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And we know that there's a generalized way to describe that.

And we go into more depth and kind of prove it in other Khan Academy videos.

In order to do this for the example of potassium-40, we know that when time is 1.25 billion years, that the amount we have left is half of our initial amount. So let's say we start with N0, whatever that might be. We know, after that long, that half of the sample will be left. Whatever we started with, we're going to have half left after 1.25 billion years. And then to solve for k, we can take the natural log of both sides.

It might be 1 gram, kilogram, 5 grams-- whatever it might be-- whatever we start with, we take e to the negative k times 1.25 billion years. So you get the natural log of 1/2-- we don't have that N0 there anymore-- is equal to the natural log of this thing.

But what's interesting is as soon as you die and you're not ingesting anymore plants, or breathing from the atmosphere if you are a plant, or fixing from the atmosphere. Once a plant dies, it's no longer taking in carbon dioxide from the atmosphere and turning it into new tissue. And this carbon-14 does this decay at a specific rate. And you say, hey, that bone has one half the carbon-14 of all the living things that you see right now.

And then you can use that rate to actually determine how long ago that thing must've died. It would be a pretty reasonable estimate to say, well, that thing must be 5,730 years old.

And it'll get a little bit mathy, usually involving a little bit of algebra or a little bit of exponential decay, but to really show you how you can actually figure out the age of some volcanic rock using this technique, using a little bit of mathematics.

It is one of the simplest examples of a differential equation.

It's just a little section of the surface of the Earth. And that carbon-14 that you did have at you're death is going to decay via beta decay-- and we learned about this-- back into nitrogen-14. So it'll decay back into nitrogen-14, and in beta decay you emit an electron and an electron anti-neutrino. But essentially what you have happening here is you have one of the neutrons is turning into a proton and emitting this stuff in the process. So I just said while you're living you have kind of straight-up carbon-14. What it's essentially saying is any given carbon-14 atom has a 50% chance of decaying into nitrogen-14 in 5,730 years.

And it has seven protons, and it also has seven neutrons. So the different versions of a given element, those are each called isotopes. So anyway, we have our atmosphere, and then coming from our sun, we have what's commonly called cosmic rays, but they're actually not rays. You can view them as just single protons, which is the same thing as a hydrogen nucleus. But every now and then one of those neutrons will bump into one of the nitrogen-14's in just the right way so that it bumps off one of the protons in the nitrogen and essentially replaces that proton with itself. But this number 14 doesn't go down to 13 because it replaces it with itself. And now since it only has six protons, this is no longer nitrogen, by definition. And that proton that was bumped off just kind of gets emitted. But this process-- and once again, it's not a typical process, but it happens every now and then-- this is how carbon-14 forms. You can essentially view it as a nitrogen-14 where one of the protons is replaced with a neutron. It makes its way into oceans-- it's already in the air, but it completely mixes through the whole atmosphere-- and the air. And plants are really just made out of that fixed carbon, that carbon that was taken in gaseous form and put into, I guess you could say, into kind of a solid form, put it into a living form. It gets put into plants, and then it gets put into the things that eat the plants. Well, the interesting thing is the only time you can take in this carbon-14 is while you're alive, while you're eating new things.

What I want to do in this video is kind of introduce you to the idea of, one, how carbon-14 comes about, and how it gets into all living things. They can also be alpha particles, which is the same thing as a helium nucleus. And they're going to come in, and they're going to bump into things in our atmosphere, and they're actually going to form neutrons. And we'll show a neutron with a lowercase n, and a 1 for its mass number. And what's interesting about this is this is constantly being formed in our atmosphere, not in huge quantities, but in reasonable quantities. Because as soon as you die and you get buried under the ground, there's no way for the carbon-14 to become part of your tissue anymore because you're not eating anything with new carbon-14.

And then either later in this video or in future videos we'll talk about how it's actually used to date things, how we use it actually figure out that that bone is 12,000 years old, or that person died 18,000 years ago, whatever it might be. So let me just draw the surface of the Earth like that. So then you have the Earth's atmosphere right over here. And 78%, the most abundant element in our atmosphere is nitrogen. And we don't write anything, because it has no protons down here. And what's interesting here is once you die, you're not going to get any new carbon-14. You can't just say all the carbon-14's on the left are going to decay and all the carbon-14's on the right aren't going to decay in that 5,730 years.