Jonathan Osbourne

**PhD., University of Maryland**

Published author

Jonathan is a published author and recently completed a book on physics and applied mathematics.

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**Carbon dating** uses an unstable isotope of carbon to find the date of dead substances. This isotope Carbon-14 has a half life of 5,700 years. The ratio of Carbon-14 remaining indicates the times since the death of a living substance. Carbon-14 only works for things between 3 and 40 thousand years old.

So let's talk about carbon dating. Carbon dating is based on an isotope of carbon, carbon 14, that's unstable. It decays with a half life of 5700 years into nitrogen 14 and electron and an electron antineutreno. So this is just an ordinary beta decay process and this carbon fourteen's half life is way way way too short for any carbon to just kind of exist naturally in the atmosphere, you'd think, not quite right.

The natural abundance is 1.3 times 10 to the -12. So that mean that 1.3 times 10 to the -12 carbon 14 atoms, exist for each and every carbon 12 atom in nature. Now carbon 12 of course is stable. So you'd think that if you got this 1.3 times 10 to the -12 carbon 14 atoms for each carbon 12 atom at some time, well then 5700 years later, half of the carbon 14 will have decayed. So now we should have less. But in fact what happens is, cosmic rays from the sun interact with the upper atmosphere and they actually create carbon 14, at this rate so that in equilibrium, 1.3 times 10 to the -12 carbon 14 atoms will exist for every carbon 12 atom. So that's taking into account all the decays and all that stuff, this is a natural abundance.

Now, I'm alive. You're alive. What do we do? We breathe. We breathe in carbon dioxide, we eat carbon, we take in carbon and so our bodies continually renewing our supply of carbon 14. So for that reason, every living thing that is interacting with its environment is expected to have this natural abundance of carbon 14. 1.3 times 10 to the -12. But when something dies, now it's not interacting with the environment anymore. It's no longer replenishing its carbon 14 supply. And that means that as time goes on, the carbon 14 abundance will decrease. So that means the carbon 14 abundance can tell us how long something's been dead. Alright. So let's see how we can use this to do a problem.

So, I've got a specimen. It's bound to have a carbon 14 ratio that's only 0.5 times 10 to the -12. Not 1.3 times 10 to the -12. And I want to know how long is it been dead. Alright. Well, here's the idea. We know that the amount at time t is equal to the initial amount times one half to the time over the half life, alright? This is our standard radioactive decay formula, always works. So the amount that we've got at our time now is 0.5 times 10 to the -12. The initial amount when he died must have been 1.3 because he was interacting with its environment. And then we have one half t over 5700 years. Alright, so that means that t is going to be, I'm just going to solve this equation real quickly, it's going to be 5700 years times the natural log of 0.5 over 1.3 divided by the natural log of one half. And if you type that in your calculator you'll find that this specimen is 700, oh sorry, 7860 years dead. Alright? So that's the way that we can do these calculations. It's always the same thing and if you're having trouble in going from this step to this step, make sure you know how to do that. We just divide. We take the natural log of both sides and then we solve for t. It's not that bad. Alright.

Let's do it a different, let's do a different one. Let's say that a specimen has been dead for 10,000 years and I want to know its carbon 14 ratio. Well, we're going to use exactly the same equation. So we'll say alright, the amount at 10,000 is equal to the initial amount that I started with 1.3 times 10 to the -12 times a half to the 10,000 divided by 5700. So you just type all that in. And when you do so, you'll end up with 0.385 times 10 to the -12. Now one thing that it's important to keep in mind about carbon dating is that this is a really small number. Alright? The abundance, the natural abundance is already very small. So if something's been dead for longer than a few carbon 14 half lives, there's not enough carbon 14 left to measure it accurately enough to really say for sure how long the thing's been dead. So, if you're trying to use his to date dinosaurs, just stop. It's not going to work. Alright? They're way too old. You can usually date something that's under about 40 or 50,000 years old using this technique. If it's older, got to use other isotopes.

And that's carbon dating.