Difficulty  
Time Required  Short (25 days) 
Prerequisites  None. 
Material Availability  Readily available. 
Cost  Low ($20  $50) 
Safety  No issues. 
Abstract
What do rocks and clocks have in common? Both keep track of time. Yes, radioactive isotopes present in rocks and other ancient material decay atom by atom at a steady rate, much as clocks tick time away. Geologists use those radioactive isotopes to date volcanic ash or granite formations like the giant Half Dome in Yosemite National Park. Anthropologists, archeologists, and paleontologists also use radioactive isotopes to date mummies, pottery, and dinosaur fossils. Does this sound abstract and complicated? It is no more complicated than playing a dice game! In this science project you will see for yourself by modeling radioisotope dating with a few rolls of the dice.Objective
Create a model of radioactive decay using dice and test its predictive power on dating the age of a hypothetical rock or artifact.
Credits
Sabine De Brabandere, Ph.D., Science Buddies
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Introduction
As humans, it seems easy for us to keep track of time lapses, as long as they range from a couple of seconds to a number of years. That is what we encounter in our daily lives, right? The Earth orbits the Sun in about one year's time, the Earth rotates on its axis every 24 hours, 60 ticks of the second hand on a clock indicates 1 minute has passed. Geologists have a much harder job keeping track of time. Studying the Earth and its evolution, they work with time scales of thousands to billions of years. Where can they find a clock to measure these huge time periods? Or on a slightly smaller scale, where can paleontologists find a clock to tell the age of fossils, or how can archeologists determine how old ancient pottery and buried artifacts are?
Geologists (along with paleontologists, archeologists, and anthropologists) actually turn to the elements for answers to their geological time questions. We and everything around us are made of atoms. Atoms are tiny. They are mostly empty space with a denser tiny area called the nucleus and a cloud of electrons surrounding the nucleus. The nucleus itself is made of protons and neutrons, collectively called nucleons. Figure 1 provides a visual representation of an atom.
Figure 1. Representation of an atom with its nucleus and an electron cloud around it. Note that, in this drawing, the nucleus is shown disproportionately large.
The number of protons within an atom's nucleus is called the atomic number. It determines the identity of the atom. The atomic number is important for locating an element on the periodic table, shown in Figure 2. You might have seen the periodic table in your science textbook or displayed on a poster in the classroom. What do you know about it?
Figure 2. Periodic table showing elements with their atomic symbol and atomic numbers.
In the periodic table, each entry represents an element. The element is listed by its atomic symbol, a one, two or threeletter long label. For example, gold's atomic symbol is Au. Above the atomic symbol, each entry lists the element's atomic number; e.g., the element gold (Au) has an atomic number of 79.
While an element always has the same atomic number, meaning it has the same number of protons in its nucleus, it can have a different number of total nucleons in its nucleus. Scientists call these different variations of the same element isotopes of each other. For example, the element potassium (which always has 19 protons in its nucleus) occurs in nature in three forms: an isotope with 39 nucleons (19 protons and 20 neutrons), one with 40 nucleons (19 protons and 21 neutrons), and one with 41 nucleons (19 protons and 22 neutrons) .
Some isotopes are radioactive. Any idea what the word radioactive means? Radioactive refers to the characteristic that these isotopes are unstable and tend to fall apart. They emit, or radiate, particles in their conversion to stability. We call this process radioactive decay. Isotopes exhibit a range of radioactive decay processes. Resources provided in the Bibliography enable you to research this topic in more detail. We will explore only the decay processes of interest to geologists.
Geologists who want to date objects are interested in the isotopes that change identity as they undergo radioactive decay. In other words, they change their number of protons during radioactive decay and turn into a different element. As an example, the potassium40 isotope (which contains 19 protons, 40 nucleons, and is represented by the atomic symbol K) will change into the argon40 isotope (which contains 18 protons, 40 nucleons, and is represented by the symbol Ar). When this happens, potassium40, which is emitting particles in its conversion to a more stable form, is called the parent isotope. The isotope that is created during the process (here argon40) is called the daughter isotope. The particles emitted in the process are what we call radiation.
