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I feel like to know whether a salt is acidic or basic, you really just have to see if it includes one of 3 cations (Li, K, Na) or one of 3 anions (Cl, Br, I) and recognize that these will not affect pH, so the other half of the molecule must be the definer of the salt. Is this an oversimplification of what needs to be done to identify a salt, or does this actually work?
Those are good ions to remember. However, keep in mind as a general rule that cations of BOTH Group 1 and Group 2 metals, plus other cations with a +1 charge are too large or have too low a charge to significantly polarize the water molecules they are surrounded by and thus do not affect pH. With reference to anions, the general rule is that the conjugate bases of strong acids practically have no effect on a solution's pH. (i.e. Cl-, Br-, I-, NO3-, ClO4-). But besides just memorizing it you can come up with this fact logically. If these anions did affect pH they would have to function as Bronsted bases, (i.e. Cl- would produce HCl) but we know that HCl is a strong acid and so by definition ionizes completely. In short, if Cl- affected pH it would form HCl, but that doesn't make sense because HCl is a strong acid which is completely ionized. This is a helpful way to think about it because it is advantageous to go beyond the memorization to understanding the logic of it in case you forget a few (i.e. your list did not include the nitrate ion or perchlorate ion for example).
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