Transcript Slide 1

Module 05-GSA-01
Understanding Radioactivity in Geology
The Basics of Decay
What does popping a bag of popcorn
teach us about radioactive decay?
Prepared for SSAC by
C E Stringer, University of South Florida - Tampa
Quantitative concepts and
skills
Exponential function
Graph, exponential
Graph, trendline
Probability
© The Washington Center for Improving the Quality of Undergraduate Education. All rights reserved. 2005
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Preview
This module is the first of a series on radioactive decay and how it is used to quantify
the age of geologic materials. This subject is fundamental to understanding the
magnitude of geologic time, the rate of geologic processes, and the quantitative
history of the Earth. The subject is a mathematical one, involving the proportional
relation between rate of reaction and quantity of reactant (“parents”); the declining
amount of the parents as expressed by an exponential decay function; and the
concept of a constant half-life.
The goal of this module is to give you a basic understanding of radioactive decay and
the mathematics used to describe it. The module is based on an analogy with the
mathematics of popping a pot of popcorn. Slides 3 and 4 give you background
information on radioactive decay, and Slide 5 introduces a problem designed to help
you understand the mathematics of the decay by means of the popcorn analogy.
Slides 5, 6, and 7 introduce Excel spreadsheets and graphs that help you solve the
problem. Slides 8, 9, and 10 summarize what you learned in your problem-solving
experience and relate the problem back to general radioactive decay processes.
Slide 11 consists of some questions that constitute your homework assignment.
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What do we mean when we say radioactive decay?
Terminology: Forms of an element with the
same atomic number but different mass
numbers (meaning they have different numbers
of neutrons) are called isotopes.
When a nuclide decomposes (or decays) to form a
different nuclide, it is called a radioisotope. The
phenomenon is called radioactivity.
When a radioisotope decays to form a
different nuclide, it also emits a
particle. The three initial types of
particles recognized were α-particles,
β-particles, and γ-radiation.
The radioisotope can also be thought of as
the “parent” and the nuclide it decays to
can be termed the “daughter.”
Here is an example of a decay equation:
Uranium is the
parent nuclide.
92 is the
atomic number
and 238 is the
mass number.
U  Th He
238
92
234
90
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The helium atom is
an a- particle.
Thorium is the daughter nuclide.
Remember that the atomic number is the number of protons in an atom’s nucleus and
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the mass number is the number of protons plus neutrons!
Radioactive Decay
When a parent decays to a daughter product,
that daughter product may decay again to a
more stable form. These transformations take
place until a stable, non-radioactive isotope is
formed. The series of reactions is referred to
as a decay series or decay chain.
There are three naturally-occurring
decay series: the U-238, Th-232, and U235 chains.
Example
The figure on the right shows the
Uranium-238 series. Uranium-238 is the
parent nuclide and Lead-206 is the
stable, final daughter nuclide. The
column on the left tells you what type of
radiation is emitted in each decay
reaction.
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From www.compumike.com/ science/halflife1.php
So, why am I supposed to be thinking about popcorn?
Suppose you put 1000 kernels of popcorn in a
popcorn popper and raise the temperature to a
constant level hot enough for the kernels to
begin popping. Each kernel of popcorn has the
potential to pop, but they don’t all begin popping
at the same time. If the heat is left at a constant
level for a long enough period of time, most of
the kernels will eventually pop but you won’t
know which one will pop at which time.
Cell C3 is the number of kernels you start
with in the popper.
Column B lists the numbers of seconds
that have passed. Remember we are
thinking in 10-second intervals.
Let’s say that each kernel has a 10% probability of
popping during any 10-second interval. What would
happen after a 10-second interval? Create the
Excel spreadsheet shown below to find out!
B
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3
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5
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Set up Column C to calculate the number of kernels remaining
unpopped after each 10-second period. Create an absolute
reference in Cell C7 by typing =$C$3. The formula in Cell C8
should be
=C7-$C$4*C7.
C
Nstart=
Ppop=
sec
0
10
20
30
40
50
D
1000
0.1 per 10 sec
Nremain
1000
900
810
729
656
590
Cell C4 is the probability that a kernel
will pop in a
10-second interval.
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What happens in the 10-second intervals after the first one?
Expand your Excel spreadsheet to find out.
First, create fifteen 10-second intervals in
Column B.
Because we assumed that our 10%
probability of popping remains the same, we
can simply copy and paste our formula from
Cell C8 down the column.
Create Column D to look at the ratio of
popped to unpopped kernels. This is
simply the number of kernels remaining
after each 10-second interval divided by
the number of kernels you started with.
