If the half-life of element X is 10 years then when will one atom decay?

Question

If I understand this correctly a single atom of element X which has a half-life of 10 years could decay into element Y in 10 years, 1 billion years, or 1 second.... right?

Answer 1

Right.

Answer 2

Yes. The half life manifests itself only statistically when you have a large number of atoms initially.

Exactly, the decay of a single atom is a quantum process: after a given time, the atom is a superposition of decayed (with some particles leaving) and not decayed and until not observed, the former's probability grows like 1-exp(lambda t). When you try to measure the decay, for example its energetic outcome, you will "kill the cat" with the given probability, observing the empirical decay law.

The other result resets the atom to undecayed state. With this particular time evolution, it does not matter how often and how randomly you make the observation. It can be even "continuous" with the same result, which is in fact the case as no one usually explicitly looks at the atoms from time to time.

Hope this helps!

Answer 3

The atom will decay when it is good and ready.  However, the chances are even that the atom will have decayed within 10 years.  It could decay in the next second, but the chances are only 2.196 chances in 1 billion (10^9) that it would do so.  

On the other hand, chances are only 1 in 1024 that the atom would last the century.  Every century thereafter would add another three zeroes (and a scratch more) to the front of the probability of the atom's still being around after that period.  The probability would be a bit worse than 1 in a billion that the atom would still be around after 3 centuries or 1 in a thousand billion billion billion that it would still be around after a millennium.  If you won a penny bet on the atom's still being around after a millennium, you would be owed far more than all of the wealth in the entire world.  (Actually collecting on such a bet would be highly problematic, to say the least!)