Introduction
Radioactive decay is a spontaneous and completely random process. There is no way to predict how long it will take a specific atom of a radioactive isotope to disintegrate and produce a new atom. The probability, however, that a specific atom will decay after a certain period of time can be simulated by studying other random processes, such as a coin toss or a “roll of the dice.”
Concepts
 Radioactive decay
 Halflife
 Firstorder rate
 Probability
Experiment Overview
The purpose of this activity is to simulate radioactive decay by studying the probability of a random process—rolling dice. The “radioactive decay” of dice will be studied having students roll 100 dice in Round 1 and separating the dice that display a specific “decay number,” for example, all dice that read six. The dice that “decayed” (landed on six) during Round 1 will then be massed and subtracted from the total mass of dice rolled. This is the mass of “nondecayed” dice remaining that will be rolled in Round 2. This process will be repeated 10 times. The “halflife” of dice will be determined by graphing the mass of dice remaining after each round.
Materials
Balance, 0.01g precision, 300g capacity Cardboard box or other container Data table transparency* Decay curve transparency* Dice, sixsided, 100* Marker, transparency Overhead projector Plastic tub, small *Materials included in kit.
Procedure
 Select two student volunteers. Have one student place the data table transparency on the overhead projector.
 Have one student tare the plastic tub on the balance, then add 100 dice to the tub. Record the initial mass of 100 dice on the overhead transparency.
 Assign a decay number. Have the students roll all 100 dice in a cardboard box or other container. Hint: Roll the dice so they all have an opportunity to have a “random” roll and all of the dice land flat. Reroll any dice that do not land flat.
 Assume that any die landing with the assigned decay number face up has “decayed.” Remove these “decayed” dice from the box and place them in the plastic tub.
 Mass the tub and dice on the balance. Record the mass of dice that decayed in the appropriate column of the data table.
 Subtract the mass of dice that decayed from the initial mass of 100 dice to determine the mass of dice remaining. Record the “Mass of dice remaining” in the data table and enter this value as the initial mass of dice for the second round.
 Repeat steps 2–6 using only the dice that did NOT decay in the previous round.
 Repeat step 7 nine more times.
 Place the decay curve transparency on the overhead projector.
 Plot the data from the data table on the decay curve transparency.
Correlation to Next Generation Science Standards (NGSS)^{†}
Science & Engineering Practices
Asking questions and defining problems Developing and using models Planning and carrying out investigations Analyzing and interpreting data Using mathematics and computational thinking
Disciplinary Core Ideas
HSPS1.A: Structure and Properties of Matter
Crosscutting Concepts
Cause and effect Patterns Scale, proportion, and quantity Systems and system models Energy and matter Structure and function
Performance Expectations
MSPS14: Develop a model that predicts and describes changes in particle motion, temperature, and state of a pure substance when thermal energy is added or removed.
Sample Data
{12640_Data_Table_3}
^{1}The mass of dice remaining will equal the mass of dice (initial) for the next round.
Answers to Questions
 Graph the results obtained for the “radioactive decay” of dice. Note: Include a point on the graph for the mass of dice “remaining” after zero rolls of the dice (Round zero).
{12640_Answers_Figure_2}
 Determine the halflife: Choose two points on the yaxis, where the first point (mass of dice remaining) is about twice as large as the second point (e.g., 200 g of dice and 100 g of dice). How many rounds are needed for onehalf of the dice to decay?
Compare the results for Round 4 (122 g of dice) and round 7 (60 g of dice). Three rounds are required for approximately onehalf of the dice to decay. The estimated halflife is three rounds.
 Verify the halflife value by choosing another set of two points on the yaxis. Is the halflife a “constant” for the decay of the dice?
Compare the results for Round 2 (180 g of dice) and round 5 (92 g of dice)—three rounds are required for approximately onehalf of the dice to decay. The halflife appears to have a constant value (t_{½} = 3 rounds). However, as the mass of dice being rolled decreases, the halflife is no longer constant. Radioactive decay is a random process. Large numbers of dice are needed to simulate a random process.
 Predict the value of the halflife with that obtained using the samesided dice but a different “decay number.” Does the halflife depend on the “decay number” that was assigned? Explain, based on probability.
The halflife for sixsided dice should not depend on the "decay number" that was selected. There is an equal probability (1/6 or 16.7%) of rolling any decay number.
 Using the value of the halflife determined in Questions 2 and 3, predict what mass of dice should have remained after 9 rounds. Compare this with the number of dice that actually remained (see the data table). What factors might account for any difference between the predicted and actual number of dice remaining after 9 rounds?
If one halflife is equal to three rounds, then 9 rounds equal three halflives. The mass of dice remaining after 3 halflives should be 33 (each arrow in the following sequence represents one halflife). Halflives
{12640_Answers_Figure_3}
Mass of dice
(228 g → 114 g → 67 g → 33 g)
The actual mass of dice remaining after 9 rounds was 30 g!
 Using the concept of halflife, predict the number of rounds that would be needed to reduce the mass of dice from 10,000 g to 625 g using 6sided dice and one “decay” number.
Each arrow in the following sequence represents one halflife (3 rounds). Four halflives (12 rounds) would be needed to reduce the mass of dice from 10,000 g to 625 g. Halflives
{12640_Answers_Figure_4}
Mass of dice
(10,000 g → 5000 g → 2500 g → 1250 g → 625 g)
Discussion
Radioactive nuclei disintegrate via different processes and at different rates. The amount of time required for different radioactive nuclei to decompose varies widely, from seconds or minutes for very unstable nuclei to a billion years or more for longlived radioactive nuclei. Polonium218, for example, emits alpha particles and decays very quickly—within minutes. Uranium238 also decays via alphaparticle production, but the decay takes place over billions of years! The relative rate of decay of different radioactive isotopes is most conveniently described by comparing their halflives. The halflife (t_{½}) of a radioactive isotope (called a radioisotope) is the amount of time needed for onehalf of the atoms in a sample to decay. Every radioisotope has a characteristic halflife which is independent of the total number of atoms in the sample. Thus, the halflife of polonium218 is about three minutes, while the halflife of uranium238 is more than 4 billion years. Regardless of the total number of atoms in a sample of polonium218, onehalf of the atoms will always “disappear” (decompose to produce other atoms) within three minutes.
To better understand halflives, consider the following example. Iodine131 is an artificial radioisotope of iodine that is produced in nuclear reactors for use in medical research and in nuclear medicine. It has a halflife of eight days. If 32 grams of iodine131 are originally produced in a nuclear reactor, after eight days only 16 grams of iodine131 will remain. After two halflives, or 16 days, only 8 grams will be left, and after three halflives (24 days) only 4 grams will be left. Every eight days, the amount of iodine131 that remains will decrease by 50%.
The process of radioactive decay may be modeled by studying a random process such as a coin toss. Imagine placing 100 coins headsup in a box to start, shaking the box, and then discarding the coins that land tailsup or “decay.” Since the probability of a specific coin “decaying” (landing tailsup) is 50%, we would predict that only 50 coins will remain in the box after the first coin toss. Repeating the coin toss with 50 coins remaining in the box should result in an additional 25 coins “decaying” in the second round, and so on (see Table 2).
{12640_Discussion_Table_2_“Radioactive Decay” of Coins}
A simulated “radioactive decay curve” obtained by graphing the data (see Figure 1) shows that the “halflife” of coins is equal to “one coin toss.” The number of coins remaining in the box decreases by 50% after each coin toss.
{12640_Discussion_Figure_1_“Radioactive decay” of coins}
