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Why is the oxalate ion a bidentate ligand? Why does it not act like a tetradentate ligand (having four oxygen atoms, with lone pairs)?

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The oxalate anion cannot be tetradentate. While there are four oxygen atoms, the ion is planar and the oxygen atoms are pointing in different directions.

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The best any metal can do is bond to two of them, although there are two bonding modes possible:

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Perhaps you were envisioning some $\kappa^4$-bonding mode that is "side-on". However, most of the nonbonding electrons (orbitals) are also in the plane.

This is not to say that oxalate cannot form four bonds to metal centers; it just cannot do so with only one metal center. Oxalate can act as a bridging ligand

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