Carbon is a radioactive isotope of carbon, containing 6 protons and 8 . The time period calculated in this example is called the half-life of carbon A quick way to calculate Half-Life is to use the expression: t12 = λ. Where λ is the decay constant and has the value of. x 10−4yr−1. You can calculate half life if you know how much of the substance is left after a certain time, though typically it works the other way - the half life.

### Calculating Half-Life - Chemistry LibreTexts

This is not a tremendous amount. 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. 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.

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. 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.

### Content - Radioactive decay and 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.

### Half-life and carbon dating (video) | Nuclei | Khan Academy

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.

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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.

**Ex: Exponential Model - Determine Age Using Carbon-14 Given Half Life**

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?

## Radioactive Half-Life Formula

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. Analysis of this ratio allows archaeologists to estimate the age of organisms that were alive many thousands of years ago.

Along with stable carbon, radioactive carbon is taken in by plants and animals, and remains at a constant level within them while they are alive. After death, the C decays and the C C ratio in the remains decreases.

Comparing this ratio to the C C ratio in living organisms allows us to determine how long ago the organism lived and died. Image used with permission CC-BY 4. C dating does have limitations. For example, a sample can be C dating if it is approximately to 50, years old. Before or after this range, there is too little of the isotope to be detected.

Substances must have obtained C from the atmosphere.

For this reason, aquatic samples cannot be effectively C dated. Lastly, accuracy of C dating has been affected by atmosphere nuclear weapons testing. Fission bombs ignite to produce more C artificially. Samples tested during and after this period must be checked against another method of dating isotopic or tree rings. To calculate the age of a substance using isotopic dating, use the equation below: Ra has a half-life of years. Radioactive Dating Using Nuclides Other than Carbon Radioactive dating can also use other radioactive nuclides with longer half-lives to date older events.

For example, uranium which decays in a series of steps into lead can be used for establishing the age of rocks and the approximate age of the oldest rocks on earth.

Since U has a half-life of 4. In a sample of rock that does not contain appreciable amounts of Pb, the most abundant isotope of lead, we can assume that lead was not present when the rock was formed. Therefore, by measuring and analyzing the ratio of U Pb, we can determine the age of the rock.

This assumes that all of the lead present came from the decay of uranium If there is additional lead present, which is indicated by the presence of other lead isotopes in the sample, it is necessary to make an adjustment. Potassium-argon dating uses a similar method. K decays by positron emission and electron capture to form Ar with a half-life of 1.

If a rock sample is crushed and the amount of Ar gas that escapes is measured, determination of the Ar