7.5 Activity and Half-life

A free GCSE Physics lesson for AQA (8463) from OBA School: activity in becquerels, random decay, half-life from graphs, and calculations after a number of half-lives. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with every video, practice quizzes and printable worksheets with mark schemes, in our free online course.

What you will learn

• Define activity and use the unit becquerel (Bq)
• Explain how random decay still gives a predictable pattern for large numbers of nuclei
• Define half-life and find it from a graph of count rate or activity against time
• Calculate the activity, count rate, mass or number of nuclei left after a whole number of half-lives
• (Higher tier) express the net decline after a number of half-lives as a ratio

Key words

• Activity: The number of unstable nuclei that decay each second.
• Becquerel (Bq): The unit of activity: 1 Bq = 1 decay per second.
• Count rate: The number of decays recorded by a detector each second or minute. It is proportional to the activity if the distance stays the same.
• Half-life: The time it takes for the number of unstable nuclei in a sample (or the count rate / activity) to halve.
• Random: Each nucleus has the same chance of decaying in a given time, but you cannot say which one will decay or when.
• Net decline (Higher): How much the activity has fallen, written as a ratio of the final value to the starting value.

Activity and count rate

The activity of a radioactive source is the number of nuclei that decay each second. It is measured in becquerels (Bq), where 1 Bq is 1 decay per second. As nuclei decay, fewer unstable nuclei are left, so the activity falls.

A Geiger–Müller tube only catches the radiation going towards it, so the count rate is less than the activity. But if the source and tube stay the same distance apart, the count rate is proportional to the activity, so it falls in the same way.

Worked example: units

A source has an activity of 3.6 kBq.
3.6 kBq = 3.6 × 1000 = 3600 Bq, so 3600 nuclei decay every second.

Answer: 3600 decays per second

Random decay: a model with dice

You cannot predict when one nucleus will decay, just as you cannot predict when one die will land on a six. But with a very large number of nuclei, the same fraction decays in each equal time interval, so the overall pattern is predictable.

In the model, each throw removes about one-sixth of the dice left, so the number falls by the same fraction each time, not the same number. It takes just under 4 throws for the number of dice to halve, whether you start from 1200 or from 600: a “half-life” of about 3.8 throws.

Key fact: Radioactive decay behaves the same way. Each nucleus has a fixed chance of decaying per second, so the number left falls by half in each equal time interval: the half-life.

Half-life from a graph

The half-life of a radioactive isotope is the average time it takes for the number of unstable nuclei in a sample to halve. Equally, it is the time for the count rate (or activity) to fall to half its starting value. Each isotope has its own half-life, from fractions of a second to billions of years.

• Read the starting value (800 counts per minute) and halve it (400).
• Go across to the curve and down to the time axis: 20 minutes.
• Check from another point: 400 → 200 also takes 20 minutes.

Exam tip: Always use count rates corrected for background. Halve from any point on the curve, not only from time zero, and take a mean of several readings for a better value.

Half-life calculations

After each half-life the amount left halves: 1 → ½ → ¼ → 1/8 → 1/16 … After n half-lives the amount left is starting amount ÷ 2ⁿ. This works for mass, number of nuclei, activity or count rate.

Worked example: mass left

A sample contains 6.4 mg of an isotope with a half-life of 8 days. How much is left after 32 days?
number of half-lives n = 32 ÷ 8 = 4
mass left = 6.4 ÷ 2⁴ = 6.4 ÷ 16 = 0.40 mg (6.4 → 3.2 → 1.6 → 0.8 → 0.4)

Answer: 0.40 mg

Worked example: how long?

The activity of a source falls from 4800 Bq to 300 Bq. Its half-life is 2.5 hours. How long did this take?
4800 ÷ 300 = 16 = 2⁴, so 4 half-lives
time = 4 × 2.5 = 10 hours

Answer: 10 hours

Exam tip: Count the halvings with an arrow chain (4800 → 2400 → 1200 → 600 → 300). It is quick and shows your method clearly.

Net decline as a ratio (Higher tier)

The net decline after n half-lives can be written as the ratio of what is left to what you started with: 1 : 2ⁿ (or the fraction 1/2ⁿ). The fraction that has decayed is 1 − 1/2ⁿ.

• Fraction left; 1 1/2; 2 1/4; 3 1/8; 4 1/16; 5 1/32; 6 1/64
• Ratio left : start; 1 1 : 2; 2 1 : 4; 3 1 : 8; 4 1 : 16; 5 1 : 32; 6 1 : 64
• Percentage left; 1 50%; 2 25%; 3 12.5%; 4 6.25%; 5 3.125%; 6 1.5625%

Worked example: net decline (Higher tier)

A sample of 5.0 × 10⁶ unstable nuclei decays for 3 half-lives.
ratio left : start = 1 : 2³ = 1 : 8, so 5.0 × 10⁶ ÷ 8 = 6.25 × 10⁵ nuclei remain
fraction decayed = 1 − 1/8 = 7/8 = 87.5%

Answer: net decline 1 : 8; 6.25 × 10⁵ nuclei left

Exam tip: To find n from a ratio, ask “2 to the power of what gives this number?” For example 1 : 32 means 2⁵ = 32, so 5 half-lives.

Common mistakes

• Thinking the whole sample decays after two half-lives. After two half-lives a quarter is still left.
• Dividing by 2 × n instead of by 2ⁿ: after 4 half-lives the amount is divided by 16, not 8.
• Using count rates that include background when finding the half-life.
• Saying half-life is “half the time it takes to decay”. It is the time for the amount (or activity) to halve.
• Thinking a radioactive source can be made to decay faster by heating it. Half-life is fixed for each isotope.

Revision questions

Try these, then check your answers below each one.

Question 1. What is the unit of activity?

Answer: becquerel

Working: The becquerel (Bq): 1 Bq = 1 decay per second.

Question 2. A source has an activity of 3200 Bq. Its half-life is 4 hours. Calculate its activity after 12 hours.

Answer: 400 Bq

Working: 12 ÷ 4 = 3 half-lives; 3200 → 1600 → 800 → 400 Bq.

Question 3. The corrected count rate from a sample falls from 1200 counts per minute to 150 counts per minute in 30 minutes. What is the half-life, in minutes?

Answer: 10 minutes

Working: 1200 → 600 → 300 → 150 is 3 half-lives. Half-life = 30 ÷ 3 = 10 minutes.

Related lessons

• Previous lesson: 7.4 More About Alpha, Beta and Gamma Radiation
• Next lesson: 7.6 Nuclear Radiation in Medicine
• In this chapter: 7.1 Atoms and Radiation
• In this chapter: 7.2 The Discovery of the Nucleus
• In this chapter: 7.3 Changes in the Nucleus

Continue in the full course

This lesson is section 7.5 (AGCPH0705) of our free GCSE Physics course for AQA. In the full course you get every video for this section, a self-marking practice quiz, a printable worksheet with a mark scheme, chapter tests, and your progress saved. Go to the full GCSE Physics course.