Half-Life

This lesson covers half-life — one of the most important concepts in radioactivity — as required by the Edexcel GCSE Physics specification (1PH0), Topic 6: Radioactivity. You need to understand the random nature of radioactive decay, the definition of half-life, how to calculate the amount of radioactive material remaining after a given number of half-lives, and how to read half-life values from decay curves.

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The Random Nature of Radioactive Decay

Radioactive decay is a random process. This means:

• You cannot predict when a particular individual nucleus will decay.
• You cannot make a nucleus decay by heating it, cooling it, applying pressure, or any other physical or chemical change.
• However, for a large number of nuclei, you can predict how many will decay in a given time.

This is similar to rolling dice — you cannot predict which specific dice will land on a six, but you can predict that roughly one-sixth of a large number of dice will.

Exam Tip: If a question asks "why can't you predict when a single nucleus will decay?", the answer is that radioactive decay is a random process. This is a key concept in the 1PH0 specification.

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What Is Half-Life?

Half-life is the time it takes for:

• The number of radioactive nuclei in a sample to halve, OR
• The activity (rate of decay / count rate) of a sample to halve.

Both definitions are equivalent and either may be used in the exam.

Key Points

• Half-life is a fixed property of a particular isotope — it does not change.
• Different isotopes have very different half-lives — from fractions of a second to billions of years.
• Half-life does not depend on the amount of material, temperature, pressure, or any external conditions.

Examples of Half-Lives

Isotope	Half-Life	Use/Context
Polonium-218	3 minutes	Very short-lived
Iodine-131	8 days	Medical tracers/thyroid treatment
Strontium-90	29 years	Nuclear waste
Carbon-14	5,730 years	Carbon dating
Uranium-238	4.5 billion years	Dating rocks

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Calculating the Remaining Amount After n Half-Lives

After each half-life, the number of radioactive nuclei (or the activity) halves:

Number of Half-Lives	Fraction Remaining	As a Power of ½
0	1 (all of it)	(½)⁰ = 1
1	½	(½)¹ = 0.5
2	¼	(½)² = 0.25
3	⅛	(½)³ = 0.125
4	1/16	(½)⁴ = 0.0625
5	1/32	(½)⁵ = 0.03125
n	1/2ⁿ	(½)ⁿ

The Formula

Fraction remaining = (½)ⁿ

Where n is the number of half-lives that have passed.

You can also find n from the total time and the half-life:

n = total time ÷ half-life

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Worked Examples

Example 1: Simple Calculation

A sample contains 800 radioactive nuclei. The half-life is 2 hours. How many radioactive nuclei remain after 6 hours?

Step 1: Find the number of half-lives. n = total time ÷ half-life = 6 ÷ 2 = 3 half-lives

Step 2: Calculate the remaining amount. After 1 half-life: 800 ÷ 2 = 400 After 2 half-lives: 400 ÷ 2 = 200 After 3 half-lives: 200 ÷ 2 = 100 nuclei

Or using the formula: 800 × (½)³ = 800 × ⅛ = 100 nuclei ✓

Example 2: Activity Calculation

A radioactive source has an initial activity of 1200 Bq. The half-life is 5 minutes. What is the activity after 20 minutes?

Step 1: n = 20 ÷ 5 = 4 half-lives

Step 2: Activity remaining = 1200 × (½)⁴ = 1200 × 1/16 = 75 Bq ✓

Example 3: Working Backwards to Find Half-Life

A sample has an initial count rate of 640 counts per minute. After 18 minutes, the count rate is 80 counts per minute. What is the half-life?

Step 1: Find the number of halvings. 640 → 320 → 160 → 80 That is 3 half-lives to go from 640 to 80.

Step 2: half-life = total time ÷ number of half-lives = 18 ÷ 3 = 6 minutes ✓

Exam Tip: When working backwards, keep halving the initial value until you reach the final value and count how many times you halved. Then divide the total time by that number. This is the most reliable method and avoids formula errors.

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Reading Half-Life from a Decay Curve

A decay curve (or half-life graph) shows how the activity or number of radioactive nuclei decreases over time. The graph has a characteristic exponential decay shape — it curves downward, approaching (but never reaching) zero.

How to Read the Half-Life from a Graph

1. Read the initial activity (or count rate) from the y-axis at time = 0.
2. Calculate half of this value.
3. Draw a horizontal line from this half-value to the curve.
4. Draw a vertical line down from where it meets the curve to the x-axis.
5. Read the time from the x-axis — this is one half-life.

Checking Your Answer

To verify, repeat the process: find half of the half-value (i.e., quarter of the original) and check that this occurs at exactly two half-lives on the x-axis.

graph TD
    A["Reading Half-Life from a Graph"] --> B["1. Read initial activity<br/>from y-axis at t = 0"]
    B --> C["2. Calculate half<br/>of the initial value"]
    C --> D["3. Draw horizontal line<br/>to the curve"]
    D --> E["4. Draw vertical line<br/>down to x-axis"]
    E --> F["5. Read the time<br/>= one half-life"]
    F --> G["6. Verify: check that<br/>¼ of original occurs at<br/>2 × half-life"]

    style A fill:#2c3e50,color:#fff
    style F fill:#27ae60,color:#fff
    style G fill:#27ae60,color:#fff

Exam Tip: Graph-reading questions are very common. Always use a ruler to draw neat horizontal and vertical lines on the graph. Take your reading from the CURVE, not from where you think it should be. If asked to find the half-life, always verify your answer using a second point on the graph.

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Carbon-14 Dating

One important application of half-life is carbon dating, used to determine the age of organic materials (things that were once alive).

How Carbon Dating Works

1. All living organisms absorb carbon-14 (a radioactive isotope) from the atmosphere during their lifetime.
2. When an organism dies, it stops absorbing carbon-14.
3. The carbon-14 in the dead organism decays with a half-life of 5,730 years.
4. By measuring the proportion of carbon-14 remaining in a sample and comparing it to the amount in a living organism, scientists can calculate how long ago the organism died.

Example

If a bone sample contains ¼ of the carbon-14 found in a living organism:

• ¼ = (½)² → 2 half-lives have passed
• Age = 2 × 5,730 = 11,460 years