Deepak K Kapur
Hi,

A radioactive substance which has a half-life of, say 5 years, will 'certainly' decay to half its amount after 5 years ( i am good at maths!).

My question is,

When radioactivity is a random process, where does this 'certainty' come into picture?

Thanks.

## Answers and Replies

Gold Member
2021 Award
radioactivity is a random process at the micro level but it is subject to statistical distributions that accumulate at the macro level.

So all that can be said about a single atom that decays is that it will decay sometime during the life of the universe and that there's a 50:50 chance that the decay will occur within the half-life period.

A kilogram of such material has so many atoms that statistically you can be sure that very close to half of them will have decayed by the end of the half-life period.

Gold Member
The "random process" follows a rule: it follows the exponential distribution.

Here is an in-depth mathematical discussion:
http://depts.washington.edu/amath/courses/423-winter-2010/note2.pdf [Broken]

As to your question about "certainly", well, this word doesn't belong in this description! And smaller the quantity of material being observed, the less certainty. That is, for large amounts the statistics predict that the observed half-life will be 5 years; but for small quantities the uncertainty becomes greater and greater.

After all, the exponential decay process has no memory ... it does not know what has happened in the past, nor what will happen in the future. But the probability is there for each and every atom.

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Homework Helper
"Certainly"= "plus or minus a few atoms"!

Hyrum
The "random process" follows a rule: it follows the exponential distribution.

Here is an in-depth mathematical discussion:
http://depts.washington.edu/amath/courses/423-winter-2010/note2.pdf [Broken]

As to your question about "certainly", well, this word doesn't belong in this description! And smaller the quantity of material being observed, the less certainty. That is, for large amounts the statistics predict that the observed half-life will be 5 years; but for small quantities the uncertainty becomes greater and greater.

After all, the exponential decay process has no memory ... it does not know what has happened in the past, nor what will happen in the future. But the probability is there for each and every atom.

To clarify:

The random process doesn't necessarily follow exponential distribution. All it follows is the probability of a given atom decaying in a certain time-frame. The exponential distribution is an effect caused by this probability and the fact that atoms are disappearing and will no longer be accounted for.

Say within time τ there is a probability of p that a given atom will decay then we can effectively conclude that after time τ the number of atoms left relative to the initial count is 1-p. Now because there are fewer atoms then the relative number of atoms left after time xτ is $$(1-p)^x.$$ And this represents the exponential decay process.

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Mentor
I simply say part is random and part is not. Which will decay next is random but the average rate of decay is not.

Mentor
A radioactive substance which has a half-life of, say 5 years, will 'certainly' decay to half its amount after 5 years

Who said/wrote 'certainly'? (If it wasn't you, that is)

Deepak K Kapur
That is, for large amounts the statistics predict that the observed half-life will be 5 years;

What is the reason for this?

'Large amount' is a relative term. When does amount become large, when we consider 100000 atoms or 100000000 etc.

Gold Member
The Casino will, "certainly" make money on roulette because of the 0 on the wheel. Individuals can still take vast levels of winnings away with them, though. There is no conflict between those two statements. You need to understand what statistics is actually telling you and appreciate that there is variation around a general trend.

Gold Member
What is the reason for this?

'Large amount' is a relative term. When does amount become large, when we consider 100000 atoms or 100000000 etc.

Look up the Law of Large Numbers ... it is a precise mathematical statement which applies in this situation.

Mentor
If we start out with N undecayed atoms, after one half-life we expect to have N/2 left. The actual number will probably not be exactly N/2 because of the randomness of the process. The statistics are exactly the same as with coin-tossing: the probability is 0.5 that any particular atom has not yet decayed, versus 0.5 for any particular coin coming up "heads."

It turns out that the standard deviation of the number left after one half-life is ##\sigma = \sqrt{N} / 2##. See for example the section "Statistical Uncertainties" in www.csupomona.edu/~pbsiegel/bio431/texnotes/chapter2.pdf‎ [Broken], in particular equation (21), ##\sigma^2 = Npq##. In this case p = q = 1/2.

Starting with 100 atoms, after one half-life you expect to have 50 ± 5, i.e. 50 ± 10% left.

Starting with 1000000 atoms, after one half-life you expect to have 500000 ± 500, i.e. 500000 ± 0.1% left.

Starting with 1020 atoms (< 0.001 mole), after one half-life you expect to have (5 x 1019) ± (5 x 109), i.e. (5 x 1019) ± 0.00000001% left.

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thegreenlaser
To put it another way, let's say you start out with 1000 atoms and the half life is 1 year. That is, there's a 50/50 chance that each atom will have decayed by the 1 year mark. After 1 year, there's still a chance that none of the atoms will have decayed, but the odds of that happening are one in 21000 or about one in 10301, which is ludicrously small. And 1000 isn't even that many atoms; in a 12 g sample of carbon you have around 6e23 atoms.

If we have a large number of atoms, then after one half-life we could find that any number of those atoms has decayed (anywhere from none to all), but probability tells us that it's extremely unlikely that the number of decayed atoms will be very far from half. Still, it's incorrect to say with 100% certainty that exactly half of the atoms will decay.