A molecule undergoes two independent first order reactions whose respective half lives are $12 \min$ and $3 \min$. If both the reactions are occurring then the time taken for the $50\%$ consumption of the reactant is ______ $\min$. (Nearest integer)

For parallel reaction

When adding the rate constants of the two reactions, we need to use the formula for combining rate constants of two reactions in parallel, which is:

$\mathrm{k}={\mathrm{k}}_1+{\mathrm{k}}_2$

where ${\mathrm{k}}_1$ and ${\mathrm{k}}_2$ are the rate constants of the individual reactions.

In this case, the half-lives of the two reactions are $12$ min and $3$ min, respectively. The rate constants of the two reactions can be calculated using the formula for half-life of a first-order reaction:

$\frac{1}{{\mathrm{t}}_{1/2}}=\frac{1}{12}+\frac{1}{3}=\frac{5}{12}$

Net ${\mathrm{t}}_{1/2}=\frac{12}{5}=2.4 \min \approx 2 \min$

Hence time taken for $50\%$ consumption of reactant will be close to $2 \min$.