What determines which specific atoms of a radioactive material decay over a period of time?

The answer is: "nothing".
In fact, it's totally random. When there is a certain amount of atoms of a radioactive element, it's not possible to predict which atom will decay first and which one will decay later. We can only predict the amount of atoms that will be left after a certain time t (if we know the half-life of the element), but we can't predict which atom will decay first.

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