Question
Obtain an expression for half-life period of zero order reaction.
✓
Answer
The rate law expression for zero order reaction is, $[ A ]_t=- kt +[ A ]_0$ where $[ A ]_0$ and $[ A ]_t$ are the concentrations of the reactant at time, $t =0$ and after time $t$ respectively, Half-life period, $t_{1 / 2}$ is the time when the concentration reduces from $[A]_0$ to $[A]_0 / 2$. i.e., at $t=t_{1 / 2},[A]_t=[A]_0 / 2$.
$\begin{aligned}
& \therefore \frac{[ A ]_0}{2}=-k t_{1 / 2}+[ A ]_0 \\
& \therefore k t_{1 / 2}=[ A ]_0-\frac{[ A ]_0}{2}=\frac{[ A ]_0}{2}
\end{aligned}$
$\therefore t_{1 / 2}=\frac{[ A ]_0}{2 k}$
Hence for a zero-order reaction, the half-life period is directly proportional to the initial concentration of the reactant.
$\begin{aligned}
& \therefore \frac{[ A ]_0}{2}=-k t_{1 / 2}+[ A ]_0 \\
& \therefore k t_{1 / 2}=[ A ]_0-\frac{[ A ]_0}{2}=\frac{[ A ]_0}{2}
\end{aligned}$
$\therefore t_{1 / 2}=\frac{[ A ]_0}{2 k}$
Hence for a zero-order reaction, the half-life period is directly proportional to the initial concentration of the reactant.
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