# Math of dating, Potassium-Argon, Rubidium-Strontium, and Uranium-Lead, all

Math Exploration: Fossil Dating

Intro

From a young age I have found
fossils everywhere: In my backyard, the playground, and all over parks. The
idea of prehistoric creatures has always fascinated me, but I could never
understand how paleontologists know how old they are. I wondered how paleontologists
were able to know how fossil relates to evolution and how they use fossils to
classify eras. This brought about the question: How can math be used to
determine the age of fossils?

In order to
answer this question, one must determine the different methods of fossil dating
and what equations are involved in these types of dating. Exponential equations
and logarithms can be used to show how fossils age over time. Paleontologists
use these equations to classify eras and show the evolution of species.

Scientists
use several methods to date fossils, mainly involving mathematical formulas.
Radiometric dating is used most often to determine the absolute date of the
fossil. There are four types of radiometric dating which use mathematical
equations to determine the exact date of fossils: Potassium-Argon Dating,

Background

In order to understand the math
behind Fossil dating, an understanding of the terminology must be collected.
The different types of dating, Potassium-Argon, Rubidium-Strontium, and
Uranium-Lead, all require different techniques and equations that can determine
the age of fossils. These methods are used to determine the age of the rocks in
which the fossil is found, therefore, the scientists can determine when the
fossil lived. Understanding how each one works and the methods used to date
fossils gives one a better understanding of the equations associated with each
technique.

Potassium-Argon
dating is a technique used for older fossils, as it can date back farther than
many methods. The radioisotope, Potassium-40 is said to dissolve in Argon-40.
By figuring the rate of decay in Potassium-40 and finding the ratio of
Potassium to Argon, one can find the date of a fossil. (Earlham 1)

Rubidium-Strontium
dating is an accurate type of dating because it has such a high half-life. As
igneous rock cools, rubidium decays, turning into strontium, therefore creating
a ratio between the two. The ratio determines how old the rock and fossil are.

dating is a more difficult type, as the two isotopes are not directly related.
Also, it is difficult to retain in rocks. In order for the isotope to become
lead, it must undergo several short transformations (Earlham 1). Despite its
difficulties, it can be a precise way to determine the age of rocks. (Kerr 789)

Aim

The
investigative nature of this work attempts to explain how paleontologists and
other scientists use mathematics to determine the exact and relative age of
fossils. By determining the different variables that can be manipulated, an
understanding of how dating works can be determined, depending on the various
types of radiometric dating. The collection of research will create an overall
better understanding of what radiometric dating is and how it is used.

Research

The following equation can be used
for each type of radiometric dating:

Figure 1 (Radiometric Time Scale, 1)

Each of the independent variables, D, P, ?,
determine what the dependent variable, t, will be. By plugging in each variable
to the set equation, one is able to find the age of the rock, and therefore the
fossil. For the sake of simplicity, in order to find the decay constant, the
bottom formula in Figure 1 will be rewritten as: . Depending on the method and presence
of particular isotopes, the equation will produce a specified number which
states the age of the rock.

The following table gives the
respected daughter products and current scientifically accepted half-lifes of
each isotope used in radiometric dating:

Figure 2 (Radiometric Time Scale, 1)

A “half-life of an isotope is defined as the amount of time
it takes for there to be half the initial amount of the radioactive isotope
present” (LeVarge 1). The respective half-lives will be used for the equation
to determine the appropriate decay constant.

Potassium-Argon
Dating

Potassium-Argon dating is an
effective method to date relatively old materials. Some dating, up to four
billion years old is obtainable. By comparing the ratio of Potassium-40 (K-40)
to Argon-40 (Ar-40) and  knowing the
decay rate of K-40, the date of the material is found.

For example, if there are 1000 atoms of K-40
(the parent atom), and 300 atoms of Ar-40 (the daughter atom), the age is
obtainable by plugging in to the equation. In order to have all the variables
required, one must first find the appropriate decay constant (?). To find this,
using a graphing calculator, use the currently accepted half-life, 1.25 billion
years as and solve:

?=

?= 0.5545177444

For the sake of this example, the solutions will be rounded
to the hundredths place. Using this number, plug in all the variables into the
equation shown in Figure 1 to find the age. Use a graphing calculator for the
most accurate results.

Therefore, the age of the fossil is 0.48 billion
years old. Because the number used to determine ? was in billions of years, the
end result is also in billions of years. This process can be used in other
methods of dating as well, using the accepted half-lives and the ratio of
parent and daughter isotopes.

Rubidium-Strontium Dating

Strontium-87 is a stable isotope
that does not change due to decay. As Rubidium-87 decays, it transforms into
Strontium-87. Therefore, the only way age can be obtained using this method is
by comparing the amount of Rubidium-87 to Strontium-87.

In
an example similar to the one found in Potassium-Argon dating, a fossil has
4000 parent atoms and 200 daughter atoms. Repeating the same steps used in the
previous example; the half-life, 48.8 billion years is used to determine ?.

?=

?= 0.142038357

Using the ? value and the number of parent and
daughter atoms, the age can be found:

The fossil’s age is 0.35 billion years old. The same process
can be used for various types of radiometric dating because each method uses a
ratio to determine its age. As long as the variables that alter the equation
are relative to the type of dating, the results will be produced for that type
of dating.

Uranium is an unstable, reactive
element. It sheds nuclear particles until it results in lead. When a mineral
grain forms, it begins absorbing lead atoms that accumulate over time. By
determining the amount of lead atoms, the date can be determined.

Once
again, the steps from the first two examples can be used to determine the age
of a exampled fossil. Using 500 parent atoms of Uranium-238 and 2000 daughter
atoms of Lead-206, use the half-life, 4.5 billion years to find ?:

?=

?= 0.1540327068

Use the number of atoms for both the daughter and
parent, as well as ?=0.15 to find the age:

The fossil is 10.73 billion years old. This may seem higher
in age than the than the other examples using other methods of dating.

This is because in this example, the
daughter atoms were higher than the parent atoms. As the parent atoms decay to
become the daughter atoms the length of its life increases, as these processes
take a while. Therefore, the higher the number of daughter atoms in comparison
to parent atoms, the older the fossil will be.

Reflection

My
collection of work, with included examples shows how math applies to real life
situations. The exponential equations that I practice in class are able to be
manipulated and incorporated into an idea that I find interest in and I want to
understand. By completing this, I am able to understand the importance of math
in everyday life. While reflecting on the work of this paper, I find that I
could change several things. I had to trust the work of others, due to the lack
of supplies and accessibility that would help me conduct my own experiment.
Therefore, any variation in their works could impact mine. However, I have
successfully proven the use of mathematics in real terms and have a better

Conclusion

By manipulating the variables in an
exponential equation, it is possible to determine the age of a fossil quickly
and accurately. The different options available for radiometric dating provide
different results which can be used in different ways to support claims. Not
only is this an example of how exponents are used to solve real world problems,
but also how formulas is an effective and fast method to getting answers.

Bibliography

Alden,

Earlham College – Geology 211 – Radiometric Dating,

Kerr,
Richard A. “Stretching the Reign of Early Animals”. Science, May
2000, Vol. 288   Issue               5467,
page 789.

LeVarge,
Sheree. “Carbon Dating.” BioMath: Carbon
Dating,
www.biology.arizona.edu/biomath/tutorials/applications/carbon.html.

Potassium-Argon Dating,
www.anth.ucsb.edu/faculty/stsmith/classes/anth3/courseware/Chronology/09_Potassium_Argon_Dating.html.

Time Scale,

Appendix

Figure 1: