Half life of multi chain decays

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The discussion focuses on understanding the half-life of multi-chain decays, particularly how it relates to the decay rates of intermediate steps. It suggests that if the decay rate of Y is significantly higher than that of X, the decay events occur closely together, leading to a total decay rate that approximates twice that of X. A differential equation can be established to analyze this relationship further. The conversation also raises questions about the measurement of activities and whether they are distinguishable. Overall, the relationship between half-lives in multi-chain decays is complex and requires careful mathematical modeling.
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Homework Statement
If we had a multi decay mechanism where say, X decayed into Y then Y decayed into Z, and we plotted activity against time of a sample of X, what would the half life of the graph represent? (Assume the half life of X to Y step is much greater than the Y to Z half life)
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I am not really too sure where to start with this one, I would guess that the half life of the graph is very similar to the half life of the X to Y step, but at the same time I am unsure of how I would prove it. Any tips or reading material? Thanks!
 
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Well, you could set up a differential equation and solve it.

A more qualitative approach would be to say: if the decay rate constant (inversely proportional to half life) of Y is much greater than that of X, then effectively X decays relatively slowly, and (almost) every time an X decays to a Y, that Y (almost) immediately decays to a Z. In other words, every time an X decays there are two decay events within a very short time of each other. So the total decay rate is (to a good approximation) twice the decay rate of X; the two graphs are the same except that one has the y coordinate multiplied by 2, and the half lives are the same.

Now do the differential equation.
 
It depends what activity is measured. Are the two activities indistinguishable?
 
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