It is now time to explore why geologists are so interested in these radioactive decay processes as a means of dating objects. But before we do, can you list some characteristics a good clock should have? Predictable, reliable... but what do these words mean? Can you describe them in more detail? Now, try to link the clock characteristics you just listed to the characteristics of radioactive decay that appeal to geologists:
 Radioactive decay processes happen at a stable measurable rate characterized by the halflife time. The halflife time is the time period after which the remainder of the parent isotopes is half of what you start out with. Do not worry if this sounds confusing; the following example will help clarify.
 The steady, atombyatom transformation of one isotope to another is not affected by any influence of the environment outside the nucleus.
 Nature offers a number of unstable isotopes with halflife times ranging from several billion years to only a tenthousandth of a second, allowing for "clocks" that can tell wide ranges of time.
Could you link these to your list of characteristics of a good clock?
This example might help clarify the processes and terms just introduced: Looking at the parent isotope potassium40 (abbreviated as K40) that decays into the daughter isotope argon40 (abbreviated as Ar40), scientists measured the halflife time to be 1.25 billion years. This means that half of the K40 atoms existing today will have made the transformation to Ar40 at some point during the next 1.25 billion years, no matter what weather they experience, pressure they undergo, or any other outside circumstances. Science cannot predict which particular K40 atom in this sample will decay and which will not during the next 1.25 billion years, but that is OK. It can predict what happens on average. It is like flipping a huge amount of coins: you know that the likelihood, or probability, is that you will end up with half of them heads up, but you have no idea which particular one will end up heads, or if even half of them will be heads for sure.
So, can radioactive isotopes be used as a clock? Can geologists say that once the amount of K40 isotopes in the sample has reduced to half its original amount, 1.25 billion years will have gone by? Yes — as long as they use a big enough sample so statistical fluctuations average out. To take it a step further, once only 1/4 of the original amount of K40 isotopes are left (half of the half left over after 1.25 billion years), geologists can say that 2.5 billion years (double the halflife time) have gone by. Now, can you predict how much time has gone by if only 1/8 is left? You can probably see now that as the sample ages, fewer and fewer parent isotopes will be present in the rock, so the rock will be less and less radioactive. Figure 3 shows a graphical representation of this example.
Figure 3. Representation of an aging rock. The radioactivity levels are indicated by wiggly arrows; green dots represent parent isotopes (here, K40) and yellow dots represent daughter isotopes present in the rock at the indicated time after the formation of the rock. Snapshots of the rock are taken after multiples of 1.25 billion years (the halflife time of the parent isotope K40).
So, how do geologists use radioactive decay as clocks to measure the age of a sample? Using a technique called radiometric dating, geologists take a sample of the material and measure the number of parent and daughter isotopes present in the sample. Adding these two values gives the original amount of parent isotopes in the sample. Geologists can then use Equation 1, referred to as the radioactive decay formula, to determine the age of a sample. Specifically, by dividing the number of parent isotopes currently left in the sample ( ) by the original amount of parent isotopes in the sample ( ), the geologists calculate a ratio termed . They can then use this ratio ( ) in Equation 1 to calculate the time since formation of the sample ( ) to determine the age of the sample.
Equation 1:

Is this radioactive decay formula intimidating? If so, try not to worry: This science project will only use its graphical representation, known as the decay curve . Coming back to our example, Figure 4 shows the decay curve for the potassium (K40) isotope. Can you figure out that the halflife time of K40 is 1.25 billion years from the graph? Can you also figure out that 1/4 of the K40 parent isotopes in the sample are left after 2.5 billion years, and only 1/8 of the K40 parent isotopes remain after 3.75 billion years? How long before all of the K40 parent isotopes decay?
