Half-Life, Mean Life, and Decay Probability
For exponential decay, T_1/2 = ln2/lambda and mean life tau = 1/lambda. The probability that a nucleus survives to time t is exp(-lambda t), and the probability it has decayed by then is 1 - exp(-lambda t).
Why this shows up in the exam
Repeated-halving problems · Converting half-life to decay constant · Decay-probability and safety-wait calculations
Learn the idea
Half-life and mean life are fixed multiples of 1/lambda, while an individual decay time remains unpredictable. Half-life describes an ensemble: after each half-life, half the current nuclei remain on average. It does not schedule a particular nucleus, which may decay at any time after preparation.
🧠 Memory hook: Half-life is 0.693 tau; it is not a deadline.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- T₁/2 = ln 2 / lambda — time for the expected surviving population to halve
- tau = 1/lambda — mean lifetime of an exponentially decaying species
- P(decay by t) = 1 - e^(-lambda t) — cumulative decay probability for one nucleus
How to approach it
- 1Use powers of two for integer half-lives
- 2Use lambda for non-integer intervals
- 3Distinguish survival probability from decay probability
Common slip-ups that cost marks
- •Setting half-life equal to mean life
- •Claiming every nucleus decays by one half-life
- •Using elapsed time instead of number of half-lives in repeated halving
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
In hydrogen, an electron transitions from n = 2 to n = 1. Using E_n = -13.6/n^2 eV, find the emitted photon energy.
More from Atoms and Nuclei
Hydrogen spectrum and spectral series
Understand the origin, calculation, and interpretation of spectral lines, series limits, wavelength and frequency relations, and transitions in hydrogen and hydrogen-like ions.
Atomic models and Bohr theory
Learn the historical development of atomic models, especially the Bohr model, and how it explains the structure, radii, and energy levels of hydrogen and hydrogen-like atoms.
Nuclear fission, fusion, and energy production
Understand the processes of nuclear fission and fusion, their energy yields, the role of binding energy, and applications like nuclear reactors and solar energy.
Mass defect, binding energy, and mass-energy equivalence
Learn how mass defect leads to nuclear binding energy, the use of E=mc², and how to calculate binding energy per nucleon and related quantities.
Nuclear size, radius, and density
Explore how nuclear size is measured, the formula for nuclear radius, its dependence on mass number, and the concept of nuclear density.
Nuclear reactions and conservation laws
Study the equations for nuclear reactions, including conservation of mass number, atomic number, and other physical quantities.