Exploration: A Logarithmic Look at Half-Life and the
Decay Rates
The Geiger Applet is designed to help students understands the process of radioactive decay, specifically the determination and uses of its half-life and decay
rate.
Prerequisites:

Students must be familiar with logarithms and natural logarithms to
answer many of these questions

Students should be able to work with graphs.

Students should be comfortable with converting the mass of a substance
to the number of atoms the substance contains.
Learning Outcomes:
By the end of this lesson, students should understand that radioactive decay's
rate can be calculated from observing it slightly and that there are many
applications for the decay rate and half-life.
If you have not already done so, open up the Geiger Counter Applet. You will
need it to answer some of the following questions.
Finding the Decay Rate and Half Life
From the list of isotopes, select Radon-222. Run the Geiger Counter.
1. a. In the graph as seen in the applet, what are the units of your dependent
and independent variables?
dependent units:
independent units:
b. What are the units of the area of your graph? If the x units are converted to
days, what are the units of the area of your graph?
c. How many counts have been detected by the Geiger counter after 11 days?
2. Consider three graphs: N (t )  N 0 2 t / h , C (t )  C0 2 t / h , D(t )  D0 2t / h and where t
is time and h is the half life (equal for all 3 formulae), and where D0  N 0  g
and C0  N0  f (f and g are constants).
a. Prove these three graphs are all of the same format and that their
slopes are all the same (hint: consider how D(t) and C(t) relate to
N(t)).
b. Suppose that, out of the total activity, the Geiger counter only
recieves 5 percent. Rewrite your N(t) formula to be a graph for the
total activity.
c. Activity can be measured as the current number of counts
multiplied by the decay constant (k). Use your formulae from (a)
and (b) to write a decay formula for the total number of counts, and
another formula for the number of counts that reach the Geiger
counter.
3. Consider a graph of "Counts that Reach the Geiger Counter versus Time"
where time is in years.
a. Use the slope tool to find the slope of your graph at 11 days. What
are the units of the slope? How would you convert this to
counts/day? Counts/s? What does the slope tell you about the
sample?
b. When tracking how the counts decay over time, what does the
decay constant represent?
c. Use the converted slope from your Geiger graph to calculate the
decay constant at 4 different points which seem to be the most
consistent with an exponential curve. Find the average.
Time
(days)
Counts
Slope
Converted Slope
(counts/day)
Average:
Decay
Constant
4. The equation used for this curve can be either N (t )  N 0 2t / h  A as seen in
the applet or N (t )  N 0 e  kt  A , which is more common, where h is the half
life and k is the decay constant.
a. Prove that the half-life can be calculated as h  ln(2) k .
b. Calculate the half life of Radon-222. Round your answer to the
nearest hundredth of a day if necessary.
c. Use the "Create a Curve Fit" to create the curve with your
calculated half-life. Is it close to the Geiger counter data?
d. Copy the data of the Geiger graph into Excel and graph it. Add an
exponential trend line. What is the k value? What is the half-life
value? Compare your values with Excel's. What are some possible
sources of error?
Summary:
It is important to realize that, while the graph's y units are the same as those for
activity, that does not mean that it is an activity-time graph. The activity is the
total number of disintegrations the sample experiences over a period of time; the
graph only shows the disintegrations which reach the Geiger counter during that
period of time. While the values on this graph are proportional to an activity-time
graph, they are not the same. The other important thing to realize is that
proportional exponential graphs will have the same slope. Because of this, we
can use the slope of the Geiger counter's graph as if it was a graph of Counts vs.
Time or Activity vs. Time.
Working with Half-Lives
From the list of isotopes, choose 5 you wish to work with for the remainder of the
lesson.
5. Using the method described above, calculate the decay constant and halflife of each isotope. Record them in the table below. Also, convert the
mass to kilograms and record it for each isotope (1 u = 1.6604E-27kg).
Isotope
Mass (u)
Mass (kg)
Decay
Constant
Half-Life
6. Suppose that you have 50 grams of each isotope.
a. How many atoms do you have?
b. Which isotope has the most activity? Explain.
c. After 20 years, which isotope has the greatest activity? What is its
activity?
7. Using Excel, find the approximate half-lives of Astatine-210 and
Protactinium-234.
a. Suppose each isotope begins with 10,000 atoms. After 3 hours,
which isotope has greater activity? Calculate the activities.
b. Which has more atoms after 10 days? Which as more mass? After
how long would the masses be equal?
8. Suppose you discover under your front step a 2 kg mass of Radium-223
(half life of 11.43 days). You decide to vacate your home until there is only
1 billionth of the Radium left, just to be safe.
a. How long will you have to wait before coming back to your home?
Give your answer to the nearest week.
b. While you are away, you learn that the Radium had been under
your door step for the past 9 months. How much of radium was
there to begin with? Round your answer to the nearest kilogram.
Summary:
Scientists have used their knowledge of half-lives and radiation for many
purposes. Through their knowledge of Carbon-14's half life, they have been able
to date many ancient artifacts by comparing the level of Carbon-14 in the artifact
to the level in a still-living object. The also have been able to use it to determine
how long after a nuclear power station has been shut down or a nuclear test has
been conducted it is safe to occupy the area. With half-lives they can find the
current amount or past amount of an isotope or determine how long it has existed
or will exist in significant amounts.