3.8.1.3
Rearranging exponential decay: use ln to find λ or t
Radioactive Decay & Half-Life — AQA A-Level Physics
- To find time t: rearrange $N = N_{0} e^{-\lambda t}$ to $t = -(1/\lambda) \ln(N/N_{0})$.
- To find λ: rearrange to $\lambda = -(1/t) \ln(N/N_{0})$.
- The same rearrangements work for $A = A_{0} e^{-\lambda t}$ and $C = C_{0} e^{-\lambda t}$.
- Always check that λ and t have consistent units. If λ is in s⁻¹, t must be in seconds.
Common Mistake
MEDIUM
Students often: Forgetting to convert 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. to seconds before calculating the decay constantThe probability of decay of a nucleus per unit time. Measured in s⁻¹..
Instead: If the question asks for λ in s⁻¹, you MUST convert 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. from years/hours/days to seconds first. 1 year ≈ 3.156 × 10⁷ s.
Instead: If the question asks for λ in s⁻¹, you MUST convert 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. from years/hours/days to seconds first. 1 year ≈ 3.156 × 10⁷ s.