I’m guessing based on what I know about NMR, but I think it’s because it relies on being able to flip the spin of a single proton. If you had an even mass number then you could have 50% of the nuclei with their spins in one direction, 50% in the other. It’s harder in that situation to flip one of the nuclei because it would go to a higher energy state.
If you have an odd number of nuclei then there is always one left over without a matching nuclei with spin in the other direction. So the amount of energy you need to flip its spin direction is less.
I actually had not heard about this feature of NMR, but it makes sense when you think about it I think…
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