Thanks for asking. This is an interesting question.
First, let's clear up one point: 13C is not radioactive, 14C is. 13C is a stable isotope with a spin-1/2 nucleus, which makes it amenable to study by nuclear magnetic resonance. 13C NMR is, in fact, an extremely common technique.
In practice, we know that 13C nuclei stay put, and do not relocate to other parts of the molecule. It may help you to consider a few points:
- Electron positions are so indeterminate because they have a high degree of kinetic energy and have very little mass (relative to protons and neutrons, at least). That's why we describe them with orbitals, which are basically positional probability distributions
- Nuclei are much heavier and have less kinetic energy, so we can know their position with much more precision
- Instead of wondering whether a 12C and a 13C nucleus could swap position (which involves two extremely unlikely events), you should consider the probability that a single neutron will spontaneously relocate to a different nucleus. While this event is much more likely, it is still extremely unlikely to happen.
- For decades people have used site-specifically-labeled compounds to study metabolic pathways and synthetic reaction mechanisms. In practice, we know that if we purchase ore create a site-specifically-labeled compound that the 13C stays in that position. If it migrated, we could observe changes in the 31C NMR spectrum over time, and we don't.
Please take a break and have some tea for a while.
Thank you for your graceful and refined answer along with your tender heart.
I have understood that people already have been watching whether any relocation of the carbon nuclei of a chemical compound in its stable condition could have happened or not, and they didn't find.
May I ask you one more question, Sir? If you are busy now, then please feel free to ignore the following.
Can we observe and follow in real time the motions of carbon atoms at the exact instant of the chemical reaction with enough time resolution power by the current NMR technique? Can we know the detailed paths which the reactant carbon atoms follow in space? To say further, can we picture the whole motions of all atoms that participate in a chemical reaction from the NMR spectrum data?
From a perfect layman