Question

Disintegration rate of a radio-active sample is \( 10^{10} \) per hour at 20 hours from the start. It reduces to \( 5 \times 10^9 \) per hour after 30 hours. Calculate the decay constant.

Hint:

The decay constant can be calculated using the ratio of disintegration rates at different times, applying the radioactive decay law.

Answer 1

Step 1: Use the radioactive decay law.
The number of disintegrations \( N \) at any time \( t \) is given by: \[ N = N_0 e^{-\lambda t} \] where \( N_0 \) is the initial disintegration rate, \( \lambda \) is the decay constant, and \( t \) is the time.
Step 2: Set up the equation.
At \( t = 20 \) hours, \( N = 10^{10} \), and at \( t = 30 \) hours, \( N = 5 \times 10^9 \). Using the decay law: \[ \frac{N(t=30)}{N(t=20)} = \frac{5 \times 10^9}{10^{10}} = e^{-\lambda (30-20)} \] \[ 0.5 = e^{-10 \lambda} \] Taking the natural logarithm of both sides: \[ \ln(0.5) = -10 \lambda \] \[ \lambda = -\frac{\ln(0.5)}{10} = 0.0693 \, \text{hr}^{-1} \]