12: Radioactivity Simulation

PURPOSE

• To simulate radioactive decay by studying a random process, specifically rolling dice.
• To better understand the concept of half-life (\( t_{1/2} \)) by observing how the number of dice remaining decreases with each round.
• To determine the "half-life" of the dice by graphing the number of dice remaining after each round.
• To study how a spontaneous and completely random process (radioactive decay) can be simulated using the probability of a specific outcome in another random process (like rolling a die on a "decay number").

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 particular atom will decay after a certain period can be simulated by studying other random processes, such as a coin toss or a “roll of the dice.”

Radioactive nuclei disintegrate through different processes at varying rates. The time required for different radioactive nuclei to decay varies widely, from seconds or minutes for highly unstable nuclei to billions of 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 (\( t_{1/2} \)) 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-life, 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. One could make the rule that a coin landing on 'heads' does not decay, but if the coin lands on 'tails', it decays. Imagine placing 100 coins heads-up in a box to start, shaking the box, and then discarding the coins that land tails or “decay.” Since the probability of a specific coin “decaying” (landing tails) 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 the table below). A simulated “radioactive decay curve” obtained by graphing the data (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.

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 by rolling 10 dice ten times in Round 1 and recording the number of dice that display a specific “decay number,” for example, all dice that read six. (Rolling 10 dice ten times is equivalent to rolling 100 dice once.) The total number of dice that “decayed” (landed on six) during Round 1 will then be counted and subtracted from the total number of dice rolled. This is the number of dice remaining that will be rolled in Round 2. This process will be repeated until no dice remain. The “half-life” of dice will be determined by graphing the number of dice remaining after each round.