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I am unable to rationalise the order of increasing bond length in $\ce{CO3^2-}$, $\ce{CO}$ and $\ce{CO2}$. Having gone through the factors affecting bond length in two different books, my approach to the problem was:

  • $\ce{CO}$ has bond order 3 while carbonate and $\ce{CO2}$ have bond order 4 so $\ce{CO}$ should have longest bond length.

    $\ce{CO2}$ is sp hybridized while $\ce{CO3^2-}$ is sp2 hybridized. Since carbonate has lesser s character and therefore it should have a greater bond length than $\ce{CO2}$

Thus, $$\ce{CO2 < CO3^2- < CO}$$

However, I am unsure about my method. What would be the best approach to determine the relative bond length order?

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Actually, in regard to bond length, you should be looking at the bond order of a single bond within the molecule. The bond order is 2, not 4, for $\ce{CO2}$, and the bond order for the carbonate ion is somewhere between 1 and 2 due to resonance. Thus, the carbonate ion has the longest bond length, followed by carbon dioxide, and finally carbon monoxide.

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