Activity and Half-Life

This lesson covers activity and half-life as required by the AQA GCSE Physics specification (4.4.2). You need to understand how the activity of a radioactive source changes over time, how to define and calculate half-life, and how to interpret decay curves.

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Activity

The activity of a radioactive source is the rate at which nuclei decay — in other words, the number of nuclear decays per second. Activity is measured in becquerels (Bq).

1 Bq = 1 decay per second

Larger units are also commonly used:

Unit	Symbol	Value
Becquerel	Bq	1 decay per second
Kilobecquerel	kBq	1,000 decays per second
Megabecquerel	MBq	1,000,000 decays per second
Gigabecquerel	GBq	1,000,000,000 decays per second

As time passes, the number of undecayed nuclei in a sample decreases. Since there are fewer unstable nuclei available to decay, the activity decreases over time. This means that radioactive sources become less active as they get older.

Exam Tip: Activity is directly proportional to the number of undecayed nuclei. As the number of undecayed nuclei halves, the activity also halves. This is the basis of half-life calculations.

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What Is Half-Life?

The half-life of a radioactive isotope is the time taken for half the unstable nuclei in a sample to decay. Equivalently, it is the time taken for the activity (or count rate) of a source to fall to half its original value.

Both definitions are equally valid, and AQA accepts either:

1. The time taken for the number of undecayed nuclei to halve.
2. The time taken for the activity (count rate) to halve.

Key Points About Half-Life

• Half-life is a fixed property of a particular isotope — it does not change over time.
• Half-life is not affected by temperature, pressure, chemical state, or any physical conditions.
• Different isotopes have very different half-lives — from fractions of a second to billions of years.

Isotope	Half-Life	Use
Polonium-218	3.1 minutes	Very short — used in research
Iodine-131	8 days	Medical treatment of thyroid problems
Cobalt-60	5.3 years	Sterilisation of equipment, cancer treatment
Carbon-14	5,730 years	Archaeological dating (carbon dating)
Uranium-238	4.5 billion years	Geological dating of rocks

Exam Tip: Learn both definitions of half-life. Some mark schemes ask for the definition in terms of undecayed nuclei; others ask for it in terms of activity. The safest approach is to state both: "Half-life is the time taken for the number of undecayed nuclei in a sample to halve, or equivalently, the time for the activity to fall to half its original value."

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Calculating Half-Life from Data

You can calculate half-life from a table of data by finding the time taken for the activity (or count rate) to drop to half its initial value.

Worked Example 1

A radioactive source has an initial activity of 800 Bq. The activity is measured at regular intervals:

Time (minutes)	Activity (Bq)
0	800
10	400
20	200
30	100
40	50
50	25

The activity drops from 800 Bq to 400 Bq in 10 minutes. It drops from 400 Bq to 200 Bq in another 10 minutes. It drops from 200 Bq to 100 Bq in another 10 minutes.

The half-life is 10 minutes.

Worked Example 2

A sample starts with 6400 undecayed nuclei. After 15 years, there are 800 undecayed nuclei. Find the half-life.

Work out how many times the number has halved:

• 6400 --> 3200 (1 half-life)
• 3200 --> 1600 (2 half-lives)
• 1600 --> 800 (3 half-lives)

So 3 half-lives have passed in 15 years. Half-life = 15 / 3 = 5 years.

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Decay Curves

A decay curve (or half-life graph) shows how the activity (or number of undecayed nuclei) changes over time. The shape is always a smooth, decreasing curve that never quite reaches zero — this is called an exponential decay curve.

To read half-life from a graph:

1. Read the initial activity from the y-axis.
2. Calculate half of this initial activity.
3. Draw a horizontal line from this half-value to the curve.
4. Draw a vertical line down from the curve to the x-axis.
5. Read off the time — this is one half-life.

You can verify your answer by checking subsequent half-lives: the time for the activity to halve again should be the same.

graph TD
    A["Reading Half-Life from a Graph"] --> B["Step 1: Read initial activity<br>from y-axis"]
    B --> C["Step 2: Calculate half<br>of initial activity"]
    C --> D["Step 3: Draw horizontal line<br>to curve"]
    D --> E["Step 4: Draw vertical line<br>down to x-axis"]
    E --> F["Step 5: Read time value<br>= one half-life"]

    style A fill:#2c3e50,color:#fff
    style B fill:#3498db,color:#fff
    style C fill:#3498db,color:#fff
    style D fill:#3498db,color:#fff
    style E fill:#3498db,color:#fff
    style F fill:#27ae60,color:#fff

Exam Tip: When reading half-life from a graph, always check your answer by finding a second half-life. If you read the half-life as 20 minutes, verify that the activity also halves between 20 and 40 minutes, and between 40 and 60 minutes. If the values are consistent, you can be confident your answer is correct.

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Calculating Remaining Activity or Nuclei

You can calculate how much activity or how many nuclei remain after a given number of half-lives.

Formula approach:

After n half-lives, the remaining quantity = initial quantity x (1/2)^n

Worked Examples

Example 1: A source has an initial activity of 1200 Bq and a half-life of 6 hours. What is the activity after 24 hours?

• Number of half-lives = 24 / 6 = 4
• After 4 half-lives: 1200 x (1/2)^4 = 1200 x 1/16 = 75 Bq

Example 2: A sample contains 5000 undecayed nuclei. The half-life is 2 days. How many undecayed nuclei remain after 10 days?

• Number of half-lives = 10 / 2 = 5
• After 5 half-lives: 5000 x (1/2)^5 = 5000 x 1/32 = 156.25, which we round to approximately 156 nuclei

Example 3: A source has an activity of 4800 Bq. After 3 half-lives, the activity is 600 Bq. Show this is correct.

• After 1 half-life: 4800 / 2 = 2400 Bq
• After 2 half-lives: 2400 / 2 = 1200 Bq
• After 3 half-lives: 1200 / 2 = 600 Bq. Correct.

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Net Decline and Count Rate

In experiments, a Geiger-Muller tube measures the count rate, which includes both the radiation from the source and background radiation. To find the true count rate from the source alone:

Corrected count rate = Measured count rate - Background count rate

You must always subtract background radiation before using data to calculate half-life.

Worked Example

Background count rate = 30 counts per minute.

Time (min)	Measured Count Rate (cpm)	Corrected Count Rate (cpm)
0	830	800
5	430	400
10	230	200
15	130	100