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Bond Order and Bond Length
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The correct order of C-O bond length among CO, $CO_3^{2-}$, $CO_2$ isAll three species exhibit resonance, and the bond length depends on the bond order: more the single bond character, more will be the bond length.
(i) CO: $:C\equiv O:$ with a minor form $:C=O:$. The C-O bond is essentially a triple bond (bond order about 3).
(ii) $CO_3^{2-}$: three equivalent resonating structures, each with one $C=O$ and two $C-O^-$ bonds. Bond order $= \dfrac{4}{3} = 1.33$.
(iii) $CO_2$: $O=C=O$ with minor forms $^-O-C\equiv O^+$ and $^+O\equiv C-O^-$. Bond order is about 2.
Bond order: $CO$ (3) > $CO_2$ (2) > $CO_3^{2-}$ (1.33)
Bond length is inversely related to bond order (about 110 pm, 115 pm and 129 pm respectively).
Hence the correct order of C-O bond length is: $CO \lt CO_2 \lt CO_3^{2-}$ -
The correct order in which the O-O bond length increases in the following is :-Bond length decreases as the bond order increases.
$O_2$: O=O, a double bond (bond order 2); bond length about 121 pm.
$O_3$: resonance between two structures, each with one O=O and one O-O bond, so each O-O bond has bond order 1.5; bond length about 128 pm.
$H_2O_2$: H-O-O-H, a single O-O bond (bond order 1); bond length about 148 pm.
Increasing bond length: $O_2 \lt O_3 \lt H_2O_2$