Batteries have electrodes in them that are in contact with a solution containing the reactive compounds. I’m saying “solution,” but that doesn’t always mean liquid. Sometimes it’ gel or even a nanopourous solid. In any case, It’s the compounds in solution that provide the electrons on one side and take up the electrons on the other. They need to be at the surface of the electrode to react. That means that the area right at the surface of the electrode can be run out of the compounds it needs. It takes time for fresh compounds to diffuse to the electrode. This effect is going to be strongest when the battery is almost dead, because the concentration of the compounds is very low.
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