Physics 001.002.012 Half-Life

Alignment

Learning Intentions

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

• Describe half-life as the time taken for half of the unstable nuclei in a radioactive sample to decay.
• Recognise that radioactive decay is random for individual nuclei but predictable for large samples.
• Represent half-life using tables, diagrams and decay curves.
• Connect half-life to the decreasing activity or number of undecayed nuclei in a sample.

Success Criteria

By the end of the lesson, students have successfully:

• Defined half-life using appropriate physics language.
• Identified the half-life of a sample from a table or graph.
• Explained why half-life does not mean all nuclei decay after two half-lives.
• Described the pattern of repeated halving in radioactive decay.

Syllabus Reference

• Unit 1: Thermal, Nuclear and Electrical Physics
• Topic 2: Ionising Radiation and Nuclear Reactions
• Science Understanding: Nuclear Model and Stability
• Describe the concept of half-life.
• Related next dot point: Solve radioactive decay problems using and other arithmetic or graphical methods.

Phenomenon

A radioactive medical tracer is injected into a patient so that doctors can image an organ. The tracer must remain active long enough to collect useful information, but it should not remain highly radioactive in the body for too long.

This creates a practical question:

Why do doctors choose radionuclides with particular half-lives?

The answer depends on the concept of half-life. Different radionuclides decay at different rates, and half-life provides a way to describe how quickly the amount of undecayed radioactive material decreases.

Key Idea

Half-life is the time taken for half of the unstable nuclei in a radioactive sample to decay.

Concept

Radioactive decay is spontaneous and random. It is impossible to predict exactly when one particular unstable nucleus will decay. However, when there are many unstable nuclei in a sample, the overall pattern of decay is predictable.

After one half-life, half of the original unstable nuclei remain.

After two half-lives, half of the remaining nuclei decay again, so one quarter of the original sample remains.

After three half-lives, one eighth of the original sample remains.

The repeated halving pattern can be described as:

where:

is the number of undecayed nuclei remaining

is the original number of unstable nuclei

is the number of half-lives that have passed

Half-life can also describe the time taken for the activity of a radioactive sample to halve, because activity depends on the number of undecayed unstable nuclei present.

Convention

The key conventions associated with half-life are below.

• Half-life is usually represented by .
• Time units depend on the radionuclide, such as seconds, minutes, days, years or millions of years.
• The number of undecayed nuclei decreases exponentially, not linearly.
• Activity is measured in becquerels, , where means one decay per second.
• On a decay graph, half-life can be found by identifying the time taken for the quantity or activity to halve.
• Each half-life halves the remaining amount, not the original amount.

Example decay sequence:

Number of half-lives	Fraction remaining	Percentage remaining

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.

• Half-life means half of the mass disappears. Actually, unstable nuclei decay into different nuclei or particles; matter is not simply disappearing.
• After two half-lives, all of the sample has decayed. Actually, two half-lives leaves one quarter of the original unstable nuclei.
• Radioactive decay happens at a steady linear rate. Actually, decay follows an exponential pattern.
• Half-life predicts when a single nucleus will decay. Actually, it describes the average behaviour of a large number of nuclei.
• A longer half-life means the material is more dangerous in every situation. Actually, risk depends on type of radiation, activity, exposure pathway, quantity, energy and biological effects.

Further Reading

• Medical imaging radioisotopes and why their half-lives matter.
• Carbon dating using carbon-14.
• Radioactive waste storage and long half-life radionuclides.
• Exponential decay graphs in nuclear physics.

Explicit Instruction

Begin with a demonstration using 100 coins, dice, counters or an online simulation.

Each coin or die represents an unstable nucleus.

For coins:

• Heads means the nucleus has decayed.
• Tails means the nucleus remains undecayed.
• Remove the decayed nuclei each round.
• Each round represents one half-life.

Students should observe that the exact number removed each round varies, but the trend is approximately halving.

Teacher explanation:

A radioactive sample contains a large number of unstable nuclei. Each nucleus has a chance of decaying, but we cannot predict exactly which nucleus will decay next. Since there are so many nuclei, the overall sample follows a predictable pattern. The time taken for the number of unstable nuclei to halve is called the half-life.

Important distinction:

Half-life is not the time for half the sample to vanish. It is the time for half of the unstable nuclei to decay into new nuclei.

If a sample begins with unstable nuclei and has a half-life of minutes:

Time	Number of half-lives	Undecayed nuclei remaining

The amount halves every minutes.

Worked Examples

Worked Example 1

A radioactive sample has a half-life of hours. Describe what this means.

