When people explain a galvanic reaction, they usually stop the explanation in the dogma that one is more reactive/electronegative/nobler then the other. But that doesn't really explain the mechanism. I mean, why there would be a electrostatic force between elements with different reactivity? What mechanism creates a potential there? Why electrons get repelled from $\ce{Zn}$ and get attracted to $\ce{Cu}$?
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1You have a net transfer of electrons from zinc to copper(II) ions. That's not the same thing as what you've written. – Zhe Jul 14 '17 at 15:54
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2There is a layering of ions and oriented water dipoles at the interfaces between water and metal. That creates a voltage difference between the electrodes, depending on the ion concentrations in the electrolyte and the polarisability of the metal atoms at the interface. Wherever the local electric field is high enough, metal ions can be formed out of the metal or coalece into it, creating a current. (My personal understanding of the process.) – Karl Jul 14 '17 at 16:34
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There's also https://chemistry.stackexchange.com/questions/20084/why-do-electrons-leave-the-zinc-in-a-galvanic-cell – Mithoron Jul 14 '17 at 22:46