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I was wondering what processes are used to select an electrolyte for a galvanic cell.

I know a zinc copper battery usually has zinc and copper sulfate electrolytes. How do these electrolytes interact wishing each half cell (i.e.: how does zinc sulfate interact with water, the anion from the salt bridge and zinc itself)?

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The point of an electrolyte is to allow ion flow between the anode and the cathode in order to balance the charges on either side so the reaction can continue. What you are looking for in a good electrolyte is that it won't react with either of your metals (i.e. won't form a precipitate), and that the ions will readily move (that's what makes soluble ionic compounds a great choice).

With $\ce{ZnSO4}$, for example, you wouldn't want to have $\ce{Pb^2+}$ ions anywhere in your cell, since $\ce{PbSO4}$ would precipitate and your concentrations of your ions would change, therefore changing your overall cell potential.

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