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Physics 001.002.013 Radioactive Decay Problems

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Physics 001.002.013 Radioactive Decay Problems

Alignment

Learning Intentions

By the end of the lesson, students will be able to:

  • Solve radioactive decay problems using .
  • Determine the number of half-lives, , from elapsed time and half-life.
  • Use arithmetic methods, such as repeated halving and fraction reasoning, to solve decay problems.
  • Use decay graphs to estimate remaining quantity, original quantity, elapsed time and half-life.
  • Recognise that number of undecayed nuclei, mass, count rate and activity can all follow the same half-life pattern when proportional to the number of radioactive nuclei.

Success Criteria

By the end of the lesson, students have successfully:

  • Identified , and in a radioactive decay problem.
  • Calculated using .
  • Substituted values into correctly.
  • Solved for remaining amount, initial amount, elapsed time or number of half-lives.
  • Estimated half-life from a graph by finding repeated halving.
  • Explained why radioactive decay produces an exponential decay curve rather than a straight-line decrease.

Syllabus Reference

  • Unit 1: Thermal, Nuclear and Electrical Physics
  • Topic 2: Ionising Radiation and Nuclear Reactions
  • Nuclear Model and Stability
  • Solve radioactive decay problems using and other arithmetic or graphical methods.

Phenomenon

A medical radioisotope is prepared in a hospital before being used in imaging. The sample is active enough at 8:00 am, but by the afternoon the activity has decreased. The sample has not been used up like fuel; instead, unstable nuclei have spontaneously decayed.

The key question is:

How can we predict how much radioactive material remains after a given time?

Radioactive decay is useful because even though we cannot predict exactly when one unstable nucleus will decay, we can predict the behaviour of a large sample using half-life.

Key Idea

Radioactive decay is modelled using repeated halving. After each half-life, half of the undecayed radioactive nuclei remain.

The main equation is:

where:

  • is the remaining number, mass, activity or count rate after decay.
  • is the original number, mass, activity or count rate.
  • is the number of half-lives that have passed.

The number of half-lives is found using:

where:

  • is the elapsed time.
  • is the half-life.

Concept

The concept and thought that best describes the cause of the phenomenon is below.

Radioactive decay is random for individual nuclei but predictable for large samples. Each unstable nucleus has a fixed probability of decaying in a given time interval. This means the same fraction of nuclei decays in each half-life, not the same number.

For example, if a sample begins with undecayed nuclei:

Half-lives passedFraction remainingNumber remaining

This repeated halving produces an exponential decay curve.

Convention

The key conventions associated with the concept and in the branch of established knowledge is below.

  • Use for the original amount.
  • Use for the remaining amount.
  • Use for half-life.
  • Use for the number of half-lives.
  • If the problem gives time, first calculate using .
  • The units for and must match.
  • Count rate and activity can be treated like when they are proportional to the number of undecayed nuclei.
  • Activity is commonly measured in becquerels, .
  • Count rate may be measured in counts per second or counts per minute.
  • On a decay graph, half-life is the time taken for the quantity to halve from any starting value.

Misconceptions

Common misconceptions students have regarding the concept when applying to various situations and solving problems. It could be a conceptual, mathematical or logical misconception.

  • Students may think radioactive decay removes the same number of nuclei each half-life rather than the same fraction.
  • Students may confuse with the number of nuclei instead of the number of half-lives.
  • Students may use directly in instead of first calculating .
  • Students may forget to convert time units before calculating .
  • Students may think a sample reaches exactly zero after many half-lives; mathematically, the model approaches zero but does not reach zero.
  • Students may think half-life depends on the original amount of material. It does not; half-life is a property of the radionuclide.
  • Students may misread a graph by measuring the time for the graph to drop by a fixed vertical amount instead of the time for it to halve.

Further Reading

  • Radioactive decay simulations using dice, coins or counters.
  • Medical imaging radioisotopes and why short half-lives are useful.
  • Radiometric dating and how half-life allows scientists to estimate age.
  • Exponential decay graphs and comparison with linear relationships.

Explicit Instruction

Teacher Explanation

Radioactive decay problems can be solved in three main ways:

  1. Formula method:

  1. Arithmetic repeated-halving method:

  1. Graphical method:

Use a decay graph to estimate:

  • the amount remaining after a time,
  • the time taken to reach a quantity,
  • the half-life by finding repeated halvings.

A recommended problem-solving method is:

  1. Write the known values.
  2. Check that the time units match.
  3. Calculate the number of half-lives using .
  4. Substitute into .
  5. Check whether the answer makes physical sense.

Worked Examples

Worked Example 1

A radioactive sample contains undecayed nuclei. Its half-life is hours. How many undecayed nuclei remain after hours?

Known:

Find .

Therefore, undecayed nuclei remain after hours.

Arithmetic check:

Worked Example 2

A radioisotope has an initial activity of . After half-lives, what is its activity?

Known:

Find .

Since activity is proportional to the number of undecayed nuclei, use the same model:

Misplaced & A &= A_{0}\left(\dfrac{1}{2}\right)^n \\ A &= 960\left(\dfrac{1}{2}\right)^3 \\ A &= 960\left(\dfrac{1}{8}\right) \\ A &= 120 \text{ Bq} \end{align}

Therefore, the activity is .

