The decay constant of a radioactive nuclide is 4.6 × $10 ^{- 3}$ $\mathrm { s } ^ { - 1 }$ What is the half-life of this isotope,in minutes?

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The half-life of a radioactive isotope is the time it takes for half of the atoms in a sample to decay. The half-life is related to the decay constant by the equation:$$t_{1/2} = \frac{\ln 2}{k}$$where $t_{1/2}$ is the half-life, and $k$ is the decay constant.In this problem, the decay constant is given as $4.6 \times 10^{-3} \mathrm{s}^{-1}$. Substituting this value into the equation above, we get:$$t_{1/2} = \frac{\ln 2}{4.6 \times 10^{-3} \mathrm{s}^{-1}} = 150.9 \mathrm{s}$$Converting this to minutes, we get:$$t_{1/2} = 150.9 \mathrm{s} \times \frac{1 \mathrm{min}}{60 \mathrm{s}} = 2.52 \mathrm{min}$$Therefore, the half-life of this isotope is 2.52 minutes, which is closest to choice A) 2.5 min.

The Decay Constant of a Radioactive Nuclide Is 4 10−310 ^{- 3}10−3

The Decay Constant of a Radioactive Nuclide Is 4 $10 ^{- 3}$