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CBSE Class 12 Physics 2019 Delhi Set 2 Paper

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Question : 6 of 6
Marks: +1, -0
(a) Write the relation between half life and average life of a radioactive nucleus.
(b) In a given sample two isotopes, AA and B are initially present in the ratio of 1:21:2. 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 2:12:1 ?
Solution:  
(a) Relation between average life and half life
(b) Finding the required time
(a) T=1λT=\frac{1}{\lambda}
Alternatively T1/2ln⁡2T1/20.69311.44T1/2\frac{ \frac{ \frac{T_{1/2}}{\ln 2} }{ \frac{T_{1/2}}{0.6931} } }{ 1.44 T_{1/2} }
(b) We have N=N0e−λ⋅tN=N_0 e^{-\lambda \cdot t}
N01N02=12,N1N2=21\frac{N_{0_1}}{N_{0_2}} = \frac{1}{2}, \frac{N_1}{N_2} = \frac{2}{1}
∴N1N2=N01N02exp⁡(−(λ1−λ2)t)\therefore \frac{N_1}{N_2} = \frac{N_{0_1}}{N_{0_2}} \exp\left(-(\lambda_1 - \lambda_2) t\right)
∴2=12exp⁡(−(λ1−λ2)t)\therefore 2 = \frac{1}{2} \exp\left(-(\lambda_1 - \lambda_2) t\right)
⇒exp⁡(−(λ1−λ2)t)=4\Rightarrow \exp\left(-(\lambda_1 - \lambda_2) t\right) = 4
⇒−(λ1−λ2)t=2ln⁡2\Rightarrow -(\lambda_1 - \lambda_2) t = 2 \ln 2
⇒−ln⁡2(160−130)t=2\Rightarrow -\ln 2 \left( \frac{1}{60} - \frac{1}{30} \right) t = 2
⇒t60=2\Rightarrow \frac{t}{60} = 2
⇒t=120 years\Rightarrow t = 120 \text{ years}
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