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# Investigating Exponential Decay (2 Favorites)

ACTIVITY in Half Lives, Radioactive Isotopes. Last updated July 1, 2020.

### Summary

In this activity, students will learn about radioactivity, exponential decay, and half-life through two hands-on experiences.

### Grade Level

High School

### NGSS Alignment

This activity will help prepare your students to meet the performance expectations in the following standards:

**HS-PS1-8:**Develop models to illustrate the changes in the composition of the nucleus of the atom and the energy released during the processes of fission, fusion, and radioactive decay.**Scientific and Engineering Practices**:- Using Mathematics and Computational Thinking
- Developing and Using Models
- Analyzing and Interpreting Data

### Objectives

By the end of this activity, students should be able to

- Explain the concepts of
*radioactivity*,*exponential decay*, and*half-life*in the field of nuclear chemistry. - Calculate the amount of radioactive sample remaining after a given number of half-lives have occurred.
- Analysis data to determine the number of half-lives that have occurred in a given scenario.
- Graph collected data and interpret the results.

### Chemistry Topics

This activity supports students’ understanding of

- Nuclear chemistry
- Half-life
- Radioactivity
- Exponential decay (alpha, beta, gamma)

### Time

**Teacher Preparation**: 10 minutes**Lesson**: 30 minutes

### Materials

Part A (per group, or group of 2):

- Blank paper
- Pencil or pen
- Scissors

Part B (per group):

- Plastic cup
- 20 Skittles
- Calculator

### Safety

- Food in the lab should be considered a chemical not for consumption.

### Teacher Notes

- During Part A of the activity, each student will need their own scissors and paper, or it can be shared between partners.
- During Part B of the activity, students need to understand that the exponential decay might not be a perfect decay graph due to the limited sample size of the Skittle “atoms”. Multiple trials may be preferred.
- Part B of the activity can be completed in pairs or small groups of 3-4 students.
- An Answer Key has been provided for teacher reference.

### For The Student

### Lesson

### Objective

To understand the concept of *half-life*.

### Background

Large atomic nuclei, with a disproportionate ratio of neutrons to protons, are unstable and will “decay" over time. When radioactive decay occurs, energy is lost and carried off by *gamma*, *alpha*, or *beta* particles. Not all atoms of a substance do this at the exact same moment, but by studying a quantity of a radioactive substance, scientists are able to determine the time it takes for half of the nuclei in a given sample to decay. This is called the *half-life*.

The parent nuclei decay into more stable “daughter” nuclei. Uranium-238 decays through a series of elements to become lead-206. Once stable, the atom is no longer considered radioactive.

If the half-life of a certain substance is 3 days and you begin with 40 grams, after 3 days you will have 20 grams of radioactive substance, and 20 grams of stable substance. After another 3 days, or the second half-life, you will have 10 grams of radioactive substance and 30 grams of stable substance. So, after 2 half-lives, you have 10g out of the original 40g, or one-fourth of your original radioactive substance.

Original amount = 40 g After one half-life, or 3 days = 40g x 1/2 = 20g After second half-life, or 6 days = 40g x 1/2 x 1/2 = 10g 30g is now stable after 6 days, but 10g is still radioactive. |

Let's see how this works with some practice.

### Materials

**Part A:**- Scissors
- 1 sheet of paper
- pencil or pen

**Part B**- Twenty Skittles
- cup
- paper towel or sheet of paper

### Part A

**Activity Procedure and Questions **

- Cut a large circle out of a piece of paper. This represents 100% of your radioactive material.
- Let's say one half-life goes by. Cut your circle in half. Label one side "decayed" and set it aside.
- Now you have “half” of your original material, or ½, or 50% that is still radioactive.
- Cut the radioactive piece in half, making it look like 2 identical wedges. This represents another half-life. Label one wedge "decayed" and set it aside.
- Now you are left with ¼ or 25% of your original material. How many half-lives have gone by? ______________
- A third half-life goes by. Demonstrate this with your radioactive wedge. What fraction of the original amount is left? Draw the wedge below:
- If a fourth half-life went by, what fraction of radioactive substance would be left?
- Demonstrate the fourth half-life by cutting the wedge in half.
- Do you see that even after 4 half-lives you still have radioactive material?
- Try this problem: You have 50 grams of a radioactive substance and two half-lives go by. How much mass of the original substance would still be radioactive?
- Try this problem: The half-life of radon-222 is 3.824 days. After what time will one fourth of the original amount of radon remain?
- Another example: The half-life of iodine-131 is 8.040 days. What percentage of an iodine-131 sample will remain after 24.12 days?

### Part B

**Activity Procedure and Questions **

- Place 20 Skittles in the cup. These are the
*parent nuclei*. - Decide if "S” side-up will be the radioactive or decayed particles. Record the designated 'radioactive' side of the candy:__________________
- Put the lid on the container (or use your hand if no lid is provided) and shake the cup so that the Skittles will change position and tumble about.
- Carefully pour the items onto a sheet of paper. Separate the particles by designated side-up.
- Put aside the particles that are “decayed”. Count the “radioactive” particles and record in the data table for “1” half-life.
- Put only the radioactive particles back into the container. Shake it, and again pour the contents onto your sheet of paper.
- Separate the particles like before, setting aside the decayed particles, and counting the radioactive particles. Record the number of radioactive particles in the data table under “2” half-lives.
- Repeat procedure 7, recording the data in the appropriate place, until there are no radioactive particles left.
- Graph the number of parent nuclei versus the number of half-lives on the graph paper provided below.
- Draw a smooth line that best fits the points. Describe what your graph looks like:
- Is there a relationship between the number of radioactive particles remaining and the number of shakes? Explain.
- Cobalt-60 has a half-life of 10.47 minutes. How many milligrams of cobalt-60 remain after 52.35 minutes if you start with 10.0mg?
- The half-life of polonium-218 is 3.0 minutes. If you start with 24.0 mg, how long will it be before only 3.0 mg remain?
- Explain the concept “half-life”:

**Data Table** (Add more columns if needed)

Shake number/ half-life |
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |

Number of radioactive nuclei remaining |
20 (starting amount) |

**Graph **(Label and number the axes)