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Standard XII

Mathematics

Rate of Change

A radioactive...

Question

A radioactive nucleus with decay constant $0.5 / s$ is being produced at a constant rate of $100 n u c l e i / s$ . If at $t = 0$ there were no nuclei, the time when there are $50$ nuclei is

1 s

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2 l n (\frac{4}{3}) s

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l n 2 s

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l n (\frac{4}{3}) s

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Solution

The correct option is B $2 l n (\frac{4}{3}) s$
Given: $λ = 0.5$
$\frac{d N}{d t} = 100 - λ N$
$\frac{d N}{100 - λ N} = d t$
Integrating :
$- \frac{1}{λ} (l n 100 - λ N)_{N 1}^{N 2} = (t)_{t 1}^{t 2}$
At $t = 0$ , $N = 0$
At $N = 50$ ,
$- 2 (l n 75 - l n 100) = t$
Hence, $t = 2 l n \frac{4}{3} s$ .

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