Half-Life and Radioactive Decay Calculations

The half-life, t₁/₂, is the time it takes for half of the radioactive nuclei present to decay. It is a constant for a given nuclide. Start with 80 g or 8 g and the half-life is the same; only the amount left differs. Ordinary temperature, pressure and bonding changes have negligible effects on most decay half-lives; electron-capture rates can show small electronic-environment effects, so independence is a very good introductory approximation rather than an absolute universal law.

A half-life is a statistical population property, not a scheduled lifetime for each nucleus. A nucleus that has survived one half-life has the same decay probability over the next interval under the constant-λ model. Large samples closely follow the exponential expectation; a tiny sample shows random fluctuations. The continuum curve approaches zero asymptotically, but a real finite sample consists of discrete nuclei and can eventually have none left.

Radioactive decay is first order, so the integrated rate law applies. First convert the half-life into a rate constant with k = 0.693 / t₁/₂, then use ln(N/N₀) = −kt. The same two equations solve for N, for t or for t₁/₂ depending on which quantity is missing.

Write f = N/N₀ = 2^{−t/t₁/₂} = e^−λt. Then t = −ln f/λ, λ = ln 2/t₁/₂ and t₁/₂ = t ln 2/[−ln f]. If 30.0% remains after 50.0 years, λ = −ln(0.300)/50.0 ≈ 0.0241 yr⁻¹ and t₁/₂ ≈ 28.8 years. The logarithm uses the fraction remaining, not percent decayed. Time and λ must use reciprocal units: a decay constant in s⁻¹ needs seconds.

Living things exchange carbon with the atmosphere and hold a steady ¹⁴C level. Once the organism dies, uptake stops and the ¹⁴C decays with a half-life of 5730 years. Measuring the fraction of ¹⁴C remaining and solving for t gives the age. Because roughly ten half-lives leaves under 0.1% of the original, the method runs out at about 50,000 years.

The simple carbon-14 calculation assumes a known initial isotope ratio and no later carbon exchange or contamination. Atmospheric ratios vary, so real radiocarbon ages require calibration rather than interpreting every idealized 5730-year calculation as a precise calendar date. Newer carbon contamination makes an old sample appear too young in the simple model; an initially low carbon-14 fraction can make it appear older. A daughter isotope or decay chain can also complicate a measured activity; identify what isotope the detector is actually counting.

A nuclide's half-life is fixed by its decay constant, independent of how large the sample is. Activity is the number of decays per unit time, A_activity=λN, so a larger sample of the same nuclide has proportionally higher activity. Both samples lose the same fraction over one half-life, but they lose different absolute numbers of nuclei.

Activity Aactivity = λN is measured in becquerels, 1 Bq = 1 decay/s. Convert a half-life in days to seconds before calculating activity in Bq. For a pure single isotope with constant detector efficiency and negligible background, activity ratios equal remaining-nucleus ratios. For a measured count rate, subtract background and account for detector efficiency, sample geometry and other emitting isotopes before making that inference. Equal masses of different radioisotopes need not have equal activities because their λ values and nuclei per gram differ.