Question 404: Atoms & Nuclei

Question

The nuclear activity of a radioactive element becomes $${\left( {{1 \over 8}} \right)^{th}}$$ of its initial value in 30 years. The half-life of radioactive element is _____________ years.

Accepted Answer

We know, $$A = {A_0}{e^{ - \lambda t}}$$
For half life
$${{{A_0}} \over 2} = {e^{ - \lambda {t_{1/2}}}}$$
$$ \Rightarrow $$ $${\lambda {t_{1/2}}}$$ = ln 2  .....(1)
And when radioactive element becomes $${\left( {{1 \over 8}} ight)^{th}}$$ of its initial value in 30 years
$${{{A_0}} \over 8} = {A_0}{e^{ - \lambda 	imes 30}} \Rightarrow \lambda  	imes 30 = \ln 8$$
$$ \Rightarrow $$ 30$$\lambda  = 3\ln 2$$
$$ \Rightarrow $$ $$\lambda  = {{3\ln 2} \over {30}}$$ .....(2)
Putting value of $$\lambda $$ in (1), we get
$${{3\ln 2} \over {30}} 	imes {t_{1/2}}$$ = ln 2
$$ \Rightarrow $$ $${t_{1/2}}$$ = 10 years

Atoms & Nuclei

The nuclear activity of a radioactive element becomes ${\left( {{1 \over 8}} \right)^{th}}$ of its initial value in 30 years. The half-life of radioactive element is _____________ years.

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