Question

The number of half-lives \((t_{1/2})\) required for a radioactive isotope to decrease to \(2\%\) of its original abundance is _________ (rounded off to two decimal places).

Hint:

For problems asking how many half-lives remain until a fraction \(f\) is left, use: \[ n = \frac{\ln(f)}{\ln(1/2)}. \] This formula avoids step-by-step decay iteration.

Answer 1

Step 1: General formula.
The fraction of the isotope remaining after \(n\) half-lives is \[ \frac{N}{N_0} = \left(\frac{1}{2}\right)^n. \] Step 2: Set the fraction equal to \(2\%\).
\[ \left(\frac{1}{2}\right)^n = 0.02. \] Step 3: Take logarithms.
\[ n \ln\!\left(\tfrac{1}{2}\right) = \ln(0.02). \] \[ n = \frac{\ln(0.02)}{\ln(0.5)}. \] Step 4: Compute.
\(\ln(0.02) \approx -3.9120,\quad \ln(0.5) \approx -0.6931.\)
\[ n = \frac{-3.9120}{-0.6931} \approx 5.64. \] \[ \boxed{5.64} \]