Worked Example 3

A sample has a half-life of days. After some time, its mass decreases from to . How much time has passed?

Known:

Find .

Use arithmetic halving:

This is half-lives.

Misplaced & t &= nT_{\frac{1}{2}} \\ t &= 3 \times 12 \\ t &= 36 \text{ days} \end{align}

Therefore, days have passed.

Graphical Method

A decay graph shows the amount remaining on the vertical axis and time on the horizontal axis.

To estimate half-life from a graph:

  1. Choose a clear starting value on the vertical axis.
  2. Find half of that value.
  3. Read across to the curve.
  4. Drop down to the time axis.
  5. The time difference is the half-life.
  6. Repeat with another halving to check consistency.

Example:

If a graph shows:

Time, Count rate

Then the half-life is because the count rate halves every .

Check for Understanding

Check 1

A sample contains undecayed nuclei. After one half-life, how many undecayed nuclei remain?

Answer:

Check 2

A sample has a half-life of hours. How many half-lives pass in hours?

Answer:

half-lives.

Check 3

A sample has an initial activity of . After half-lives, what is its activity?

Answer:

.

Investigation (Alternative to Explicit)

Hypothesis

If a large number of dice are used to model radioactive nuclei, then the number of dice remaining after each roll will decrease exponentially because approximately the same fraction is removed each trial.

Data Collection

Equipment:

  • dice, coins or counters
  • Cup or container
  • Data table
  • Graph paper or spreadsheet

Method using dice:

  1. Each die represents one radioactive nucleus.
  2. Roll all dice.
  3. Remove every die that lands on . These represent nuclei that have decayed.
  4. Count the dice remaining.
  5. Record the trial number and number remaining.
  6. Repeat until only a small number remain.
  7. Plot number remaining against trial number.

Suggested data table:

Trial numberNumber remaining

Analysis

Students:

  • Plot number remaining on the vertical axis.
  • Plot trial number on the horizontal axis.
  • Draw a smooth curve of best fit.
  • Estimate the trial number where the sample falls from to .
  • Estimate the trial number where the sample falls from to .
  • Compare these estimates to decide whether the half-life is approximately constant.

Guiding questions:

  • Is the graph linear or curved?
  • Does the same number decay each trial?
  • Does approximately the same fraction decay each trial?
  • Why is the graph smoother when more dice are used?
  • How is the dice model similar to radioactive decay?
  • How is the dice model limited?

Evaluation

Strengths:

  • The model shows random decay of individual particles.
  • The model shows predictable patterns for large samples.
  • The graph resembles exponential decay.

Limitations:

  • Dice have a probability of per roll, not necessarily per trial.
  • Real radioactive decay is not caused by rolling or external action.
  • Small samples produce more random variation.
  • Real half-life is measured using time, not trial number.

Problems

The following problems are designed to develop arithmetic, algebraic and graphical methods for radioactive decay.

Problem 1

A radioactive sample contains undecayed nuclei. How many remain after half-life?

Problem 2

A radioactive sample contains undecayed nuclei. How many remain after half-lives?

Problem 3

A radioisotope has a half-life of hours. How many half-lives pass in hours?

Problem 4

A sample has an initial activity of . Its half-life is hours. What is its activity after hours?

Problem 5

A radioactive sample has a mass of . After half-lives, what mass remains?

Problem 6

A radioisotope has a half-life of days. How long does it take for a sample to decrease from to ?

Problem 7

A sample decreases from undecayed nuclei to undecayed nuclei. How many half-lives have passed?

Problem 8

A sample has an initial count rate of . After minutes, its count rate is . What is the half-life?

Problem 9

A graph of radioactive decay shows that the activity decreases from to between minutes and minutes. It then decreases from to between minutes and minutes. Estimate the half-life.

Problem 10

A student says, “After two half-lives, all of the radioactive material has decayed because each half-life removes half.” Explain the error in this statement.

Problem 11

A sample has half-life hours and initial activity .

Complete the table:

TimeNumber of half-livesActivity

Problem 12

A sample has an initial activity of and a half-life of days. Estimate the activity after days.

Problem 13

A radioactive isotope used in imaging has a half-life of hours. A hospital receives a sample with activity at am. Estimate its activity at pm on the same day.

Problem 14

A decay graph shows that the count rate falls from to over minutes. Estimate the count rate after minutes.

Problem 15

A sample has an initial mass of and a half-life of years. Use the formula method to determine the mass remaining after years.

Followup

Self-check

Students should be able to answer the following questions:

  • Can I identify whether the question gives , , , or ?
  • Can I calculate the number of half-lives using ?
  • Can I use correctly?
  • Can I solve the same problem by repeated halving?
  • Can I estimate half-life from a graph?
  • Can I explain why decay is exponential?
  • Can I explain why a radioactive sample does not halve by the same absolute amount each time?

Exit Ticket

  1. A sample has and hours. Find after hours.
  2. A sample decreases from to . How many half-lives have passed?
  3. A graph shows that a sample falls from counts per minute to counts per minute in minutes. What is the half-life?

Answers:

  2. half-lives
  3. minutes

Next Topic

Energy in nuclear reactions: electron volts, joules, mass defect and the mass-energy equivalence relationship.

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