Name
Date
Understanding Half-lives
Radioactive decay is a random process. However, for a given
type of nucleus, the probability that the nucleus will decay in a
given time period is constant. That means that within a system
of nuclei, the average number of nuclei decaying in a given time
period is proportional to the number of nuclei in the system.
In these computer activities, unstable parent nuclei decay into
stable daughter nuclei. You will explore how the decay
probability affects the rate at which the parent nuclei decay and
the rate at which daughter nuclei are produced.
1. Start the Java applet “Two-State Radioactive Decay”
(http://www.opensourcephysics.org/document/ServeFile.cfm?ID=10576&DocID=2067&Attachment=1).
2. In this simulation the unstable parent nuclei are red. They decay into the stable blue daughter
nuclei. The probability with which parent nuclei decay in a certain amount of time is called
the decay probability k. To get a feel for how the simulation works, start the decay process
for the default settings by clicking the play button in the lower left corner of the simulation.
Stop the simulation after most of the parent nuclei have decayed. Try it again, changing the k
value and make sure you hit the return button after changing the k value.
3. Reset the k value to 0.5 and hit the return button. Reset the time to zero (click on the double
left arrow at the bottom left) and record all of the initial date on the screen in the table below
for Trial 1. When about 95% of the parent nuclei have decayed, stop the simulation and
record all of the information about the system in the final column below.
Quantity
Trial 1
Trial 2
Trial 3
(Step 3, pick k value)
(Step 4, double k value) (Step 5, half k value in Trial 1)
initial
final
initial
final
initial
final
N active
k
N stable
t
t
4. If you double the decay probability, how does the system behavior change? Reset the time
and then double the value for k. Remember to push the return key after you update the k
value. Record the initial values then let the system run until you reach the same number of
active nuclei as in your first trial. Stop the simulation and record all of the information for
the final system.
5. What if you reduce the decay probability? How does the system behavior change? Reset the
time and then reduce the value for k to ½ of its value in Trial 1. Record the initial values
then let the system run until you reach the same number of active nuclei as in your first trial.
Stop the simulation and record all of the information for the final system.
6. What do you observe about the behavior of the system? How does it depend on the decay
probability?
7. It is sometimes easier to notice patterns in the behavior of the system if the information is
presented in a graphical form. Check the “plot” box on the bottom right corner of the applet
and a window labeled “Two-State Decay Plot” will pop up. Move it to the side of the main
window. Reset the k value to what you used in Trial 1 and the time and let the simulation
run. The red curve will show the number of parent nuclei present at a given time instant
while the blue curve will show the number of daughter nuclei present. Stop the simulation at
the same time you did in Trial 1 above.
8. Describe the shapes of the two curves in the plot window.
9. The half-life for a radioactive nucleus is the amount of time it takes for ½ of the nuclei in the
system to decay. Using the plot, determine the half-life for Trial 1.
10. Often, the age of a sample is estimated by determining the amount of parent and daughter
nuclei in the sample and then estimating how much time must have passed in order for the
system to reach that state. For example, if the sample is made up of 60% parent nuclei and
40% daughter nuclei, how much time has elapsed since the first parent nuclei began to decay
(or since the simulation started)? Use your graph to estimate the time instant at which you
would find 60% of your sample being parent nuclei and 40% being daughter nuclei.
11. Repeat Trial 2 and describe how the two curves in the plot window differ from the curves in
Trial 1. How do the half-lives compare?
12. Repeat Trial 3 and compare your plots and half-lives to those from Trials 1 and 2.
13. You can change other features in the system. Does changing any of the other parameters in
the system affect the shape of the graphs or the half-lives? Systematically change the
parameters, taking care to record your settings, results and observations. Which factors, if
any, affect the half-life of the system?
Mathematical Extension:
Carry out a mathematical analysis of your data to determine the half-life and decay probability of
your sample.
1. Reset the time and parameters to match one of your previous trials and record which trial you
will be working with. Then, check the “table” box in the lower right corner of the main
applet window. A window labeled “Two-State Decay Table” will appear. Start the
simulation and the data table will fill with the number of parent and daughter nuclei at each
time step in the simulation as well as the number of parent nuclei decaying in each step. Stop
the simulation at the same time you did in the original trial.
2. Copy the data from the table window (click in the window, type Ctrl-a, then Ctrl-c). Then
open excel and paste the copied data into a new worksheet. Graph the number of parent
nuclei vs the time. What type of functional relationship (such as linear, quadratic, power,
exponential, or logarithmic) does there appear to be between the number of parent nuclei and
the time? Calculate a line of best fit (regression or trendline) using your expected functional
relationship. Does your line of best fit displayed on the graph match your data well?
3. The line of best fit is a decaying exponential, which can be written as N(t) = N0e-kt where k is the
decay constant. How does the fit prediction compare to the value of k used in the simulation?
Nuclear Forensics Extension:
In many systems, the unstable parent nucleus decays
to an unstable daughter nucleus. This daughter
nucleus decays to a third stable nucleus, called the
granddaughter. As in the simpler two-nucleus
system, the probability of decay is a constant, so the
number of parent nuclei and daughter nuclei
decaying in a given instant is proportional to the
number of each one present at a given instant.
However, the number of daughter nuclei is
increasing due to parent nuclei decays while it is
also decreasing due to its own decay. The
mathematical model for this process is much more
complicated than that of the two nucleus system. However, using a computer simulation you can
explore the way these systems behave.
1. Start the Java applet “Three-State Radioactive Decay”
(http://www.opensourcephysics.org/document/ServeFile.cfm?ID=10577&DocID=2069&Attachment=1).
As in the previous simulation, the parent (unstable) nuclei are red. They decay into green
daughter nuclei with a decay probability of k1. The daughter nuclei decay into blue
granddaughter nuclei with a probability of k2. To get a feel for how the simulation works,
start the decay process for the default settings by clicking the play button in the lower left
corner of the simulation. Stop the simulation after most of the parent nuclei have decayed.
Click on the plot window to graphically review the behavior of the system. How do the
parent and daughter curves compare to those in the two-state system? How does the
granddaughter curve compare the to the parent and daughter curves in the two-state system?
2. What values of k1 and k2 would make the system behave the same as in your first trial for the
two-state system? Try your idea and see if it works. Record your trial values and
observations about the graphs. If your first trial doesn’t work, try again.
3. The default setting has k1=2*k2. What if k1=k2? Reset the system and run the new
simulation. Record your observations below.
4. What if k1=k2/2? Reset the system and run the new simulation. Record your observations
below.
5. Often, the age of a sample is estimated by determining the amount of parent and daughter
nuclei in the sample and then estimating how much time must have passed in order for the
system to reach that state. For example, if the sample has twice as many parent nuclei as
daughter nuclei, how much time has elapsed since the first parent nuclei began to decay (or
since the simulation started)? Use your graph or data table from step 4 to estimate this
number.
6. Suppose at that instant, there were approximately ½ as many granddaughter nuclei as
daughter nuclei. Is that consistent with what your model predicts for the number of
granddaughter nuclei? What could this mean about the sample and your comparisons to the
model?