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I think we always use molarity within ICE tables since molarity gives the concentration in the solution (moles per liter), whereas moles would just give the amount of substance without taking into account the volume of where it is in and therefore not qualifying as a 'concentration' value. Im pretty sure all problems give you the information needed to convert to molarity, so thats another clue to convert and then use molarity!
I've seen ICE tables use moles on a Khan Academy video on acid base titrations. Personally, if they give you a volume, its probably a hint to use molarity. Dr. Lavelle's future lectures on acids and bases may shed more light on this.
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