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Half-life: Calculating Radioactive Decay and Interpreting Decay Graphs

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  1. 0:05 Introduction
  2. 1:11 Half-Life
  3. 1:37 Half-Life Example
  4. 3:32 Simulating Half-Life
  5. 5:17 Scaling Up
  6. 6:05 Lesson Summary
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Taught by

Kristin Born

Kristin has an M.S. in Chemistry and has taught many at many levels, including introductory and AP Chemistry.

What causes a radioactive particle to decay? We'll never really know, but our best guess lies in probability. In this lesson, we are going to focus on the half-life, a way of measuring the probability that a particle will react.

Introduction

Imagine you're getting settled in to watch the new action film at your local theater. You have a big tub of popcorn on your lap, and you're sitting back and watching as the previews begin. About 15 minutes later, the previews finish up and you notice half of your popcorn is gone! It must have been good. The movie starts and you slow down your eating a little, but 15 minutes after the movie has started, you have eaten half of what you had left and are down to a quarter of your popcorn. This continues for the rest of the movie until all of your popcorn is gone.

If we were to graph your popcorn eating during the movie, it may look something like this. You may notice a few things about this graph. First, your popcorn eating did not happen at a steady pace. If that were the case, it would look more like a straight line. What it shows is that you ate faster at the beginning than at the end, because more popcorn is consumed in the first 15 minutes than in the second 15 minutes. The second thing you may notice is that every 15 minutes you eat half of what you had.

Half-Life

Decay of an atom can be predicted by using its half-life.
Half-Life Example

This phenomenon takes place every day in many chemical reactions and nuclear reactions, and it is called the half-life, which is the amount of time it takes for half of a sample to react. Your popcorn had a half-life of 15 minutes, meaning that every 15 minutes, half of it will get eaten. Just like your popcorn, radioactive particles have half-lives.

A Half-Life Example

Let's take an example. Say we have a bunch of cobalt-60 atoms. Cobalt-60 decays down to nickel-60 during beta decay. But how do we know when the cobalt-60 atoms are going to decay? Are they all going to decay at once or randomly? Initially it may seem like atoms decay randomly, but their probability of decaying can be predicted using an atom's half-life.

It turns out that the half-life of cobalt-60 is about 5.27 years. That means if I start out with 16 cobalt-60 atoms, and I wait 5.27 years, I will probably be left with eight cobalt-60 atoms and eight nickel-60 atoms. Then, if I wait 5.27 more years, half of the eight cobalt-60 atoms that were left should decay, giving me only four cobalt-60 atoms and a total of 12 nickel-60 atoms. If I wait 5.27 years after that, half of the cobalt that remained will decay into nickel-60, giving me 14 atoms of nickel-60 and only two atoms of cobalt-60. If I wait another 5.27 years, half of those two remaining atoms, so one atom, should decay, giving me a total of 15 nickel-60 atoms and one lonely little cobalt-60 atom. Now we can never have half of an atom, so what happens next? This is where probability makes more of a presence. If I wait 5.27 more years, there is a 50% chance that the one remaining cobalt-60 atom will decay. It either will, or it won't. If it does, then all the atoms will have decayed. If it doesn't, then it has a 50% chance of decaying in the next 5.27 years.

What graphing a half-life usually looks like
Half-Life Graph

Simulating Half-Life

One thing that is really neat about half-life is that it can be simulated very easily. All you need is a bunch of pennies (or any other type of coin) and a clock or stopwatch. We are going to assign a half-life of 30 seconds to this made-up radioactive element. First, write down how many coins you have. This will be our initial amount. Now wait 30 seconds. As you're waiting, shake up the coins in a container or in your hands so they get all mixed up. When the 30 seconds have passed, dump your coins on a table and remove all the 'heads'. This should leave about half your coins. The 'heads' represent 'decayed' atoms, and they will no longer be part of our sample. Write down how many remain, and shake the remaining coins for another 30 seconds. Again, after 30 seconds have passed, dump your coins on a table and remove all the 'heads'. Write down how many remain, and continue this process until all your coins have 'decayed'. Notice at the end how you can never have half of a coin, so when you're down to just one or two coins, you can see the probabilistic nature of them decaying. You may end up shaking one coin for many 30-second half-lives because it just won't decay! Or you may be down to four coins, and all of them come up as heads.

You should quickly see that there is no way of being able to predict exactly what will happen during each shake, but you can make good predictions about what should happen. In fact, if you were to graph your results by putting the time on the x-axis and the number of 'radioactive' coins on the y-axis, you should almost always end up with a graph that looks like this.

Scaling Up

Materials can be dated by using the half-life of carbon-14.
Carbon Dating Image

In nuclear reactions, most likely instead of starting out with 16 atoms or even 100 atoms, you would be measuring the amount you have in grams (or some other mass unit). The same rules still apply: If I start out with 20 grams of carbon-14 and wait about 5,730 years (the half-life of carbon-14), around 10 grams will remain and 10 grams will have been converted into nitrogen-14 (that's the product of a carbon-14 atom that underwent beta decay). This is how carbon dating works, and it's used to determine how old an artifact is. Scientists measure how much carbon-14 is left in a sample, and they are able to estimate how many half-lives it went through. This will allow them to get an approximate idea of how old the material is.

Lesson Summary

You are surrounded by radioactive isotopes. However, many don't pose much of a threat because they have such long half-lives. The half-life of a radioactive isotope is the time it takes for half of the sample to react, or decay. This time can range anywhere from a portion of a second to thousands of years depending on the identity of the starting isotope.

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