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A battery is dead when the system is at equilibrium. This occurs when there is a buildup of charge in the cathode and the anode, so there will no longer be any flow of charge. This can be avoided by having something like a porous disk to maintain a neutral state in the anode and cathode.
For a galvanic cell, a battery is also dead when you run of out metal in the anode. If Zn has to give off 2 electrons to become Zn2+ to power the battery, then your battery will die when you run out of Zn metal, since all you are left with is Zn2+ ion.
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