Are you interested in using it for some particular purpose? I've used it before as an electrolyte to help in de-rusting a chain I needed and it actually worked pretty well.
2H2O à O2 +4H+ + 4e-
And at the cathode (which electrons flow out of) water will be reduced, to form hydrogen gas. If you also connect a rusted piece of metal at this point, for example, it will compete for electrons, and so besides hydrogen gas you will get your reduced metal, say, iron.
H2O + 2e- à H2 + 2OH-
If you add those two half reactions up by crossing out the same species on opposite sides of the arrow, you'll end up with
2H2O à 2H2 + O2 which is the overall reaction.
In addition to the bicarbonate being slowly neutralized, the Na+ ions from the sodium bicarbonate will probably grab some of the OH- ions yielding sodium hydroxide, and over time the solution will become more basic.
But as for other half reactions, there are none for the bicarbonate system because it is not an oxidation/reduction reaction (note the +4 formal charge of carbon in carbon dioxide, bicarbonate, carbonate, and carbonic acid). It's all just an acid/base and decomposition reaction.
I know carbonic acid, carbonate, bicarbonate, and carbon dioxide all exist in some crazy dynamic equilibrium, which is still studied to this day, because I guess the mechanism for carbonic acid decomposition into CO2 and water still isn't fully understood. (but water greatly increases the decomposition rate - which is why it is so hard to "observe" it in its natural habitat, so to speak)
To be honest electrochemistry isn't my best area, and the more I read about it after you asked this question, the more I realized how crazy even something as seemingly "simple" as electrolysis of water + an electrolyte can be.
There are so many insane things going on down there at the atomic level, ions zipping from one electrode to the other, electrons being stolen and donated everywhere. Crazy stuff man!
Grab a piece of pH paper and take a few drops of water with a pipette from close to the two electrodes and check them out.
I think it's because of bicarbonate's crazy equillibriums with CO2, H2O, and H2CO3 that it's used in our body to control blood pH.
Sorry that's the best I can come up with at the moment
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