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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Because the larger the set of random events, the more predictable the outcome of their total is. If you were to flip a coin, and get a penny every time you got heads, and lose a penny every time you got tails, the longer you repeated this bet, the more likely you’ll be at or near zero pennies.
One kilogram of pure Unranium 238 has about 2,529,240,000,000,000,000,000,000 molecules in it. Each one of those molecules has a tiny chance of decaying, in any given second. But that tiny chance, repeated that many times, is going to start averaging out pretty quickly.
As for why they measure decay in terms of ‘half-life’, it’s because that, too, is because after decaying away half the Uranium, you’ll be left with 1,264,620,000,000,000,000,000,000 molecules of U238, and 1,264,620,000,000,000,000,000,000 molecules of something else. That process will continue until all of the moleules have decayed into a stable isotope.
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