Question

Define half life of first order reaction. Obtain the expression for half life and rate constant of the first order reaction.

Hint:

India's environmental stance involves balancing growth with ecological sustainability.

Answer 1

Step 1: Definition of Half-Life: The half-life (\( t_{1/2} \)) of a first-order reaction is the time required for the concentration of the reactant to decrease to half its initial value.
Step 2: Rate Constant Expression: For a first-order reaction, rate = \( k[A] \). The integrated rate law is: \[ \ln \frac{[A]_0}{[A]} = kt, \] where \( [A]_0 \) is the initial concentration, \( [A] \) is the concentration at time \( t \), and \( k \) is the rate constant. Rearranging gives: \[ k = \frac{1}{t} \ln \frac{[A]_0}{[A]}. \]
Step 3: Half-Life Expression: At half-life, \( [A] = \frac{[A]_0}{2} \). Substitute into the integrated rate law: \[ \ln \frac{[A]_0}{[A]_0/2} = kt_{1/2} \quad \Rightarrow \quad \ln 2 = kt_{1/2} \quad \Rightarrow \quad t_{1/2} = \frac{\ln 2}{k} \approx \frac{0.693}{k}. \]