JEE MAIN - Chemistry (2022 - 29th July Evening Shift - No. 19)
Assuming $$1 \,\mu \mathrm{g}$$ of trace radioactive element X with a half life of 30 years is absorbed by a growing tree. The amount of X remaining in the tree after 100 years is ______ $$\times\, 10^{-1} \mu \mathrm{g}$$.
[Given : ln 10 = 2.303; log 2 = 0.30]
Answer
1
Explanation
$$t=\frac{1}{\lambda} \ln \left(\frac{a}{a-x}\right)$$
$$ \begin{aligned} &\Rightarrow100=\left(\frac{30}{\ln 2}\right)\left[\ln \left(\frac{1}{w}\right)\right] \\ &\Rightarrow{\left[\frac{100 \times \log 2}{30}\right]=\log \left(\frac{1}{w}\right)} \\ &\Rightarrow1=\log \left(\frac{1}{w}\right) \\ &\Rightarrow\frac{1}{w}=10 \\ &\text { So } w=0.1 \mu g \end{aligned} $$
$$ \begin{aligned} &\Rightarrow100=\left(\frac{30}{\ln 2}\right)\left[\ln \left(\frac{1}{w}\right)\right] \\ &\Rightarrow{\left[\frac{100 \times \log 2}{30}\right]=\log \left(\frac{1}{w}\right)} \\ &\Rightarrow1=\log \left(\frac{1}{w}\right) \\ &\Rightarrow\frac{1}{w}=10 \\ &\text { So } w=0.1 \mu g \end{aligned} $$
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