The bond length of a molecule depends on the bond order, which is related to the number of bonds between the atoms. For molecules and ions with resonance, the bond length will also depend on how much the resonance delocalizes the electron density over the bond.
### Bond Length in CO:
In CO (carbon monoxide), there is a triple bond between carbon and oxygen. The bond order is 3, indicating a strong bond. The triple bond leads to a short bond length because the bond order is high and the atoms are more strongly held together.
### Bond Length in CO2−3:
In the carbonate ion (CO2−3), the carbon-oxygen bond has a bond order of 1.33 due to resonance between the two C−O bonds. The presence of multiple resonance structures distributes the electron density over the three oxygen atoms, which decreases the bond strength and results in a longer bond compared to CO.
### Bond Length in CO2:
In CO2 (carbon dioxide), the bond order between carbon and each oxygen is 2 because of the double bonds. Therefore, the bond length in CO2 is longer than in CO but shorter than in CO2−3, as a double bond is weaker than a triple bond and stronger than a single bond.
Thus, the correct order of bond lengths is:
\[
\boxed{CO<CO_3^{2-}<CO_2}
\]