I know that I can make a solution of sodium hydroxide and zinc filler and use that to put a thin zinc plate on things but I want to know if I can do this with other metals like copper.
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Not as well as zinc. Zinc reacts with sodium hydroxide to form zinc hydroxide, an insoluble salt. But excess sodium hydroxide reacts with zinc hydroxide to form zincate ion. The amphoteric nature of zinc hydroxide is why zinc is soluble in sodium hydroxide solution. But copper(II) hydroxide is not as amphoteric as zinc hydroxide, that means it does not dissolve as well. However, if you add ammonium hydroxide, then it is different because ammonia molecules can surround the copper cation and make copper hydroxide soluble.