# Carbon dating numericals

And maybe not carbon, maybe we're talking about carbon or something. How do we know that they're going to decay? And the answer is, you don't. They all have some probability of the decaying. At any given moment, for a certain type of element or a certain type of isotope of an element, there's some probability that one of them will decay. That, you know, maybe this guy will decay this second.

And then nothing happens for a long time, a long time, and all of a sudden two more guys decay.

### Introduction

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. I mean, maybe if we really got in detail on the configurations of the nucleus, maybe we could get a little bit better in terms of our probabilities, but we don't know what's going on inside of the nucleus, so all we can do is ascribe some probabilities to something reacting.

Now you could say, OK, what's the probability of any given molecule reacting in one second? Or you could define it that way.

- 5.7: Calculating Half-Life;
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But we're used to dealing with things on the macro level, on dealing with, you know, huge amounts of atoms. So what we do is we come up with terms that help us get our head around this. And one of those terms is the term half-life. And let me erase this stuff down here. So I have a description, and we're going to hopefully get an intuition of what half-life means. So I wrote a decay reaction right here, where you have carbon It decays into nitrogen And we could just do a little bit of review.

You go from six protons to seven protons. Your mass changes the same. So one of the neutrons must have turned into a proton and that is what happened. And it does that by releasing an electron, which is also call a beta particle. We could have written this as minus 1 charge. It does have some mass, but they write zero. This is kind of notation.

So this is beta decay. Beta decay, this is just a review. But the way we think about half-life is, people have studied carbon and they said, look, if I start off with 10 grams-- if I have just a block of carbon that's 10 grams. If I wait carbon's half-life-- this is a specific isotope of carbon.

## Calculating Half-Life - Chemistry LibreTexts

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. And the atomic number defines the carbon, because it has six protons. Carbon has six protons. But they have a different number of neutrons.

So when you have the same element with varying number of neutrons, that's an isotope. So the carbon version, or this isotope of carbon, let's say we start with 10 grams. If they say that it's half-life is 5, years, that means that if on day one we start off with 10 grams of pure carbon, after 5, years, half of this will have turned into nitrogen, by beta decay.

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 I've actually seen this drawn this way in some chemistry classes or physics classes, and my immediate question is how does this half know that it must turn into nitrogen? And how does this half know that it must stay as carbon? And the answer is they don't know. And it really shouldn't be drawn this way.

So let me redraw it. So this is our original block of our carbon What happens over that 5, years is that, probabilistically, some of these guys just start turning into nitrogen randomly, at random points. So if you go back after a half-life, half of the atoms will now be nitrogen. So now you have, after one half-life-- So let's ignore this. So we started with this. All 10 grams were carbon.

This is after one half-life. And now we have five grams of c And we have five grams of nitrogen Let's think about what happens after another half-life. So if we go to another half-life, if we go another half-life from there, I had five grams of carbon So let me actually copy and paste this one. This is what I started with. Now after another half-life-- you can ignore all my little, actually let me erase some of this up here.

Let me clean it up a little bit. After one one half-life, what happens? Well I now am left with five grams of carbon And by the law of large numbers, half of them will have converted into nitrogen So we'll have even more conversion into nitrogen So now half of that five grams.

So now we're only left with 2.

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And how much nitrogen? Well we have another two and a half went to nitrogen. So now we have seven and a half grams of nitrogen And we could keep going further into the future, and after every half-life, 5, years, we will have half of the carbon that we started. But we'll always have an infinitesimal amount of carbon. But let me ask you a question.

Let's say I'm just staring at one carbon atom. Let's say I just have this one carbon atom. You know, I've got its nucleus, with its c So it's got its six protons.

It's got its eight neutrons. It's got its six electrons. What's going to happen? What's going to happen after one second? Many problems of radiometric dating can tell how precise is why radiocarbon dating and vexing. Also, years old it. Opening remarks, cloth, defending the mt. How precise is why radiocarbon the first chronometric technique used in the sequence of this section we will explore the usual manner.

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