Theme E — Nuclear and Quantum Physics
E.3.6 Decay Constant and Half-Life — IB Physics HL Revision Notes & Topical Questions
5 min read 9 parts Lesson 23 of 45 1 quiz
Curriculum Equation Key قوانين الرقم الهيدروجيني والاتزان الأيوني - pH & Acid-Base Equilibrium FormulasFast Exam Solution
pH = -log[H⁺] | pOH = -log[OH⁻] | pH + pOH = 14 | [H⁺][OH⁻] = 10⁻¹⁴
Word Equation & Conditions: الرقم الهيدروجيني = -لوغاريتم تركيز أيون الهيدروجين | pH + pOH = 14
Location & Specifications: الاتزان الأيوني، كيمياء 3 ثانوي، التحصيلي، و AP Chemistry.
Lesson summary
E.3.6 Decay Constant and Half-Life — IB PHYSICS HL notes: Define the decay constant, use A = λN and T½ = ln 2/λ and find half-life from decay curves and log graphs.Reason — decay is random so the average number of decays per unit time is usedDefinition — the probability that an i…
- Large samples
- Minus sign
- Inverse proportion
- Molar mass
- Constant half-life
- Background
Section 1 of 9The decay constant
Reason
- Reason decay is random so the average number of decays per unit time is used
Definition
- Definition the probability that an individual nucleus will decay per unit time
Unit
- Unit s−1 or per minute or per year
Size
- Size a large λ means rapid decay and a short half-life
Property
- Property it depends on the nuclide only, not on N, temperature or chemical state
Large samples
- Large samples it is meaningful only for a large number of nuclei
Section 2 of 9Activity equation
Equation
- A = ΔN/Δt = −λN, so A = λN in magnitude
Symbols
- A in Bq, λ in s−1 and N the number of undecayed nuclei
Meaning
- Meaning the greater the decay constant or number of nuclei, the greater the activity
Minus sign
- N decreases with time
Rate
- Rate dN/dt = −λN, the rate is proportional to the amount present
Section 3 of 9Linking λ and half-life
1
Start from N = N0e−λt.
2
Put N = ½N0 at t = T½ to get ½ = e−λT½.
3
Take natural logs: ln(½) = −λT½.
4
Use ln 2 = −ln(½) to get T½ = ln 2/λ ≈ 0.693/λ.
Inverse proportion
- Inverse proportion a shorter half-life means a larger decay constant and faster decay
Product
- λT½ = 0.693 always
Units
- Units if T½ is in years, λ is in years−1, so convert to s−1 before using A = λN in becquerel
Section 4 of 9Number of nuclei
Moles
- Moles n = m/M with M in g mol−1
Nuclei
- N = (m/M) × NA with NA = 6.02 × 1023 mol−1
Molar mass
- Molar mass about A in g mol−1 for a nuclide of mass number A
Combine
- Combine with A = λN to find the activity of a sample
Section 5 of 9Half-life from graphs
Decay curve
- Decay curve draw a line at half the original value and read the time, repeating at other starting values
Constant half-life
- Constant half-life each halving takes the same time
Log graph
- Log graph ln N = −λt + ln N0 so ln N against t has gradient −λ and intercept ln N0
Counts
- Counts count rate is proportional to activity and to N so the graphs have the same shapes
Background
- Background subtract it before plotting
Section 6 of 9Checking answers
Short half-life
- Short half-life a large λ and very high activity for a few nuclei, and the sample is quickly used up
Long half-life
- Long half-life a tiny λ, around 10−17 s−1 for 109 years
Route
- Route find N from the mass, λ from the half-life, then A = λN
Section 7 of 9More from the specification
More from the specification
Decay probability and strong force
- Decay probability and strong force radioactive decay constants describe spontaneous tunneling rates overcoming residual nuclear strong force attraction.
Decay Constant (λ) Defined
- The radioactive decay constant λ represents the probability per unit time of the decay of a given radioactive nucleus, expressed in reciprocal seconds (s⁻¹).
Exponential Law of Radioactive Decay
- The decrease in the number of radioactive nuclei N in a sample follows the exponential decay equation N = N₀ * e^(-λt), with activity decreasing according to A = A₀ * e^(-λt).
Relationship Between Half-Life and Decay Constant
- Half-life t₁/₂ is related to the decay constant by the mathematical equation t₁/₂ = ln(2) / λ ≈ 0.693 / λ, independent of sample size or ambient conditions.
Binding Energy and Stability
- The binding energy per nucleon reflects nuclear stability;
- unstable parent nuclides with excess protons or neutrons emit alpha, beta,
- or gamma radiation to achieve higher binding energy.
Binding energy
- The minimum energy required to separate the nucleons of a nucleus to infinity.
Alpha particle
- Helium nucleus, charge +2.
Alpha decay
- Loss of 2 protons and 2 neutrons.
Ionising radiation
- Radiation with enough energy to remove electrons from atoms
Penetrating power
- How far radiation travels through a material.
Radiation
- Transfer by electromagnetic waves, requiring no medium
Radioactive
- Describes a nucleus that decays spontaneously and randomly.
Proton
- Positive particle in the nucleus, relative charge +1 and relative mass 1.
Neutron
- Neutral particle in the nucleus, relative mass 1.
Isotope
- Atom of the same element with a different number of neutrons.
Isotopes
- Nuclei with the same proton number but different neutron numbers.
Source
- Reservoir at TH providing heat QH
Electron
- Negatively charged particle in an atom.
Section 8 of 9Key terms
| Term | Meaning |
|---|---|
| Decay constant | The probability per unit time that an individual nucleus will decay. |
| Activity | The number of nuclei decaying per second, A = λN. |
| Half-life | The time for the number of undecayed nuclei or the activity to fall to half. |
| Becquerel | One decay per second. |
| Average decay rate | The mean number of nuclei expected to decay per unit time. |
Section 9 of 9Exam technique
1
Using half-life in years and quoting activity in Bq without converting to seconds.
2
Using the mass instead of the number of nuclei N in A = λN.
3
Thinking a larger λ means a longer half-life — they are inversely proportional.
4
Using log10 2 instead of ln 2 in T½ = ln 2/λ.
5
Believing λ changes as the sample decays — it is constant.
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