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How can we determine if a molecule acts as an acid or a base if there are no hydrogens present in the formula? Is it solely based on the metal or nonmetal present or are there other ways of identifying its characteristics in an aqueous solution?
You can look at the lewis definition of an acid or base. Since this definition deals with the number of lone pairs present on an atom, you can look at whether or not the molecule would accept or donate electrons. If it look like it would accept electrons, such as BF3 which does not have a complete octet, it is an acid.
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