Half life of multi chain decays

[heroslayer99]

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)

Relevant Equations

x

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!

 

[mjc123]

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.

 

[haruspex]

It depends what activity is measured. Are the two activities indistinguishable?