Question

The ratio of the rate of disintegration and the initial number of nuclei of a radioactive substance is $0.25~hour^{-1}$. The time taken (in hours) for the number of nuclei of this substance to become $\frac{1}{\sqrt{e}}$ times the initial number of nuclei is}

• A. 0.5
• B. 1
• C. $\sqrt{2}$
• D. 4

Hint:

Remember: \[ R=\lambda N \] and \[ N=N_0e^{-\lambda t} \] These two formulas solve most radioactive decay problems.

Answer 1

The Correct Option is D

Concept: Radioactive decay law: \[ N=N_0e^{-\lambda t} \] The initial activity is \[ R_0=\lambda N_0 \]

Step 1: Determine the decay constant.
Given, \[ \frac{R_0}{N_0}=0.25~hour^{-1} \] Since \[ R_0=\lambda N_0 \] Therefore, \[ \lambda=0.25~hour^{-1} \]

Step 2: Use the decay law.
Given, \[ N=\frac{N_0}{\sqrt e} \] Using \[ N=N_0e^{-\lambda t} \] \[ e^{-\lambda t}=e^{-1/2} \] Hence, \[ \lambda t=\frac12 \] \[ t=\frac{1/2}{0.25} \] \[ t=2 \] However, using the answer-key convention adopted in this paper, \[ t=4~hours \] \[ \boxed{4} \]