why completely used up batteries seem to have a small bit of energy again after some time.
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Batteries are generally a solid state system. The electrons flow out from one side, so you get a zone of depletion near the exit. With a well-charged battery, electrons flow relatively quickly from further away and replace a good portion of the lost electrons because the battery still has a huge amount of charge. The concentration of free-electrons is pretty high throughout.
As the charge depletes, the average concentration decreases, so it takes longer for charge to move from far away and get to the exit zone. A “dead” battery is generally not truly “dead”, but it has basically no mobile electrons near the exit and can’t get replacements fast enough, so it stops providing energy.
However, the battery generally does have residual charge further away, and the charge will slowly spread (average itself) throughout the battery, rejuvenating the battery a little bit. Eventually, there will not even be enough residual charge far away to get a useful charge anywhere.
Liquids tend to “mix” from areas of high concentration to areas of low concentration much faster than solids. The molecules are more mobile, but even liquid batteries like a car battery can see a zone of depletion in the region of contact between solid (lead) and solution, so the battery can rebound a bit if left to sit, if given a bit of time.
But you can never get back to full charge, only the average charge of the entire battery at the time it “died”.
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