Question

Two radioactive materials \(A\) and \(B\) having decay constant ' \(7 \lambda\) ' and ' \(\lambda\) ' respectively, initially have same number of nuclei. The time taken to have the ratio of number of nuclei of material B to that of A as ' e ' is

• A. \(\frac{1}{\lambda}\)
• B. \(\frac{1}{6\lambda}\)
• C. \(\frac{1}{7\lambda}\)
• D. \(\frac{1}{8\lambda}\)

Hint:

N = N0 e^(-lambda t) for each sample; set the ratio N_B/N_A equal to e.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
\(N_A=N_0e^{-7\lambda t}\) and \(N_B=N_0e^{-\lambda t}\).

Step 2: Ratio
\(\dfrac{N_B}{N_A}=e^{-\lambda t+7\lambda t}=e^{6\lambda t}\)

Step 3: Calculation
\(e^{6\lambda t}=e\Rightarrow6\lambda t=1\Rightarrow t=\dfrac{1}{6\lambda}\), option (B).

Final Answer:
The required time is 1/(6 lambda). \[ \boxed{\text{(B) }\dfrac{1}{6\lambda}} \]