To determine the half-life of the radioactive element, a student plots a graph of ℓn \left|\right. \frac{dN \left(\right. t \left.\right)}{dt} \left|\right. versus t. Here \frac{dN \left(\right. t \left.\right)}{dt} is the rate of radioactive decay at time t. If the number of radioactive nuclei of this element decreases by a factor of p after 4.16 years, the value of p is

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To determine the half-life of the radioactive element, a student plots a graph of $ℓn | \frac{dN (t)}{dt} |$ versus t. Here $\frac{dN (t)}{dt}$ is the rate of radioactive decay at time t. If the number of radioactive nuclei of this element decreases by a factor of p after 4.16 years, the value of p is

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$\frac{dN}{dt} = N_{0} e^{- λt}$

$ℓn (\frac{dN}{dt}) = ℓn N_{0} - λt$

$4 = {ℓnN}_{0}$

$3 = 4 - λ \times 2$

$λ = 0 .5 / year$

$N = N_{0} e^{- λt}$

$\frac{N}{N_{0}} = e^{- λt} = e^{- 2 .08} = \frac{1}{e^{2 .08}} = \frac{1}{8} = 8$

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