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Theme E — Nuclear and Quantum Physics

E.3.5 Activity and Half-Life — IB Physics HL Revision Notes & Topical Questions

5 min read 8 parts Lesson 22 of 45 1 quiz

📚 IB Physics HL Hub / Topical Revision
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Curriculum Equation Key Radioactive Decay & Half-Life FormulasFast Exam Solution
N(t) = N₀ × (1/2)t / t½ | λ = ln(2) / t½ ≈ 0.693 / t½ | E = Δm × c²
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Lesson summary

E.3.5 Activity and Half-Life — IB PHYSICS SL notes: Define activity and half-life and use them to predict how a radioactive sample decays.Activity = the number of nuclei in a sample that decay per unit time.Its SI unit is the becquerel (Bq), where 1 Bq is one decay per second.Act…

  • Activity and count rate
  • It is a constant for a given isotope
  • It is impossible to predict when one nucleus decays
  • The same fall is seen at every stage
  • After two half-lives one quarter remains
  • Fraction decayed

Part 1 of 5Half-life

1Half-life
Half-life the time taken for half of the undecayed nuclei of a nuclide to decay, or for the activity of the source to fall to half.
  • The time for the corrected count rate to halve is also the half-life.
2It is a constant for a given isotope
  • It is a constant for a given isotope it does not depend on the number of nuclei, temperature or chemical state.
3It is impossible to predict when one nucleus decays
  • It is impossible to predict when one nucleus decays but the behaviour of a large sample can be predicted from its half-life.
4
Half-lives range from a fraction of a second to billions of years.
  • Technetium-99m is about 6 hours and iodine-131 about 8 days.
  • Carbon-14 is about 5700 years and uranium-235 about 704 million years.
  • Uranium-238 is about 4.5 billion years.

Part 2 of 5Pattern of decay over half-lives

1
Each half-life halves the number of undecayed nuclei and the activity.
  • After n half-lives the proportion remaining = (1/2)n.
  • The number of half-lives is n = elapsed time divided by the half-life.
2The same fall is seen at every stage
  • The same fall is seen at every stage the time to go from 100% to 50% equals the time to go from 50% to 25%.
Half-lives passedProportion remainingActivity
01A0
11/2A0/2
21/4A0/4
31/8A0/8
41/16A0/16
1After two half-lives one quarter remains
  • After two half-lives one quarter remains not zero.
2Fraction decayed
1 minus the fraction remaining — so after 5 half-lives 31/32 has decayed.

Part 3 of 5Decay curves and measuring half-life

1
Measure the initial activity A0 of the sample.
2
Measure how the activity (or corrected count rate) changes with time.
3
Subtract the background count rate from every reading.
4
Plot corrected count rate against time and draw a smooth best-fit curve.
5
Read the time for the value to fall to half; repeat from several starting values and take the mean.
Decay curves and measuring half-life
The curve is exponential
  • The curve is exponential it becomes less steep but never reaches zero.
If background is not subtracted, the curve levels off at a non-zero value and the half-life is wrong.
Real data scatter about the curve
  • Real data scatter about the curve because decay is random.

Part 4 of 5Why half-life matters in practice

Why half-life matters in practice
Half-life decides how long a source stays useful or hazardous.
  • A medical tracer needs a half-life of hours so that it is gone from the body quickly.
  • Waste with a half-life of thousands of years needs very long storage.
For equal numbers of nuclei, the shorter half-life gives the larger activity
  • For equal numbers of nuclei, the shorter half-life gives the larger activity but the source decays away sooner.
The number of undecayed nuclei halves every half-life
  • The number of undecayed nuclei halves every half-life but the mass of the sample does not, because the stable daughter nuclei remain.

Part 5 of 5More from the specification

More from the specification
Decay constant and mechanical centripetal force
  • Decay constant and mechanical centripetal force unstable radioisotopes decay exponentially over time, providing a radiometric means independent of external force.
Radioactive Activity and Half-Life
  • The activity of a radioisotope decreases exponentially over time according to A = A0 * e^(-lambda * t), halving with each half-life interval t1/2 = ln(2) / lambda.
Alpha, Beta, and Gamma Radiation
  • Unstable parent nuclei decay by emitting alpha particles, beta particles (electrons or positrons),
    • (2 protons and 2 neutrons)
  • or de-exciting via high-energy gamma rays.
Nuclear Binding Energy and Mass Defect
  • The difference between the mass of an intact nucleus and the combined mass of its constituent protons and neutrons converts to binding energy via Einstein's relation Delta E = Delta m * c2.
Nuclear Stability and Particle Balance
  • The ratio of neutrons to protons determines radioactive decay modes,
  • as unstable nuclides emit radiation to achieve a more favorable binding energy per nucleon.
Ionising radiation
  • Radiation with enough energy to remove electrons from atoms
Penetrating power
  • How far radiation travels through a material.
Nuclear notation
  • Symbol showing mass number and atomic number.
Beta-plus decay
  • Proton changes to a neutron with emission of a positron and a neutrino.
Decay equation
  • An equation in nuclide notation with balanced nucleon and proton numbers.
Radiation
  • Transfer by electromagnetic waves, requiring no medium
Proton
  • Positive particle in the nucleus, relative charge +1 and relative mass 1.
Neutron
  • Neutral particle in the nucleus, relative mass 1.
Electron
  • Negatively charged particle in an atom.
Binding energy
  • The minimum energy required to separate the nucleons of a nucleus to infinity.
Background radiation
  • Radiation that is always present in the environment.
Alpha particle
  • Helium nucleus, charge +2.
Alpha decay
  • Loss of 2 protons and 2 neutrons.

Key terms

TermMeaning
ActivityNumber of nuclei decaying per unit time; unit becquerel (Bq).
Becquerel (Bq)One decay per second.
Half-lifeTime for half the undecayed nuclei, or the activity, to fall by half.
Exponential decayDecay in which the rate is proportional to the amount present.
Count rateRate of pulses registered by a detector, a fraction of the activity.
Corrected count rateCount rate with the background count rate subtracted.

Avoid these mistakes

1
Believing all nuclei have decayed after two half-lives — one quarter remains.
2
Reading half-life from a graph not corrected for background — subtract background first.
3
Thinking half-life depends on the starting amount or temperature — it is constant for an isotope.
4
Confusing activity (decays per second) with count rate (detected counts) — the detector sees only a fraction.
5
Using the mass of the whole sample to track decay — only the undecayed radioactive nuclei halve.

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