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I know that electronegative metals like copper or silver can electroplate out of an aqueous solution in electrolysis. However, I've also noticed $\ce{ZnSO4_(aq)}$ can electroplate out when electrolyzed, yet Zinc is more electropositive than hydrogen. This doesn't happen with other electropositive ionic solutions like $\ce{Al(C2H3O2)3_(aq)}$.

Why does this happen, and why is it limited only to Zinc?

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It's too simple to think of this situation in terms of electronegativity and it's certainly not limited to zinc. If you look at a table of standard electrode potentials, you can see that the reduction of many metal ions like $\ce{Zn^2+,Pb^2+,Cd^2+,}\textrm{ etc.}$ have standard potentials below that of the reduction of hydronium (0 V) and should be much less favourable than making hydrogen gas if this were the only consideration.

However, in real electrochemical cells, it generally takes a certain amount of extra voltage to get a certain reaction to occur, known as the overpotential. Like all chemical reactions, redox reactions have a certain activation energy separate from the energy change between the reactants and products, e.g. elemental gold is far lower in energy than a gold solution, but a gold solution will not immediately reduce spontaneously because there is an activation barrier to the process, an extra potential must be applied to make it happen at a reasonable rate. Similarly, though the standard potential for reducing hydronium is 0 V, in practice it may not be observed until much more negative applied voltages.

This overpotential for hydrogen evolution is strongly dependant on the type of electrode used. For the reaction to occur, hydronium ion must adsorb to the surface. In electrodes like platinum, this interaction is stabilized so the overpotential is relatively small and it's thus quite difficult to deposit metals like zinc onto platinum. For this reason, other electrodes must be used for working with aqueous solutions at very negative potentials. Glassy carbon, mercury, and bismuth electrodes are often used for this as hydronium is not readily adsorbed to their surfaces, giving large overpotentials for hydrogen evolution. These types of electrodes may be useful down to -1 V or so and more exotic materials like boron-doped diamond electrodes, even more negative.

However, no electrode can prevent hydrogen evolution if the potential is negative enough. Zinc is about the most easily oxidized metal that is practical to electroplate from aqueous solution. Some metals are so easily oxidized that they are basically impossible to plate in aqueous solutions which is why one generally cannot electroplate from aqueous aluminum solutions or solutions of alkali metals (things like sodium and potassium are so easily oxidized that they react violently with water to form hydrogen gas).

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