Numerical Decay Models and Radioactive Dating

For radioactive decay, the expected rate of change is −λN. Over a small step Δt, estimate the change using the value at the start of the step. The next expected population is N−λNΔt. In a spreadsheet, place time in one column and use the previous population cell in the next-row formula.

For a discharging capacitor use the same update with λ replaced by 1/RC and N replaced by charge or voltage. The model predicts a mean smooth trend; individual radioactive decays remain random. A simulated population need not be rounded to an integer when it represents an expectation.

The exact exponential multiplies the value by the same ratio over each equal time interval: exp(−λΔt). The Euler update uses 1−λΔt, which is close only for a sufficiently small step. A large step can overestimate the decline or even produce a non-physical negative population.

Repeat with half the step size and compare at the same final time. Convergence towards a stable result supports the numerical approximation; it does not prove the physical assumptions. A dice simulation removes dice according to a fixed probability each trial and shows random scatter around an expected geometric decline.

For a suitable radioactive isotope, compare present activity with a justified estimate of initial activity. Carbon dating assumes an appropriate initial carbon-14 proportion and that after death the sample is approximately closed to further carbon exchange. Contamination, changes in initial isotope proportion and background subtraction affect the inferred age.

Use an exponential or count half-lives when the activity ratio is simple. A measured total detector rate must be corrected for background before it is used as the isotope’s activity proxy. Old samples with a weak signal can have large relative uncertainty.

Try each problem before opening the worked solution. State the physical model, show substitutions with consistent units, and explain what your answer means.