Question

For a radioactive material, half-life is 10 minutes. If initially there are 600 number of nuclei, the time taken (in minutes) for the disintegration of 450 nuclei is

• A. 20
• B. 30
• C. 10
• D. 15

Answer 1

The Correct Option is A

Method 2: Logarithmic Approach

Given:

• \( N_0 = 600 \), \( N = 150 \), \( T = 10 \) minutes

Decay law:

\[ N = N_0 \left(\frac{1}{2}\right)^{t/T} \]

Take natural log on both sides:

\[ \ln\left(\frac{N}{N_0}\right) = \frac{t}{T} \ln\left(\frac{1}{2}\right) \]

Substitute values:

\[ \ln\left(\frac{150}{600}\right) = \frac{t}{10} \ln\left(\frac{1}{2}\right) \]

\[ \ln\left(\frac{1}{4}\right) = \frac{t}{10} \ln\left(\frac{1}{2}\right) \]

Since \( \ln(1/4) = 2 \ln(1/2) \):

\[ 2 \ln\left(\frac{1}{2}\right) = \frac{t}{10} \ln\left(\frac{1}{2}\right) \]

Cancel common term:

\[ \frac{t}{10} = 2 \Rightarrow t = 20 \text{ minutes} \]

Final Answer: 20 minutes