The Math of Numb3rs Bones of Contention
After Charlie examines the notebook, he realizes that the equations date a new Using carbon dating vocabulary, this says that the half life is now 1 unit of time. The same phenomenon really precludes using traditional radio-carbon dating over the last 50 years, however, prior to this it appear that the. To find the percent of Carbon 14 remaining after a given number of years, type in The halflife of carbon 14 is ± 30 years, and the method of dating lies in.
Carbon has six protons. But they have a different number of neutrons. So when you have the same element with varying number of neutrons, that's an isotope. So the carbon version, or this isotope of carbon, let's say we start with 10 grams. If they say that it's half-life is 5, years, that means that if on day one we start off with 10 grams of pure carbon, after 5, years, half of this will have turned into nitrogen, by beta decay.
And you might say, oh OK, so maybe-- let's see, let me make nitrogen magenta, right there-- so you might say, OK, maybe that half turns into nitrogen. And I've actually seen this drawn this way in some chemistry classes or physics classes, and my immediate question is how does this half know that it must turn into nitrogen?
And how does this half know that it must stay as carbon? And the answer is they don't know. And it really shouldn't be drawn this way. So let me redraw it. So this is our original block of our carbon What happens over that 5, years is that, probabilistically, some of these guys just start turning into nitrogen randomly, at random points.
So if you go back after a half-life, half of the atoms will now be nitrogen.
Decay graphs and half lives article (article) | Khan Academy
So now you have, after one half-life-- So let's ignore this. So we started with this. All 10 grams were carbon. This is after one half-life. And now we have five grams of c And we have five grams of nitrogen Let's think about what happens after another half-life.
So if we go to another half-life, if we go another half-life from there, I had five grams of carbon So let me actually copy and paste this one. This is what I started with. Now after another half-life-- you can ignore all my little, actually let me erase some of this up here. Let me clean it up a little bit. After one one half-life, what happens? Well I now am left with five grams of carbon And by the law of large numbers, half of them will have converted into nitrogen So we'll have even more conversion into nitrogen So now half of that five grams.
- How is carbon dating done?
- Half-life and carbon dating
- Carbon 14 dating 1
So now we're only left with 2. And how much nitrogen?
Well we have another two and a half went to nitrogen. So now we have seven and a half grams of nitrogen And we could keep going further into the future, and after every half-life, 5, years, we will have half of the carbon that we started.
But we'll always have an infinitesimal amount of carbon. But let me ask you a question. Let's say I'm just staring at one carbon atom. Let's say I just have this one carbon atom. You know, I've got its nucleus, with its c So it's got its six protons.
It's got its eight neutrons. It's got its six electrons. What's going to happen? What's going to happen after one second? Well, I don't know. It'll probably still be carbon, but there's some probability that after one second it will have already turned into nitrogen What's going to happen after one billion years?
Well, after one billion years I'll say, well you know, it'll probably have turned into nitrogen at that point, but I'm not sure.
This might be the one ultra-stable nucleus that just happened to, kind of, go against the odds and stay carbon So after one half-life, if you're just looking at one atom after 5, years, you don't know whether this turned into a nitrogen or not.
Now, if you look at it over a huge number of atoms. I mean, if you start approaching, you know, Avogadro's number or anything larger-- I erased that. I don't know which half, but half of them will turn into it. So you might get a question like, I start with, oh I don't know, let's say I start with 80 grams of something with, let's just call it x, and it has a half-life of two years.
I'm just making up this compound. And then let's say we go into a time machine and we look back at our sample, and let's say we only have 10 grams of our sample left. And we want to know how much time has passed by.
So 10 grams left of x. How much time, you know, x is decaying the whole time, how much time has passed? Well let's think about it.
We're starting at time, 0 with 80 grams. After two years, how much are we going to have left? We're going to have 40 grams. So t equals 2. But after two more years, how many are we going to have? We're going to have 20 grams.
Numb3rs 210: Bones of Contention
So this is t equals 3 I'm sorry, this is t equals 4 years. And then after two more years, I'll only have half of that left again. So now I'm only going to have 10 grams left.
And that's where I am. And this is t equals 6. So in the real world, looking at a sample like say a bone dug up by an archaeologist, how do we know how much carbon 14 we started with? That's actually kind of cool.
It's a semi-long story, so bear with me. In the atmosphere, cosmic rays smash into normal carbon 12 atoms in atmospheric carbon dioxideand create carbon 14 isotopes. This process is constantly occurring, and has been for a very long time, so there is a fairly constant ratio of carbon 14 atoms to carbon 12 atoms in the atmosphere. Now living plants 'breathe' CO2 indiscriminately they don't care about isotopes one way or the otherand so while they are living they have the same ratio of carbon 14 in them as the atmosphere.
Animals, including humans, consume plants a lot and animals that consume plantsand thus they also tend to have the same ratio of carbon 14 to carbon 12 atoms. This equilibrium persists in living organisms as long as they continue living, but when they die, they no longer 'breathe' or eat new 14 carbon isotopes Now it's fairly simple to determine how many total carbon atoms should be in a sample given its weight and chemical makeup.
And given the fact that the ratio of carbon 14 to carbon 12 in living organisms is approximately 1: In actually measuring these quantities, we take advantage of the fact that the rate of decay how many radioactive emissions occur per unit time is dependent on how many atoms there are in a sample this criteria leads to an exponential decay rate. We have devices to measure the radioactivity of a sample, and the ratio described above translates into a rate of