Why can't free electron flow in water to conduct electricity?

This question is related to how batteries work. In a battery, the electrons can only flow in conductors, because they can't flow in the solution.

my own thoughts on the matter:

• I believe it is related to the "energy" of electrons.

• "flow" isn't related to "energy" of electrons quite well. As gold and copper are good conductors not because they have low electron negativity, their electrons are quite low in energy in the potential wells of the sea of nucleus. they are good conductors because, within the deep well, they are "free" to flow.

• in water, in order to be free, the electron needs to be so energetic, it has to be above the potential well of water molecules to be free in water. In that case, I don't think the electrons will be free either, the H+ in water should capture those high energy electrons and become gas, as H+ has very high electron negativity. However, In this case, I can't explain how does O- do with his extra electron.

• I am aware of the fact that, electrons "flows" in snail pace, it is the "pushing" between electrons (like domino) that conducts electricity

The metals conduct electricity because they have a high concentration of "free" electrons in them. These free electrons exists even in absence of current, they don't need to be sent or injected into the metal. Water does not have free electrons. All electrons are bound to water molecules or to $$OH^-$$ ions. In electrolites there are other ions but still no free electrons.

The reason for some materials having or not having free electrons (condoctors versus insulators) resides in both chemistry and the structure of the material. It's not a simple function of just one parameter (see diamond versus graphite conductivity).

If you somehow "inject" electrons into an insulator (like water) they will not make it conductor. They may attach to the ions already present or otherwise interact with the molecules of the insulator, creating a static charge which eventually will discharge into the air or surrounding bodies.

And lastly, the "pushing" between electrons is not relevant to electrical conduction/ Actually it can be neglected. After all this is why we can get a basic model of conduction based on free electron gas assumption. The interactions that matter are the interaction of the electrons with the electric field (created due to the power source) and with the lattice of positive ions (one of the mechanisms responsible for the electrical resistance).

• @nasu just to confirm: [1] free means electrons can move between nucleus (metal) or move between lattice (semi conductors) or really free (a gas of electron) [2] not due to pushing: so conduction in microscopic view, could be due the the absence of other electrons in the direction of conduction (an hole moves in a semiconductor, or this electron moves left because the electron on this electron's left have moved left already leaving a vacancy) – eliu May 31 '19 at 18:16
• @nasu [3] I personally disagree on the electric field in conductor interpretation, superconductor requires no field for electron to continue to circulate, voltage does not drop in a conductor, it drops a lot a resistor, yet the current flow is the same. – eliu May 31 '19 at 18:19