Radioactive material gets a bad rap, what with radiation and fallout and nuclear waste and all. But it offers some practical uses. One of the coolest (OK, maybe the coolest) is using radioactive carbon to determine the age of old bones or plants. To understand this, you must first understand radioactivity and decay.
When an element undergoes radioactive decay, it creates radiation and turns into some other element. Of course, the best way to understand something is to model it, because the last thing you want to do at home is experiment with something radioactive. Here are two ways to model radioactive decay.
Before doing any modeling, you must first understand one key idea: Each atom in a sample of material has an essentially random chance to decay. The rate of decay depends upon the number of atoms you have. This means that as more of these atoms decay you have a lower rate of radioactive decay. I know can be hard to wrap your head around, so let's model it with a six-sided die.
Start with 100 objects. You can use Lego bricks, pennies, beans---anything you can easily count. Then find a six-sided die. You will roll it for each of the 100 objects. If you roll a one, then that object decays and turns into something else. Every time you roll a one, put that object into a separate pile. Count the remaining objects and repeat the process until half of them have decayed. This is called the half-life---the amount of time required for one-half of a given number of atoms to disintegrate.
My son and I used colorful plastic tiles. We only had 80 though.
It took a while, but we finally got pretty close to 40 tiles left. The plot of the number of tiles as a function of the number of turns looks like this:
Notice that in the first run, 11 tiles decayed. The last run started with 49 tiles and only six decayed. So you can indeed see that the number of objects that decay depends upon the number of objects you have. But it's hard to see this with so few tiles. What if I start with 1,000 tiles? It would be a huge hassle to roll a die 1,000 times. Instead, let's write a computer program.
You can easily write a Python program to simulate rolling a die 1,000 times. Here's the basic outline of the code:
- Create a bunch of some thing, in this case, yellow spheres. These represent atoms. It doesn't actually matter that I used spheres. They're just easy to draw in Python.
- Generate a random number for each sphere. This simulates rolling a die.
- If that random number is less than some value (pick one), the sphere decays.
- Count the number of decayed spheres and do it all over again.
Here is the program. Press play to run it and click the pencil to edit or review the code. Notice the visual display of the spheres and a graph below that.
I thought about making the yellow spheres turn another color (to more accurately represent radioactive decay), but making them vanish more closely mimics the die-rolling exercise. Now for some homework. You might need to modify the code to find the answers, but don't worry. You can't break it. If you mess it up beyond repair, just reload the page and start over.
- Moving your mouse over the graph reveals the time and atom (sphere) count. No, this isn't a question, but you'll need this below.
- How many atoms decay in the first round (from t=0 to t=1)? How many atoms decay from t=20 to t=21?
- This first calculation goes through 25 runs. Based on the graph, at what time should all the atoms decay?
- Start with 2,000 atoms. At what time do only 1,000 atoms remain? This is the half-life.
- Start with 4,000 atoms (you can do this by changing line 8 in the code to 4000). What is the half-life?
- Suppose you want the atom to decay with a die roll of one or two. This means that the decay rate is 2/6 instead of 1/6. What would happen to the half-life?
If you want one more homework question, I have one. You can derive this if you like, but here is the mathematical model for the decay of some atoms.
In this expression, N0 represents the starting number of atoms, r is the probability that something decays (per second) and t is the time (in seconds). Do the data above agree with this mathematical model?
I consider carbon dating one of the coolest applications of radioactive decay. You probably know about it from paleontology. Suppose you find some old bones. Of course the first question you might have about these bones is how old they are. You can determine that with carbon dating. OK, technically carbon dating doesn't tell you the age of the bones, but rather when the animal they come from stopped breathing.
Carbon dating relies upon the presence of carbon-14, an isotope of carbon. To understand an isotope, you need to know a little about the structure of atoms. Atoms, of course, are made of three things: electrons, protons and neutrons. If the atom is neutral, it has the same number of electrons and protons. If you start with the simplest element, you have one proton and one electron. You know this element as hydrogen. Add one neutron and you have hydrogen-2, an isotope.
The most common form of carbon is carbon-12. It has six neutrons, six protons and six electrons. It is stable and does not decay. You see a lot of carbon-12 in atmospheric carbon dioxide. Some of that carbon-12 gets exposed to cosmic radiation and turns into carbon-14, which has eight neurons. Carbon-14 is radioactive, with a half-life of 5,700 years.
So what makes this useful? Plants. Plants take in carbon dioxide during photosynthesis, and end up with some small amount of carbon-14. Animals eat those plants, and then other animals eat those animals, and soon everything has some amount of carbon-14. And when any of those things dies, they stop taking in carbon-14. (Yes, I know, they stop taking in everything, but I'm only interested in carbon-14 here.) The carbon-14 within that plant or animal begins to decay. By measuring the relative abundance of carbon-14 vs. carbon-12 (and, technically, carbon-13), you can work backward to figure out when that plant or animal stopped taking in fresh carbon-14. In other words, you know when it died.
Want an example? I'll make up something similar to carbon dating so you can see what is going on. I'll build a model using a lot of spheres. Most of them are yellow, but 20 percent of them are blue (OK, technically, they're cyan). The blue spheres are radioactive, and decay at the same rate I used in the example above. Click play to see what that might look like.
Again, I made radioactive spheres disappear when they decayed. This is fine, because when carbon-14 decays, it produces nitrogen-14. It's no longer carbon. But you could imagine that if you knew that the sample started with 20 percent blue spheres and you knew their half-life, then you could determine the age by examining one frame from the animation. This is exactly how carbon dating works, but with dinosaurs instead of models.
