Q104 / 150 · MHT CET 2023 (12 May Shift 2) · Physics

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

  • A
    $\frac{\mathrm{N}_1 \mathrm{~T}_1}{\mathrm{~N}_2 \mathrm{~T}_2}$
  • B
    $\frac{\mathrm{N}_2 \mathrm{~T}_2}{\mathrm{~N}_1 \mathrm{~T}_1}$
  • Correct answer
    C
    $\frac{\mathrm{N}_1 \mathrm{~T}_2}{\mathrm{~N}_2 \mathrm{~T}_1}$
  • D
    $\frac{\mathrm{N}_1 \mathrm{~N}_2}{\mathrm{~T}_1 \mathrm{~T}_2}$

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: $$ \begin{aligned} & \frac{\mathrm{A}_1}{\mathrm{~A}_2}=\frac{\lambda_1 \mathrm{~N}_1}{\lambda_2 \mathrm{~N}_2} \\ & \frac{\mathrm{A}_1}{\mathrm{~A}_2}=\frac{\frac{\ln 2}{\mathrm{~T}_1} \mathrm{~N}_1}{\frac{\ln 2}{\mathrm{~T}_2} \mathrm{~N}_2} \\ & \frac{\mathrm{A}_1}{\mathrm{~A}_2}=\frac{\mathrm{N}_1 \mathrm{~T}_2}{\mathrm{~N}_2 \mathrm{~T}_1} \end{aligned} $$

Hence, the correct answer is option: 'C'.

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