Exploring Radioactive Decay:
An Attempt to Model the Radioactive Decay of the Carbon-14 Isotope used in
Radiocarbon Dating through a Dice Simulation
An Internal Assessment:
In the International Baccalaureate Diploma Subject of Mathematics HL
Candidate Name: Eric Todd
Candidate Session Number: 001395 - 0167
Hillcrest High School
Midvale, Utah, United States of America
School Code: 001395
Supervisor: Mr. Ken Herlin
Submission Date: January 23, 2014
Examination Session: May 2014
Word Count: 2994
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Introduction
When my class studied the Anasazi and other Ancestral Pueblo peoples, as part of the
state curriculum of Utah, I was intrigued that scientists could determine how old pieces of
pottery from different civilizations were, through the use of carbon dating (en.wikipedia.org/
wiki/Radiocarbon_dating). At the time, the concept of radiocarbon dating was foreign to me, but
was still an interesting topic that I wanted to understand because at the time it was introduced to
me, I could not see how the chemical makeup of an object could be used to determine its age. I
studied the subject on and off through materials I could obtain at my local library, but my interest
in carbon dating was again sparked a couple years later in my physics class when we discussed
radioactive decay and its ability to be modeled by real life situations, and was even furthered by
our discussion of carbon dating and radioactive decay in HL math. It is with this peaked interest
that I am now attempting to explore the topic of radioactive decay and radiocarbon dating.
This exploration attempts to model radioactive decay through a dice simulation, which
model will be used to be compared with the modeling of the radioactive decay of the carbon-14
isotope that is used in radiocarbon dating. I have chosen to use a dice simulation because dice are
a classic example of probability, which can be used for simple decay, and because I have always
had an interest in dice probability and gaming. Though my interest in dice probability and roleplaying games that are dependent on dice does not specifically apply to what I am exploring, I
still think it will make my findings more interesting to me personally. By mirroring the
modelling of radioactive decay that is used in radiocarbon dating, I hope to explore the concepts
of half-lives and decay constants in radioactive decay in context with the models.
How Radiocarbon Dating Works
Radiocarbon dating was developed by W.F. Libby, E.C. Anderson, and J.R. Arnold in
1949 as a method of estimating the age of old organic material (Higham – c14dating.com).
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Carbon dating uses an unstable isotope of carbon, carbon-14. While an organism is alive, carbon14 is decaying into more stable isotopes, like nitrogen-14, through beta decay, while the
organism absorbs more carbon-14 to keep a natural balance. When an organism dies, the rate of
decay of carbon-14 decreases logarithmically (archserve.id.ucsb.edu) and therefore is able to be
modeled as an exponential decay curve.
The amount of Carbon-12 in an organism, a stable isotope of carbon, stays constant
during the life of, and after the death of an organism because of the stable nature of that isotope.
The natural levels of carbon-12 and carbon-14 in the atmosphere have been recorded and
calculated so that the ratios of carbon-14 atoms present compared to carbon-12 atoms present in
organic materials can be used in the process of radiocarbon dating (ncsu.edu). The half-life of
carbon-14 is 5730 years, which means that it will take 5730 years, on average, for half of the
initial substance present to decay (Long – parks.ca.gov). Although the ratio of carbon-14 to
carbon-12 atoms is only a linear relationship, I am looking to explore modeling the radioactive
decay of the carbon-14 isotope, which is an exponential relationship.
Deriving an Equation to Model Exponential Decay
When discussing radioactive decay, any exponential decay rate can be modeled by the
general exponential decay equation shown below:
The change in the amount of radioactive substance remaining, dN, with respect to the
change in time, dt, of substance decaying can be modeled by the decay constant, λ, multiplied by
the amount of radioactive substance remaining, N. Since the rate of change is a decay rather than
exponential growth, the expression is given a negative sign.
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I then separated the variables, by multiplying by dt and dividing by N. I did this so that
the expression can be solved by integration for the amount of substance remaining, N, as a
function of time, t, so that I can calculate the half-life of the substance later on.
