Question

If the time taken for a radioactive substance to decay 8% to 77% is 12 minutes, then the half life of the substance in minutes is

• A. 24
• B. 18
• C. 12
• D. 6

Hint:

Use the relation \( \frac{N}{N_0} = e^{-\lambda t} \) and \( T_{1/2} = \frac{\ln 2}{\lambda} \) to connect decay percentage and half-life.

Answer 1

The Correct Option is D

Step 1: Use the formula for radioactive decay.
\[ N = N_0 e^{-\lambda t} \Rightarrow \frac{N}{N_0} = e^{-\lambda t} \] Step 2: Given \( N/N_0 = 0.77 \) and \( t = 12 \) min, solve for \( \lambda \): \[ 0.77 = e^{-\lambda \cdot 12} \Rightarrow \ln(0.77) = -12\lambda \Rightarrow \lambda = -\frac{\ln(0.77)}{12} \] Step 3: Use half-life formula \( T_{1/2} = \frac{\ln 2}{\lambda} \): \[ T_{1/2} = \frac{\ln 2}{\frac{-\ln(0.77)}{12}} = \frac{12 \ln 2}{-\ln(0.77)} \] Step 4: Calculate numerically: \[ \ln 2 \approx 0.693, \ln(0.77) \approx -0.261 \Rightarrow T_{1/2} = \frac{12 \times 0.693}{0.261} \approx \frac{8.316}{0.261} \approx 6.0 \, \text{min} \] Step 5: Select the correct option.
The half-life of the substance is approximately 6 minutes, matching option (4).