# How Old Is That Rock? Roll the Dice & Use Radiometric Dating to Find Out

 Difficulty Time Required Short (2-5 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

### MLA Style

Science Buddies Staff. "How Old Is That Rock? Roll the Dice & Use Radiometric Dating to Find Out" Science Buddies. Science Buddies, 26 Mar. 2013. Web. 30 Aug. 2014 <http://www.sciencebuddies.org/science-fair-projects/project_ideas/Geo_p044.shtml>

### APA Style

Science Buddies Staff. (2013, March 26). How Old Is That Rock? Roll the Dice & Use Radiometric Dating to Find Out. Retrieved August 30, 2014 from http://www.sciencebuddies.org/science-fair-projects/project_ideas/Geo_p044.shtml

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Last edit date: 2013-03-26

## 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; the second hand of a clock ticks every minute. 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 below 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 below. 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 three-letter 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 below 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 potassium-40 isotope (which contains 19 protons, 40 nucleons, and is represented by the atomic symbol K) will change into the argon-40 isotope (which contains 18 protons, 40 nucleons, and is represented by the symbol Ar). When this happens, potassium-40, 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 argon-40) 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:

1. Radioactive decay processes happen at a stable measurable rate characterized by the half-life time. The half-life 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.
2. The steady, atom-by-atom transformation of one isotope to another is not affected by any influence of the environment outside the nucleus.
3. Nature offers a number of unstable isotopes with half-life times ranging from several billion years to only a ten-thousandth 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 potassium-40 (abbreviated as K-40) that decays into the daughter isotope argon-40 (abbreviated as Ar-40), scientists measured the half-life time to be 1.25 billion years. This means that half of the K-40 atoms existing today will have made the transformation to Ar-40 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 K-40 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 K-40 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 K-40 isotopes are left (half of the half left over after 1.25 billion years), geologists can say that 2.5 billion years (double the half-life 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 below 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, K-40) 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 half-life time of the parent isotope K-40).

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 below, 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 ( $N$ ) by the original amount of parent isotopes in the sample ( $N_{0}$ ), the geologists calculate a ratio termed ${N}/{N_{0}}$ . They can then use this ratio ( ${N}/{N_{0}}$ ) in Equation 1 to calculate the time since formation of the sample ( $T$ ) to determine the age of the sample.

Equation 1:

 $N$ [Please enable JavaScript to view equation] : number of parent isotope currently left in the sample $N_{0}$ [Please enable JavaScript to view equation] : original number of parent isotope in the sample $T$ [Please enable JavaScript to view equation] : time since formation $T_{1/2}$ [Please enable JavaScript to view equation] : half life time of the parent isotope

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 below shows the decay curve for the potassium (K-40) isotope. Can you figure out that the half-life time of K-40 is 1.25 billion years from the graph? Can you also figure out that 1/4 of the K-40 parent isotopes in the sample are left after 2.5 billion years, and only 1/8 of the K-40 parent isotopes remain after 3.75 billion years? How long before all of the K-40 parent isotopes decay?

 Figure 4. An example of the decay curve of potassium (K-40). 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 half-life 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
• Unstable
• Parent isotope
• Daughter isotope
• Half-life time
• Probability
• Statistical fluctuations
• 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 half-life times compare? How are they similar and how are they different?

## Bibliography

Examples of objects dated by radiometric dating can be found at the following site:

The following article provides a real-world example of radiometric dating:

