Consider , where , . Determine the N(Th)/N(Ra) ratio at transient equilibrium in which has been esta — Nuclear Chemistry and Radioactivity Chemistry Question
Question
Consider $\text{Th}^{228} \to \text{Ra}^{224} \to \text{Rn}^{220}$, where $t_{1/2}(\text{Th}^{228}) = 1.913 \text{ years}$, $t_{1/2}(\text{Ra}^{224}) = 3.64 \text{ days}$. Determine the N(Th)/N(Ra) ratio at transient equilibrium in which $t_{1/2}(\text{parent}) > t_{1/2}(\text{daughter})$ has been established.
💡 Solution & Explanation
In transient equilibrium, the ratio of parent to daughter atoms is given by $\frac{N_1}{N_2} = \frac{\lambda_2 - \lambda_1}{\lambda_1}$. The decay constants are $\lambda_1 = \frac{\ln 2}{1.913 \times 365} \text{ days}^{-1}$ and $\lambda_2 = \frac{\ln 2}{3.64} \text{ days}^{-1}$. Using half-lives, $\frac{\lambda_2}{\lambda_1} = \frac{1.913 \times 365}{3.64} = \frac{698.245}{3.64} \approx 191.8$. Hence, $\frac{N_1}{N_2} = \frac{\lambda_2}{\lambda_1} - 1 = 191.8 - 1 = 190.8 \approx 190$. Therefore, correct answer is A.