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I’m sure there is an easy answer to this, but for some reason I can’t wrap my head around how a sample of an element can ever decay all the way to zero, when measured in half lives. It seems like you could always split a number in half, it would just be infinitesimally small.

In: Chemistry

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Because in reality, a half life is actually a measure of statistical probability.

At any given time, any one atom of an unstable isotope can or cannot decay. The chance of it decaying is what determines the half-life.

The only reason half-life’s are so stable and reliable is because you are talkin about insanely large populations of atoms. Billions of billions of billions of them. Theoretically, every single atom in a mass of unstable matter could all undergo decay at the same time, but that is so unlikely we don’t really consider it possible.

The half life is the time over which each *individual* atom has a *50% chance* of decaying. *As a result of that*, a sample of a very large number of atoms will decrease by half in that time, but that isn’t what it actually means.

If you have 1 atom, there is a 1/2 chance it will decay after one half life, and a 1/2 chance it won’t. There is no half decay.

And if you have 2 atoms, there is a 1/4 chance they both will decay, a 1/2 chance one will decay, and a 1/4 chance neither will. It’s just like flipping coins.

No that is not what half life means(lol you are accidentally half correct) let me explain

When a radioactive material starts decaying, initially it’s decays very rapidly but soon it starts slowing down and it keeps on slowing down and decays very slowly.

Like some radioactive material will decay to half in about 20 years the the next half will decay in 1 million years.

Now 1 million years is a not and no human has conducted that experiment and that is why we measure the decay in half life as it is measurable in human life time.

In order to understand this look at the graph of radioactive decay and learn about the properties of that curve.

Half life isn’t perfect… If we had an infinite number of atoms then half of them would theoretically decay within the half life, however with a finite number of atoms more than half of them can decay within said half life as its just repeated probability…

Here is an example, take 100 **fair** coins and place them face up on a cookie sheet…. Dump them on the floor, now remove any coins that are face down. The face up coins you can put back on the cookie sheet and repeat and count how many repetitions it takes for there to be no coins left. Then repeat this a few times…. The coins, when doing this, have a theoretical half life of 1 iteration (dumping coins) meaning that each time you will expect about 50% of the coins to be taken away… However sometimes more than 50% are taken away and sometimes less than 50% will be taken away. If you increase the number of coins to 1000 then the ratio will move closer to 50% (say it might be 506 heads up and 494 heads down)… Increase it again to say 100,000 coins and you might see something like 50,012 up and 49,988 down, still its getting closer and closer to being 50-50. When you hypothetically increase it to an infinite number of coins the ratio will be 50-50

That is the analogy for half life… Now if I have a 10kg rock of uranium, then I will have something like 10^22 uranium atoms so the ratio is going to be very very close to 50-50. This is also a good introduction to the laws of large numbers.

Edit: bolded the fair in “fair coins” to emphasize that.

The half-life is a friendly rewording of “the time in which a single atom has a 50% chance of decaying”. And this time exists for a single atom, and this time exists for every atom in a heaping pile, and in theory it’s the same time unless you make a critical mass or something.

And if you divide the half-life by ln(2), you get the [average lifetime](https://en.wikipedia.org/wiki/Exponential_decay#Mean_lifetime) of an atom. (The average lifetime can be more statistically useful than the 50% time.)

But remember you can be 100% sure that your coin is 50/50 fair, yet you can never be 100% sure that 8 coinflips will return 4 heads. In the same way, you can never be sure that all atoms in your sample will decay away. There is a probability that some will remain, and in a million billion zillion years, it will decay infinitesimally close to zero, but it will never be certain.

The half life is random event for each atom, and if you had say a million atoms and a very long half life then it would be unlikely ever to decay to a perfect zero, but then also it would not be radioactive since the remaining atoms wouldn’t be decaying. https://youtu.be/AaDwk8UCrew

Half-life doesn’t mean that half the element will be gone, although that is a consequence.

Half-life means that each individual atom has a 50% chance of decaying in that span of time. Eventually, you get down to the last atom, which has a 50% chance of being gone after one half-life, 75% after two, 87.5% after three, etc. You can never say when it will reach zero, but it will eventually happen.

Because the quantum of an element is 1 atom. You cannot divide a single atom of an element and still have that same element.

Half-life decay is a measure of a large population of atoms. Once that last atom decays, there is no more population.