Radioactive Decay & Half-Life

Random and spontaneous nature of decay, activity, decay constant, exponential decay, half-life, carbon dating and radioisotope applications.

Spec Points Covered
  • Explain why radioactive decayThe spontaneous and random disintegration of an unstable nucleus, emitting radiation (alpha, beta, or gamma) to become more stable. is random and spontaneous, and state the evidence for this.
  • Define activityThe number of nuclear decays per unit time. Measured in becquerels (Bq), where 1 Bq = 1 decay per second. and the decay constantThe probability of decay of a nucleus per unit time. Measured in s⁻¹., and apply $A = \lambda N$.
  • Use $N = N_{0} e^{-\lambda t}$ and equivalent forms for activityThe number of nuclear decays per unit time. Measured in becquerels (Bq), where 1 Bq = 1 decay per second. and count rate.
  • Derive and apply the half-lifeThe time taken for half the number of radioactive nuclei in a sample to decay, or for the activityThe number of nuclear decays per unit time. Measured in becquerels (Bq), where 1 Bq = 1 decay per second. to halve. equation t_{1/2} = ln 2 / λ.
  • Determine half-lifeThe time taken for half the number of radioactive nuclei in a sample to decay, or for the activity to halve. from decay curves and logarithmic graphs.
  • Apply radioactive decayThe spontaneous and random disintegration of an unstable nucleus, emitting radiation (alpha, beta, or gamma) to become more stable. equations to carbon dating, smoke detectors and radioisotope powerThe rate of energy transfer. Measured in watts (W). systems.
Notes
01 Radioactive decay is random and spontaneous Radioactive decay 3.8.1.3 → 02 Activity is the number of decays per second: A = λN Activity (A) 3.8.1.3 → 03 The number of undecayed nuclei falls exponentially: N = N₀\(e^{−λt}\) $N = N_0 e^{-\lambda t}$ 3.8.1.3 → 04 Finding N from mass using the Avogadro constant Avogadro constant (Nₐ) 3.8.1.3 → 05 Half-life is the time for the number of undecayed nuclei to halve Half-life (t₁/₂) 3.8.1.4 → 06 Half-life can be found from decay curves and log graphs 3.8.1.4 → 07 Carbon-14 dating works for samples between 500 and 60,000 years old Radiocarbon dating 3.8.1.4 → 08 Uranium-lead dating determines the age of rocks over billions of years 3.8.1.4 → 09 Radioisotope power systems convert decay heat to electricity 3.8.1.4 → 10 Rearranging exponential decay: use ln to find λ or t 3.8.1.3 →
Σ Key Equations Full Reference →
On Data Sheet
Not on Data Sheet
Activity equation
$$A = \lambda N$$
  • Where:
    • $A$ = activity (Bq)
    • $λ$ = decay constant (s⁻¹)
    • $N$ = number of undecayed nuclei
Also written as A = −ΔN/Δt. The minus sign indicates N is decreasing.
Half-life equation
$$t_{1/2} = \frac{\ln 2}{\lambda}$$
  • Where:
    • $t₁/₂$ = half-life (s)
    • $λ$ = decay constant (s⁻¹)
Derived from N = N₀\(e^{−λt}\) by setting N = ½N₀. ln 2 ≈ 0.693.
Exponential decay of nuclei
$$N = N_0 e^{-\lambda t}$$
  • Where:
    • $N$ = number of undecayed nuclei at time t
    • $N₀$ = initial number of nuclei
    • $λ$ = decay constant (s⁻¹)
    • $t$ = time (s)
Can substitute A for N (activity) or C for N (count rate).
Number of nuclei from mass
$$N = \frac{m \times N_A}{M}$$
  • Where:
    • $N$ = number of nuclei
    • $m$ = mass of sample (g)
    • $N_A$ = Avogadro constant = 6.02 × 10²³ mol⁻¹
    • $M$ = molar mass (g mol⁻¹)
Molar mass M is numerically equal to the nucleon number A.
Logarithmic form (for straight-line graphs)
$$\ln N = \ln N_0 - \lambda t$$
  • Where:
    • $ln N$ = y-axis value
    • $t$ = x-axis value (time)
    • $−λ$ = gradient
    • $ln N₀$ = y-intercept
Compare with y = mx + c. Gradient gives −λ, y-intercept gives ln N₀.
Rearranged for time
$$t = -\frac{1}{\lambda} \ln \left( \frac{N}{N_0} \right)$$
  • Where:
    • $t$ = time elapsed (s)
    • $λ$ = decay constant (s⁻¹)
    • $N$ = remaining nuclei
    • $N₀$ = initial nuclei
Works with A/A₀ or C/C₀ in place of N/N₀.
Q Retrieval Practice All 15 Questions →
Q1. Define radioactive decayThe spontaneous and random disintegration of an unstable nucleus, emitting radiation (alpha, beta, or gamma) to become more stable. and state its two key properties.
  • The spontaneous disintegration of a nucleus to form a more stable nucleus by emitting radiation.
  • It is random (equal probability for each nucleus, unpredictable) and spontaneous (unaffected by external conditions).
Q2. Define the decay constantThe probability of decay of a nucleus per unit time. Measured in s⁻¹. λ and state its unit.
  • The probability that an individual nucleus will decay per unit of time.
  • Unit: s⁻¹.
Q3. State the equation relating activity, decay constantThe probability of decay of a nucleus per unit time. Measured in s⁻¹. and number of nuclei.
  • A = λN.
  • Activity (Bq) = decay constant (s⁻¹) × number of undecayed nuclei.
Q4. Write the exponential decay equation for the number of undecayed nuclei.
N = N₀\(e^{−λt}\), where N₀ is the initial number of nuclei, λ is the decay constant, and t is time.
Q5. State two other quantities that follow the same exponential decay law as N.
  • Activity: A = A₀\(e^{−λt}\).
  • Count rate: C = C₀\(e^{−λt}\).
  • Both are proportional to N.