For my own convenience before integration, I will rewrite the expression above as: since
the antiderivative of is ln| |:
Now, taking the antiderivative of both sides, I got the following:
∫( )
=
∫(
)
| |
I subtracted the constant C, that resulted on the left side of the expression, for
convenience in the same step so that I could get to an equation that models radioactive decay
more quickly. I then exponentiated this expression to separate N, in order to create an equation
that models the amount of substance remaining after any given time of decay, and I got:
| |
Since e is the base of ln, the expression can be simplified below:
| |
Using the properties of exponents we learned in class, the right side of the equation can
be separated into two bases of e being multiplied together, since two bases multiplied together
add their exponents.
| |
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The absolute value of the amount of substance remaining can be understood to mean that
the expression could be either negative or positive, as long as its value is the same magnitude
away from zero. This will be expressed below:
The amount of substance present initially, when t = 0, is the same as the part of the
expression modeled by
, as shown evaluated below:
( )
( )
=
( )
So, the value of
(
) =
( )
( ) =
( )
can be expressed as N0, because N0 represents the initial amount of
substance, when no time has passed, when t is equal to zero, which is the same as N (0). This
then completes the derivation of the equation that will model my radioactive decay simulation:
The variables will be defined in respect to the dice simulation with N representing the
number of dice remaining at any given time; t. N0 represents the initial number of dice, before
any decaying occurs. e is the base of the natural logarithm, and it indicates that the decay is
exponential in nature. λ is the decay constant, which is proportional to the rate of decay. t
represents the time over which the decay occurs, which in the dice simulation is the number of
rolls. In order to create a model of exponential decay similar to carbon-14 through the
simulation, I will try to apply the same restrictions to the simulation as exist with carbon-14.
Determining the Half-Life of Carbon-14, and its Decay Constant, λ
The half-life of a radioactive substance can be determined when the amount of substance
remaining is half of its initial amount. In this case, the half life of carbon-14 can be modeled
using the exponential decay equation above for the solving below:
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Now, solving algebraically for the half-life, divide out
of both sides:
=
, and then taking the natural log
=
Using properties of logarithms, I am going to rewrite this expression for personal
convenience so that it can be seen how I simplified it more clearly:
=
( )
=
In this equation, t represents the half life of any given radioactive substance, and it is
proportional to
, which relies on the decay constant of the substance. I am specifically
interested in the isotope of carbon-14, and am going to use this equation to evaluate its decay
constant using its researched half-life of 5730 years (Long – parks.ca.gov).
The Dice Simulation
In order to model how radioactive decay of carbon-14 may look, I will be performing a
simulation using dice. The chance that a carbon-14 atom will decay is a constant value.
However, the probability of its half-life is based on an average calculation which does not
guarantee that there are “exactly one-half of the atoms remaining”, after the half-life, “only
approximately, because of the random variation in the process” (en.wikipedia.org/wiki/Half-life).
Without a known decay constant in the dice simulation, one will have to be calculated
using its half-life, which will be defined as the number of rolls it takes to decay half of the initial
amount of dice present before decay began. In the simulation, I will define decay to be a roll of
one or two. I will roll the dice, with an initial amount of 50 dice, until all of the initial dice have
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decayed, so that the half-life can be determined, and from that, the decay constant. The graph of
radioactive decay modeled by the dice simulation will then be compared to that of carbon-14.
Figure 1: This is a screenshot of the second roll of the first trial of my dice simulation of
radioactive decay. I used this random dice roller to roll fifty dice, and then roll the
subsequent number of dice after removing those that were decayed for each roll. In the
first trial, the first roll of fifty dice yielded seventeen dice that decayed which left thirtythree dice remaining. This screenshot shows the roll of thirty-three dice, and 8 dice are
shown as decaying after the second roll, which leaves 25 dice remaining to decay on the
next roll.
# Rolls
0
1
2
3
4
5
6
7
8
9
10
11
#Present
Trial 1
Trial 2
Trial 3
50
50
50
33
27
30
25
19
20
18
13
15
13
10
10
8
5
7
5
3
5
3
2
4
2
1
3
2
1
2
0
0
2
N/A
N/A
0
Table 1: This table contains the data for three trials of
the dice simulation of radioactive decay. The data was
collected using a random dice simulation online, which
is shown in the figure above (random.org/dice). This is
raw data that will be used to model a graph of
radioactive decay through the dice simulation that will
be compared to the radioactive decay model of carbon14. The half-life can be determined by finding the
number of rolls it took to decay half of the initial dice
present. With the determined half-life, the decay
constant can also be found using the relationship
.
