaction?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → Infinite time is taken for 100% completion of the reaction.
For a first order reaction, Rate \( = k[\mathrm{R}] \), and the integrated rate equation is:
\( \displaystyle t = \frac{2.303}{k}\log\frac{[\mathrm{R}]_0}{[\mathrm{R}]} \)
For 100% completion \( [\mathrm{R}] \) would have to become zero. Then \( [\mathrm{R}]_0/[\mathrm{R}] \) becomes infinitely large, and so does its logarithm:
\( \displaystyle [\mathrm{R}] \rightarrow 0 \;\Rightarrow\; \log\frac{[\mathrm{R}]_0}{[\mathrm{R}]} \rightarrow \infty \;\Rightarrow\; t \rightarrow \infty \)
In each half-life the amount left is halved, but it never becomes zero. So a first order reaction takes infinite time for 100% completion. Its half-life, found by putting \( [\mathrm{R}] = [\mathrm{R}]_0/2 \), is independent of the initial concentration:
\( \displaystyle t_{1/2} = \frac{2.303}{k}\log 2 = \frac{0.693}{k} \)
Why the other options are incorrect
Option (1) Half life is directly proportional to the initial concentration of reactant. — Incorrect. This holds for a zero order reaction, \( t_{1/2} = [\mathrm{R}]_0/2k \).
Option (2) Half life is inversely proportional to the initial concentration of reactant. — Incorrect. This holds for a second order reaction, \( t_{1/2} = 1/k[\mathrm{R}]_0 \).
Option (4) Rate of the reaction is proportional to the square of initial concentration of reactant. — Incorrect. That is the rate law of a second order reaction; a first order rate depends on the first power of the concentration.