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NEET 2008
Exam
NEET
Year
2008
Subject
Physics
Chapter
Nuclear Physics

NEET 2008 · Physics · Nuclear Physics

Two radioactive materials X_1 and X_2 have decay constants 5 λ and...

Two radioactive materials X1X_1 and X2X_2 have decay constants 5λ5 \lambda and λ\lambda respectively. If initially they have the same number of nuclei, then the ratio of the number of nuclei of X1X_1 to that of X2X_2 will be 1e\frac{1}{e} after a time

Options

  1. Aλ\lambda
  2. B12λ\frac{1}{2} \lambda
  3. C14λ\frac{1}{4 \lambda}
  4. Deλ\frac{e}{\lambda}

Correct answer

C. 14λ\frac{1}{4 \lambda}

Step-by-step solution

If NN is the number of radioactive nuclei present at some instant, then N=N0e−λtN=N_0 e^{-\lambda t} The constant N0N_0 represents the number of radioactive nuclei at t=0t=0 Now, N1N2=e−λ1te−λ2tN1N2=e−5λte−λt=e−4λtN1N2=1e\begin{aligned} & \frac{N_1}{N_2}=\frac{e^{-\lambda_1 t}}{e^{-\lambda_2 t}} \\ & \frac{N_1}{N_2}=\frac{e^{-5 \lambda t}}{e^{-\lambda t}}=e^{-4 \lambda t} \\ & \frac{N_1}{N_2}=\frac{1}{e} \end{aligned} but N1N2=1e\frac{N_1}{N_2}=\frac{1}{e} (as provided) Therefore, 1e=1e4λt\frac{1}{e}=\frac{1}{e^{4 \lambda t}} or 4λt=1ort=14λ4 \lambda t=1 or t=\frac{1}{4 \lambda}
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