MHT CET · Physics · Nuclear Physics
Two different radioactive elements with half-lives ' \(\mathrm{T}_{1}\) ' and ' \(\mathrm{T}_{2}\) ' have undecayed
atoms ' \(\mathrm{N}_{1}{ }^{\prime}\) and \({ }^{\prime} \mathrm{N}_{2}{ }^{\prime}\) respectively, present at a given instant. The ratio of their
activities at this instant is
- A \(\frac{\mathrm{T}_{1} \mathrm{~T}_{2}}{\mathrm{~N}_{1} \mathrm{~N}_{2}}\)
- B \(\frac{\mathrm{N}_{1} \mathrm{~N}_{2}}{\mathrm{~T}_{1} \mathrm{~T}_{2}}\)
- C \(\frac{\mathrm{N}_{1} \mathrm{~T}_{1}}{\mathrm{~N}_{2} \mathrm{~T}_{2}}\)
- D \(\frac{\mathrm{N}_{1} \mathrm{~T}_{2}}{\mathrm{~N}_{2} \mathrm{~T}_{1}}\)
Answer & Solution
Correct Answer
(D) \(\frac{\mathrm{N}_{1} \mathrm{~T}_{2}}{\mathrm{~N}_{2} \mathrm{~T}_{1}}\)
Step-by-step Solution
Detailed explanation
Activity, \(\mathrm{R}\left(=\frac{\mathrm{dN}}{\mathrm{dt}}\right)=\lambda \mathrm{N}\)
Decay constant \(\lambda=\frac{\log _{e} 2}{\mathrm{~T}}\)
\(\therefore\) Activity \(R=\frac{\left(\log _{\mathrm{e}} 2\right) \mathrm{N}}{\mathrm{T}}\)
\(\therefore \mathrm{R}_{1}=\frac{\left(\log _{e} 2\right) \mathrm{N}_{1}}{\mathrm{~T}_{1}}, \mathrm{R}_{2}=\frac{\left(\log _{e} 2\right) \mathrm{N}_{2}}{\mathrm{~T}_{2}}\)
For two elements \(\frac{R_{1}}{R_{2}}=\frac{N_{1}}{T_{1}} \times \frac{T_{2}}{N_{2}}=\left(\frac{N_{1}}{N_{2}}\right)=\left(\frac{T_{2}}{T_{1}}\right)\)
Decay constant \(\lambda=\frac{\log _{e} 2}{\mathrm{~T}}\)
\(\therefore\) Activity \(R=\frac{\left(\log _{\mathrm{e}} 2\right) \mathrm{N}}{\mathrm{T}}\)
\(\therefore \mathrm{R}_{1}=\frac{\left(\log _{e} 2\right) \mathrm{N}_{1}}{\mathrm{~T}_{1}}, \mathrm{R}_{2}=\frac{\left(\log _{e} 2\right) \mathrm{N}_{2}}{\mathrm{~T}_{2}}\)
For two elements \(\frac{R_{1}}{R_{2}}=\frac{N_{1}}{T_{1}} \times \frac{T_{2}}{N_{2}}=\left(\frac{N_{1}}{N_{2}}\right)=\left(\frac{T_{2}}{T_{1}}\right)\)
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