Half-Life

This lesson explains the concept of half-life, how to calculate it from graphs and data, and why radioactive decay is a random process. Half-life is a key concept in the Radioactivity section of the Edexcel GCSE Combined Science specification (1SC0).

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What Is Half-Life?

The half-life of a radioactive isotope is the time taken for:

• half of the radioactive nuclei in a sample to decay, or
• the count rate (activity) from a sample to fall to half its original value.

Both definitions are equivalent and either can be used in an exam answer.

Term	Definition
Activity	The rate at which a radioactive source decays, measured in becquerels (Bq). 1 Bq = 1 decay per second.
Count rate	The number of decays detected per second (or per minute) by a detector such as a Geiger-Müller tube.
Half-life ($t_{1/2}$t1/2​)	The time for the activity (or number of undecayed nuclei) to halve.

Exam Tip: Make sure you can state both definitions of half-life — some mark schemes require the definition in terms of undecayed nuclei, others in terms of count rate.

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Radioactive Decay Is Random

Radioactive decay is:

• Random — you cannot predict which particular nucleus will decay next.
• Spontaneous — decay is not affected by external conditions such as temperature, pressure or chemical bonding.

Because decay is random, the half-life is a statistical average — it applies to a large number of nuclei, not to an individual atom.

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Decay Curves

A decay curve is a graph of activity (or count rate, or number of undecayed nuclei) against time. It has a characteristic shape:

graph LR
    A["High activity<br/>(many undecayed nuclei)"] --> B["Activity halves<br/>after 1 half-life"]
    B --> C["Halves again<br/>after 2 half-lives"]
    C --> D["Halves again<br/>after 3 half-lives"]
    D --> E["Continues decreasing<br/>(never reaches zero)"]

The curve is an exponential decay — the activity never quite reaches zero.

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Reading Half-Life from a Graph

To find the half-life from a decay curve:

1. Read the initial activity from the y-axis.
2. Calculate half of the initial activity.
3. Find this value on the y-axis and read across to the curve.
4. Read down to the x-axis (time axis) to find the half-life.
5. For accuracy, repeat by finding the time for the activity to halve again (e.g. from half to a quarter of the original) and check the time interval is the same.

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Calculating Half-Life from Data

Method 1: Halving the Count Rate

Time (minutes)	Count rate (Bq)
0	800
2	400
4	200
6	100
8	50

The count rate halves every 2 minutes, so the half-life is 2 minutes.

Method 2: Using the Number of Half-Lives

If you know the starting activity and the final activity, you can work out how many half-lives have passed:

$\text{Number of half-lives} = \frac{\log(\text{initial activity} / \text{final activity})}{\log 2}$Number of half-lives=log2log(initial activity/final activity)​

Or simply keep halving the initial activity until you reach the final activity and count the steps.

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Worked Example 1

A radioactive sample has an activity of 6400 Bq. After 15 hours the activity is 400 Bq. Calculate the half-life.

Step 1 — Find the number of halvings:

$6400 \rightarrow 3200 \rightarrow 1600 \rightarrow 800 \rightarrow 400$6400→3200→1600→800→400

That is 4 half-lives.

Step 2 — Calculate the half-life:

$t_{1/2} = \frac{15}{4} = 3.75 \text{ hours}$t1/2​=415​=3.75 hours

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Worked Example 2

A radioactive isotope has a half-life of 8 days. A sample initially contains 12 000 undecayed nuclei. How many undecayed nuclei remain after 24 days?

Step 1 — Number of half-lives:

$\frac{24}{8} = 3 \text{ half-lives}$824​=3 half-lives

Step 2 — Halve three times:

$12\,000 \rightarrow 6000 \rightarrow 3000 \rightarrow 1500$12000→6000→3000→1500

After 24 days, 1500 undecayed nuclei remain.

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Worked Example 3

A sample has an initial count rate of 480 counts per minute (after background correction). The half-life is 6 hours. What is the count rate after 18 hours?

Number of half-lives: $18 / 6 = 3$18/6=3.

$480 \rightarrow 240 \rightarrow 120 \rightarrow 60 \text{ counts per minute}$480→240→120→60 counts per minute

Exam Tip: If the question mentions a background count rate, you must subtract it from the measured count rate before doing your half-life calculations. Background radiation is always present and is not from the source being studied.

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Background Radiation

Background radiation is the low-level radiation that is always present in the environment. Sources include:

Source	Example
Natural (most of it)	Radon gas from rocks, cosmic rays from space, food and drink, rocks and soil
Man-made (small contribution)	Medical X-rays, nuclear power, nuclear weapons testing (historical)

When measuring the activity of a source, you should always subtract the background count rate to get the corrected activity.

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Half-Lives of Different Isotopes

Different isotopes have very different half-lives:

Isotope	Half-life	Use
Polonium-218	3 minutes	(Short-lived; research)
Iodine-131	8 days	Medical diagnosis (thyroid)
Cobalt-60	5.3 years	Cancer treatment
Carbon-14	5730 years	Carbon dating (archaeology)
Uranium-238	4.5 billion years	Dating rocks and the Earth

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Common Misconceptions

Misconception	Correction
After two half-lives, all the radioactive atoms have decayed	After two half-lives, one quarter of the original atoms remain
Half-life means half the atoms disappear	The atoms don't disappear — they decay into a different element or isotope
You can speed up or slow down radioactive decay	Radioactive decay is not affected by temperature, pressure, or chemical reactions
The count rate reaches zero after a few half-lives	The count rate approaches zero asymptotically — it never actually reaches zero

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Summary