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What does popping a bag of popcorn teach us about radioactive decay?

The Basics of Decay

Prepared for SSAC byC E Stringer, University of South Florida - Tampa

© The Washington Center for Improving the Quality of Undergraduate Education. All rights reserved. 2005

Understanding Radioactivity in Geology

Module 05-GSA-01

Quantitative concepts and skillsExponential functionGraph, exponentialGraph, trendlineProbability

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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.

Preview

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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.

The radioisotope can also be thought of as the “parent” and the nuclide it decays to can be termed the “daughter.”

Remember that the atomic number is the number of protons in an atom’s nucleus and the mass number is the number of protons plus neutrons!

HeThU 42

23490

23892

Here is an example of a decay equation:

Uranium is the parent nuclide.

92 is the atomic number and 238 is the mass number.

Thorium is the daughter nuclide.

The helium atom is an particle.

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.

What do we mean when we say radioactive decay?

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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 U-235 chains.

From www.compumike.com/ science/halflife1.php

ExampleThe 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.

Radioactive Decay

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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.

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 C D23 Nstart= 10004 Ppop= 0.1 per 10 sec56 sec Nremain7 0 10008 10 9009 20 810

10 30 72911 40 65612 50 590

Cell C3 is the number of kernels you start with in the popper.

Cell C4 is the probability that a kernel will pop in a 10-second interval.

Column B lists the numbers of seconds that have passed. Remember we are

thinking in 10-second intervals.

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.

So, why am I supposed to be thinking about popcorn?

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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.

B C D E F23 Nstart= 10004 Ppop= 0.1 per 10 sec56 sec Nremain N/N07 0 1000 1.0008 10 900 0.9009 20 810 0.81010 30 729 0.72911 40 656 0.65612 50 590 0.59013 60 531 0.53114 70 478 0.47815 80 430 0.43016 90 387 0.38717 100 349 0.34918 110 314 0.31419 120 282 0.28220 130 254 0.25421 140 229 0.22922 150 206 0.206

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?

What happens in the 10-second intervals after the first one?

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B C D E F23 Nstart= 10004 Ppop= 0.1 per 10 sec56 sec Nremain N/N07 0 1000 1.0008 10 900 0.9009 20 810 0.81010 30 729 0.72911 40 656 0.65612 50 590 0.59013 60 531 0.53114 70 478 0.47815 80 430 0.43016 90 387 0.38717 100 349 0.34918 110 314 0.31419 120 282 0.28220 130 254 0.25421 140 229 0.22922 150 206 0.206

y = 1000e -0.0105x

R 2 = 1

0100200

300400500600700

800900

1000

0 50 100 150

Time (sec)

Nu

mb

er P

aren

ts R

emai

nin

g

Create a graph by plotting the seconds on the x-axis and number of kernels remaining (or parents) on the y-axis

Insert a trendline and use the option tab to display the line equation and correlation coefficient.

The half-life of the popcorn is the time at which half of the kernels remain unpopped. Use your trendline equation to determine the

half-life of our popcorn sample set.

Looking at Popcorn Popping Graphically

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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-t

where:No is the starting number of kernels. 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.

The Popcorn Popping Function

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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!

You might be asking yourself, “Why are we STILL thinking about popcorn?”

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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 ).

The decay constant () is related to the half-life (t1/2) of the nuclide by the following equation:

= ln(2)/t1/2

We will look at the derivation of this equation in the next module.

Conclusion

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“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.”

http://serc.carleton.edu/quantskills/methods/quantlit/RadDecay.html

See Also –

http://serc.carleton.edu/quantskills/activities/popcorn.html

References

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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 1/2-life, 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?

End-of-module assignments