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I have a simple theoretical question regarding half-life decay graphs for radioactive substances.

If the graph plots activity versus time (not mass versus time), then what physical quantity can possibly be deduced from the graph? I'm tempted to think that "half-life" is a quantity that can be deduced... but I'm not sure if that qualifies as a "physical" quantity?

I also know that the number of counts is directly proportional to the amount of radioactive substance available. I don't have the mass given in the graph, but just have the counts and the time. Can we somehow deduce the the amount of substance available?

The graph given in the problem is very simple, it has only counts plotted against time.

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Yes you can compute the total number of nuclei from such a graph.

Basically, if you plot # of decays per second on the Y axis, and time on the X axis, then the area under the curve (if your curve ran all the way to infinite time) would be "all the nuclei" since they all decay exactly once.

If you have just a short time segment, you can estimate the half life (for example, by plotting as a lin-log plot and fitting a straight line). When you have the half life (or the 1/e time) it is trivial to compute the total number of nuclei from the initial activity.

If we describe the activity at time $t$ as

$$A_t = A_0 e^{-t/\tau}$$

then we can find $A_0$ directly from the plot; we find $\tau$ as the slope from the lin-log plot; and the total area (total number of atoms that will eventually decay) is

$$\int_0^\infty A(t) dt = \frac{A_0}{\tau}$$

Note than in the above, $\tau$ is the 1/e time. It is related to the half life by

$$\tau = \frac{t_{\frac12}}{\ln 2}$$

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