# Carbon dating nuclear equation Contents:

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

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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. 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. 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? Well, I don't know. It'll probably still be carbon, but there's some probability that after one second it will have already turned into nitrogen What's going to happen after one billion years?

## What is the nuclear equation for the decay of carbon-14?

Well, after one billion years I'll say, well you know, it'll probably have turned into nitrogen at that point, but I'm not sure. This might be the one ultra-stable nucleus that just happened to, kind of, go against the odds and stay carbon So after one half-life, if you're just looking at one atom after 5, years, you don't know whether this turned into a nitrogen or not. Now, if you look at it over a huge number of atoms. I mean, if you start approaching, you know, Avogadro's number or anything larger-- I erased that.

I don't know which half, but half of them will turn into it. So here's our electron and an electron ejected from the nucleus is called a beta particle. We could put a beta here, and it's an electron, so a negative one charge, and then a zero here. If a beta particle is ejected from the nucleus of a thorium, so we're starting with thorium, this nucleus ejects a beta particle, so we go ahead and put a beta particle in here, so zero and negative one, what else is produced here? What else do we make? Well, once again, the number of nucleons is conserved, so I have nucleons on the left, I need on the right.

I have a zero here, so I need nucleons. Charge is also conserved, so I have 90 positive charges on the left, I have 90 protons.

On the right, I have a negative charge here, so I have a negative one charge, and so I must need 91 positive charges, because 91 positive charges and one negative charge gives me 90 positive charges on the right. So I need an atomic number of If you look at the periodic table, and you find the atomic number of 91, you'll see that this is protactinium. So we're going to make protactinium here, so Pa. What is happening in beta decay?

Let's look at it in a little bit more detail. We already talked about the number of protons, so we have 90 protons on the left, how many neutrons do we have? Well, minus 90, minus 90 gives us the number of neutrons. On the right, we have 91 protons, how many neutrons do we have? Well, that'd be minus So minus 91 gives us neutrons.

So we went from neutrons on the left to neutrons on the right, and we went from 90 protons on the left, to 91 protons on the right. So we lost a neutron, and we gained a proton. You could think about the neutron turning into a proton, and this is an oversimplified way of thinking about it. Let's go ahead and write that down here. So a neutron turning into a proton. So a neutron has no charge, so we put a zero here. And a neutron is a nucleon, so we put a one right here. So a neutron is turning into a proton, so let's go ahead and write our proton here.

A proton has a plus one charge, and it's a nucleon so we put a one here. When we think about what else is made, we know that nucleons are conserved, so we have one nucleon on the left, one nucleon on the right. Therefore, we would have a zero here.

## A world of ibohyhozeq.tk standard model. Beta decay.

In terms of charge, if we have zero charge on the left, plus one on the right, we need negative one right here. This of course represents the electron, so this is the electron that's ejected from the nucleus. This is our beta particle. And also actually, something else is produced. You're also going to make an anti-neutrino, and that's just really not part of this video, so we'll just ignore it for now. So a neutron has turned into a proton, and we're also getting a beta particle ejected from the nucleus. When this conversion, this process is actually governed by the weak force, the weak interaction, so there's a lot of stuff going on in the nucleus which we just won't get into in this video.

## Writing nuclear equations for alpha, beta, and gamma decay

The important thing is to be able to look at a nuclear equation, recognize it as beta decay, and be able to write everything in your nuclear equation. Let's do one more type of decay. This is gamma decay. Gamma rays are given off, and a gamma ray has no charge and no mass; it's pretty much just energy, if you think about it. These are pretty easy decay problems. Let's start with technetiumm, and the m right here stands for metastable, which means a nucleus in its excited state, so a nucleus in its excited state, so it has more energy.

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4. It's going to give off a gamma ray, so let's go ahead and draw in our gamma ray here, so zero and zero.