Question

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)

Hint:

Always combine the reciprocal contributions of individual half-lives to find the effective rate.

Answer 1

Correct Answer: 2

Step 1: Relationship Between Rate Constants and Effective Half-Life

For independent first-order reactions, the effective rate constant (\( k_\text{eff} \)) is given by:

\[ k_\text{eff} = k_1 + k_2 \]

The effective half-life (\( t_\text{eff} \)) is related to the individual half-lives (\( t_1 \) and \( t_2 \)) as:

\[ \frac{1}{t_\text{eff}} = \frac{1}{t_1} + \frac{1}{t_2} \]

Step 2: Substitute the Given Half-Lives

Substitute \( t_1 = 12 \, \text{min} \) and \( t_2 = 3 \, \text{min} \):

\[ \frac{1}{t_\text{eff}} = \frac{1}{12} + \frac{1}{3} \]

Convert to a common denominator:

\[ \frac{1}{t_\text{eff}} = \frac{1}{12} + \frac{4}{12} = \frac{5}{12} \]

Step 3: Calculate \( t_\text{eff} \)

Solve for \( t_\text{eff} \):

\[ t_\text{eff} = \frac{12}{5} = 2.4 \, \text{min} \]

Round to the nearest integer:

\[ t_\text{eff} = 2 \, \text{min} \]

Final Answer:

The time taken for 50% consumption of the reactant is: 2 minutes.