Figure 4. An example of the decay curve of potassium (K40). This figure also illustrates how to use a decay curve to figure the time since formation, if the fraction of parent isotope remaining in the sample is known. The red lines show how to obtain the halflife time, or the time after which half of the parent isotopes have decayed. The green and pink lines guide you to the time after which only 1/4 and 1/8, respectively, of the parent isotopes remain. The arrows indicate how to read the graph, starting from a fraction of parent isotope remaining via a horizontal line to a point on the curve, and then vertically down to a time on the time axis.
Does this still seem a bit abstract? This geology science project will guide you through the process of radiometric dating, enabling you to explore and fill in the blanks. It explains how to create a model of radioactive decay using dice. The model will behave the same way as isotopes in nature in important ways. You will create a decay curve for your hypothetical rare isotope, and use it to estimate the time since formation of hypothetical samples created by a friend.
Terms and Concepts
 Atoms
 Nucleons
 Atomic number
 Periodic table
 Element
 Isotopes
 Radioactive
 Unstable
 Radioactive decay
 Parent isotope
 Daughter isotope
 Halflife time
 Probability
 Statistical fluctuations
 Radiometric dating
 Decay curve
Questions
 What are some important characteristics of isotope decay that make them interesting to geologists?
 How many isotopes (parent and daughter isotopes together) are present in a rock at any given time if, at formation, that rock had 1 trillion isotopes?
 Would you choose the same radioactive isotope to date material expected to be about 10,000 years old as material that is expected to be billions of years old?
 How do the decay curves of different isotopes with different halflife times compare? How are they similar and how are they different?
Bibliography
 Goldman, M. (n.d.). Isotopes and radioactivity. Physics 2000. Retrieved July 17, 2012, from http://www.colorado.edu/physics/2000/isotopes/index.html
 Nuclear Science Division  Lawrence Berkeley National Laboratory. (2007, March 30). The ABC's of nuclear science. Retrieved February 18, 2013, from http://www.lbl.gov/abc/Basic.html#Nuclearstructure
 Kids.Net.Au. (2013). Radiometric dating. Retrieved February 18, 2013, from http://encyclopedia.kids.net.au/page/ra/Radiometric_dating
Examples of objects dated by radiometric dating can be found at the following site:
 U.S. Geological Survey. (2001, June 13). Radiometric time scale . USGS. Retrieved January 22, 2013, from http://pubs.usgs.gov/gip/geotime/radiometric.html
The following article provides a realworld example of radiometric dating:
 Grens, K. (2013, March 1). Coral clocks . The Scientist. Retrieved March 8, 2013, from http://www.thescientist.com/?articles.view/articleNo/34463/title/CoralClocks/
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Materials and Equipment
 Sixsided dice (100), can be purchased online from Amazon.com or at a boardgame shop or the toy section of a large department store.
 Stickers (100), small enough to fit on one side of a die
 Pot big enough to hold all 100 dice
 Sticky note
 Plastic bag to hold all the dice
 Permanent marker
 Lab notebook
 Graphing paper and pencil or pen or a graphing computer program
 Partner (volunteer)
 Colored (i.e., not black) pen or thin marker
 Paper and pen
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I Did This Project! Please log in and let us know how things went.Experimental Procedure
Creating a Radioactive Decay Model and Plotting the Decay Curve
In this part of the science project, you will make a model to explore radiometric dating. The model uses 100 sixsided dice, where each die represents one isotope in a radioactive sample used for dating. You will roll the dice to represent one unit of time passing, during which the parent isotopes have a chance to decay into the daughter isotopes. How much of a chance? Or, in other words, what is the probability of decay? One in six! Why? You will put a sticker on one side of the dice and if a die lands with the sticker facing up, this will represent that isotope decaying into the daughter isotope. If the sticker is not facing up, it means that the isotope has not decayed yet, so further rolls of the dice will decide when this parent isotope decays. You will collect the daughter isotopes in a separate bag so they can no longer decay and only use the remaining parent isotopes in the following roll. Table 1 lists the relation between model and real life.
