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But we know that the amount as a function of time-- so if we say N is the amount of a radioactive sample we have at some time-- we know that's equal to the initial amount we have.
In the last video, we give a bit of an overview of potassium-argon dating.
In this video, I want to go through a concrete example.
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.
So we got the natural log of 1 over 1 plus 0.01 over 0.11 over negative k. We're just dividing both sides of this equation by negative k. So let's take the natural log of our previous answer. If you saw a sample that had this ratio of argon-40 to potassium-40, you would actually be able to do that high school mathematics.