Isotope 1 is continuously produced and it then undergoes radioactive decay. The decay products build up and, if radioactive, 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. The production rate of isotope 1 is fixed at 30 particles per minute. For a constant production rate, the number of isotopes 1, 2, and 3 would reach a steady-state; in this simulation, however, the production rate of isotope 1 is shut off after a time selected by a slider. Because isotope 4 is stable, the number of isotope 4 particles would continuously increase if the production rate were not shut off; by default it is not shown in the graph so that the behavior of the other isotopes can be more readily observed, but it can be added to the plot with the Isotope 4 checkbox.
Select the Example button to see the decay chain for 211Pb, continuously produced at 20 particles per minute.
The number of particles of each isotope is found by numerically integrating the coupled first-order rate equations (RK4), starting from zero particles of isotope 1:
$$\frac{dN_1}{dt} = P - \lambda_1 N_1$$\(P\) = production rate of isotope 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), \((t_{1/2})_i\) is the half-life (min), and \(P\) is the production rate of isotope 1 (30 particles/min). Because isotope 1 is continuously produced and isotope 4 is stable, the total \(N_1+N_2+N_3+N_4\) increases with time, but the increase stops when the production rate of isotope 1 is shut off.
This simulation was created in the Department of Chemical and Biological Engineering at University of Colorado Boulder for LearnChemE.com using Claude AI by John L. Falconer in collaboration with Martha A. Grover. It was prepared with financial support from the National Science Foundation (DUE 2336987 and 2336988). Address any questions or comments to LearnChemE@gmail.com.