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Figure 2: This graph shows the
radioactive decay curve for
carbon-14, whose half-life is
5730 years (Cash – uclmail.net).
This is the control model that
will be compared to the
simulation models. It shows
activity level of carbon-14 over
time, indicating decay.
Number of Dice
Remaining, N
Dice Simulation, Trial 1
N = 52.358e-0.385t
60
40
20
0
0
2
4
6
8
10
12
Roll Number, t
Figure 4: This graph shows the
radioactive decay curve for the
second dice simulation. This
model, like all of the dice
simulation models, shows the yaxis on a scale of N instead of
Number of Dice
Remaining, N
Dice Simulation, Trial 2
N = 48.299e-0.452t
60
40
20
0
0
2
4
6
8
10
12
Roll Number, t
Number of Dice
Remaining, N
Dice Simulation, Trial 3
N = 42.694e-0.344t
60
40
20
0
0
2
4
6
8
10
Roll Number, t
Figure 3: This graph shows the
radioactive decay curve for the
first trial of the dice simulation.
The half-life for this trial is 2
rolls, and the equation for the
model is shown on the graph.
This model is a similar shape of
decay compared to the carbon-14
model, but shows y-axis values
instead of percentages.
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while the carbon-14 model does
use the percentage valued y-axis.
Figure 5: This graph shows the
radioactive decay curve modeled
by the third dice simulation. It
has the worst initial amount
predicted out of the three in
comparison to the actual value of
50 initial dice. The y-axis uses a
scale of N values instead of as
the carbon-14 model uses.
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Half-Life and the Decay Constant
The half-lives are so different in nature that I will compare the decay constants of the
radioactive decay of carbon-14 and of that modeled by the dice simulation instead.
The Decay Constant of Carbon-14
Using the half-life of 5730 years, the decay constant can be found using the relationship
determined above:
.
( )
( ) =
=
( )
=
Above shows the solving for the decay constant of carbon-14 to three significant figures.
The Decay Constant of the Dice Simulation
Now, solving for the decay constant of the dice simulation, I first had to determine the
half-life. In order to find the half-life produced by the dice simulation, I did the average half-life
of the three trials to find a general half-life for all dice simulations. The respective half-lives for
each trial are 2, 1.43, and 1.5, which I found using the amount of dice decayed after half, or more
than half the dice were gone and comparing it to the number of rolls it took for that number of
dice to decay. Averaging the half-lives gave me 1.64 rolls to three significant figures. Now,
using the half-life of the dice simulation, I calculated the decay constant using the same
relationship as before:
( )
=
( ) =
( )
=
The decay constant calculated using the dice models of radioactive decay is very different
than the decay constant for carbon-14’s radioactive decay. One likely reason for the difference is
the difference in half-lives, because years cannot equate to the number of rolls of dice, or it could
also be due to the chosen probability of decay for the dice compared to the probability of carbon14 decaying.
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Conclusion: Limitations, Implications and Extensions
When I started exploring radioactive decay to see if I could model the same isotope that
is used in radiocarbon dating through a dice simulation, I soon discovered that it is very difficult
to replicate radioactive decay without a radioactive substance. However I did find that it was
possible to create a model of a similar form to that of carbon-14’s model. Though, through my
dice simulation I was able to recreate models similar to a radioactive decay graph, the dice were
what ultimately limited my ability to replicate models in regards to the radioactive decay of
carbon-14.
The probability of decay for the dice was per roll of each die, since a roll of 1 or 2
would mean that it decayed, out of a total of 6 possible rolls per die. Each roll was independent
of the other dice rolled and without replacement, as when a die decayed it was taken out of the
total number of dice. Compared to the probability that carbon-14 decays, the dice simulation is
limited to a minimum chance of decay at , with intervals only able to increase at values of as
well, which further limited the dice simulation’s accuracy in, and ability to replicate the
conditions needed to model the radioactive decay of carbon-14 because changes in decay are less
than after 9 half-lives or more (archserve.id.ucsb.edu). However, despite the resulting model
seeming negligent due to this limitation, I still feel accomplished because I was able to create a
model similar to that of the radioactive decay of carbon-14, even though I wasn’t able to
replicate it.
