## Half Life Accuracy?

I was debating a friend and we started running numbers for the statistical accuracy of half-lives. I am a numbers guy(Accountant/Statistical Analysis), not a chemist. He is a scientist and his wife is a chemical engineer. Here is my question:

If we have been studying half lives of radioactive particles for less than 200 years, how do we know exactly what they are? We came up with 1.1764e-7 for the percentage of time we have studied the half life of Potassium. That is assuming 200 years of study/1.7billion years for the half-life. In the statistical world, that is not enough to even discuss, however I understand that we have to use what we have. These where just quick numbers, so please correct errors. What are we missing? Why are they so sure these half-lifes are correct? Thanks.

ps- Yes, I googled "half life accuracy, radiometric dating" and used the search function :)
 Admin You don't measure half life by waiting, but by measuring activity of the sample. Knowing how many atoms fissioned in a given period of time you can calculate half life - and the accuracy of the result of calculation depends on the accuracy of the activity measurement.
 If I have a small bowl of potassium and a large bowl, wouldn't they both lose half of their respective potassium in the same amount of time to decay? I was under the impression it is a set rate, thus we would need to be able to watch a specific amount of this decay occur to come up with a rate. Of course, assuming there was a known amount of argon in the first place. Keep the explanation coming please. I want to have a firm grasp on this.

## Half Life Accuracy?

Sounds like you don't know what it means to measure activity.

Decay is described by the formula:

$$N(t)=N_0e^{-\frac t \tau}$$

where N0 is the initial amount of a substance (at t=0) and τ is the half life. Imagine you were observing a sample of known mass (thus known N0) for time t. During this time N0-N(t) atoms decayed. We can count these decays using some kind of radiation measuring device (like Geiger counter) - that means we measured sample activity. We know initial number of atoms (N0), we know number of decays (N0-N(t)), we know how long we observed the sample (t). Plug everything into the equation - there is only one unknown, half time, which can be easily solved for. We can measure all three parameters (mass, time, activity) with a very high accuracy and precision, so the error of the half life determination is usually pretty small.

Sure, the more active the sample, the better (unless it fissions too fast, which is a different problem). But you don't have to wait for billions of years.

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hi mram10!

if the half-life is H, then after time 2H only 1/4 is left, after 4H, 1/16 is left and so on

we can also go the other way … after time H/2, 1/√2 is left, after time H/4, 1/√√2 is left, and so on

so we don't need to wait for the half-life, we can wait for any time, and then do a calculation

the number of decays of a piece of potassium can be counted accurately enough to find the half-life to 4 or 5 significant figures

 Quote by mram10 If I have a small bowl of potassium and a large bowl, wouldn't they both lose half of their respective potassium in the same amount of time to decay? I was under the impression it is a set rate, thus we would need to be able to watch a specific amount of this decay occur to come up with a rate. Of course, assuming there was a known amount of argon in the first place. Keep the explanation coming please. I want to have a firm grasp on this.
 Makes more sense. After a little more reading, it is clear that decay is an average based on the law of large numbers, not a constant, meaning each atom decays at a different rate. As for the T in the equation, it seems like 10 years of observing and testing a known quantity would still render a large margin of error when we are using millions and billions of years. I will continue to study this, but please keep the basic analogies coming to aid in my understanding. You guys have been very helpful thus far.

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 Quote by mram10 Makes more sense. After a little more reading, it is clear that decay is an average based on the law of large numbers, not a constant, meaning each atom decays at a different rate. As for the T in the equation, it seems like 10 years of observing and testing a known quantity would still render a large margin of error when we are using millions and billions of years. I will continue to study this, but please keep the basic analogies coming to aid in my understanding. You guys have been very helpful thus far.
We would not agree that "each atom decays at a different rate". Each atom decays at a different time, maybe. You cannot have a rate of something that happens instantaneously only once. You can have a time in which it has half a chance of happening.

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 Quote by Borek You don't measure half life by waiting, but by measuring activity of the sample. Knowing how many atoms fissioned in a given period of time you can calculate half life - and the accuracy of the result of calculation depends on the accuracy of the activity measurement.
...which makes accuracy a function of the size of the sample and decay rate. Fortunately, atoms are small so you can watch a lot at once. Ie, if the half life is one year and you watch 100 atoms, you'll only see 50 decay events. The key to accuracy is seeing enough decay events that the rate becomes regular.

This is also the same statistical problem that exists with neutrino detection.

 Quote by russ_watters ...which makes accuracy a function of the size of the sample and decay rate. Fortunately, atoms are small so you can watch a lot at once.
Exactly. Question was about potassium, which makes dealing with large samples pretty easy. 40 g sample of potassium contains about 6x1019 atoms of K-40.

For some other isotopes it won't be that easy.
 Good discussion guys. How on earth can they watch a sample size that large?? Also, if a small amount like that has that many atoms to watch, does that make an even better case for lack of statistical sample size?? Just spit balling here, but every time I talk with my friend, we come up with more questions. I guess that is the nature of science though.

 Quote by mram10 How on earth can they watch a sample size that large??
Where do you see a large sample? 40 g of metallic potassium is a two inch diameter ball.

 Also, if a small amount like that has that many atoms to watch, does that make an even better case for lack of statistical sample size??
No idea what you mean. You don't watch each atom separately. You put sensors around the sample and count emitted beta particles (it is more complicated, but let's not muddy water now). And it is not a problem to weight the sample with 10-6 accuracy, so that's how precisely you know number of atoms.
 Borek, Thank you. for taking the time again to explain this. "Large sample" was referring to the number of atoms associated with 40g of potassium. By your response, it is not considered to be a large number of atoms to keep track of. I would love to hear how they measure the beta particles. I have been watching online courses to try and understand chemistry better. If you don't want to type that much, please give me a good link that explains it easily :) Thanks.
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 Thanks for the link Russ. Another "dumb" question: Have we been able to observe a chunk of new radioactive material(short half life) from inception for a significant amount of time to verify that the rate is the same from inception till decay is complete?
 Admin Many of the isotopes (including those used routinely in medicine) have half lives measured in hours or days, and they were researched thoroughly. The earliest samples of artificial isotopes (if they still exist) are zillions of half times old.
 Great info. What kind of this would effect the half lives of isotopes? Would heat increase or decrease the half life? Would atmospheric pressure? Water? Etc.
 Admin Very high temperatures could change the half life, but we are talking about tens if not hundreds of thousands Kelvins minimum. As far as I know no other factors can change half life - with one exception. Electron capture (which is sometimes treated as a variant of a beta decay) depends on the electron density around nucleus, so speed of decay can be a function of what is the compound in which the isotope is present. But it is not very common. Edit: thousands added.