Radioactive Isotope Decay Model  
Part of the Model  What It Scientifically Represents 
Die  Isotope 
Total number of dice  Total amount of isotopes (parent and daughter together) in the sample 
Number of dice in pot  Number of parent isotopes in the sample 
Number of dice in bag labeled "Daughter isotopes in sample"  Number of daughter isotopes in the sample 
Die after roll  side facing up has no sticker  Isotope remaining a parent isotope 
Die after roll  side facing up has a sticker  Isotope decaying into a daughter isotope 
Chance of a die landing with the sticker facing up  Chance for a parent isotope to decay to the daughter isotope within the next time unit 
Rolling the dice  One unit of time passing (in this case, the time in which 1/6 of the isotopes decay). At zero rolls, the material has just formed. 
 First you will prepare the model.
 Take each die, one at a time, and place a sticker on one side of each die.
 Place all 100 dice, with stickers on one side, in your pot.
 Write "Parent isotopes in sample" on a sticky note and affix it to the pot.
 Using the permanent marker, write "Daughter isotopes in sample" on the plastic bag.
 Next, in your lab notebook, create a data table like Table 2. You will record your results in this data table.
Time  Number of Parent Isotopes Left  Fraction of Parent Isotopes Remaining  
Trial 1  Trial 2  Trial 3  Average Value  
0  100  100  100  100  1 
1  
2  
3  
... 
 Collect data for a decay of 100 isotopes over time and record your results in the data table. This table will be used later to graph its decay curve and determine its halflife time.
 Start by writing down "100" for the "Number of parent isotopes left" in your data table for Trial 1 (you will use 100 dice in your sample). For this first roll the time will be 1.
 Roll the dice out on the floor or table.
 Remove from the sample all the dice with the "daughter isotope" sticker facing up. These represent the isotopes that decayed during this given time unit. Collect them in the bag labeled "Daughter isotopes in the sample."
 Count the number of parent isotopes (dice) remaining on the floor.
 Note this number in your data table under "Number of parent isotopes left" for the following time slot.
 Place all the remaining dice (parent isotopes only) in your pot marked "Parent isotopes in sample" with the sticky note.
 Repeat steps 3.b.–3.f. until all parent isotopes have decayed (i.e., there are no dice remaining to be put in the pot).
 Note that, at any given time during the process, the number of parent isotopes (dice in your pot) plus the number of daughter isotopes (dice in your bag) adds up to 100, which is the initial number of parent isotopes in the sample. Atoms neither disappear nor are they created; they just change identity.
 Repeat step 3 at least two more times for a total of at least three trials.
 Knowing that the decay of an isotope (a dice rolling and showing one particular side facing up) is a statistical process, do you expect variations between the values obtained in your different tests? Can you observe these variations?
 Calculate the average of the number of parent isotopes left for each elapsed time and write it down in the "Average Value" column of your data table.
 Calculate the fraction of parent isotopes remaining using the average numbers obtained in step 5 and write it down in the "Fraction of Parent Isotopes Remaining" column of your data table.
 For example, if your average value of parent isotopes left after one roll (time is 1) is 85, the fraction of parent isotopes left would be 85 divided by 100, or 0.85.
 Your data table should now be completely filled in and ready to use in making a decay curve graph.
Graph the Decay Curve and Determine the HalfLife Time
In this part of the science project, you will create a graph of the decay curve of your isotope and use your curve to determine the halflife time of your isotope. Remember, the halflife time of an isotope is the time it takes for half of the initial amount of isotopes to decay. You will then compare the halflife time you obtained using your data to the predicted halflife time using probability. How close will your halflife time be to the calculated one?
