Select the correct relation of half life for a zero order reaction.

The correct answer is Option (1) → \(t_{0.5}=\frac{[R]_0}{2k}\)

The half-life, \( t_{1/2} \) (here written \( t_{0.5} \)), is the time in which the concentration of a reactant falls to half its initial value. For a zero order reaction the concentration falls linearly with time, and the integrated rate equation gives:

    \( \displaystyle k = \frac{[\mathrm{R}]_0 - [\mathrm{R}]}{t} \)

At \( t = t_{1/2} \), \( [\mathrm{R}] = \frac{1}{2}[\mathrm{R}]_0 \):

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

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

So for a zero order reaction the half-life is directly proportional to the initial concentration and inversely proportional to the rate constant.

Shortcut: A half-life must have the unit of time. For a zero order reaction k is in mol L⁻¹ s⁻¹, so \( [\mathrm{R}]_0/k \) is in seconds, while Option (2) gives s⁻¹ and Option (4) gives mol L⁻¹. Of Options (1) and (3), only Option (1) has the factor 2 that comes from “half”.

Why the other options are incorrect

• Option (2) \(t_{0.5}=\frac{2k}{[R]_0}\) — Incorrect. It is the inverse of the correct expression and has the unit s⁻¹, not a time.

• Option (3) \(t_{0.5}=\frac{[R]_0}{k}\) — Incorrect. This is the time for the whole reactant to be used up (putting [R] = 0), twice the half-life.

• Option (4) \(t_{0.5}=\frac{[R]_0}{2}\) — Incorrect. It contains no rate constant, so it is a concentration, not a time.

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