Question

{"given":"Decay constants: $\\lambda_1 = 10\\lambda$ (for $X_1$), $\\lambda_2 = \\lambda$ (for $X_2$). Initial nuclei: $N_1(0) = N_2(0) = N_0$. Condition: $\\dfrac{N_1}{N_2} = \\dfrac{1}{e}$.","key_observation":"By the radioactive decay law, $N = N_0 e^{-\\lambda t}$. The ratio $N_1/N_2 = e^{-10\\lambda t}/e^{-\\lambda t} = e^{-9\\lambda t}$. Setting this equal to $1/e = e^{-1}$ gives $9\\lambda t = 1$.","option_analysis":[{"label":"(A)","text":"$t = \\dfrac{1}{10\\lambda}$","verdict":"incorrect","explanation":"This gives $e^{-9\\lambda \\cdot \\frac{1}{10\\lambda}} = e^{-9/10} \\neq e^{-1}$. The exponent does not equal $-1$."},{"label":"(B)","text":"$t = \\dfrac{1}{11\\lambda}$","verdict":"incorrect","explanation":"This gives $e^{-9\\lambda \\cdot \\frac{1}{11\\lambda}} = e^{-9/11} \\neq e^{-1}$. The exponent does not equal $-1$."},{"label":"(C)","text":"$t = \\dfrac{11}{10\\lambda}$","verdict":"incorrect","explanation":"This gives $e^{-9\\lambda \\cdot \\frac{11}{10\\lambda}} = e^{-99/10} \\neq e^{-1}$. The exponent does not equal $-1$."},{"label":"(D)","text":"$t = \\dfrac{1}{9\\lambda}$","verdict":"correct","explanation":"Substituting: $e^{-9\\lambda \\cdot \\frac{1}{9\\lambda}} = e^{-1} = \\dfrac{1}{e}$. This satisfies the given condition exactly."}],"answer":"(D)","formula_steps":[]}