Select the correct statement.

The correct answer is Option (1) → For the zero order reaction, the half-life of a reaction is directly proportional to the initial concentration of the reactant, whereas for the first order reaction, the half-life is independent of the initial concentration of the reacting species.

The half-life, \( t_{1/2} \), is the time in which the concentration of a reactant falls to half its initial value. Putting \( [\mathrm{R}] = [\mathrm{R}]_0/2 \) in each integrated rate equation gives its dependence on \( [\mathrm{R}]_0 \).

Zero order:

    \( \displaystyle k = \frac{[\mathrm{R}]_0 - [\mathrm{R}]}{t} \Rightarrow t_{1/2} = \frac{[\mathrm{R}]_0 - [\mathrm{R}]_0/2}{k} = \frac{[\mathrm{R}]_0}{2k} \)

First order:

    \( \displaystyle k = \frac{2.303}{t}\log\frac{[\mathrm{R}]_0}{[\mathrm{R}]} \Rightarrow t_{1/2} = \frac{2.303}{k}\log 2 = \frac{0.693}{k} \)

Why the other options are incorrect

• Option (2) For the zero order reaction, the half-life of a reaction is inversely proportional to the initial concentration of the reactant, whereas for the first order reaction, the half-life is independent of the initial concentration of the reacting species. — Incorrect. For a zero order reaction \( t_{1/2} = [\mathrm{R}]_0/2k \) is directly, not inversely, proportional to \( [\mathrm{R}]_0 \); an inverse dependence belongs to a second order reaction.

• Option (3) The half-life of a reaction is directly proportional to the initial concentration of the reacting species for both zero and first order reactions. — Incorrect. It is right for zero order, but the first order half-life, \( 0.693/k \), contains no concentration term.

• Option (4) The half-life of a reaction is inversely proportional to the initial concentration of the reacting species for both zero and first order reactions. — Incorrect. It is wrong for both orders: neither half-life is inversely proportional to \( [\mathrm{R}]_0 \).

NCERT (2026-27 edition):
Half-Life of a Reaction