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Standard electrode potentials are reported for materials that are not usable as actual electrodes in water because they react spontaneously and rapidly. https://www.engineeringtoolbox.com/electrode-potential-d_482.html

Some metals, like zinc, are useful as cell electrodes if pure, but if not so pure, exhibit cathodic areas that cause the metal to corrode and produce hydrogen without delivering useful external current. One way to counteract that was to coat the surface with a tiny bit of mercury, which apparently coated the cathodic sites but permitted the zinc to migrate thru as if the mercury were not there.

Other metals, like aluminum, develop a film which inhibits corrosion by developing a resistance to atom/ion movement. It has been suggested to me that the actual electrode reaction on aluminum in water is production of hydrogen, so that the half-cell using an aluminum electrode in water is the hydrogen electrode, because bare aluminum reacts immediately with water. This would be analogous to the zinc anode with surface impurities (cathodic sites), except that we do not consider zinc to be producing a hydrogen electrode, just corroding.

it seems that there is considerable usage of metals like zinc, aluminum and magnesium for cathodic protection under conditions where no hydrogen is expected, so it is reasonable to visualize the protection to be developed by reaction of the metal, possibly thru a partially protective film which is equivalent to a cell resistance rather than changing the chemistry to a hydrogen electrode. https://performancemetals.com/pages/sacrificial-anodes-faqs

Is there any evidence that metals like aluminum (or zinc or any other metals) could support a hydrogen electrode like platinum or palladium do? The standard half-cell with Pt or Pd is in 1 M HCl, which would not be appropriate for Al, Zn or Mg. Would any other conditions be better?

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Aluminum cannot be used easily as support for a hydrogen electrode, because it is always covered by a thin, continuous and insulating layer of aluminum oxide. As this layer is an electrical insulator, its resistance is high, and ions like $\ce{H+}$ have some difficulty in crossing it. So the measured potential of hydrogen measured on such a protected electrode is not a good measurement of the redox hydrogen potential. It corresponds to an important overpotential, which is to be avoided if you want to measure or determine a redox potential.

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