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. \(1/\lambda\)
• B. \(1/6\lambda\)
• C. \(1/7\lambda\)
• D. \(1/8\lambda\)

Hint:

Write the number of nuclei of each material using N = N0 e^(-decay constant x time) and set the ratio equal to e.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
Radioactive decay follows \(N = N_{0}e^{-\lambda t}\). Both materials start with the same \(N_{0}\).

Step 2: Write each number
\(N_{A} = N_{0}e^{-7\lambda t}\) and \(N_{B} = N_{0}e^{-\lambda t}\).

Step 3: Take the ratio
\(\dfrac{N_{B}}{N_{A}} = e^{-\lambda t + 7\lambda t} = e^{6\lambda t}\).

Step 4: Calculation
Given \(\dfrac{N_{B}}{N_{A}} = e\), so \(6\lambda t = 1\) and \(t = \dfrac{1}{6\lambda}\).

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