Final answer:
The balanced half-reaction at the cathode in a galvanic cell consisting of an iron nail wired to zinc and copper is Cu2+ (aq) + 2e → Cu(s), where copper ions are reduced to solid copper.
Step-by-step explanation:
In the described galvanic cell where an iron nail is wired to zinc and connected in a cell with copper, the balanced half-reaction happening at the cathode involves the reduction of copper ions to solid copper:
The zinc acts as the anode where the oxidation takes place, as stated in the half-reaction:
The zinc electrode is the anode and thus the negative electrode, while the copper electrode is the cathode and the positive electrode. This half-reaction is already balanced, with one copper ion (Cu2+) accepting two electrons to form solid copper (Cu(s)), and there is no need to balance mass or charge further for this half-reaction.