Solution:

The half-life is hours, so every hours half of the remaining unstable nuclei decay.

If the sample starts with of its unstable nuclei:

After hours, remains.

After hours, remains.

After hours, remains.

Therefore, the statement “half-life is hours” means it takes hours for the number of undecayed unstable nuclei, or the activity, to halve.

Worked Example 2

A sample starts with unstable nuclei. Its half-life is days. Describe the amount remaining after days.

Solution:

First identify the number of half-lives:

half-lives

Now halve the remaining amount three times:

After days, unstable nuclei remain.

This is not because the sample loses nuclei every days. The decrease gets smaller each half-life because each half-life halves the remaining amount.

Worked Example 3

A decay graph shows that a radioactive sample decreases from to in minutes. Describe the half-life.

Solution:

Half-life is the time taken for the activity to halve.

The activity decreases from to .

Since is half of , the half-life is minutes.

If another minutes passes, the activity would halve again from to .

Check for Understanding

Check 1

A radionuclide has a half-life of seconds. What does this mean?

Expected response:

It means that every seconds, half of the remaining unstable nuclei in the sample decay.

Check 2

A sample has unstable nuclei. After one half-life, how many unstable nuclei remain?

Expected response:

unstable nuclei remain.

Check 3

A sample has a half-life of days. A student says, “After days, all of the nuclei will have decayed.” Explain the mistake.

Expected response:

days is two half-lives. After one half-life, half remains. After two half-lives, half of that remaining amount remains. Therefore, one quarter of the original unstable nuclei remains, not zero.

Investigation (Alternative to Explicit)

Hypothesis

If a large number of random objects represent unstable nuclei, then the number remaining after each trial will approximately halve, modelling radioactive half-life.

Data Collection

Equipment:

• coins, counters or dice
• Container or tray
• Results table
• Graph paper or spreadsheet

Method using coins:

1. Count coins. Each coin represents one unstable nucleus.
2. Shake and tip the coins onto the tray.
3. Remove all coins that land heads-up. These represent decayed nuclei.
4. Count the remaining tails-up coins. These represent undecayed nuclei.
5. Record the number remaining after round .
6. Repeat until only a few coins remain.
7. Each round represents one half-life.
8. Repeat the experiment as a class and compare results.

Results table:

Round / Half-life	Number remaining

Analysis

Students plot number remaining on the vertical axis and number of half-lives on the horizontal axis.

Students should identify:

• The graph decreases quickly at first.
• The graph becomes less steep over time.
• The number remaining approximately halves each round.
• Different groups get slightly different values because the model includes randomness.
• The class average should show a smoother exponential decay pattern.

Discussion questions:

1. Why do different groups get different results?
2. Why does the class average better represent radioactive decay?
3. Why does the graph curve rather than form a straight line?
4. What does each round represent?
5. What does each coin represent?

Evaluation

Strengths:

• The model shows randomness in individual decay events.
• The model shows predictable behaviour for a large sample.
• The model shows repeated halving.

Limitations:

• Real nuclei are far smaller and more numerous.
• Real decay is not caused by shaking or physical movement.
• Real radionuclides have fixed half-lives, while this model uses rounds as an analogy.
• A small number of coins gives more variable results than a real radioactive sample.

Problems

The following problems are designed to check conceptual understanding of half-life before formal radioactive decay calculations.

1. Define half-life in one sentence.

2. A radionuclide has a half-life of hours. Describe what happens to the number of undecayed nuclei every hours.

3. A sample begins with unstable nuclei. Complete the table.

Number of half-lives	Undecayed nuclei remaining

4. A sample has an activity of . After one half-life, what is its activity?

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

6. A radionuclide has a half-life of years. How many half-lives pass in years?

7. A graph shows that activity decreases from to in minutes. What is the half-life?

8. Explain why radioactive decay is described as random but half-life is still useful.

9. A student says, “A sample with a half-life of hour is completely gone after hours.” Rewrite this statement correctly.

10. Compare two radionuclides:

    • Radionuclide A has a half-life of minutes.
    • Radionuclide B has a half-life of years.

    Describe which decays faster and explain how you know.

Followup

Self-check

Students should be able to answer:

• Can I define half-life without using the word “disappear”?
• Can I explain why half-life is a repeated halving process?
• Can I identify one half-life from a graph?
• Can I explain why the decay of one nucleus is random but the decay of a large sample is predictable?
• Can I distinguish between number of nuclei remaining and number of nuclei decayed?

Next Topic

Solve radioactive decay problems using and other arithmetic or graphical methods.