 Having the collected data for your isotope decay organized in your data table, it is time to graph the decay curve. Get a pen and graphing paper ready, or ready a particular graphing tool on your computer if you are familiar with one. (For an example of a decay curve, see Figure 4 in the Introduction in the Background tab.)
 The decay curve has the elapsed time (i.e., the number of times you rolled the dice) on the xaxis. Determine the range of your xaxis based on the data in your data table, divide your xaxis in appropriate equallength units so all the values fit on your axis, and add reference numbers and labels.
 The decay curve has the fraction of parent isotopes remaining in your sample represented on the yaxis. Determine the range of your yaxis based on the data in your data table, divide your axis in appropriate equal length units, and add reference numbers and labels.
 Create the graph by plotting all the data points and connecting them by a continuous line, as shown in Figure 4. There you have it— the decay curve of your isotope.
 For any graph you make, always check if you have labeled your axes and added reference numbers.
 Now determine the halflife time from your decay curve. Figure 4 in the introduction can provide clues on how to read the halflife time from the decay curve. Once you have found it, label the halflife time on your decay curve similarly to how it is labeled in Figure 4, using a different color pen.
 Next you will calculate the halflife time of your particular isotope based on the probability that each isotope will decay within a unit of time passing. Table 3 is partly filled in to show you how to get started doing this. Make a data table like Table 3 in your lab notebook and completely fill it in.
 After each roll, based on probability, 1/6 of the parent isotopes will decay to daughter isotopes and 5/6 will remain parent isotopes. Look at how these calculations have been made in Table 3 for the first few rolls.
 For example, of the 100 parent isotopes you started out with, after the first roll 17 (1/6 of 100) are expected to decay into daughter isotopes and 83 (5/6 of 100) are expected to remain parent isotopes.
 In the second roll, 1/6 of the 83 left after the first roll will decay, creating 14 daughter isotopes, and 5/6 of 83, or 69, will remain parent isotopes and be left to decay in future rolls of the dice.
 Continue these calculations and fill out the data table in your lab notebook until no parent isotopes remain.
 Also be sure to fill out the "Fraction of Parent Isotopes Remaining" column.
 After how many rolls are you likely to have approximately 50 parent isotopes remaining, or after how many rolls will the fraction of parent isotopes remaining be equal to 0.5? This is your calculated halflife time.
 After each roll, based on probability, 1/6 of the parent isotopes will decay to daughter isotopes and 5/6 will remain parent isotopes. Look at how these calculations have been made in Table 3 for the first few rolls.
Time (number of rolls)  Number of Parent Isotopes before Roll  Number of Parent Isotopes Decayed into Daughter Isotopes This Roll  Number of Parent Isotopes Remaining  Fraction of Parent Isotopes Remaining 
0  100  0  100  1 
1  100  100 X 1/6 = 17  100 X 5/6 = 83  0.83 
2  83  83 X 1/6 = 14  83 X 5/6 = 69  0.69 
3  69  69 X 1/6 = ...  69 X 5/6 = ...  ... 
4  ...  ...  ...  ... 
5  ...  ...  ...  ... 
 How does your calculated halflife time compare with the halflife time read from the decay curve? If they are different, why do you think they are? How do you think you could make your collected data even closer to the calculated halflife time? Hint: Think about statistical fluctuations, which are discussed in the Introduction in the Background tab.
Do the Decay Test! Can You Amaze Your Partner?
It this section, you will ask a volunteer partner to roll the 100 sixsided dice, simulating the decay of isotopes in your sample just as you did to collect data for the decay curve. Your partner decides after how many rolls of the dice he or she would like to stop. Your partner will hand you over the bag of daughter isotopes and the pot of parent isotopes when they have finished. Your task is to use the sample (bag with the daughter isotopes and pot with the parent isotopes) and then estimate the number of times your partner rolled the dice (or the elapsed time of your sample).
 Following are the things your partner should do:
 Have a paper and pen handy.
