Physics · 11 / 12
Radioactive decay: halve each half-life
Final-year concepts · Worked example and 4 exercises with solutions, at your pace.
The principle, simply
Individual decay is random. An exponential model predicts the average remaining in a large population of identical nuclei. Each half-life halves the previous population, not the initial one. Activity counts decays per second in becquerels.
- 01Initial population
- 02Time / half-life
- 03Repeated factor 1/2
- 04Expected remainder
WORKED EXAMPLE
From principle to reasoning
For 1,000 nuclei and a half-life of 5 time units, expected counts are 500 at t = 5, 250 at t = 10 and 125 at t = 15. Plot 1000 × (1/2)^(x/5), with x representing time.
Solve a problem in this subject
- Draw the system
Define the studied object, reference frame and positive axis. A force diagram shows external interactions acting on the system.
- Sort quantities
Separate data and unknowns; convert units before substitution. 36 km/h = 10 m/s, so never insert 36 into a formula expecting SI speed.
- Write the model
Choose a law matching assumptions: constant acceleration, thin lens, ohmic resistor. Isolate the unknown symbolically before rounding.
- Interpret
Check dimensions and signs. Negative velocity gives direction on the chosen axis. Discuss scale and model limitations.
YOUR TURN
01 · Try before checking
What fraction remains after four half-lives?
One hint
Multiply by 1/2 four times.
Reveal and understand the solution
(1/2)⁴ = 1/16 = 6.25%. It does not predict when an individual nucleus decays.
Compare method, conditions and answer; identify the exact part to revisit.
IMPROVE THROUGH PRACTICE
Three additional challenges with solutions
Start with the first challenge. Write your method before opening a hint; use the solution to find the first step that needs work.
Start with 800 nuclei. After two half-lives, how many remain on average?
A hint to get started
Each half-life halves the amount.
Step-by-step solution
- After one half-life: 400.
- After two: 200.
- This is a model average; individual decays are random.
Which step did you miss? Retry without help, then explain why the method works.
For N(t) = N₀e^(−λt), relate λ and half-life T.
A hint to get started
Set N(T)/N₀ = 1/2.
Step-by-step solution
- e^(−λT) = 1/2.
- −λT = −ln(2).
- T = ln(2)/λ; λ has inverse-time units.
Which step did you miss? Retry without help, then explain why the method works.
An isotope has a 4 h half-life. When does it reach 1/8 of its initial amount?
A hint to get started
Express 1/8 as a power of 1/2.
Step-by-step solution
- 1/8 = (1/2)³.
- Three half-lives are required.
- Time: 3 × 4 = 12 h.
Which step did you miss? Retry without help, then explain why the method works.
The trap to avoid
A half-life is not the time until all nuclei disappear.
Without notes, can you…
- Explain the principle in your own words?
- Rebuild the diagram and its connections?
- Solve the exercise and justify the method?
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