If you could watch an individual water molecule, about once in 10 hours you'd see it do this!

As it bounces around, every so often it hits another water molecule hard enough enough for one to steal a hydrogen nucleus - that is, a proton - from the other!

The water molecule with the missing proton is called a hydroxide ion, OH⁻. The one with an extra proton is called a hydronium ion, H₃O⁺.

This process is called the 'autoionization' of water. Thanks to this, roughly one in ten million molecules in a glass of water are actually OH⁻ or H₃O⁺, not the H₂O you expect.

And this gives a cool way for protons to move through water. Let's watch it!

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How does electrical current move through water? Unless it's really pure water, current is mainly carried by ions like Na⁺ and Cl⁺. Pure water is a much worse conductor. But it can still conduct a bit of electricity thanks to the mechanism shown here!

In this animated gif made by Mark Petersen, a positively charged proton gets passed from one molecule to another. This is called the 'Grotthuss mechanism' because Theodor Grotthuss proposed this theory in his paper “Theory of decomposition of liquids by electrical currents” back in 1806. It was quite revolutionary at the time, since ions were not well understood.

Something like this theory is true. But in fact all the pictures I've shown so far are oversimplified! A hydronium ion is too powerfully positive to remain a lone H₃O⁺. It usually attracts a bunch of other water molecules and creates a larger structure!

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@johncarlosbaez And the move of Na+, Cl- are purely mechanical?. I mean they simply move in the space. No other mechanisms? At what speed?

It should be compared with what happens in a wire. For a better understanding.