A single radioactive-series decay chain begins with an initial amount of only a parent nuclide, which decays to a second, which decays to a third, etc., each with its respective decay constant. The set of solutions for the time dependence of the amount of each member of the series is known as the Bateman equations.^8,9 An elegant derivation of the general solution for each series member has been recently presented by Pressyanov.¹⁰

For a series of three nuclides, the third of which is stable (₃ = 0), the time dependence for the third member is found to be

This third member is represented in our experiment by a third plastic bottle without an outlet. We model the time dependence of its weight as

where W_3BG is the weight of the empty third bottle. The time dependence for the weight of the second bottle is modeled by Eq. (3) and is written as

where W_2BG is the starting weight of the second bottle.

Figure 3 shows the experimental apparatus, where three plastic bottles were mounted on three separate force probes, each of which was connected to the same interface. The water-filled bottle #1 was then allowed to leak into bottle #2, which had a slightly larger decay constant. This translates into having a slightly larger stirrer diameter, as seen from Eq. (1). Bottle #2 then filled bottle #3. The time dependence for the weight of each bottle was measured using the Logger Pro software and the data were modeled by Eqs. (2), (8), and (7) for bottle #s 1, 2, and 3, respectively.

Figure 3.

Figure 4 presents both data and best-fit graphs for all three bottles. The overall qualitative agreement between data and best fit for all three bottles is much better than we had anticipated and demonstrates the practicality of applying Solver for such a complex situation. The best-fit values for the interrelated adjustable parameters W₁₀, ₁, W_1BG, ₂, W_2BG, and W_3BG were provided by minimizing the sum of squares of differences between data and model values for all three data sets and relevant equations combined.

Figure 4.