is a principal hub of the Libre Texts project, which is a multi-institutional collaborative venture to develop the next generation of open-access texts to improve postsecondary education at all levels of higher learning.The Libre Texts approach is highly collaborative where an Open Access textbook environment is under constant revision by students, faculty, and outside experts to supplant conventional paper-based books. So you might get a question like, I start with, oh I don't know, let's say I start with 80 grams of something with, let's just call it x, and it has a half-life of two years. So what we do is we come up with terms that help us get our head around this. So I wrote a decay reaction right here, where you have carbon-14. So now you have, after one half-life-- So let's ignore this. I don't know which half, but half of them will turn into it. And then let's say we go into a time machine and we look back at our sample, and let's say we only have 10 grams of our sample left. The carbon-14 isotope would vanish from Earth's atmosphere in less than a million years were it not for the constant influx of cosmic rays interacting with molecules of nitrogen (N) into organic compounds during photosynthesis, the resulting fraction of the isotope 14C in the plant tissue will match the fraction of the isotope in the atmosphere.After plants die or are consumed by other organisms, the incorporation of all carbon isotopes, including 14C, stops.The stable form of carbon is carbon 12 and the radioactive isotope carbon 14 decays over time into nitrogen 14 and other particles.

When an organism dies it ceases to replenish carbon in its tissues and the decay of carbon 14 to nitrogen 14 changes the ratio of carbon 12 to carbon 14.

So with that said, let's go back to the question of how do we know if one of these guys are going to decay in some way. That, you know, maybe this guy will decay this second. Remember, isotopes, if there's carbon, can come in 12, with an atomic mass number of 12, or with 14, or I mean, there's different isotopes of different elements. So the carbon-14 version, or this isotope of carbon, let's say we start with 10 grams. Well we said that during a half-life, 5,740 years in the case of carbon-14-- all different elements have a different half-life, if they're radioactive-- over 5,740 years there's a 50%-- and if I just look at any one atom-- there's a 50% chance it'll decay. Now after another half-life-- you can ignore all my little, actually let me erase some of this up here. So we'll have even more conversion into nitrogen-14. So now we're only left with 2.5 grams of c-14. Well we have another two and a half went to nitrogen. So after one half-life, if you're just looking at one atom after 5,740 years, you don't know whether this turned into a nitrogen or not.

Radiocarbon dating (usually referred to simply as carbon-14 dating) is a radiometric dating method.

And so, like everything in chemistry, and a lot of what we're starting to deal with in physics and quantum mechanics, everything is probabilistic. So one of the neutrons must have turned into a proton and that is what happened. And you might say, oh OK, so maybe-- let's see, let me make nitrogen magenta, right there-- so you might say, OK, maybe that half turns into nitrogen. And over 5,740 years, you determine that there's a 50% chance that any one of these carbon atoms will turn into a nitrogen atom. And we could keep going further into the future, and after every half-life, 5,740 years, we will have half of the carbon that we started. Now, if you look at it over a huge number of atoms. But after two more years, how many are we going to have? So this is t equals 3 I'm sorry, this is t equals 4 years.

And maybe not carbon-12, maybe we're talking about carbon-14 or something. And then nothing happens for a long time, a long time, and all of a sudden two more guys decay. And the atomic number defines the carbon, because it has six protons. If they say that it's half-life is 5,740 years, that means that if on day one we start off with 10 grams of pure carbon-14, after 5,740 years, half of this will have turned into nitrogen-14, by beta decay. What happens over that 5,740 years is that, probabilistically, some of these guys just start turning into nitrogen randomly, at random points. So if we go to another half-life, if we go another half-life from there, I had five grams of carbon-14. So now we have seven and a half grams of nitrogen-14. This exact atom, you just know that it had a 50% chance of turning into a nitrogen.