Question

In a radioactive sample the decrease in the number of active nuclei at any time is directly proportional to?

• A. Number of radioactive nuclei present in the starting
• B. Time elapsed from the start time
• C. Number of radioactive nuclei present at that particular instant
• D. A small time interval from that particular instant.

Answer 1

Answer: (C), (D)

A small time interval from that particular instant.

The phenomena of radioactivity is not fully understood mathematically. We say it’s a first order reaction but that’s not a mathematical finding. It’s a game of pure probability theory.
What do we mean by this is that, we know that a particular radioactive nucleus will decay, but when will it decay is not known.
So, if you are left with a nucleus then surely you can't answer the perfect time of its decay but if left for a longer "duration", it has a larger "Probability" of decaying.
Also, if you are left with more number of radioactive nuclei there will be a "probability" that "one of them" or "few of them" might decay.
The math we will develop is based on above two statements.
The number of nuclei that will decay will be directly proportional to:
1. No. of active nuclei present at that particular instant and
2. The time interval of observation.
Mathematically,
$\Delta N\propto N\Delta N\propto \Delta t$
Writing in differential form;
$dN\propto Ndt$