Question

In a radioactive sample \( 2 \times 10^8 \) nuclei reduce to \( 10^8 \) nuclei in 15 minutes, then the half-life of the sample in minutes is:

• A. 5
• B. 10
• C. 15
• D. 30

Hint:

The half-life of a substance can be determined by solving the exponential decay equation based on the given data.

Answer 1

The Correct Option is B

Step 1:
The decay of a radioactive substance follows the relation: \[ N = N_0 \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}} \] where \( N \) is the remaining number of nuclei, \( N_0 \) is the initial number of nuclei, \( t \) is the time elapsed, and \( T_{1/2} \) is the half-life.
Step 2:
We are given that \( N_0 = 2 \times 10^8 \), \( N = 10^8 \), and \( t = 15 \) minutes. Substituting these values: \[ 10^8 = 2 \times 10^8 \left( \frac{1}{2} \right)^{\frac{15}{T_{1/2}}} \] \[ \frac{1}{2} = \left( \frac{1}{2} \right)^{\frac{15}{T_{1/2}}} \] Thus, the half-life is \( T_{1/2} = 10 \) minutes.