Two different radioactive elements with half lives ' $\mathrm{T}_1$ ' and ' $\mathrm{T}_2$ ' have undecayed…

Two different radioactive elements with half lives ' $\mathrm{T}_1$ ' and ' $\mathrm{T}_2$ ' have undecayed atoms ' $\mathrm{N}_1$ ' and ' $\mathrm{N}_2$ ' respectively present at a given instant. The ratio of their activities at that instant is
  1. $\frac{\mathrm{N}_1 \mathrm{~T}_1}{\mathrm{~N}_2 \mathrm{~T}_2}$
  2. $\frac{\mathrm{N}_2 \mathrm{~T}_2}{\mathrm{~N}_1 \mathrm{~T}_1}$
  3. $\frac{\mathrm{N}_1 \mathrm{~T}_2}{\mathrm{~N}_2 \mathrm{~T}_1}$
  4. $\frac{\mathrm{N}_1 \mathrm{~N}_2}{\mathrm{~T}_1 \mathrm{~T}_2}$

Solution

$\therefore \quad$ Activity is given as: $\begin{aligned} \therefore \quad \mathrm{A} & =\lambda \mathrm{N} \\ \lambda & =\frac{\ln 2}{\mathrm{~T}} \end{aligned}$ $\therefore \quad$ Ratio of two different radioactive elements will be: $\begin{aligned} & \frac{\mathrm{A}_1}{\mathrm{~A}_2}=\frac{\lambda_1 \mathrm{~N}_1}{\lambda_2 \mathrm{~N}_2} \\ & \frac{\mathrm{A}_1}{\mathrm{~A}_2}=\frac{\frac{\ln 2}{\mathrm{~T}_1} \mathrm{~N}_1}{\frac{\ln 2}{\mathrm{~T}_2} \mathrm{~N}_2} \\ & \frac{\mathrm{A}_1}{\mathrm{~A}_2}=\frac{\mathrm{N}_1 \mathrm{~T}_2}{\mathrm{~N}_2 \mathrm{~T}_1} \end{aligned}$

Asked in: MHT CET 2023 (12 May Shift 2)

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