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)

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Solution

Q.
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) [2023]

Ans.
(2)

$K_{1} = \frac{\ln 2}{12}, K_{2} = \frac{\ln 2}{3}$

$K = K_{1} + K_{2} = \frac{\ln 2}{12} + \frac{\ln 2}{3}$

$\frac{\ln 2}{t} = \frac{\ln 2}{12} + \frac{\ln 2}{3} \Rightarrow \frac{1}{t} = \frac{1}{12} + \frac{1}{3}$

$\frac{1}{t} = \frac{5}{12} \Rightarrow t = \frac{12}{5} = 2 min$

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