define activity and decay constant, and recall and use A = λN
Radioactive Decay – Cambridge IGCSE/A‑Level (9702) – Syllabus 23.2
1. Key Definitions (Syllabus requirement)
- Activity (A) – the number of nuclear decays that occur in a sample per unit time.
Units: becquerel (Bq) where 1 Bq = 1 decay s⁻¹.
Older unit: 1 curie (Ci) = 3.7 × 10¹⁰ Bq. - Decay constant (λ) – the probability that a single nucleus will decay in a given second.
It is a *probability per nucleus per unit time*; units are s⁻¹.
Related quantity: mean lifetime (τ) = 1/λ (seconds).
2. Fundamental Relationship
The activity of a radioactive sample containing N undecayed nuclei is directly proportional to both the decay constant and the number of nuclei:
A = λ N
Derivation (link to the exponential law):
- The rate of change of the number of nuclei is dN/dt = –λ N.
- By definition, activity is the magnitude of the decay rate: A = –dN/dt.
- Substituting the differential equation gives A = λ N.
3. Origin of the Exponential Decay Law
Radioactive decay is a random, memory‑less (Poisson) process. Each nucleus has the same constant probability λ of decaying in any infinitesimally short time interval Δt, independent of what has happened before.
dN/dt = –λ N
Integrating from t = 0 (where N = N₀) to a later time t gives the exponential law:
N(t) = N₀ e–λt
Key points:
- All nuclei act independently – the behaviour of one nucleus does not affect another.
- The probability λ is constant; it does not change with time or with the number of remaining nuclei.
- The negative sign indicates that the number of nuclei decreases with time.
4. Half‑Life and Its Connection to λ
The half‑life, t½, is the time required for half of the original nuclei to decay.
Setting N(t½) = N₀/2 in the exponential law gives
t½ = (ln 2)/λ ≈ 0.693 / λ
Because τ = 1/λ, the relationship can also be written as:
t½ ≈ 0.693 τ
Practical use:
- If λ is known, calculate the half‑life directly with the formula above.
- If the half‑life is given, first find λ using λ = ln 2 / t½ before applying A = λN.
5. Sample Calculations (Applying A = λN)
- Finding the activity from N and λ
Given: N = 2.0 × 1020 nuclei, λ = 5.0 × 10‑4 s⁻¹
Solution:A = λN = (5.0 × 10⁻⁴ s⁻¹)(2.0 × 10²⁰) = 1.0 × 10¹⁷ Bq
- Determining the number of undecayed nuclei from A and λ
Given: A = 3.0 × 106 Bq, λ = 2.0 × 10‑3 s⁻¹
Solution:N = A/λ = (3.0 × 10⁶ Bq) / (2.0 × 10⁻³ s⁻¹) = 1.5 × 10⁹ nuclei
- Using a half‑life to find activity
Given: half‑life t½ = 30 min, sample contains N = 4.0 × 1012 nuclei.
Step 1 – Convert time to seconds: 30 min = 1800 s.
Step 2 – Find λ: λ = ln 2 / t½ = 0.693 / 1800 s ≈ 3.85 × 10⁻⁴ s⁻¹.
Step 3 – Calculate activity: A = λN = (3.85 × 10⁻⁴ s⁻¹)(4.0 × 10¹²) ≈ 1.54 × 10⁹ Bq.
6. Units Summary
| Quantity | Symbol | SI Unit | Typical Symbol in Calculations | Notes |
|---|---|---|---|---|
| Activity | A | becquerel (Bq) | s⁻¹ | 1 Bq = 1 decay s⁻¹ = 3.7 × 10⁻¹¹ Ci |
| Decay constant | λ | second⁻¹ (s⁻¹) | s⁻¹ | λ = ln 2 / t½ = 1 / τ |
| Mean lifetime | τ | second (s) | s | τ = 1 / λ |
| Number of nuclei | N | dimensionless (count) | – | Often expressed as moles: N = n NA |
| Half‑life | t½ | second (s) | s | t½ = ln 2 / λ ≈ 0.693 τ |
7. Checklist – Alignment with Syllabus 23.2
| Syllabus Requirement | How the Notes Meet It |
|---|---|
| Definition of activity and its unit (Bq) | Section 1, plus conversion to curies. |
| Definition of decay constant and its unit (s⁻¹) | Section 1, with explicit statement of “probability per nucleus per unit time” and link to mean lifetime. |
| Fundamental relationship A = λN | Section 2, including a short derivation from dN/dt = ‑λN. |
| Exponential decay law N(t)=N₀e⁻λt | Section 3, expanded explanation of the Poisson‑process nature and a decay‑curve sketch. |
| Half‑life formula t½ = ln 2 / λ | Section 4, with additional link to mean lifetime (t½ ≈ 0.693 τ) and a “plug‑in” reminder. |
| Use of A = λN in calculations | Section 5 – three worked examples, the third combines a half‑life, λ and activity. |
| Clear units table | Section 6 – reorganised table separating SI units, calculation symbols and useful notes. |
8. Key Points to Remember (Exam‑Style)
- Activity is directly proportional to both λ and N: A = λN.
- λ is a probability per nucleus per second; a larger λ means a more rapidly decaying (more radioactive) sample.
- The exponential law arises because each nucleus has a constant, independent chance of decaying in any short interval (Poisson process).
- Half‑life and decay constant are inversely related: t½ = ln 2 / λ (≈ 0.693 τ).
- Always convert time‑related quantities to seconds before using the formulas.
- When a half‑life is given, first calculate λ (λ = ln 2 / t½) and then use A = λN.