So if you want to think about the total number of potassiums that have decayed since this was kind of stuck in the lava. And we learned that anything that was there before, any argon that was there before would have been able to get out of the liquid lava before it froze or before it hardened. So maybe I could say k initial-- the potassium initial-- is going to be equal to the amount of potassium 40 we have today-- 1 milligram-- plus the amount of potassium we needed to get this amount of argon We have this amount of argon 0.

The rest of it turned into calcium And this isn't the exact number, but it'll get the general idea. And so our initial-- which is really this thing right over here. I could call this N0. This is going to be equal to-- and I won't do any of the math-- so we have 1 milligram we have left is equal to 1 milligram-- which is what we found-- plus 0. And then, all of that times e to the negative kt. And what you see here is, when we want to solve for t-- assuming we know k, and we do know k now-- that really, the absolute amount doesn't matter. What actually matters is the ratio. Because if we're solving for t, you want to divide both sides of this equation by this quantity right over here.

So you get this side-- the left-hand side-- divide both sides. You get 1 milligram over this quantity-- I'll write it in blue-- over this quantity is going to be 1 plus-- I'm just going to assume, actually, that the units here are milligrams. So you get 1 over this quantity, which is 1 plus 0. That is equal to e to the negative kt. And then, if you want to solve for t, you want to take the natural log of both sides. This is equal right over here. You want to take the natural log of both sides.

So you get the natural log of 1 over 1 plus 0.

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And then, to solve for t, you divide both sides by negative k. So I'll write it over here. And you can see, this a little bit cumbersome mathematically, but we're getting to the answer. So we got the natural log of 1 over 1 plus 0. Well, what is negative k? We're just dividing both sides of this equation by negative k. Negative k is the negative of this over the negative natural log of 2 over 1.

And now, we can get our calculator out and just solve for what this time is. And it's going to be in years because that's how we figured out this constant. So let's get my handy TI First, I'll do this part. So this is 1 divided by 1 plus 0. So that's this part right over here. That gives us that number. And then, we want to take the natural log of that.

So let's take the natural log of our previous answer. So it's the natural log of 0. It gives us negative 0. I'll draw that in yellow. So then you have the Earth's atmosphere right over here.

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Let me write that down, atmosphere. And I'll write nitrogen. Its symbol is just N. And it has seven protons, and it also has seven neutrons. So it has an atomic mass of roughly Then this is the most typical isotope of nitrogen.

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And we talk about the word isotope in the chemistry playlist. An isotope, the protons define what element it is. But this number up here can change depending on the number of neutrons you have. So the different versions of a given element, those are each called isotopes. I just view in my head as versions of an element. 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.

They can also be alpha particles, which is the same thing as a helium nucleus.

And there's even a few electrons. 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. So 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 we don't write anything, because it has no protons down here. Like we had for nitrogen, we had seven protons. So it's not really an element. It is a subatomic particle. But you have these neutrons form.

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And every now and then-- and let's just be clear-- this isn't like a typical reaction. But every now and then one of those neutrons will bump into one of the nitrogen'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. So let me make it clear. So it bumps off one of the protons. So instead of seven protons we now have six protons. But this number 14 doesn't go down to 13 because it replaces it with itself. So this still stays at And now since it only has six protons, this is no longer nitrogen, by definition.

This is now carbon. And that proton that was bumped off just kind of gets emitted. So then let me just do that in another color.

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And a proton that's just flying around, you could call that hydrogen 1. Ar dating problems with embryonic stem cell research by.

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