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

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 ?

• You asked a LOT of questions in one posting. – David White May 7 '20 at 18:17

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.

• Hmm , interesting workaround. I wonder how the observation was first made. Against $N$ or against $t$. – Kantura May 8 '20 at 20:27

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.

• I see. So let's say I have a chunk of radioactive material ''A'' which decays to ''B''. To do this experiment you couldn't just use a solid chunk of material. You would first have to break it into dust and use chemical and physical means to remove the material ''B'' every so often. So you would allow it to sit (decay for some time), then remove the material ''B'', quickly measure the $\frac{dN}{dt}$ and mass, repeat. Would you agree with this rough understanding ? – Kantura May 8 '20 at 13:32
• That's my understanding. You could then fit your data to the equations supplied by QuantumApple. – R.W. Bird May 8 '20 at 16:10