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 role-

playing 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, carbon-

14 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 , and then taking the natural log

of both sides:

=

=

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 thirty-

three 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.

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 carbon-

14. 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

.

#Present

# Rolls Trial 1 Trial 2 Trial 3

0 50 50 50

1 33 27 30

2 25 19 20

3 18 13 15

4 13 10 10

5 8 5 7

6 5 3 5

7 3 2 4

8 2 1 3

9 2 1 2

10 0 0 2

11 N/A N/A 0

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N = 52.358e-0.385t

0

20

40

60

0 2 4 6 8 10 12Nu

mb

er

of

Dic

e

Re

mai

nin

g, N

Roll Number, t

Dice Simulation, Trial 1

N = 48.299e-0.452t

0

20

40

60

0 2 4 6 8 10 12Nu

mb

er

of

Dic

e

Re

mai

nin

g, N

Roll Number, t

Dice Simulation, Trial 2

N = 42.694e-0.344t

0

20

40

60

0 2 4 6 8 10 12Nu

mb

er

of

Dic

e

Re

mai

nin

g, N

Roll Number, t

Dice Simulation, Trial 3

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.

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.

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 y-

axis on a scale of N instead of

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 carbon-

14 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 half-

lives. 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/>.