In a previous lecture, we talked about radioactive decay chains, and the Bateman equations are a…

In a previous lecture, we talked about radioactive decay
chains, and the Bateman equations are a general analytical solution to the
isotopic concentrations in these decay chains. Suppose that there are three
isotopes in a decay chain that we will call isotopes A, B, and C. Part 1: Write
down an explicit equation for their time-dependent concentrations assuming that
their half-lives are 1 day, 3 days, and 2 days, respectively, and
that the initial concentration of A at t = 0 is Ao. Assume that the concentrations
of the other two isotopes (B and C) in the decay chain are 0 at t = 0. Part 2:
Do any of these three isotopes have a “bump” in their concentration curves as a
function of time? If so, which ones have these bumps, and when do these bumps
occur? Part 3: By what factor do we multiply the half-lives of each of these
elements to get their average life? Part 4: What are the decay constants (in
inverse days) for each of the isotopes that we have used in this example?

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