Right I see what you are saying now. I admit I only took a cursory glance the first time around and assumed C13 had some long-term radioactive decay, maybe a lot longer than C14, which is about 5700 years. However on double-checking it appears C13 is an entirely stable isotope. I thought that was a bit weird since I had never heard that C13 had decay but there you go.
Incidentally it has nothing to do with what plants do to the C14 other than they simply store it since your link talks about chemical synthesis which is quite distinct from its nuclear counterpart. To create C14 needs high-energy cosmic rays, so nothing changes it in the actual plant itself. You can burn it or freeze it and the half-life remains exact since the nuclear process is on an entirely different energy level.
So this is what I'm now wondering about.
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The “natural” C13 content of CO2 is just over 1.1%. In contrast, the C13 content of the CO2 produced by burning of fossil fuels is claimed to be slightly smaller - just under 1.1%.
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So I agree this is odd and other than nuclear decay I can't see how this is claimed so maybe someone should ask him. However I still find it an interesting line to see if isotopes can give us some date information on atmospheric carbon. It's a valid line of enquiry.