define activity and decay constant, and recall and use A = λN

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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.
Sketch of N versus t showing exponential decline
Typical decay curve: N (vertical) vs. time t (horizontal). The curve follows N = N₀e⁻λt.

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)

  1. 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
  2. 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
  3. 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.
Suggested diagram: a simple decay chain showing a parent nucleus (P) → daughter nucleus (D) + emitted radiation (α, β, or γ). The arrow is labelled with the decay constant λ, and a note indicates that the activity of the parent is A = λN.

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