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The electrons in a metallic conductor, tend to drift at abysmally slow speeds, but still a current is able to flow through it, because the net movement is in the direction of the applied emf.

How is this possible in liquids.The ions themselves move(if an electrolyte is present).So, if current through a liquid travels instantaneously across terminals, does that mean a whole ion has zapped it's way across, or a net drift of ions occurs, as in a metal?

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What always "moves" at the speed of light is the electromagnetic field. That's the displacement current term in Maxwell's equations. The ions and dipoles in the liquid will then follow this field and generate a field of their own. Since ion movement is so slow, it won't make any relevant contribution at high frequencies. What we observe in an electrolyte is therefor the displacement current between the plates at the highest frequencies, then the contribution of dipoles from both the solvent and the dissolved ionic species that rotate locally in the field and then, at the lowest frequencies, the drift of ions. The matter is complicated by the immediate formation of double layers on the electrodes and the electrochemical potentials that develop (which we use for batteries and chemical sensors).

Ions do, by the way, not move alone. An ion is usually surrounded by a solvent shell: https://en.wikipedia.org/wiki/Metal_ions_in_aqueous_solution, so the dynamics would be even more complex than that of electrons in metals.

In general, electrochemistry is a whole field of its own and there are some very interesting and useful effects in these systems that can be researched with impedance spectroscopy, i.e. the measurement of the different contributions to the effective conductivity of ionic liquids at different frequency and time scales.

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