What does N/N0 tell you about the rate of
popcorn popping over time?
B
2
3
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5
6
7
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C
Nstart=
Ppop=
sec
0
10
20
30
40
50
60
70
80
90
100
110
120
130
140
150
D
E
F
1000
0.1 per 10 sec
Nremain
1000
900
810
729
656
590
531
478
430
387
349
314
282
254
229
206
N/N0
1.000
0.900
0.810
0.729
0.656
0.590
0.531
0.478
0.430
0.387
0.349
0.314
0.282
0.254
0.229
0.206
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Looking at Popcorn Popping Graphically
Create a graph by plotting the seconds on
the x-axis and number of kernels remaining
(or parents) on the y-axis
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C
Nstart=
Ppop=
sec
0
10
20
30
40
50
60
70
80
90
100
110
120
130
140
150
D
E
F
1000
0.1 per 10 sec
Nremain
1000
900
810
729
656
590
531
478
430
387
349
314
282
254
229
206
N/N0
1.000
0.900
0.810
0.729
0.656
0.590
0.531
0.478
0.430
0.387
0.349
0.314
0.282
0.254
0.229
0.206
Number Parents Remaining
B
Insert a trendline and use the option tab to
display the line equation and correlation
coefficient.
1000
900
800
700
600
500
400
300
200
100
0
y = 1000e - 0.0105x
2
R = 1
0
50
100
150
Time (sec)
The half-life of the popcorn is the time at
which half of the kernels remain unpopped.
Use your trendline equation to determine
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the half-life of our popcorn sample set.
The Popcorn Popping Function
In the last slide, we determined the function that
describes the number of our unpopped popcorn
kernels over time:
y = 1000e-0.0105x
The general version of this function is:
N = Noe-lt
where:
No is the starting number of kernels.
l is the probability any given kernel will pop.
t is time.
N is the number of unpopped kernels at time t .
We said earlier in the module
that our probability of a kernel
popping is 10% in a ten-second
period.
If this is the case, then why is the λ in
our equation 0.0105 instead of
0.0100?
Start thinking about your answer to this
question and we will explore it in depth
in the next module.
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You might be asking yourself, “Why are we STILL thinking about popcorn?”
Interesting question.
Just as a kernel pops into a
piece of popcorn…
So does a radioactive atom
of a parent isotope decay to
a radiogenic atom of the
daughter isotope.
In each case, we don’t know exactly when the individual (kernel,atom) will convert (pop, decay).
What we know is only the probability that it will occur in the next time interval!
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Conclusion
Popping popcorn is analogous to radioactive decay in the sense
that there is a certain probability that a nucleus will decay or that
a kernel of corn will pop in any given time interval.
This probability is consistent over time and is also known
as the decay constant (commonly denoted as l).
The decay constant (l) is related to the half-life (t1/2) of the
nuclide by the following equation:
l= ln(2)/t1/2
We will look at the derivation of this equation in the next module.
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References
“Clocks in Rocks” is a vital element of how we know what we know about geologic
time. There are many excellent references on radiometric dating and its context. We
particularly recommend G. Brent Dalrymple (2004), Ancient Earth, Ancient Skies: The
Age of the Earth and its Cosmic Surroundings, Stanford University Press, 247 pp.
See particularly, Chapter 4: “Clocks in Rocks: How Radiometric Dating Works.”
See Also –
http://serc.carleton.edu/quantskills/methods/quantlit/RadDecay.html
http://serc.carleton.edu/quantskills/activities/popcorn.html
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End-of-module assignments
1. Answer the question on Slide 7: determine the half-life of our popcorn sample
using the trendline equation.
2. If you had a kernel population of 3742 and a popping probability of 6%, how
would your results differ? Change your values and hand in this spreadsheet
with a graph of the new example.
3. We have discussed the half-life of our popcorn kernels. Thinking in the same
way, what do you think the third-life of the kernels is? (Hint, N is defined as the
number of kernels remaining at time, t, and so the 1/3-life is longer than the 1/2life, because 1/3<1/2). Report a general definition as well as a numerical
value.
4. Suppose you have a population of 2000 radon-222 (222Rn) atoms. The
probability that 222Rn will decay in a one-day period is .211 or 21.1%. How
many atoms of 222Rn will remain after 30 days? What is the half-life of 222Rn?
Turn in your spreadsheet that you use to arrive at your answer.
5. How is radioactive decay similar to popcorn popping? How is it different?
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