Question

A first order reaction takes $40$ min for $30$% decomposition. Calculate $t_{1/2}$.

Answer 1

Let $a$ M be the initial concentration. After $40$ minutes, the concentration is $a-\frac{30a}{100}=0.70a$.
$k=\frac{2.303}{t}log\frac{[A{]}_o}{\left[A\right]}$
$k=\frac{2.303}{40}log\frac{a}{0.70a}$
$k=8.92\times {10}^{-3}/min$
The half life period, $t_{1/2}=\frac{0.693}{k}=\frac{0.693}{8.92\times {10}^{-3}/min}=77.7$ min.