Batch decay is the radioactive decay of a given amount of pure parent material. The decay products build up and, if radioactive, will later decay with time. In this simulation isotope 1 decays to isotope 2, which decays to isotope 3, which decays to the stable isotope 4. The half-lives of the three decay processes are selected with the sliders.
Select the Example button to see the decay chain for 211Pb.
The number of particles of each isotope is found by numerically integrating the coupled first-order rate equations (RK4), starting from 1000 particles of isotope 1:
$$\frac{dN_1}{dt} = -\lambda_1 N_1$$ $$\frac{dN_2}{dt} = \lambda_1 N_1 - \lambda_2 N_2$$ $$\frac{dN_3}{dt} = \lambda_2 N_2 - \lambda_3 N_3$$ $$\frac{dN_4}{dt} = \lambda_3 N_3$$The decay constant of each step is obtained from its half-life:
$$\lambda_i = \frac{\ln 2}{(t_{1/2})_i}$$Here \(N_i\) is the number of particles of isotope \(i\), \(\lambda_i\) is the decay constant (min−1), and \((t_{1/2})_i\) is the half-life (min). Because isotope 4 is stable, the total \(N_1+N_2+N_3+N_4\) stays constant at 1000.
This simulation was created in the Department of Chemical and Biological Engineering at University of Colorado Boulder for LearnChemE.com by John L. Falconer using Claude AI. It was prepared with financial support from the National Science Foundation (DUE 2336987 and 2336988). Address any questions or comments to LearnChemE@gmail.com.