Key Equations

Radioactive Decay & Half-Life — AQA A-Level Physics

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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.
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.
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.
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.
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).
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.
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