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Here is a sketch of the functioning cell:
Zinc electrode (Zn) | Zinc sulfate solution (Zn^2+) || Copper (2) sulfate solution (Cu^2+) | Copper electrode (Cu)
At the anode (negative electrode), oxidation occurs, giving up electrons:
Zn(s) → Zn^2+(aq) + 2e^-
At the cathode (positive electrode), reduction occurs, accepting electrons:
Cu^2+(aq) + 2e^- → Cu(s)