Question

The rate constant for a first order reaction is 60 sec\(^{-1}\). How much time will it take for the reactant to reduce to 1/16th of its initial concentration?

Hint:

Use \( t=\frac{2.303}{k}\log\frac{[A]_0}{[A]} \) with ratio 16, or note that 1/16 is four half-lives.

Answer 1

Step 1: Use the half-life route. Reducing to \(1/16\) means \(\dfrac{1}{16}=\left(\dfrac{1}{2}\right)^{4}\), i.e. 4 successive half-lives.
Step 2: For a first order reaction the half-life is constant: \(t_{1/2}=\dfrac{0.693}{k}=\dfrac{0.693}{60}=0.01155\ s\).
Step 3: Total time \(=4\times t_{1/2}=4\times 0.01155\ s\).
Step 4: \(t=0.0462\ s\).
\[ \boxed{t \approx 0.046\ \text{s}} \]