Since naturally occurring uranium (U-238) has a half life of 4.5 billion years, then it means half of the uranium on earth has decayed into lead by now. But why only half, and why that specific half? What was special about the particles that did decay? Were they different in some way?
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The main thing you’re missing is the impossibly huge number of radioisotopes in a sample. In grams of uranium you’d have about 10 with 21 zeros after it nuclei. At that point, it’s not about individual atoms, you’re safely in the realm of statistics.
If each atom has a certain probability to decay, and that probability is truly random and independent of all the other atoms, then the sample as a whole will exhibit exponential decay. You can calculate the time by which half of the atoms will have decayed, and you have no idea which ones, but probabilities are basically facts when you have a billion trillion of something.
Once you have half of them left, you still have to wait just as long for half of what’s left to decay. You’re starting with half as many, but you’ll get half the decays because there are half as many, so the half life is the same.
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