Question

Derive the relation between half-life period and rate constant for a first-order reaction.

Hint:

In first-order reactions, the half-life remains constant, which is a characteristic feature in processes such as radioactive decay and drug metabolism.

Answer 1

For a first-order reaction, the rate law is: \[ \text{Rate} = k [A] \] The integrated rate law is: \[ [A] = [A]_0 e^{-kt} \] At the half-life \( (t_{1/2}) \), when \( [A] = \frac{[A]_0}{2} \), we have: \[ \frac{[A]_0}{2} = [A]_0 e^{-k t_{1/2}} \] Taking the natural logarithm on both sides: \[ \ln \frac{1}{2} = -k t_{1/2} \] \[ t_{1/2} = \frac{0.693}{k} \] Therefore, for first-order reactions, the half-life is independent of the initial concentration.