Question

Starting from 100 g of a radioactive substance, 2.5 g was left after 5 years. If its radioactive decay follows first order kinetics, calculate: (i) Rate constant for the decay of the radioactive substance (ii) The amount of substance left after one year (iii) The time required for half of the substance to decay

Hint:

Radioactive decay always follows first order kinetics. Half-life is independent of initial amount and depends only on the rate constant.

Answer 1

Step 1: For a first order reaction: \[ \ln\left(\frac{N_0}{N}\right) = kt \] Given: \[ N_0 = 100\ \text{g}, \quad N = 2.5\ \text{g}, \quad t = 5\ \text{years} \]
Step 2: Calculate the rate constant \(k\): \[ \ln\left(\frac{100}{2.5}\right) = k \times 5 \] \[ \ln(40) = 5k \] \[ k = \frac{3.689}{5} = 0.738\ \text{year}^{-1} \]
Step 3: Amount left after 1 year: \[ N = N_0 e^{-kt} \] \[ N = 100 \, e^{-0.738 \times 1} \] \[ N \approx 100 \times 0.478 = 47.8\ \text{g} \]
Step 4: Time required for half of the substance to decay (half-life): \[ t_{1/2} = \frac{0.693}{k} \] \[ t_{1/2} = \frac{0.693}{0.738} \] \[ t_{1/2} \approx 0.94\ \text{years} \]