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)

Consider two independent first-order reactions with half-lives $$t_{1/2,1} = 12$$ min and $$t_{1/2,2} = 3$$ min. Their corresponding rate constants are $$k_1 = \frac{\ln 2}{12}$$ and $$k_2 = \frac{\ln 2}{3}$$, and the effective rate constant for the combined process is $$k_{eff} = k_1 + k_2 = \frac{\ln 2}{12} + \frac{\ln 2}{3} = \frac{\ln 2 + 4\ln 2}{12} = \frac{5\ln 2}{12}$$.

From this effective rate constant, the half-life for 50% consumption follows as: $$t_{1/2} = \frac{\ln 2}{k_{eff}} = \frac{\ln 2}{\frac{5\ln 2}{12}} = \frac{12}{5} = 2.4 \text{ min} \approx 2 \text{ min}$$

Therefore, the time required for 50% consumption of the reactant is 2 min when rounded to the nearest integer.