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?
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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.
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/
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
A half-life is a mathematical result if a percentage of the atoms decays in a unit of time, so the number you looses depends on the number you have.
It gets simpler to understand if you look at growth and the time it takes to double If you have a bank account where you put in 10000 today and after a day you get 10005 because of interest. You can use that information to determine how long it takes to double of the interest in constant.
10005/10000= 1.0005 which is a 0.05% interest per day. So you have 10000*1.0005 after one day 10000*1.0005*1.0005 after two. You can rewrite it as 10000*1.0005^n for n day.
Now the problem is to solve 10000*1.0005^n = 10000*2 => n≈1386.64
You simplify it and remove the 10000 part and just solve 1.0005^n = 2 This simplification shows the time to double the money is independent of the amount of money you start with.
1386.64/365 = 3.799 years which is a bit less than 3 years and 10 months.
0.05% interest per day is equal to 20% interest per year.
So by just observing the interest during the day, we can determine the time it takes for the money to double.
You can do the say with decay. You start with 10000 atoms and after one day 10 have decayed (10000-10)/10000 =0.999. Then the problem is to solve 0.999^n =1/2 =>n≈692.8 which is a bit less the 1 year and 11 months.
This calculation for decay is a bit of a simplification, it gets better if you calculate the instant rate and not look at it in the resolution of just days. The point is still shown you can calculate the half-life from the current activity. 10000 atoms is also a very small amount and real observation of most elements will have billions of atoms.
So you just need to know the number of radioactive atoms you have and the number of decays you get right not to calculate how long the half-life is
The half-life isn’t the time it takes one atom to decay. Decay is randomly happening all the time, even in relatively stable isotopes.
The half-life represents the average rate of decay. It is the time, on average, that it takes for half of the atoms in a pure sample to decay.
If you have, say, a 2kg sample of U-238, there are a *very* large number of atoms in that sample (many orders of magnitude more than a billion) and at least some of them will be decaying – we can detect the radiation from those decay events that do occur effeciently enough to calculate the time it would take for half of that sample to decay.
Hi!
The math is easy.
You don’t have to way for 50% to be gone (1/2 life)
[This page has the math](https://www.cuemath.com/half-life-formula/)
Basically for every radioactive substance there is a “rate of decay” number that makes the formula look like simple.
If you want to figure out the time for 0.0000001 percent to decay, the math is similar with just different numbers.
These elements with long half lives they just get the decay rate and then recalculate for 1/2 life.
Basically the same way I can determine roughly how many times your heart will beat in a day without measuring it all day: measure it for a short time, then multiply to scale it up to a full day.
Half-life is how long it would take for half of a sample to decay, but you can measure how long it takes for much less than half to decay and then just scale/multiply that out to how long it will take for half to be gone.
The other big thing is samples have a LOT of atoms. Even very-rare decay events are happening frequently, because there’s just so many atoms.
EXAMPLE: Uranium-238 has a half life of 4.5 billion years. But **1 pound of Uranium-238 has 273,126,530,000,000,000,000,000,000 atoms.** **There has to be millions of atoms decaying every day for half of them to be gone in “*****only*****” 4.5B years**. Much more than millions actually, so you’d use a much smaller sample than a pound of U-238. You can easily measure the decays for a few minutes or an hour or a day, and then you know the rate and can calculate how long it will take for half to be gone.
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