Name
Date
Understanding Half-lives
Radioactive decay is a random process. In this activity different colored disks represents
different nuclie. Whether or not these nuclei decay depends on what colored area they fall onto
when poured into a container. Since the landing of disks on the patterned surface is random in
the same way that radioactive decay is random, the behavior of these two systems is very similar.
 A disk landing on a like-colored surface decays into its daughter nuclide.
 A disk landing on different colored surfaces does not decay.
 Disks that do not match the grid colors do not ever decay and represent stable nuclei.
In this investigation, you will drop red “nuclei” into the box lid. Each unstable red nucleus will
decay into a stable blue nucleus whenever it lands in a red region.
Materials:
 ~100 blue disks
 ~100 red disks


Cup
Box lid
 Red colored paper pieces (small rectangles or circle)
 Glue stick
Procedure:
1. Use the glue stick to glue red paper pieces to the inside of the box lid. Keep the pieces far
enough apart so that a red disk cannot ever touch two different pieces of paper.
2. Count your red disks and record how many you have. Then you and your teammates should
carry out a few trial “spills” of the cup into the lid. You will need to determine the following
for your team so that you consistently carry out the experiment:
 What method of spilling leads to what appears to be the most random distribution of
disks?

How should you determine if a red disk has decayed? Must it be completely in a colored
region or just near a region?

What other procedural issues you should settle before beginning?
3. On a separate piece of paper, create a data table to record the spill number, the number of
decays that occurred during that spill, the number of unstable nuclei in the system after that
spill and the number of stable nuclei in the system. What must you “measure” or count to
determine:
 the number of decays that occur during a single spill?

the number of unstable nuclei in the system after a single spill?

the number of stable nuclei in the system after a single spill?
Spill Number
Number of decays
during the spill
Number of unstable
nuclei after the spill
Number of stable
4. In a radioactive decay experiment, you typically measure the activity of the sample at a given
instant of time. Which of your data columns are comparable to time and activity? Explain.
5. Now conduct your experiment by repeating the following steps 15 times:
 Shake the cup and pour out the disks.
 Remove each red disk that lands on a red area and replace each one with a blue disk,
keeping track of how many red disks you remove. Record the spill number, the number
of red disks removed, the total number of red disks and the total number of blue disks.
 Pick up the blue and red disks in the box lid and place them in the cup.
6. Create a graph with two data sets on it. One data set will have the number of blue disks on
the vertical axis; the other plot will have the number of red disks on the vertical axis. For
both plots, the horizontal axis will be the number of times the cup was poured out. What is
the “half-life” of your unstable nucleus? How do you know this?
7. Compare your graph with the graphs from other groups. Are all of the graphs identical? Did
all the groups get the same half-life for the red disks? If not, why might they be different?
What was different about their system or technique compared to yours?
8. Suppose you obtain a sample of your red and blue nuclei and determine that 1/3 of the nuclei
are red and 2/3 of the nuclei are blue. How old is your sample? In other words, if your
system started with only red nuclei, how many tosses occurred before you obtained your
sample?
Mathematical Extension:
Carry out a mathematical analysis of your data to determine the half-life and decay probability of
your sample.
1. Review the graph you created above. What type of functional relationship (such as linear,
quadratic, power, exponential, or logarithmic) does there appear to be between the number of
disks and the number of pours? Use a graphing program, such as Excel or a TI calculator, to
calculate a line of best fit (regression or trendline) using your expected functional
relationship. Does your line of best fit match your data well?
2. The line of best fit is a decaying exponential, which can be written as:
N(t) = N0e-kt
where k > 0
After one half life, t1/2, what is the value of N(t)?
3. Use the equation N(t 1 / 2 )  N0 e kt1 / 2 and the value of N(t1/2) you got in the previous step to
derive a symbolic equation that shows the relationship between t1/2 and k. That is, write t1/2
in terms of k.

4. Use the equation you derived in the previous step and your line of best fit to determine the
“half life” for your data. That is, determine the number of pours it takes for half of the blue
disks to “decay”.
5. What is the probability that a disk will land in a colored grid square? What does this imply
that the “half life” (number of pours) should be? Does this match the value you calculated in
the previous step? If not, why not?