Does a battery last as long when it is recharged?

Ni/Cd or Nickel Cadmium rechargeables and many of the metal hydride batteries recrystallize as they heat and cool due to the action of the electric charge. This affects their ability to store charge. Lead acid batteries develop sulfur based corrosion on their metal plates and this affects their ability to recharge. Lithium batteries don't seem to have this effect to the same degree, but do eventually "age" as in this article (see link below).

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The short answer is yes, batteries like almost everything else have a limited lifespan. Different batteries have different lifespans and care instructions, so pay attention and choose a battery product with the best lifespan where it matters.