Hello, If we suspected something was wrong with our battery electrolyte, what type of measurements would we do?

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Newt Ball

I assume a lead acid battery. You certainly should have told us everything about the battery, with the question. In a lead acid battery, state of charge is indicated by electrolyte density, measured by hydrometer. Same hydrometer reading in all cells is important. If you have access to inter-cell terminals, then you may balance state of charge, of the individual cells. If you don't have terminal access, then a mild overcharge of the whole battery may improve balance somewhat.

If we are talking about a lithium ion battery, DO NOT OVERCHARGE, EVER!!