Half-Life and Evidence

Objective: Calculate expected remaining nuclei.

Half-life is the time for half of a large population of unstable nuclei to decay on average, assuming a constant decay probability per unit time. The ideal model N = N₀(1/2)^(t/T½) predicts expected amount after elapsed time t. After three half-lives, one eighth of the original undecayed nuclei remain; a 80 g pure sample would have an expected 10 g of parent nuclide, ignoring any material loss and accounting separately for decay products. Individual nuclei do not have scheduled death times. Remaining activity also halves under the model if the nuclide and detection conditions are unchanged.

Objective: Interpret a parent isotope measurement.

Radiometric dating estimates elapsed time by comparing a radioactive parent nuclide with a reference amount or daughter product under a suitable system model. For carbon-14, exchange with the environment changes after an organism dies; a sample with one quarter of its initial carbon-14 activity has undergone about two half-lives under the ideal model. That interpretation requires assumptions about initial composition, contamination, closed-system behavior, calibration, and the relevant time range. Different isotopes suit different materials and timescales. Dating a rock and dating once-living material often require different isotopic systems. A numerical result should include measurement uncertainty and a stated model.

Objective: Use repeated counts and background subtraction.

Radioactive counting is statistical: repeated counts over equal intervals fluctuate around an expected rate. Counts from a sample plus background should be compared to background-only counts measured under similar conditions; subtracting mean background estimates net sample counts. For example, 120 counts in one minute with sample and 20 background counts in one minute suggest about 100 net counts per minute, with uncertainty from both measurements. A plot of remaining activity against time often has an exponential shape, while a constant fraction lost per equal time interval is the hallmark of the model. Instrument efficiency and measurement time affect inference. A single count difference is not automatically meaningful without uncertainty.