Consider Th^228 -> Ra^224 -> Rn^220, where t_1/2(Th^228) = 1.913 years, t_1/2(Ra^224) = 3.64 days. D — Nuclear Chemistry and Radioactivity Chemistry Question

💡 Solution & Explanation

Step 1 - Transient Equilibrium Condition In a decay chain $A \xrightarrow{\lambda_A} B \xrightarrow{\lambda_B} C$, **transient equilibrium** is established when $t_{1/2}(A) > t_{1/2}(B)$ (parent half-life is longer than daughter's but comparable). At transient equilibrium, the ratio of numbers of atoms is: $$\frac{N_A}{N_B} = \frac{\lambda_B - \lambda_A}{\lambda_A} = \frac{\lambda_B}{\lambda_A} - 1 = \frac{t_{1/2}(A)}{t_{1/2}(B)} - 1$$ Step 2 - Convert Half-Lives to Common Units (days) $$t_{1/2}(\text{Th}^{228}) = 1.913\ \text{yr} \times 365\ \text{days/yr} = 698.25\ \text{days}$$ $$t_{1/2}(\text{Ra}^{224}) = 3.64\ \text{days}$$ Step 3 - Calculate N(Th)/N(Ra) $$\frac{N(\text{Th})}{N(\text{Ra})} = \frac{t_{1/2}(\text{Th})}{t_{1/2}(\text{Ra})} - 1 = \frac{698.25}{3.64} - 1 = 191.83 - 1 = \boxed{190.83 \approx 190}$$ Step 4 - Evaluate Options - **(A) 190**: Matches. **Correct.** - **(B) 380**: Twice the answer — arises if the factor of $-1$ is neglected and an extra factor of 2 is added. Incorrect. - **(C) 95**: Half the correct answer. Incorrect. - **(D) 5727**: Arises from using years instead of days without conversion. Incorrect. $$\boxed{\text{Answer: A — }N(\text{Th})/N(\text{Ra}) \approx 190}$$