Based on my simulation I would argue that half-life is not always certain. The half-life of
a radioactive substance is an average value, which allows for uncertainty of exactness for any
model of radioactive decay. In the three trials of my dice simulation, I saw that half-life was not
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always a constant value for the same conditions, and therefore cannot be used as the sole
predictor of decay.
Limitations also exist in regards to the accuracy of the radioactive decay of carbon-14
used in radiocarbon dating. Radiocarbon dating is not accurate past 50,000 years old because of
the limiting factor of half-life, “regardless of the sample size” (archserve.id.ucsb.edu). Since the
ratio of carbon-14 to carbon-12 in the atmosphere has not been constant over time, further
limitations on the accuracy of radiocarbon dating have arisen (archserve.id.ucsb.edu). Based on
this evidence, my exploration of trying to model radioactive decay similar to that of carbon-14
through a dice simulation is also limited because I did not take into account what ratio I was
trying to model, which has changed over time. In further exploration of radioactive decay, I
would try to consider the changing ratio, as well as use a medium of decay that allows smaller
probability of decay to make the simulation closer to the conditions actually faced by carbon-14
in the environment. I am also interested in looking into chain decay, which is a topic related to
radioactive decay that came up often when I was researching for this exploration, and perhaps
trying to model it using a new method that will require additional research.
In comparison to the dice simulation, the radioactive decay curve would not be accurate
in predicting the amount of dice remaining past 15 rolls (calculated at 9 half-lives using 1.64
rolls as the base). This would imply that the model of radioactive decay both the dice simulation
and carbon-14 are not as accurate as I once thought, and that if the model is not a good indicator
of age outside of a specific range of half-lives, then perhaps neither model is a good indicator of
age within the specified range of half-lives either.
I learned a lot about radioactive decay and radiocarbon dating as I explored modeling the
radioactive decay curve of carbon-14 through a dice simulation. Though the simulation did
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produce models similar to that of carbon-14’s model, the dice turned out to be the limiting factor
in model construction due to their limited ability to produce a probability of decay similar to that
of carbon-14.
I think the most important thing that I learned however, is that radiocarbon dating is only
accurate to an extent. While I believed before my investigation that radiocarbon dating could
determine the age of almost anything, my observation now, as I learned through my dice
simulation, is that half-life plays a large role in determining what can be dated, and it’s often
different than the predicted average value which limits radiocarbon dating to a boundary of halflives. I really enjoyed exploring radioactive decay in connection to dice, and I want to look into
other mediums of decay that could possibly provide a better model in the future. I also want to
explore the limitations that dice place on role-playing games, now that I have discovered those
they create on modeling radioactive decay.
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Works Cited
Cash, David. "Welcome to the Chemistry Articles Page of David Cash Ph.D. (Professor,
Mohawk College of Applied Arts and Technology, Retired)." David Cash's Chemistry
Articles Page. N.p., n.d. Web. 27 Jan. 2014.
<http://www.uclmail.net/users/dn.cash/articles.html>.
"Half-life." Wikipedia. Wikimedia Foundation, 16 Jan. 2014. Web. 22 Jan. 2014.
<http://en.wikipedia.org/wiki/Half-life>.
Higham, Thomas. "The method." The method. N.p., n.d. Web. 22 Jan. 2014.
<http://www.c14dating.com/int.html>.
Long, Kelly. "Why Is Radiocarbon Dating Important To Archaeology?." Why Is Radiocarbon
Dating Important To Archaeology?. N.p., n.d. Web. 23 Jan. 2014.
<http://www.parks.ca.gov/?page_id=24000>.
"Radiocarbon Dating." Radiocarbon Dating. N.p., n.d. Web. 22 Jan. 2014.
<http://archserve.id.ucsb.edu/courses/anth/fagan/anth3/Courseware/Chronology/08_Radi
ocarbon_Dating.html>.
"Radiocarbon Dating." Radiocarbon Dating. N.p., n.d. Web. 22 Jan. 2014.
<http://www.ncsu.edu/project/archae/enviro_radio/overview.html>.
"Radiocarbon dating." Wikipedia. Wikimedia Foundation, 15 Jan. 2014. Web. 22 Jan. 2014.
<http://en.wikipedia.org/wiki/Radiocarbon_dating>.
"True Random Number Service." RANDOM.ORG. N.p., n.d. Web. 23 Jan. 2014.
<http://www.random.org/dice/>.
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