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ngl kinda seems like a r/GCSE question lol but the H+ ions are attracted by the charge through the water to the cathode where they gain electrons from the current to form H and covalent lay bond with other Hs form H2, and the rest of the water is a left as a hydroxide ion, OH-, which is turns into water and oxygen through the reaction: 4OH- -> H2O + O2 +2e-, so the oxygen is formed where the OH- ions react and the hydrogen is formed where the hydrogen ions gain electrons to become stable hydrogen molecules

>I mean, the water molecule is split somewhere

I think you are getting hung up thinking of it as the water molecule. Keep in mind water is a sea of molecules, with a lot of exchanges going on:

H2O ⇌ H^+ + OH^-

OH^- + H2O ⇌ H2O + OH^-

H^+ + H2O ⇌ H3O^+

H3O^+ + H2O ⇌ H2O + H3O^+

So when the hydrogen in a water molecule near the cathode accepts an electron to start forming H2, the remaining OH^- doesn't itself have to migrate to the anode to be oxidized to O2. It just has to exchange with a neighboring water molecule's hydrogen, so now there is a new OH^- a little closer to the anode. So you end up with H^(+)'s being transferred along through the aqueous medium, something like a bucket brigade, to keep supplying H^+ to the cathode and OH^- to the anode, but not necessarily the same H or O in each transfer.

H3O^+ + H2O + H2O + H2O ->

H2O + H3O^+ + H2O + H2O ->

H2O + H2O + H3O^+ + H2O -> -> ->

H2O + H2O + H2O + OH^- ->

H2O + H2O + OH^- + H2O ->

H2O + OH^- + H2O + H2O -> -> ->

In effect you have H^(+)'s being passed along in one direction, and OH^(-)'s being passed along in the other direction.

(Edit: italicized the initial H and O)

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I must say, that is an exceptionally good answer -- good both for experts and beginners.

amazing, thank you very much! so like a row of people carrying sandbags and giving them to the person next to them?

Very much so, in that adding a sandbag to one end of the chain ultimately results in a sandbag, albeit a different one, being delivered to the other end of the chain. And unlike a Newton's Cradle (with simultaneous transfers), this is a much less orderly process. Sometimes a sandbag gets passed back before going on, or someone has to wait for a sandbag, etc. This makes it a much slower process than say electrical conductance in a wire.

thank you very much i always thought of water as a pot of rice, of fixed, unchangable, determined molecules. so "water" is not just a mix of water, but it's own dynamic system comparable to how in metal electrons just fly around, just that not the electrons but also entire molecules of H and O just swap back and forth.

thank you so much for taking the time to explain it to me!

> i always thought of water as a pot of rice, of fixed, unchangable, determined molecules

Well you aren't wrong, it all depends on the timescale you are thinking about.

Water molecules are constantly in motion. This is what 1 picosecond of movement in a water droplet at 0 C looks like. They do move around a lot, and a lot of collisions take place in between even diffusion controlled reactions like H-abstraction (timescale 10^(-9) sec).

But relative to molecular vibrations, for instance, they are barely moving at all. Bond stretching vibrations (O-H stretch timescale ~10^(-13) sec), are orders of magnitude faster than collisions. And electronic transitions are several orders of magnitude faster than that (timescale ~10^(-15) sec), so thinking of the atoms and molecules as "fixed" in space would be an excellent picture of them on that timescale. Thinking about an electrical conductor, instead of an H^+ transfer medium, for instance.