Question

What is half-life for a chemical reaction? Show that the half-life for a first order reaction is independent of the initial concentration of the reactants.

Hint:

First-order reactions always have constant half-life, while zero and second order reactions depend on initial concentration.

Answer 1

Step 1: Definition.
The half-life (\(t_{1/2}\)) of a reaction is the time required for the concentration of the reactant to become half of its initial concentration. Step 2: Integrated rate law for first order reaction.
For a first order reaction: \[ k = \frac{2.303}{t} \log \frac{[R]_0}{[R]} \] Step 3: Apply half-life condition.
At \(t = t_{1/2}\), \([R] = \frac{[R]_0}{2}\). \[ k = \frac{2.303}{t_{1/2}} \log \frac{[R]_0}{[R]_0/2} \] \[ k = \frac{2.303}{t_{1/2}} \log 2 \] Step 4: Simplification.
Since \(\log 2 = 0.3010\), \[ k = \frac{2.303 \times 0.3010}{t_{1/2}} \] \[ k = \frac{0.693}{t_{1/2}} \] \[ t_{1/2} = \frac{0.693}{k} \] Step 5: Independence of concentration.
From the equation, \(t_{1/2}\) depends only on the rate constant \(k\) and not on the initial concentration \([R]_0\). Thus, for first-order reactions, half-life is independent of concentration. Conclusion:
The half-life of a first order reaction is given by: \[ \boxed{t_{1/2} = \dfrac{0.693}{k}} \] It is independent of initial concentration.