Ac^227 has a half-life of 22 years with respect to radioactive decay. The decay follows two parallel — Nuclear Chemistry and Radioactivity Chemistry Question

💡 Solution & Explanation

Step 1 - Parallel (Branched) Radioactive Decay When a nuclide decays through two parallel paths simultaneously, the total decay constant $\lambda_{\text{total}}$ is the sum of the individual decay constants: $$\lambda_{\text{total}} = \lambda_{\text{Th}} + \lambda_{\text{Fr}}$$ The fraction of each path equals the percentage yield. Step 2 - Calculate Total Decay Constant $$\lambda_{\text{total}} = \frac{\ln 2}{t_{1/2}} = \frac{0.693}{22\ \text{yr}} = 3.15 \times 10^{-2}\ \text{yr}^{-1}$$ → **(A) is correct.** Step 3 - Individual Decay Constants $$\lambda_{\text{Th}} = 2\%\ \text{of}\ \lambda_{\text{total}} = 0.02 \times 3.15 \times 10^{-2} = 6.3 \times 10^{-4}\ \text{yr}^{-1}$$ → **(B) is correct.** $$\lambda_{\text{Fr}} = 98\%\ \text{of}\ \lambda_{\text{total}} = 0.98 \times 3.15 \times 10^{-2} = 3.087 \times 10^{-2}\ \text{yr}^{-1}$$ → **(C) is correct.** Step 4 - Evaluate Option D: Mass Ratio After 22 Years The **molar ratio** of Th-227 to Fr-223 produced is 2:98 = 1:49 (from the branching fractions). However, both daughter nuclides are themselves radioactive and decay further with time. After 22 years: - Th-227 has a short half-life (18.7 days) and will have almost completely decayed away. - Fr-223 has a half-life of 22 min and will also have decayed. So the **mass ratio** of Th-227 to Fr-223 remaining after 22 years is NOT 1:49. The ratio changes continuously as daughters decay. Also, the mass ratio ≠ molar ratio (masses are 227 u vs 223 u). → **(D) is incorrect.** This is the only false statement. $$\boxed{\text{Answer: D — Statement D is incorrect}}$$