 Place all the dice in the pot.
 Roll the dice out on the floor or table.
 Remove all the dice with the "daughter isotope" sticker facing up from the sample and place them in the bag labeled "Daughter isotopes in the sample."
 Mark a tally on the paper indicating the number of times the dice have been rolled.
 Place all the remaining dice (parent isotopes) in the pot labeled "Parent isotopes in the sample."
 Repeat steps 1.c.–1.f. until your partner decides to stop.
 Once your partner stops, ask him or her to give the bag and pot back to you— but do NOT allow your partner to tell you how many times he or she rolled the dice at this point.
 Following are the things you should do:
 In your lab notebook, make a data table like Table 4.
 Count the number of parent isotopes remaining in the sample (number of dice in the pot) and write it down in your data table.
Trial Number Number of Parent Isotopes in Sample Fraction of Parent Isotopes Remaining Predicted Time Lapse Based on Decay Curve Predicted Time Lapse Based on Probability Actual Time Lapses (Number of Tallies on the Paper) 1 2 3
 Calculate the fraction of parent isotopes remaining and write it down in your data table.
 Use your decay curve to estimate the number of times your partner rolled the dice (the elapsed time since formation of your sample) and write it down in your data table.
 Repeat step 2.d. but this time use the data table you created based on probability, the one similar to Table 3.
 Ask your partner to see his or her tallies, then count them and write them down in your data table.
 Analyze your data:
 Compare the last three columns (predicted time lapse based on your decay curve, predicted time lapse based on probability, and actual time lapse) of your data table. How accurate are your predictions? Was one of your predicted timelapse methods more accurate than the other? Why do you think this is so?
 See if you can observe trends in your accuracy: Are your estimations more accurate when the real time lapse is short, long, or somewhere in the middle?
 Can you find parameters that influence your accuracy? How do you think you could make your predictions more accurate?
 After all this work, do you see how geologists used their creativity and ingenuity to find accurate "clocks" in their quest to date ancient material? Did it surprise you how a statistical process like radioactive decay—where you cannot predict what will happen with individual isotopes—still lets you deduce specific information?
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I Did This Project! Please log in and let us know how things went.Variations
 Use different dice (fivesided, eightsided, tensided, etc.) or a coin to set up your model. Would you expect any differences in the decay probability, the decay curve, the halflife time, the accuracy of time estimations, etc.?
 Glue stickers on additional sides of your sixsided dice to set up your model, but be consistent with all of the dice (e.g., put stickers on three sides of all of your dice). Would you expect any differences in the decay probability, the decay curve, the halflife time, the accuracy of time estimations, etc.?
 This science project used a sample size of 100 dice. As a variation, do the experiment again with a different sample size (e.g., 200, 150, 50, or 25 dice total). How does the sample size affect the accuracy of the decay curve and time readings? If you use more dice, is it more or less accurate at telling time than when you used 100 dice? What about when you use fewer dice?
 In this science project, you compare the halflife time read from the decay curve with the calculated halflife time. As a variation, do the comparison for different fractions of the initial amounts of parent isotopes remaining (e.g., 1/4, 3/4, 1/8, etc.) See if you can find and explain trends in accuracy (e.g., graph readings for smaller fractions remaining are more or less accurate).
 Study how sample size affects the accuracy of the estimations by allowing your partner to choose how many dice he or she likes to start out with (i.e., letting your partner choose the sample size). In this variation, you do not change the sample size to graph the decay curve or make your probability data table, only the test sample involving a partner changes. When trying to figure out how many rolls your partner has made, be sure to start with the number of parent isotopes that he or she decides to use in the sample size.
If radioactive decay processes intrigue you, the following two project ideas might grab your attention:
 Particles in the Mist: See Radioactive Particles Decay with Your Own Cloud Chamber!
 Watching Nuclear Particles: See Background Radiation Zoom Through A Cloud Chamber
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