A radioactive isotope sample has a decay constant \(\lambda = 2.31 \times 10^{-4}\) year-1. Initially the sample contains \(8000 \times 10^{15}\) atoms. The sample decays to 25% of its initial value. The number of years it takes to decay to this stage is closest to ___
GATE XE - 2026,
Answer: B
The time is closest to 6000 years.
Radioactive decay follows \(N = N_{0}e^{-\lambda t}\). Decaying to 25% means \(N/N_{0} = 1/4\):
\[ \begin{aligned} t &= \frac{\ln 4}{\lambda} = \frac{1.386}{2.31 \times 10^{-4}} \\ &\approx 6000\ \text{years} \end{aligned} \]
Equivalently, the half-life is \(0.693/\lambda = 3000\) years, and falling to a quarter takes two half-lives. 3000 is one half-life, the time to 50%. The initial number of atoms does not matter.
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