Half-life Simulation

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
• Half-life
• First-order 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 “non-decayed” dice remaining that will be rolled in Round 2. This process will be repeated 10 times. The “half-life” of dice will be determined by graphing the mass of dice remaining after each round.

Materials

Procedure

1. Select two student volunteers. Have one student place the data table transparency on the overhead projector.
2. 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.
3. 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. Re-roll any dice that do not land flat.
4. 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.
5. Mass the tub and dice on the balance. Record the mass of dice that decayed in the appropriate column of the data table.
6. 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.
7. Repeat steps 2–6 using only the dice that did NOT decay in the previous round.
8. Repeat step 7 nine more times.
9. Place the decay curve transparency on the overhead projector.
10. Plot the data from the data table on the decay curve transparency.

Student Worksheet PDF

12640_Student1.pdf

12640_Teacher1.pdf

Teacher Tips

• For an interesting post-lab discussion, demonstrate decay using two decay numbers or even three decay numbers rather than just one. (If the assigned decay numbers are 1 and 6, discard all of the dice that land on either of these numbers.) Compare the shape of the decay curve for two or three decay numbers versus that for just one decay number. All of the decay curves will have the same general shape, but the rate of decay will be greater—and the half-life will be smaller—as the number of decay numbers increases.
• The rate of decay will decrease, and the half-life will increase, as the number of sides on the dice increases (assuming only one number is chosen for the “decay number”). See Table 1 for theoretical half-life values (number of dice rolls) for multifaceted dice as a function of the number of decay numbers assigned. A lab kit using these multifaceted dice (Radioactivity Half-Life Simulation, Flinn Catalog No. AP6721) is available from Flinn Scientific.
  {12640_Tips_Table_1_Theoretical Half-Life Values for the Decay of Multifaceted Dice}
  This activity is an excellent opportunity for students to develop spreadsheet skills. The data can be compiled in a Microsoft™ Excel™ spreadsheet, decay curves created from the data, and cell equations created to calculate the various half-lives.
• Radioactive decay is a first-order process—the rate of decay depends on the initial number of atoms or nuclei in the sample (No). After any time t has elapsed, the number of nuclei remaining in the sample may be calculated using Equation 1, where k is the rate constant for radioactive decay. The half-life is the time needed for one-half of the nuclei in a sample to decay. Thus, t½ is equal to the time elapsed when N = 0.5 No. Equation 1 can be rearranged to give the equation for the half-life (Equation 2):
  {12640_Tips_Equation_1}
  ln(0.5 N_o/N_o) = ln(0.5) = –kt½
  {12640_Tips_Equation_2}

Sample Data

¹The mass of dice remaining will equal the mass of dice (initial) for the next round.

Answers to Questions

1. 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}
2. Determine the half-life: Choose two points on the y-axis, 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 one-half 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 one-half of the dice to decay. The estimated half-life is three rounds.

3. Verify the half-life value by choosing another set of two points on the y-axis. Is the half-life 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 one-half of the dice to decay. The half-life appears to have a constant value (t_½ = 3 rounds). However, as the mass of dice being rolled decreases, the half-life is no longer constant. Radioactive decay is a random process. Large numbers of dice are needed to simulate a random process.

4. Predict the value of the half-life with that obtained using the same-sided dice but a different “decay number.” Does the half-life depend on the “decay number” that was assigned? Explain, based on probability.

   The half-life for six-sided 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.

5. Using the value of the half-life 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 half-life is equal to three rounds, then 9 rounds equal three half-lives. The mass of dice remaining after 3 half-lives should be 33 (each arrow in the following sequence represents one half-life).
   Half-lives

   {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!
6. Using the concept of half-life, predict the number of rounds that would be needed to reduce the mass of dice from 10,000 g to 625 g using 6-sided dice and one “decay” number.

   Each arrow in the following sequence represents one half-life (3 rounds). Four half-lives (12 rounds) would be needed to reduce the mass of dice from 10,000 g to 625 g.
   Half-lives

   {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 long-lived radioactive nuclei. Polonium-218, for example, emits alpha particles and decays very quickly—within minutes. Uranium-238 also decays via alpha-particle 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 half-lives. The half-life (t_½) of a radioactive isotope (called a radioisotope) is the amount of time needed for one-half of the atoms in a sample to decay. Every radioisotope has a characteristic half-life which is independent of the total number of atoms in the sample. Thus, the half-life of polonium-218 is about three minutes, while the half-life of uranium-238 is more than 4 billion years. Regardless of the total number of atoms in a sample of polonium-218, one-half of the atoms will always “disappear” (decompose to produce other atoms) within three minutes.

To better understand half-lives, consider the following example. Iodine-131 is an artificial radioisotope of iodine that is produced in nuclear reactors for use in medical research and in nuclear medicine. It has a half-life of eight days. If 32 grams of iodine-131 are originally produced in a nuclear reactor, after eight days only 16 grams of iodine-131 will remain. After two half-lives, or 16 days, only 8 grams will be left, and after three half-lives (24 days) only 4 grams will be left. Every eight days, the amount of iodine-131 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 heads-up in a box to start, shaking the box, and then discarding the coins that land tails-up or “decay.” Since the probability of a specific coin “decaying” (landing tails-up) 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).

A simulated “radioactive decay curve” obtained by graphing the data (see Figure 1) shows that the “half-life” of coins is equal to “one coin toss.” The number of coins remaining in the box decreases by 50% after each coin toss.