Question

Consider the following bond lengths: \(\begin{array}{lllll}\mathrm{C}-\mathrm{O} & 143 \mathrm{pm} & \mathrm{C}=\mathrm{O} & 123 \mathrm{pm} & \mathrm{C} \equiv \mathrm{O} & 109 \mathrm{pm}\end{array}\) In the \(\mathrm{CO}_{3}^{2-}\) ion, all three \(\mathrm{C}-\mathrm{O}\) bonds have identical bond lengths of \(136 \mathrm{pm}\). Why?

Short answer

The identical bond lengths of 136 pm for all three C-O bonds in the CO₃²⁻ ion can be explained by the resonance hybrid structure, in which the bond character is distributed uniformly across all three C-O bonds. The average bond length calculated from the given single and double bond lengths (along with equal contributions from the resonance structures) also equals 136 pm, thus validating that resonance leads to equal bond lengths in the carbonate ion.

Step-by-step solution

Draw the Lewis Structure of CO₃²⁻ Ion

(Write the content here) First, draw the Lewis structure for the carbonate ion (CO₃²⁻). In the carbonate ion, carbon has 4 valence electrons, and each oxygen atom has 6 valence electrons, totaling 24 valence electrons. Add two more electrons for the −2 charge. With a total of 26 valence electrons, the carbonate ion looks like this: \[ O=C=O^{-} \oplus O^{-}-C-O^{-} \oplus O^{-}-C-O^{-} \] It can be seen that there are three single-double bonded oxygen atoms in resonance with the central carbon atom.

Analyze the Resonance Structures

Resonance occurs when there is more than one possible Lewis structure with the same arrangement of atoms but different electron assignments to different positions (different bond orders). In the case of the carbonate ion, there are three resonance structures: \[ O = C - O^{-} \oplus O^{-} - C = O \oplus O^{-} - C - O^{-} \] In each resonance structure, there are two single bonds and one double bond. However, none of these individual resonance structures are an accurate representation of the true structure of the carbonate ion. Instead, the actual ion is a combination of the three resonance structures, meaning that the bond character is distributed uniformly across all three C-O bonds.

Calculate the Average Bond Length

To understand the identical bond lengths of 136pm in the carbonate ion, consider the resonance hybrid structure, which is an average of the bond lengths in the different resonance structures. Since there is one double bond and two single bonds in each resonance structure, their average bond lengths can be calculated as follows: \[ \frac{1 \times 123 \operatorname{pm} ( {C} = {O}) + 2 \times 143 \operatorname{pm}( {C} - {O})}{1 + 2} = \frac{123 + 286}{3} = 136 \operatorname{pm} \] Hence, the average bond length of the carbonate ion's C-O bonds is 136 pm.

Conclusion

The identical bond lengths of 136 pm for all three C-O bonds in the CO₃²⁻ ion can be explained by the resonance hybrid structure, in which the bond character is distributed uniformly across all three C-O bonds. The average bond length calculated from the given single and double bond lengths also equals 136 pm, thus validating that resonance leads to equal bond lengths in the carbonate ion.

Key concepts

Carbonate Ion

The carbonate ion, \(\text{CO}_3^{2-}\), is an interesting molecule chiefly because of its resonance structures. In terms of chemical composition, it contains one carbon atom and three oxygen atoms. It also carries a negative two charge.
Carbonate is often encountered in everyday substances, such as baking soda and limestone.
What makes the carbonate ion particularly noteworthy is how it exhibits resonance. In resonance, the real structure is a blend of multiple possible Lewis structures. Each of these resonance forms has different electron distribution, but the position of atoms remains the same throughout. This contributes to the stability and unique properties of the ion.
In the case of the carbonate ion, the structure is influenced equally by three resonance forms that contribute to its stability and rationalize its identical bond lengths.

Average Bond Length

Average bond length is crucial in understanding how the bond structure appears in molecules that exhibit resonance, such as the carbonate ion. When we say average bond length, we are referring to the mean length of a bond that averages over multiple resonance structures.

Imagine each C-O bond in carbonate ion contributes to an overall structure that is influenced by both single and double bond characteristics. In the example of the \( \text{CO}_3^{2-} \) ion, there is a resonance between different bond orders: usually two shorter double bonds and one longer single bond. These combine to form an averaged bond length which is intermediate in character.
The calculated value of an average bond length helps in explaining the physical reality observed, which in this case, is 136 pm for each C-O bond due to contributions from single and double bonds in the resonance structures.

Lewis Structure

Lewis structures are a vital tool for depicting chemical bonding and molecular geometry. They help visualize the arrangement of valence electrons around atoms in a molecule, like the carbonate ion.
In the case of the \(\text{CO}_3^{2-} \) ion, the Lewis structure addresses how electrons are shared between atoms. Carbon starts with four valence electrons, and each oxygen has six. By drawing the structure, you arrange a total of 24 valence electrons, with an additional two electrons accounting for the charge, leading to a total of 26 valence electrons.
Lewis structures also guide us to understand possible resonance forms by revealing possible placements of double or single bonds, and possible resonance shifts as seen in \( \text{CO}_3^{2-} \). This information is essential as it lays the groundwork for more nuanced concepts such as resonance hybrids, bond order, and average bond length.

Bond Order

Bond order quantifies the number of valence electron pairs shared between two atoms. It impacts both the stability and bond length observed. Typically, a higher bond order means a stronger, shorter bond.

Given the resonance structures possible for the carbonate ion \( \text{CO}_3^{2-} \), the bond order becomes an average rather than a definitive value for each bond. Each C-O bond in carbonate resonates between single and double bond character. Consequently, the bond order across these three bonds can be calculated by averaging the bond orders from each resonance structure.
In carbonate, there are effectively four bonds distributed over three positions, leading to a bond order of \(\frac{4}{3}\) for each C-O connection. This fractional bond order helps explain why the bond lengths are equal and shorter than a typical single bond but longer than a double bond.