Activity and half-life Edexcel International A Level Physics revision
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In plain words
Nobody can say when one particular unstable nucleus will decay. It might be in the next second or in a thousand years. But with billions of them the numbers become very predictable: in any fixed stretch of time the same fraction always decays. The time for half of them to go is the half-life.
5 things to know
- Radioactive decay is spontaneous: it is not affected by anything outside the nucleus, such as temperature or pressure. It is also random: you cannot predict when a particular nucleus will decay, though each has the same chance of decaying in a given time. The count rate fluctuates, and that is the evidence for randomness.
- Activity A is the number of decays per second, in becquerels (Bq). The decay constant λ is the probability that a nucleus decays in unit time. A = λN, where N is the number of undecayed nuclei.
- The number of undecayed nuclei falls exponentially: N = N₀e^(−λt). Activity and count rate follow the same equation.
- Half-life is the time for the number of undecayed nuclei, or the activity, to halve. λ = ln 2 ÷ half-life = 0.693 ÷ half-life.
- Taking logs gives ln A = ln A₀ − λt, so a graph of ln A against t is a straight line with gradient −λ.
Worked example
A sample has a half-life of 8.0 days and an activity of 1600 Bq. Find its decay constant, and its activity after 20 days.
- λ = 0.693 ÷ 8.0 = 0.0866 per day.
- A = A₀e^(−λt) = 1600 × e^(−0.0866 × 20) = 1600 × e^(−1.73).
- = 1600 × 0.177 = 280 Bq.
Tips and tricks
- Keep the units of time the same in λ and in t. To use A = λN with A in becquerels, λ must be in s⁻¹.
- For a whole number of half-lives there is no need for the exponential: just halve, and halve again.
It lands in your notebook with its questions as flashcards.
Activity and half-life: 5 questions and answers
These are the quiz’s questions. Do the quiz first, then come back here for the ones that got you.
How is the activity of a sample related to the number N of undecayed nuclei in it?
The decay constant is the chance of decay for each nucleus per unit time.
What fraction of a radioactive sample is left undecayed after three half-lives?
½ × ½ × ½.
What does "spontaneous" mean for radioactive decay?
Heating or squeezing a source doesn't change its decay.
How is the decay constant related to the half-life?
ln 2 is 0.693.
An isotope has a half-life of 10 minutes. What is its decay constant?
0.693 ÷ 10.
Quiz
5 questions
Tap an answer and you’ll see straight away whether it’s right, and why.
Worksheet
5 questions, 11 marks. Write your answers on paper, then check them.
Activity and half-life
Edexcel International A Level Physics WPH · 11 marks · papermunch.org
Name ______________________________ Date ______________
A source has a half-life of 5.0 hours and an activity of 800 Bq. Find its activity after 15 hours.[2]
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100 Bq. Three half-lives: 800 → 400 → 200 → 100.
An isotope has a half-life of 6.0 hours. Find its decay constant in s⁻¹.[2]
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3.2 × 10⁻⁵ s⁻¹. 0.693 ÷ (6.0 × 3600).
A sample contains 4.0 × 10¹⁵ undecayed nuclei, and its decay constant is 2.0 × 10⁻⁶ s⁻¹. Find its activity.[2]
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8.0 × 10⁹ Bq.
State what is meant by saying that radioactive decay is random and spontaneous.[2]
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Random: it is impossible to predict which nucleus will decay next, or when a particular nucleus will decay. Spontaneous: the decay is not affected by outside conditions such as temperature or pressure.
Describe how to find the half-life of a source from measurements of its count rate.[3]
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Measure the background count rate and subtract it from every reading. Record the corrected count rate at regular times and plot it against time. Read off the time for the count rate to halve, at several places on the curve, and take the mean. (Or plot ln of the count rate against time: the gradient is −λ, and the half-life is 0.693 ÷ λ.)
Answers: Activity and half-life
- 1. 100 Bq. Three half-lives: 800 → 400 → 200 → 100.
- 2. 3.2 × 10⁻⁵ s⁻¹. 0.693 ÷ (6.0 × 3600).
- 3. 8.0 × 10⁹ Bq.
- 4. Random: it is impossible to predict which nucleus will decay next, or when a particular nucleus will decay. Spontaneous: the decay is not affected by outside conditions such as temperature or pressure.
- 5. Measure the background count rate and subtract it from every reading. Record the corrected count rate at regular times and plot it against time. Read off the time for the count rate to halve, at several places on the curve, and take the mean. (Or plot ln of the count rate against time: the gradient is −λ, and the half-life is 0.693 ÷ λ.)