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When a battery is subject to overcharging or overdischarging (including attempts to recharge a primary cell), it may vent hydrogen. It seems that every type of electrochemical battery I've read about or used specifically emits hydrogen under these circumstances.

Why is hydrogen emitted in all cases, and not some other gas or chemical? What role does hydrogen play in electrochemical batteries that other chemicals can't?

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Lead storage batteries use an electrolyte of sulfuric acid and water. On overcharging you are electrolysing the water with emission of $\ce{H2}$ in one electrode and $\ce{O2}$ in the other.

Other batteries use molecules with H and Li. The idea is to use light elements so you can accumulate more charge for the weight.

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