The bond length is inversely proportional to the bond order. A higher bond order means a stronger and shorter bond.

Step 1: Determine the bond order for CO.
In carbon monoxide (CO), there is a triple bond between carbon and oxygen. The bond order is 3.

Step 2: Determine the bond order for CO₂.
In carbon dioxide (CO₂), there are two double bonds between carbon and oxygen. The bond order for each C-O bond is 2.

Step 3: Determine the average bond order for CO₃^-2.
In the carbonate ion (CO₃^-2), there is resonance. The structure consists of one C=O double bond and two C-O single bonds, which are delocalized over the three C-O bonds. The total number of bonds is 4 (one double + two single) distributed over 3 positions. Therefore, the average bond order is $\frac{4}{3}\approx 1.33$.

Step 4: Compare the bond orders and determine the bond lengths.
Bond order of CO = 3
Bond order of CO₂ = 2
Bond order of CO₃^-2 = 1.33

Since bond length is inversely proportional to bond order, the order of bond lengths will be the reverse of the order of bond orders.
Order of bond orders: CO > CO₂ > CO₃^-2
Order of bond lengths: CO < CO₂ < CO₃^-2

Option Analysis:

• A) $\mathrm{CO}<{\mathrm{CO}}_2<{\mathrm{CO}}_3^{-2}$: This order correctly reflects the increasing bond length as the bond order decreases (3 < 2 < 1.33).
• B) ${\mathrm{CO}}_2<{\mathrm{CO}}_3^{-2}<\mathrm{CO}$: Incorrect, as CO has the shortest bond length.
• C) $\mathrm{CO}<{\mathrm{CO}}_3^{-2}<{\mathrm{CO}}_2$: Incorrect, as CO₂ has a shorter bond length than CO₃^-2.
• D) ${\mathrm{CO}}_3^{-2}<{\mathrm{CO}}_2<\mathrm{CO}$: Incorrect, as CO₃^-2 has the longest bond length.

Correct Answer: (A)