While studying radioactivity I found that even the most radioactive substances i.e substances with the shortest half lives do not completely degenerate.

Suppose there is a 1 mole sample of an element X whose half life is 1 day. But as the degeneration equation is exponential in nature even after 1 million years some amount of X would remain in the sample.

What i want to know is how come certain atoms of an element possess such stability that they do not generate after a million years, while some atoms taken form the same sample degenerate after 1 day.

Is it because they have some different particles in their nucleus or something ?

What is the role of quantum tunnelling ?

-
Possibly relevant: physics.stackexchange.com/a/109899/44126 – rob Jun 3 '14 at 5:43
But my question is WHY? why does some nuclei decay and some doesnt? – Avik Jun 3 '14 at 5:48
@Avik Hey Avik, I share the same interests as you and I am too in high school. Can you provide your email id or some sort of contact? It's nice to interact with people who share the same interest as yours. – Yashbhatt Jun 4 '14 at 5:37
sure my email is avik.mukherjee.hk96@gmail.com :) – Avik Jun 4 '14 at 6:57

Let me suggest an analogy.

Suppose you take a mole, $6 \times 10^{23}$, of coins, all head up, and you start flipping them. Any coin that comes down tails is discarded. After the first flip we'll lose (about) 50% of the coins. after the second flip only 25% will be left, after the third flip only 12.5% and so on. So we'll start losing coins very rapidly, but because we had so many coins there's a fair chance at least one coin will still be there after 79 flips (because $2^{79} = 6 \times 10^{23}$).

Was the coin that came up heads 79 times in a row any different to the other coins? No, it was just pure chance. Could we have predicted which coin was going to survive 79 flips? No, because all the coins are identical.

This is what is going on with your radioactive decay. We start with a mole of atoms, all identical, and every atom has a probability to decay within one half life. Half the atoms will decay in one half life and half will be left. two half lives later 25% will be left and so on, just like the coins.

And just like the coins the atoms that manage to survive many half lives are no different to all the other atoms. It's just pure chance that some survive a long time and some don't.

Response to comment:

Avik makes the good point that I've assumed all the coin flips are random. In principle they aren't, since if we knew exactly how the force was applied to the coin we could calculate it's trajectory and predict which side it would land on. This may be hard to do in practice, but in principle it's possible.

In the case of radioactive decay the nucleus has a lower energy state available (the decayed state) but there is a potential barrier that it has to climb over to decay. The nucleus doesn't have enough energy to climb the barrier, but it can get through the barrier by quantum tunnelling. Unlike the coins, quantum tunnelling is (as far as we know) a truly random process. That means no matter how carefully we measure the state of the nucleus we will never be able to predict whether it will tunnel through the barrier or not in some pre-determined time.

I should also mention, because it's all part of the same physics, that the uncertainty principle means we can never measure the state of a nucleus precisely anyway.

-
But if all the coins were thrown in the same manner..then their spins would be identical and they would all come either heads or tails. Its just that we do not apply same force while tossing the coins and this is why we get separate outcomes. – Avik Jun 3 '14 at 6:56
@Avik: I've edited my answer to respond to your comment. – John Rennie Jun 3 '14 at 7:07
can you explain the tunnelling phenomenon in layman terms? – Avik Jun 3 '14 at 9:31
@Avik: that's really a new question. However I suggest you do some Googling for quantum tunnelling before post a new question otherwise your question is likely to be a bit vague. – John Rennie Jun 3 '14 at 9:54
okay i will do it. @john will you please answer my gravity question too? – Avik Jun 3 '14 at 10:30

The phenomenon of radioactive decay is a quantum mechanical phenomenon. At the level of atoms and nuclei the only outcome of calculations give us probabilities, the probability distribution is the exponential decay that you quote.

Probabilities mean the same in quantum mechanics and in classical mechanics and in everyday life. How come that when the lottery is drawn specific numbers come up, and not others? It is a random throw. The same with the nuclei, they decay randomly according to the probability distribution .

-
So there is no speciality about the nucleus that remain stable ? – Avik Jun 3 '14 at 5:53
No, actually they are indistinguishable except through quantum numbers look here hyperphysics.phy-astr.gsu.edu/hbase/nuclear/halfli2.html#c3 "The half-life is independent of the physical state (solid, liquid, gas), temperature, pressure, the chemical compound in which the nucleus finds itself, and essentially any other outside influence" – anna v Jun 3 '14 at 5:56
"indistinguishable except through quantum numbers" please explain – Avik Jun 3 '14 at 5:58
A nucleus with spin up can be distinguished from a nucleus with spin down, for example because of the spin orientation, but if they have all the quantum numbers the same, for example in a crystal lattice, then they have no separate identifying features, if they are sitting at the same energy levels in the lattice. If they are sitting in different energy levels that would distinguish them in a set :energy level1 versus energy level2. That would not affect the lifetime. – anna v Jun 3 '14 at 6:13
lifetimes can change only under very special cases physics.stackexchange.com/questions/8745/… – anna v Jun 3 '14 at 6:16