PHY 1407– Conceptual Physics II

A Simple Model of Radioactive Decay and Half-life

 

Leader: _________________________          Recorder: __________________________

Skeptic: _________________________         Encourager: ________________________

 

Materials

M&M'sÔ

SkittlesÔ

2 x Clean Paper Plates

Clean Styrofoam cup

Laptop (for graphs)

 

Introduction

      In this simple exercise, we will explore a simple model of radioactive decay where M&M'sÔ will represent the "radioactive" substance.

 

Procedure

1.  Count out about 50 M&M'sÔ and verify that each one has an M on one side.  Record the initial number of M&M'sÔ  as the number remaining for step 0 in the table below.

2.  Shake the M&M'sÔ in the cup and toss them onto a paper plate.  Remove the M&M'sÔ that have the M showing and count the number remaining. Record in the row labeled step 1 the number of M&M'sÔ remaining and the cumulative number removed.  (For the first step the cumulative number removed is simply the number removed. For later steps you add the number removed in that step to the previous cumulative number removed.)

3.  Repeat the trials until all the M&M'sÔ are gone.  Note that you may need more or less than number of steps provided in the table.  You may eat the M&M'sÔ if you so desire.

 

Step

Number Remaining

Number Removed

Cumulative Number removed

0

 

0

0

1

 

 

 

2

 

 

 

3

 

 

 

4

 

 

 

5

 

 

 

6

 

 

 

7

 

 

 

8

 

 

 

 

Use LoggerPro or Excel to construct a properly labeled graph of the number of M&M'sÔ vs. step.  Print and attach the graph to the report.

 

Q1)  Describe the shape of the graph.

The number of remaining M&M'sÔ  is an example of an exponential decay.  How rapidly something decays can be described in terms of a half-life.  The half-life is defined as the time it takes for the amount of radioactive substance to decay to half.  Note the half-life does not refer only to the initial amount.  If there is initially 1.0 gram then the half-life is the time for the substance to decay to 0.5 g.  But the remaining 0.5 g will decay to 0.25 g in a time equal to the half-life and that will decay to 0.125 g in a time equal to the half-life and so on.

 

Q2)  Approximately what fraction of the initial amount of M&M'sÔ were left after the first step?

 

 

Q3)  Approximately what fraction of the M&M'sÔ  remaining after the first step were left after the second step?

 

 

Q4)  Approximately what fraction of the M&M'sÔ  remaining after the second step were left after the third?

 

 

P5)  If there are 10 M&M'sÔ  initially, predict approximately how many will remain after 1 step.

 

 

Place 10 M&M'sÔ  in the cup and toss them on the plate.  Remove the ones with the m’s facing up, and count the remaining M&M'sÔ .

 

Q6)  Was your prediction, p5), approximately correct.  Note it may not be exactly correct because of random chance.

 

 

Q7)  In terms of number of steps, what is the "half life" of the M&M'sÔ  for this experiment?

 

 

Q8) In each half life, half of the remaining amount is reduced.  From the initial amount, approximately what fraction will remain after 1 step?

 

 

 

Q9)  From the initial amount, approximately what fraction will remain after 2 steps?

 

 

Q10)  From the initial amount, approximately what fraction will remain after 3 steps?


Part 2 – Improving the Statistics

You probably noticed that your data may not look exactly like the smooth graph of an exponential decay that you would see in a textbook.  This is a perfectly reasonable observation since an exponential curve assumes a very large number of nuclei to begin with.  We can improve the appearance of our graph by starting with many more M&M’sÔ.  However, instead of redoing the experiment, we can obtain the same effect simply by combining the data from all of the groups.  Add your data to the table on the chalk board at the front of the classroom. 

Step

Number M&MÔs Remaining

For Group 1

Number M&MÔs Remaining

For Group 2

Number M&MÔs Remaining

For Group 3

Number M&MÔs Remaining

For Group 4

Number M&MÔs Remaining

For Group 5

Number M&MÔs Remaining

For Group 6

Total for each step

0

 

 

 

 

 

 

 

1

 

 

 

 

 

 

 

2

 

 

 

 

 

 

 

3

 

 

 

 

 

 

 

4

 

 

 

 

 

 

 

5

 

 

 

 

 

 

 

6

 

 

 

 

 

 

 

7

 

 

 

 

 

 

 

8

 

 

 

 

 

 

 

Once all of the groups have listed their data, sum the total for each step.  Construct a properly labeled graph of the Number M&MÔs remaining vs. step for the combined data from all of the groups.  Print and attach your graph to the report.

 

Q11)  Does the combined data seem to give a smoother exponential curve?

 

 

 

Q12)  Collecting data from a larger sample is sometimes referred to as “improving the statistics”.  Describe the effects on the data form using a larger number of data points.

 

 

 

Part 3 – What happens to a Nuclei when it Decays?

 

Q13)  What do you think?  Does a nucleus completely disappear when it decays?  Explain.

 

 

 

 

In this part, we will investigate what happens to a nucleus when it decays.  The simple answer is that a nucleus doesn’t disappear when it decays.  Instead it turns into various products including different nuclei and other particles.  Repeat the procedure for part 1, except when you remove an M&MÔ replace it with a skittle.  Record both the number of M&M'sÔ and the number of SkittlesÔ for each step.

 

Step

Number M&MÔs Remaining

Number of Skittles Ô Remaining

0

 

0

1

 

 

2

 

 

3

 

 

4

 

 

5

 

 

6

 

 

7

 

 

8

 

 

 

On the same axis, make a properly labeled graph of  both the number of M&M'sÔ and the number of SkittlesÔ versus step number.

 

Q14)  Compare the Skittles column to the cumulative Number removed column from part 1.  Do they seem to behave the same way?  Why?

 

 

 

Q15)  For your data from part 1, what is true about the sum of the number remaining and the cumulative number removed?

 

 

Q16)  What is true about the sum of the number of remaining  M&M'sÔ  and SkittlesTM at each step for this part of the activity?

 

 

Q17)  What conservation law is approximately shown by this sum?