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(a) Write the relation between half-life and average life of a radioactive nucleus.

(b) In a given sample two isotopes A and B are initially present in the ratio of $1:2$ . Their half-lives are $60$ years and $30$ years respectively. How long will it take so that the sample has these isotopes in the ratio of $2:1$ ?

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The half-life of a radioactive nucleus is given by the relation

$T_{1/2}=\frac{ln2}{\lambda}$

Average life

$\tau=\frac{1}{\lambda}$

Therefore

$T_{1/2}=\tau ln2$

(b) we have no. of particle $A$ with $N_{o}$ and the particle $B$ with $2N_{o}$ .

And note; Half-life of $A$ is $60$ years and $30$ years of $B$ .

Decay constant of $\lambda_A=\frac{0.693}{60}$

Decay constant of $\lambda_B=\frac{0.693}{30}$

By using the radioactive decay equation,

$N(t)=N_{o}e^{-\lambda t}$

Initially

$N_{0A}=\frac{N_{0B}}{2}$

We have to find t when

$N_{0A}=2{N_{0B}}$

Which implies that

$\\2N_{0B}=\frac{{N_{0B}}}{2}e^{-\lambda _A t}\\4=e^{-\lambda _A t}\\ln\frac{1}{4}={\lambda _A t}\\\\ln\frac{1}{4}={\frac{ln2}{60} t}\\-2ln2={\frac{ln2}{60} t}$

$\\t=120 \ \text{years}$

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rishi.raj

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