How do we determine half life for elements that have half life in billions of years

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It’s been little over 100 years since we discovered radioactivity. How then do we know the half life of elements that have half lives in hundreds plus years?

In: 21

It’s easily calculated if we know how many atoms we have and can count how many of those disintegrate every second.

In some cases it’s not that easy. Bismuth-209 was found th have the longest half life ever discovered, and it took a detector cooled to near absolute zero to count out 128 alpha particles over 5 days: https://physicsworld.com/a/bismuth-breaks-half-life-record-for-alpha-decay/

It’s a prediction, which we made with math.

Essentially, all the ones short enough for us to observe follow the same formula, and there’s no reason to think there’s a fundamental difference, so we apply that formula to the other atoms too.

And then from what observations we do have, like comparing carbon dating (which uses the formula) to other methods of measuring the age of rocks, it all lines up.

Scale.

Sure Uranium 238 has a half life of 4.5 billion years, but you’re not staring at one atom waiting for it to decay – that could take an eternity.

Instead you slap a brick of U-238 on the table, and it contains 10^24 atoms.

Any one atom may have a vanishingly small 0.00000000001% chance decay today, but you have 1000000000000000000000000 atoms.

So you’ll record *millions* of decay events today, and can use that value to calculate the half life.

Remember what *half*-life means. It’s the average time for *half* a sample to decay. We use half-life because the decay is exponential, so we can’t meaningfully measure the “full life.” After X amount of time, half the sample will have decayed, and after another X time, half of what was left will have decayed, etc.

So now that we’ve refreshed our memories on half-life:

It takes hundreds of years for *half* the sample to decay. But *some* of the sample will decay in a few years. So maybe we can’t directly measure the half-life, but we can measure the 0.01%-life. Then we can use that to calculate the half-life.

By measuring how quickly it decays

You don’t need to wait for it to get to half to know what the half life is, once you have a few data points you can do the calculation

If you’ve got a hunk of an element and a special sensor set up to capture and count each time it decays(measure the gamma/beta/alpha particles) then you just need a couple data points far enough apart to get a reliable answer

If our chunk creates 1 trillion decay events in the first hour we’re measuring it, then the next day at the same time its down to 999,999,998 decay events, then the next day we’re down to 999,999,996 decay events, you can run the numbers and determine it has a half life of around 950,000 years. More measurements and more time will let you get a more precise answer, but in 48 hours you determined an answer of 950,000 years