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There's a common belief that once a car battery dies, after it gets jump started, it will never be as good as before, and will likely die again. I'm curious whether this is a myth, or if not - what's the science behind this. I have noticed previously that when a car battery first died, it started having more issues starting up in cold winters, but am unsure whether this is just a coincidence.

From my understanding, which is a from rather introductory electromagnetism course, is that in a simple circuit, a capacitor can be discharged and charged without any harm to it's capacitance.

I currently have a completely dead/flat 12V 8.5A lead acid battery from a motorcycle that wasn't used for a year. It seems it's not as straight forward as charging it with a trickle charger, because if it's below 3V, this particular charger won't detect it, so will require jump starting it to atleast 3V+.

The manual for the trickle charger also had stated, "Unless the battery was rapidly discharged... most 12V lead acid batteries that are at a state of charge that is less than 9V is likely to be worn out or defective". I'm aware that lead acid batteries loses effectiveness over time (I'm assuming, that equates to losing capacitance?), so it can't be as good as a new one.

But disregarding the wear/tear time factor, does having a battery that went flat actually damaged the battery internally vs. if this battery was used consistently? I'm debating between trying to save this dead battery (if even possible) vs. buying a new one.

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This is a major annoyance with "smart" chargers in that they follow certain recommendations and refuse to charge a battery that is too dead. You can probably revive the battery with a bench power supply to the point where the charger will work. Old fashioned chargers, which consist of little more than a transformer and rectifier, will also work.

According to this website, deep discharge (below 10.5V for a 12V battery) causes actual damage (sulphation of the plates) which is irreversible:

Lead sulfate (sulfation) now coats most of the battery plates. Lead sulfate is a soft material, which can is reconverted back into lead and sulfuric acid, provided the discharged battery is immediately connected to a battery charger. If a lead acid battery is not immediately recharged, the lead sulfate will begin to form hard crystals, which can not be reconverted by a standard fixed voltage (13.6 volts) battery converter/charger.

This is certainly my experience with automotive batteries in practice, especially in a cold climate.

So I think, even though you could probably salvage the battery and get a bit more life out of it, it would be better to replace it so you don't get stranded somewhere.

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