I don't know why I'm asking or why you'd want to but I'm just curious.
Every resource I read about electrolysis says the reaction is $\ce{2H_2O (l) \rightarrow 2H_2 (g) +O_2 (g)}$.
Can you electrolyse a single molecule?
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$\begingroup$ In chemical equations you have stoichiometric coefficients not numbers of molecules in reaction mixture. $\endgroup$– MithoronCommented Jul 23, 2016 at 19:38
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In a traditional experimental setup this would never happen, as there are always several water molecules. But lets just think about it for the sake of it.
If you wanted to design an experiment where you only have one water molecule, you would have to get two electrodes extremely close together. Lets say you find a way to levitate the molecule between the two electrodes. You might be able to then apply a voltage and tear the molecule apart, forming one H$_2$(g) molecule at the cathode. The oxygen radical would then most likely immediately react with the anode to which it is attracted by the Lorentz force and form an oxide.
So the result of electrolysing several single molecules of water might result in gaseous hydrogen and no oxygen, but a buildup of oxide at the anode.
However, having two electrodes so close together would probably create a series of problems. Not only to hold the single water molecule in place (which could be achieved by optical levitation) but because quantum mechanical effects such as tunneling currents would occur. Probably there are several more complications that would by all means make it nearly impossible to actually observe this. The knowledge gain derived from such an experiment would likely not justify the effort and expenses.
EDIT
I just revisited this question and thought about it again: it's not absolutely necessary that the two H atoms would form an H$_2$ molecule. Depending on the way the cathodes are formed, the electric field might not pull the charged particles together, but to a collision with something else in the vicinity; the cathode or any ambient gas. Furthermore, my notion above that "H$_2$(g)" would form is a bit nonsensical, as a single molecule cannot really be attributed a state of matter (i.e., being gaseous in this case).
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$\begingroup$ I wonder if something of the sort hasn't been performed using an AFM tip and a molecule of water adsorbed on a surface. Chemical reactions have been triggered like that, previously. $\endgroup$ Commented Jul 23, 2016 at 13:31
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$\begingroup$ @NicolauSakerNeto I do also wonder that. Rather than an AFM I could actually imagine something like that happening with an STM (tunneling current rather than force). Atoms most certainly have been rearanged with STMs, see the famous IBM logo image. Not sure wheter molecules have been "torn apart" or electrolysed in that way though. Maybe somebody knows a reference. $\endgroup$– tipaviCommented Jul 23, 2016 at 17:13