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There is a question on my assignment about stoichiometry. I spent 2 whole days searching and review this topic but still clueless.

Reaction in lithium ion battery $$ \ce{Li(s) + CoO2(s) → LiCoO2(s)} \, . $$ When the battery is recharged, electrical energy will reverse the reaction. How do the relatives amounts of lithium and cobalt oxide change as the battery is recharged and how?

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About maximum $20~\%$ (as I remember, perhaps more now) of the $\ce{Li}$ goes (as $\ce{Li+}$) out of the structure of $\ce{LiCoO2}$, through the elctrolyte, and into the electrode on the other side of the battery. That electrode consists mainly of graphite, which intercalates the $\ce{Li}$ in it's layers in a quasi-metallic form.

So it is $\ce{Li_{1-x}CoO2 + x~Li}$ after charging. $\ce{LiCoO2}$ has a special structure that does not break down when some $\ce{Li}$ is missing, except if you keep a charged battery at over $\mathrm{50~^\circ C}$ or so.

By the way, the related wikipedia article tells you all this. Two days of researching? ;-)

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