Question

{"given":"Half-life of $^{131}\\text{I}$: $T_{1/2} = 8$ days. The number of undecayed nuclei at time $t$ is given by $$N = N_0 e^{-\\lambda t}$$ where $\\lambda = \\frac{\\ln 2}{T_{1/2}}$ is the decay constant.","key_observation":"Radioactive decay is a purely probabilistic (random) process. The exponential decay law $N = N_0 e^{-\\lambda t}$ never reaches exactly zero, meaning decay can continue up to $t \\to \\infty$. Any individual nucleus has a non-zero probability of decaying at any moment after $t = 0$.","option_analysis":[{"label":"(A)","text":"no nucleus will decay before t = 4 days","verdict":"incorrect","explanation":"Since decay is random, there is a non-zero probability that any nucleus decays immediately after $t = 0$. We cannot assert that no decay occurs before $t = 4$ days."},{"label":"(B)","text":"no nucleus will decay before t = 8 days","verdict":"incorrect","explanation":"Even though 8 days is one half-life, individual nuclei can decay at any time. In fact, on average half the sample decays within the first 8 days."},{"label":"(C)","text":"all nuclei will decay before t = 16 days","verdict":"incorrect","explanation":"The exponential decay function $N = N_0 e^{-\\lambda t}$ asymptotically approaches zero but never reaches it, so it is impossible to assert all nuclei will decay by any finite time."},{"label":"(D)","text":"a given nucleus may decay at any time after t = 0","verdict":"correct","explanation":"Radioactive decay is a random process governed by probability. A given nucleus has a non-zero probability of decaying at any time $t > 0$, from immediately to theoretically $t \\to \\infty$."}],"answer":"(D)","formula_steps":[]}