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You can determine why carbonato can be either monodentate or bidentate by looking at its Lewis structure. In its Lewis structure, the C atom in the middle forms one double bond with an O atom and forms two single bonds with the remaining two O atoms. The two single C-O bonds include O atoms that have lone pairs that can form coordinate covalent bonds with a transition metal ion. CO3 2- can then become monodentate by creating one coordinate bond using one oxygen atom's lone pair, or become bidentate by creating two coordinate bonds using both the oxygen atoms with lone pairs. Carbonato cannot form a coordinate covalent bond using the O atom in the double C-O bond because oxygen is most stable when it has two bonding pairs and two non-bonding pairs. That O atom is already in a favorable state and would not likely want to form a coordinate bond.
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