In a radioactive decay series in which a parent nuclide (subscript 1) decays into a daughter nuclide (subscript 2) and having decay constants ₁ and ₂, respectively, the number of daughter nuclei at time t is given by the well-known result⁷

where N₁₀ is the number of parent nuclei at t = 0, the time at which no daughter nuclei are present. For the readily available ¹³⁷Cs beta decay to an excited state of ¹³⁷Ba (respective half-lives of 30.2 y and 2.55 m), the much longer parent half-life means that ₂>>₁, so Eq. (3) may now be written as

where the final equilibrium value of N₂ is written as N_2f = N₁₀₁/₂.

A constant water flow obtained directly from a faucet was used to fill a leaking reservoir as a model of this radioactive decay of a long-lived parent to a short-lived daughter. This fluid flow experiment may now be modeled as Eq. (4) plus a constant amount of background radiation. The weight of the leaking reservoir as a function of time was again obtained with the Logger Pro software and the data were fit to the equation

where W_2f = W₁₀₁/₂ and W_2BG is the initial weight of the fluid and bottle before filling begins. Figure 2 presents both data and a graph of Eq. (5) using the best-fit values for the adjustable parameters of W_2f, ₂, and W_2BG. Sliders for these parameters were again employed for instructional purposes, and the final best-fit values were also obtained by the use of the Solver add-in. This fit is in excellent agreement with the data for this example of secular equilibrium, and we disagree with the statement by Greenslade² that the "analogue breaks down for large times." Although at this point we are only interested in a qualitative agreement between the data and the mathematical model, all best-fit adjustable parameter values are consistent with direct measurements of corresponding quantities.

Figure 2.