At radioactive equilibrium, a sample contains two radioactive elements and in the atomic ratio . If — Nuclear Chemistry and Radioactivity Chemistry Question

💡 Solution & Explanation

At steady state (radioactive equilibrium), the rate of disintegration of parent $A$ equals the rate of disintegration of daughter $B$: $N_A \lambda_A = N_B \lambda_B$. Expressing decay constant in terms of half-life gives $\frac{N_A}{T_{1/2(A)}} = \frac{N_B}{T_{1/2(B)}}$. Rearranging for $T_{1/2(B)}$ gives $T_{1/2(B)} = T_{1/2(A)} \times (\frac{N_B}{N_A}) = (2 \times 10^{10}\text{ years}) \times (\frac{1}{3.1 \times 10^9}) = \frac{20}{3.1} \approx 6.45\text{ years}$.