I’ve been reading about HL, but I still don’t get why would you use it. So, if half life of coffee is 5h, how that info is relevant when we know that full life is 10 (roughly)? On top of that, how do you get the half-life of a material other than waiting to be completely ‘dead’ and say, ok full life is X, the half life is X/2.
Also, let’s take uranium which in Earth’s crust has a half-life of almost 4.5 billion years.. how did we get this number?
Thank you!
In: Chemistry
We use half-lives when the decay rate is random but follows a predictable distribution. We don’t actually know when the “full-life” is, because it’s impossible to predict thanks to the process being random, but we can use the average time of decay to predict [a curve](https://ohiostate.pressbooks.pub/app/uploads/sites/205/2019/08/hypothetical-distribution-half-lives-.png) which tells us when half of the remaining substance will have decayed. And then it takes another half-life for half of that remaining half to decay, and so on.
So for Uranium-238 for example the process of decay for every single atom is random. In a given sample, some of the atoms will decay tomorrow, some won’t decay for the next 10 billion years. But if you have 1 kg of Uranium that’s 2.53×10^24 atoms, a very very large number, so even with the very small chance for any given atom to decay in any given second, you can still observe thousands and thousands of decays over any measurable period of time. Maybe millions, I don’t know, I can’t math. But you can do this experiment and observe how many decays your 1 kg experiences over a set period of time and from that extrapolate the half life
>So, if half life of coffee is 5h, how that info is relevant when we know that full life is 10 (roughly)?
It’s important because the “full life” isn’t 10h
Each 5h period halves the amount that is not yet decayed, so if t=0 is 100% then t=5h is 50% and t=10h is 25% and t=15h is 12.5% etc
It’s a curve that trends towards zero but it’s not a straight line. That’s the critical point
There is no full-life concept. Half-life is just the time it takes for half the material to decay. So if 1lb of matter has a half-life of an hour, in one hour there would only be half a pound left. Next hour there only be half of that left so a quarter pound. Your body has noticeable reactions to caffeine at and above 30mg. If the half-life of caffeine in your system is 6hours, and you took 100mg. It would take about 12hours(100mg hour0, 50mg hour6, 25mg hour12) for your body to stop being affected by it. As far as discovering half life you just monitor the average rate of decay and do the math to see how much time it takes for half of it to be gone.
If half life is 5h then full life is not 10h. It is always true that half is gone by the half life, no matter how much you start with. Say you drink 100mg of caffeine, about a cup. After 5h you would have 50mg left in your body. But this would be equivalent to drinking half a cup. So after another 5h, 10h after drinking the coffee, you would have 25mg of caffeine left in your body. After 15h you would have 12.5mg and so on. Mathematically the caffeine never have a full life as you theoretically never get to zero. It only have a half life.
You do not have to wait a full half life in order to measure the half life. You can measure a shorter time and then expect that the rate remains the same. For example if you drink 100mg of caffeine, and then we wait an hour and measure that you have 87mg of caffein left in your body. We see that 13% of your caffeine is metabolised every hour so that means that by hour 2 you would have 75mg of caffeine, by hour 3 you would have 65mg. Notice how the amount of caffeine metabolised every hour goes down, because 13% of 100mg is 13mg while 13% of 75mg is only 10mg. By hour 4 you would have 57mg and finally by hour 5 you would have 50mg, half of what you started with.
We can do the same with Uranium. Measure the exact ratio of Uranium isotopes in some ore and then leave it for a year and measure it again. You would notice a tiny decrease in the amount of radioactive Uranium isotopes. You can then expect this to continue for 4.5 billion years and then you have your half life.
>So, if half life of coffee is 5h, how that info is relevant when we know that full life is 10 (roughly)?
It is not 10 hours.
Lets start with 1 shot of coffee. That’s around 20mg of caffine in your blood.
5 hours after drinking it there is 10mg of caffine in your blood.
10 hours after there is 5mg of caffine in your blood.
15 hours after there is 2.5mg
and so on.
It’s a half life because after the amount keeps halving every time, it doesn’t go down at the same rate from 100% to 0.
