Tried looking it up and it got complicated real quick.
In: Physics
short answer: its *always* splitting and fissioning, but its a question how fast its happening. we can count the rate at which it splits naturally, and based on that, we can get a estimate of how long it would take for half a given sample to fission and spilt, which is 4.5 billion years.
its like being able to say “this person can walk at 4 miles an hour. ergo, he would take 100 hours of non stop walking to cover 400 miles”. we are extrapolating based on known data.
One nice model to think about radioactive decay is to imaging having a massive bucket of dice. You roll the whole bucket and then remove any dice that landed on a “1,” then collect all the dice back in the bucket and repeat.
If you were rolling a bunch of standard 6-sided dice then you’d expect 1/6 of them to roll a “1” on any given roll. After one roll you have 5/6 of the original number of dice (83%), after two you have 25/36 (69%), three rolls reduces this to 57%, and a fourth roll takes things to 48%. You could keep doing this over and over again and the number of remaining dice would get lower and lower. Eventually each die will roll a “1,” but there’s no guarantee of how long it’ll take and the more dice you start with the more likely it is that one of the dice has a super long lucky streak.
However, regardless of the number of dice you start out with it’ll always be about four rolls to halve the number of dice remaining, at least on average (statistical abnormalities are more common with smaller numbers of dice, but roll a bucket of a billion dice and the results are likely to be pretty close to average). We could say that these six-sided dice have a half life of about four rolls and we could even start describing things in terms of fractional rolls if we’re not too afraid of logarithms (the actual half life of D6 dice is about 3.80 rolls).
By comparison, we might repeat the experiment with 20-sided dice. As before, only a 1 results in the die being removed. Here we see that after one roll we still have 95% of the dice remaining. 10 rolls in and we still have about 60% sticking around, and somewhere around 13-14 rolls we finally hit the halfway point. Thus we see that a D20 has a much longer half life than a D6.
In both of these setups we arrived at a half life by looking at how many dice we returned to the bucket and kept going until the bucket was half as full as it started. However, suppose you start with a bucket with a million dice inside and you don’t know how many sides they have. You roll them and find that 10 dice turned up with a 1. You don’t have to put all the dice back into the bucket to have a good idea about how many sides these dice must have–it’s probably about 100,000, to give ten rolls of 1 out of a million attempts. As soon as you know how many sides the dice have it’s just a matter of some arithmetic to find out how many rolls it would take to come down to half the original population–about 69,000 in this case. Here it’s a lot easier to count the 10 dice we removed than to count the 999,990 that we put back in the bucket.
Turning back to radioactive half lives, much of the same logic still applies. Of course, radioactive decay doesn’t happen in discrete steps like the dice throwing game, but it does follow similar probabilistic patterns. For some fast-decaying isotopes it’s sufficient to just start with a sample and wait until some of it has decayed. You could wait for half of it to be gone, or with a more precise scale you could get away with letting just 1% decay, or less.
However, for a slowly-decaying isotope like U-238 waiting for even 1% of a sample to decay is wildly impractical, so instead of “tracking the dice in the bucket” we “count the ones that were removed.” This could take the form of putting a known quantity of the isotope into a position where we can count the decays. Even with a 4.5 billion year half life there will still be some decays–a billion atoms is nothing, and a billion billion is just starting to get into units that make sense at a macroscopic scale (that would be about half a milligram of U-238). In fact, this decay counting approach is only attractive for long lived isotopes since the decays quickly get too numerous to count for something that decays faster.
It‘s the duration until half of the sample decayed. You can measure the rate without having to wait the whole half life. It‘s like the speed in your car, you don‘t need to wait an hour to know how many miles/hour you are driving
You have a chunk of U238 and half of it will have decayed in 4.5 billion years. So 1/4 decays in 2.25 billion years 1/8 in 1 billion years 1/16 in 500 million years 1/32 in 250 million years 1/64 in 125 million… 1/(2^(50)) = ~ 1/(1.26×10^(15)) in 4×10^(-6) years = ~ 2 minutes. The chunk of U238 contains around 10^(23) atoms so 1/(2^(50)) of that is ~10^(7) number of atoms.
Since atoms are plenty in the sample with a half life of 4.5 billion years you still get (with this naive estimation) 10 million or so decays under about 2 minutes. So you can measure for some trivial amount of time and get the decay rate of the matterial. Thats how many decays per unit time tend to happen. Even though the decay rate tells you how many decays per unit time happen on average with things like this the law of large numbers work well so the fluctuations around the measured averages is negligible. Half life is what you get when either you take N atoms and ask how long do you have to wait on average to get N/2 or because N isn’t really a factor here you can say how long do you have to wait for one atom to have a 50% chance of being not decayed (or decayed its 50-50). Its just when you have N many atoms the probabilities turn to frequencies.
So with N many atoms 4.5 billion years of half life isn’t that long but sometimes matterials can have insanely long half life and in that case you need to run an experiment for a year or two to observe a dozen or so decays in a reasonable sized sample.
Simple answer: half-life is a speed measure. It could be quarter-life or one-billionth-life. Calculate one and multiply like you convert mph to kmph.
You can tell a turtle would take 10 years to circle the world without having to sit there for 10 years and watch it. You just track the progress. Half-life doesn’t mean it doesn’t do anything for a billion year and then half overnight.
The reason is basically that there are LOADS of atoms of stuff in a human-scale lump of something, and we are really good at measuring tiny amounts of stuff too. In a gram of some Uranium salt there are enough atoms (something like 2×10^(21) of them) that hundreds will decay each second despite this incredibly long half-life (I think somewhere around 1000 per second from some rough and maybe wrong calculations), and if we leave a bunch of Uranium salt alone for a few days that is quite a lot of atoms, whose presence we can pick up even at that tiny concentration. By measuring exactly what amount of new elements are produced in what time period, we can work out the decay rate of the Uranium atoms.
It’s the same question when an advert says a watch loses one second every 30 years and the watch just came out. How do they know this when the watch isn’t 30 years old!?! They don’t wait 30 years. They have extremely accurate clocks and see how much the watch varies after a few seconds or minutes.
So if the watch loses 0.00000000000579174282808486 seconds every minute, then you can say the watch loses 1 seconds every 30 years.
Similarly for Uranium. The half life is how long it takes for 50% of the uranium to mutate. So just look at how much of the uranium mutates in 1 minute (or whatever) and extrapolate and assume that it is totally stochastic process.
If you have a giant tank of water with a tap at the bottom constantly poring out water, you could calculate at what point it would be empty by measuring the flow rate
The basic answer is a kilogram of uranium contains something like 10^24 atoms. That’s a million billion billion.
It’s enough that even an individual atom only has a 50% chance of splitting over 4.5 billion years, there are enough atoms splitting over a reasonable amount of time that you can measure thr decays and use math to figure out the rest.
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