## Materials and Equipment

• Six-sided dice (100), can be purchased online from Amazon.com or at a board-game 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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## 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 six-sided 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 below 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.
Table 1. Table showing how parts of the isotope decay model match up scientifically with the parts of radioactive isotope decay.
1. First you will prepare the model.
1. Take each die, one at a time, and place a sticker on one side of each die.
2. Place all 100 dice, with stickers on one side, in your pot.
3. Write "Parent isotopes in sample" on a sticky note and affix it to the pot.
4. Using the permanent marker, write "Daughter isotopes in sample" on the plastic bag.
2. Next, in your lab notebook, create a data tFable like Table 2 below. 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 ...
Table 2. In your lab notebook, create a data table like this one to record your results. You will record the number of parent isotopes left in your sample after each time unit passes (i.e., after each roll of dice) for three different trials and average the values over the three trials. The data table also lists the fraction of parent isotopes remaining, which is the number of remaining isotopes (averaged over the three tests) divided by the original number of parent isotopes in the sample (100 in this case). The data collected will be used to graph the decay curve of this particular isotope.
1. 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 half-life time.
1. 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.
2. Roll the dice out on the floor or table.
3. 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."
4. Count the number of parent isotopes (dice) remaining on the floor.
5. Note this number in your data table under "Number of parent isotopes left" for the following time slot.
6. Place all the remaining dice (parent isotopes only) in your pot marked "Parent isotopes in sample" with the sticky note.
7. Repeat steps b to f until all parent isotopes have decayed (i.e., there are no dice remaining to be put in the pot.)
8. 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.
2. Repeat step 3 at least two more times for a total of at least three trials.
1. 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?
3. 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.
4. 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.
1. 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.
5. 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 Half-Life 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 half-life time of your isotope. Remember, the half-life time of an isotope is the time it takes for half of the initial amount of isotopes to decay. You will then compare the half-life time you obtained using your data to the predicted half-life time using probability. How close will your half-life time be to the calculated one?

1. 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.)
1. The decay curve has the elapsed time (i.e., the number of times you rolled the dice) on the x-axis. Determine the range of your x-axis based on the data in your data table, divide your x-axis in appropriate equal-length units so all the values fit on your axis, and add reference numbers and labels.
2. The decay curve has the fraction of parent isotopes remaining in your sample represented on the y-axis. Determine the range of your y-axis based on the data in your data table, divide your axis in appropriate equal length units, and add reference numbers and labels.
3. 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.
4. For any graph you make, always check if you have labeled your axes and added reference numbers.
2. Now determine the half-life time from your decay curve. Figure 4 in the introduction can provide clues on how to read the half-life time from the decay curve. Once you have found it, label the half-life time on your decay curve similarly to how it is labeled in Figure 4, using a different color pen.
3. Next you will calculate the half-life time of your particular isotope based on the probability that each isotope will decay within a unit of time passing. Table 3 below is partly filled in to show you how to get started doing this. Make a data table like Table 3 below in your lab notebook and completely fill it in.
1. 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 below for the first few rolls.
1. 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.
2. 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.
2. Continue these calculations and fill out the data table in your lab notebook until no parent isotopes remain.
1. Also be sure to fill out the "Fraction of Parent Isotopes Remaining" column.
3. 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 half-life time.
 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 ... ... ... ...
Table 3. In your lab notebook, make a data table like this one to calculate the number of parent isotopes remaining in a sample over time and determine the half-life time of your isotope based on probability.
1. How does your calculated half-life time compare with the half-life 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 half-life time? Hint: Think about statistical fluctuations, which is 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 six-sided 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).

1. Following are the things your partner should do:
1. Have a paper and pen handy.
2. Place all the dice in the pot.
3. Roll the dice out on the floor or table.
4. 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."
5. Mark a tally on the paper indicating the number of times the dice have been rolled.
6. Place all the remaining dice (parent isotopes) in the pot labeled "Parent isotopes in the sample."
7. Repeat steps 1c through 1f until your partner decides to stop.
8. 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.
2. Following are the things you should do:
1. In your lab notebook, make a data table like Table 4 below.
2. 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
Table 4. Create a data table in your lab notebook like this one to record your results.

3. Calculate the fraction of parent isotopes remaining and write it down in your data table.
4. 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.
5. Repeat step 2d but this time use the data table you created based on probability, the one similar to Table 3.
6. Ask your partner to see his or her tallies, then count them and write them down in your data table.
1. 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 time-lapse methods more accurate than the other? Why do you think this is so?
2. 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?
3. Can you find parameters that influence your accuracy? How do you think you could make your predictions more accurate?
4. 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?

## Variations

• Use different dice (five-sided, eight-sided, ten-sided, etc.) or a coin to set up your model. Would you expect any differences in the decay probability, the decay curve, the half-life time, the accuracy of time estimations, etc.?
• Glue stickers on additional sides of your six-sided 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 half-life 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 half-life time read from the decay curve with the calculated half-life 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:

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