To understand what’s going on, imagine simultaneously flipping 100 coins. Any coin that lands heads you remove and flip the remaining coins again. After the first flip you’ll have around 50 coins left, after the 2nd flip you’ll have 25 and so on. In this case the coins have a half-life of 1 flip.
>Also, let’s take uranium which in Earth’s crust has a half-life of almost 4.5 billion years.. how did we get this number?
For this you can measure how radioactive a sameple of a substance is, use that to calculate the rate that radioactive atoms are individually decaying, then use that to calculate how long it would take half the atoms in the sample you have to decay.
Radioactive material can decay at any point in time, it’s random when exactly it decays, but we know about how likely it is to decay. What you have to understand is that the rate of decay depends on the concentration, so the less particles you have left, the less decays you get. So you don’t have a linear curve where there’s zero particles left after 2 half lives. After 2 half lives theres 1/4 the original particles, and then 1/8 and so forth. You never actually get to 0 you only get closer and closer(at least looking at it mathematically, in real life at some point the last particle will decay at some point).
How do we get the number? There’s two ways basically, one is to measure the radioactivity over a long period of time, then you can see how it slows down over time and calculate the half live. The second is to get a precise measurement of how much material you have, and then find the radioactivity, then you can calculate the half live based on that.
Let’s say that in the normal environment, there is one part of isotope A, and one of A’. Animals and plants while alive ingest these isotopes and so have a 1:1 ratio of them in their system.
After they die, they stop ingesting anything at all. The isotopes keep decaying and as A’ decays much faster than A, the ratios change.
In the environment however, the ratio remains 1:1 due to the processes forming the isotopes.
After time we measure the ratios. If the ratio has skewed 2:1 in favour of A, then we know one half-life of A’ has passed. Different ratios indicate different time spans.
Generally speaking when isotopes are used for dating, two or more different isotopes with different half-lives are measured to be as accurate as possible.
Remember this process is exponential, so from 1kg of isotope, a half-life passes and you have 0.5kg left. Another half life and you have 0.25 kg, then 0.125 and so on. You don’t get to zero.
To measure say Uranium you take a known amount of a known isotope, you know the radiation it emits, you measure the radiation and you say ‘in time *x* it emitted *y* radiation from *z* moles of isotope’ and you have everything you need to calculate the half life. Just like you don’t need to weigh an entire mountain if you know its size and density already, a measured sample is enough.
Full life is not half life x 2. Full life is in theory infinity, in pratice the last decay happens at some discrete point but you cant predict when it happens. For coffee with a half life of 5 hours after 10 hours not 100% is gone but 75%. Not half of the original amount decay after half live but half of the current amount. So if you start out with 100 of something with a half live of 5h after 5h you have 50. After 10h you have 25. After 15h you have 12 or 13. After 20 you have 6-8 and so on.
For Uranium you just take something like a kg of it and wait for a day to detect how much decayed. You can then say it would take x billion years for half to decay.
> Also, let’s take uranium which in Earth’s crust has a half-life of almost 4.5 billion years.. how did we get this number?
The decay of each individual atom is random, but if you have enough then you get predictable patterns. You can e.g. get a sample of 10^20 = 100000000000000000000 uranium atoms. You measure that around 500 of them decay each second. You might have 520 decays in a specific second, or 490, or whatever, but the average will be 500. That means 0.00…5% of them decay each second and 99.99…5% stay. You can use that information to extrapolate how much uranium will be left at any point in the future. Half of it will be left after 4.5 billion years, a quarter will be left after 9 billion years, 1/8 after 13.5 billion years and so on.
As for how we reach a half-life we haven’t had time to literally confirm:
You can take a known amount of a substance, know roughly how many atoms are in it. I’ll make up some really dumb numbers:
10,000 thousand atoms
Then you watch and track how many decay in an hour. Do that a bunch and you have an average. Let’s say you see only 5 every hour.
That means that each atom has a 1/2000 chance of decaying every hour.
To have 5000 atoms left (half-life) you have to figure out how long on average it would take 5000 atoms with a 1/2000 chance per hour to decay, which is just algebra and then you have your answer.
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