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A radio-active elements has half-life of 15 years. What is the fraction that will decay in 30 years?
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Verified Answer
The correct answer is:
$0.75$
Given, half-life, $T_{1 / 2}=15$ years
Time, $t=30$ years
$\therefore$ Number of half-life, $n=\frac{t}{T}=\frac{30}{15}=2$
The number of nuclei left undecayed in 30 years or 2 half-lives is
$\begin{aligned}
\qquad N=N_{0}\left(\frac{1}{2}\right)^{n} \Rightarrow \frac{N}{N_{0}} &=\left(\frac{1}{2}\right)^{2}=\frac{1}{4} \\
\therefore \text { Fraction of decayed element } &=\left(1-\frac{N}{N_{0}}\right)=1-\frac{1}{4} \\
&=\frac{3}{4} \text { or } 0.75
\end{aligned}$
Time, $t=30$ years
$\therefore$ Number of half-life, $n=\frac{t}{T}=\frac{30}{15}=2$
The number of nuclei left undecayed in 30 years or 2 half-lives is
$\begin{aligned}
\qquad N=N_{0}\left(\frac{1}{2}\right)^{n} \Rightarrow \frac{N}{N_{0}} &=\left(\frac{1}{2}\right)^{2}=\frac{1}{4} \\
\therefore \text { Fraction of decayed element } &=\left(1-\frac{N}{N_{0}}\right)=1-\frac{1}{4} \\
&=\frac{3}{4} \text { or } 0.75
\end{aligned}$
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