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  2. (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
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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 ?

 

 

 

 
 
 
 
 

Answers (1)
R rishi.raj

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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