How do we observe that $\frac{dN}{dt} \propto N$?

Question

I was learning how do derive the radioactive decay formula which starts with the experimental observation that the rate of decay is directly proportional to the amount of isotope present. i.e. $\frac{dN}{dt} \propto N$

So you would graph $\frac{dN}{dt}$ on the y-axis and $N$ on the x-axis. You would observe that for all isotopes a straight line graph results.

My question is related to doing the actual experiment. You could use a GM counter for $\frac{dN}{dt}$ but I don't know how $N$ would be measured.

I guess you would need an isotope with a short half-life to start with and a very accurate weighing scales.

For alpha decay does the alpha particle always leave the isotope and hence not contribute to the weight of the isotope remaining ? If it does always leave the initial chunk then the amount of isotope remaining in the chunk can be calculated. If some of the alpha particles get trapped within the chunk then we cannot use the weighing scales to keep track of N.

What about beta and gamma decay ? As the isotope decays the initial chunk does not get any lighter.

How would you measure how much isotope is remaining ?

Answer 1

I don't know if this counts as an answer, but $\frac{dN}{dt} \propto N \Leftrightarrow N(t) = N_0 e^{-t/\tau}$. This is a mathematical equivalence. So instead of plotting $\frac{dN}{dt}$ vs $N$, you could just plot $\log \left(\frac{dN}{dt} \right)$ vs $t$ and check that it follows a straight line.

Answer 2

I'm guessing: Start with a pure sample of your isotope. After some time, do a chemical (or physical) separation of the isotope and the decay product. Then weigh them separately.