Carbon dating algebra
And I want to know, how much do I have after, I don't know, after years? How much do I have?
Well I just plug into the formula. N of is equal to the amount that I started off with, grams, times e to the minus 1. So what is that? So I already have that 1. So let me say, times equals-- and of course, this throws a negative out there, so let me put the negative number out there. So there's a negative.
And I have to raise e to this power.
Exponentials and Logarithms
So this is equal to N of The amount of the substance I can expect after years is equal to times e to the minus 0. And let's see, my calculator doesn't have an e to the power, so Let me just take e. I need to get a better calculator. I should get my scientific calculator back. But e is, let's say 2. So this is equal to grams. So just like that, using this exponential decay formula, I was able to figure out how much of the carbon I have after kind of an unusual period of time, a non-half-life period of time.
Let's do another one like this. Let's go the other way around. Let's say, I'm trying to figure out. Let's say I start off with grams of c And I want to know how long-- so I want to know a certain amount of time-- does it take for me to get to grams of c? So, you just say that grams is how much I'm ending up with.
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It's equal to the amount that I started off with, grams, times e to the minus k. And now we solve for time. How do we do that? Well we could divide both sides by What's divided by ? So you get 0. You take the natural log of both sides. You get the natural log of 0.
And so t is equal to this divided by 1. So the natural log, 0. So let me do the math. So if you have 0. This is all a negative number. Oh, I'll just divide it by this, and then just take the negative of that. Equals that and then I have to take a negative. So this is equal to 1, years to get from to grams of my substance. This might seem a little complicated, but if there's one thing you just have to do, is you just have to remember this formula.
And if you want to know where it came from, watch the previous video. For any particular element you solve for this k value. And then you just substitute what you know, and then solve for what you don't know.
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. 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.
Introduction to exponential decay
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. And a proton that's just flying around, you could call that hydrogen 1. And it can gain an electron some ways. If it doesn't gain an electron, it's just a hydrogen ion, a positive ion, either way, or a hydrogen nucleus.
But this process-- and once again, it's not a typical process, but it happens every now and then-- this is how carbon forms. So this right here is carbon You can essentially view it as a nitrogen where one of the protons is replaced with a neutron.
And what's interesting about this is this is constantly being formed in our atmosphere, not in huge quantities, but in reasonable quantities. So let me write this down. And let me be very clear. Let's look at the periodic table over here.
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So carbon by definition has six protons, but the typical isotope, the most common isotope of carbon is carbon So carbon is the most common. So most of the carbon in your body is carbon But what's interesting is that a small fraction of carbon forms, and then this carbon can then also combine with oxygen to form carbon dioxide.
And then that carbon dioxide gets absorbed into the rest of the atmosphere, into our oceans. It can be fixed by plants. When people talk about carbon fixation, they're really talking about using mainly light energy from the sun to take gaseous carbon and turn it into actual kind of organic tissue.
Carbon 14 Dating - Math Central
And so this carbon, it's constantly being formed. It makes its way into oceans-- it's already in the air, but it completely mixes through the whole atmosphere-- and the air. And then it makes its way into plants.