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In other words: What is the purpose of the solution that the electrodes are immersed in, in a galvanic cell, and is it possible to generate a current without any solution?

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The solution provides ions to the electrodes and a solution where the oxidized anode material can dissolve when it looses it's electron.

When the anode electrode oxidizes it looses it's electron which moves through the wire to the cathode electrode and the resulting ionized anode material dissolves in the solution.

The electron moving through the wire then combines with the cations in the cathode solution so the cations are then reduced to a metal.

So the solution gives something to be reduced at the cathode and something where the oxidized anode material can dissolve into or form a salt with.

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