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
Say you have 100 coins in front of you on a table. You flip all of them and discard every coin that comes up tails, keeping the heads. Then you take the remaining coins and start over, repeating the same procedure for another round, and then another, and so on.
After one round, you will have *approximately* 50 coins left. Not exactly, because it’s random, but chances are it will be close to 50, since every coin has a 1-in-2 probability of landing on tails. Similarly, if you start the second round with 50 coins, you’ll have roughly 25 coins left at the end of it, and so on.
This means the half-life of your coins is a single round of this silly game. In every round, you lose (on average) one half (50%) of your coins. However, **this does not mean that you lose all your coins in two rounds**. After all, we already said that after round 2, you’ll likely still have around 25 coins left. And it will take several more rounds to lose them all. After round 3, you’ll have about 12.5 of them (that is, if you played this game many many times, and every time you noted down how many coins you had left after round 3, it would average out to 12.5). After round 4, you have about 6.25, then 3.125 after round 5, and so on. On average, you can expect to lose your last coin in round 8, most of the time that you play this game (but sometimes you lose it earlier and other times later).
So you see that the “full life” of your initial 100 coins is a lot longer than twice the half life. Not only that, but if you started with more coins, then it would take longer still. With an initial stack of 1000 coins, you need another 3 rounds or so before you lose your last coin (on average). If you start with 10,000 coins, you’re up to 14 rounds. Even though all this time, the half life of your coins is still one single round.
Half-lives are useful to describe scenarios like this, where every element in a population has some chance of decaying, or vanishing, or otherwise being removed from the population, over a given interval. This is very different to things that decay, or diminish, or shrink (or whatever) at a fixed rate. If it takes me 10 minutes to empty half of my pool, then it probably takes me another 10 to empty the other half. If it takes me 3 days to finish a loaf of bread, it takes me 1.5 days to finish half a loaf. For those situations, it makes no sense to talk about a “half-life”, and that’s why we don’t do that – we just talk about whatever the (average) rate is at which something gets used up.
As for how you measure half-life, you don’t have to wait for all of your initial atoms to decay, or even for half of them to do so. For many elements, that could take millions of years. Instead, you can use math to work it out more quickly. Suppose, for instance, that in our game we used biased coins that came up heads 84% of the time. I could find out that probability pretty accurately just by observing one round of the game, if we started with enough coins (e.g. start with 10,000, have 8423 left after round 1, then my estimate of the probability is 84.23%). From that probability, I can extrapolate that it would take about 4 rounds (0.84^4 ≈ 0.5) to lose half the coins, and so that tells me the half-life. Similarly, starting with a sample of many atoms, you just need to measure for long enough to get a precise idea of the decay probability over a given time interval, and you can then extrapolate that to find the half-life. Half-life is just a convenient, standard way to express the probability that atoms will decay over a given time span, and you can convert between the two as needed.
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