Question

Consider the following bond lengths: \(\begin{array}{cccccc}\mathbf{C}-\mathbf{O} & 143 \mathrm{pm} & \mathbf{C}=\mathbf{O} & 123 \mathrm{pm} & \mathbf{C} \equiv \mathbf{O} & 109 \mathrm{pm}\end{array}\) 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 $136 \mathrm{pm}$ in the $\mathrm{CO}_{3}^{2-}$ ion can be explained by resonance structures. There are three equivalent resonance structures for the CO₃²⁻ ion, and the true bond length is an average of the bond lengths in these structures. Since all structures contribute equally, the bond lengths lie between the lengths for single and double bonds. This explains why the bond length in the CO₃²⁻ ion is different from the given bond lengths for single, double, and triple C-O bonds. An estimated average bond length of $133 \,\mathrm{pm}$ is close enough to the given bond length of $136 \,\mathrm{pm}$, supporting the resonance explanation.

Step-by-step solution

Understanding Resonance Structures

Resonance structures occur when there are multiple ways to represent a molecule with Lewis structures, and none of them capture the true nature of the molecule. In reality, the molecule exists as an average of all these structures with the true bond being more stable than any individual resonance structures.

Drawing the Lewis structures for CO₃²⁻

Given that the ion has a 2- charge and carbon makes four bonds while oxygen makes two bonds and two lone pairs, we can draw three different resonance structures for the CO₃²⁻ ion. In each structure, one of the oxygen atoms forms a double bond with the carbon, while the other two oxygen atoms form single bonds with the carbon atom and have an extra lone pair. The negative charges reside on the oxygen atoms that form single bonds.

Averaging the Bond Lengths

The true bond length of the C-O bonds in CO₃²⁻ is an average of the bond lengths in the resonance structures. As all resonance structures are equivalent, the bond lengths in CO₃²⁻ should be identical and lie between the lengths for single and double bonds.

Calculating the Average Bond Length

To estimate the average bond length, we can use the bond length for a single C-O bond (143 pm) and the bond length for a double C=O bond (123 pm). Given that there are three equivalent structures, which contribute equally to the overall bond, we can calculate the average bond length as follows: \( \text{Average C-O bond length} = \dfrac{(\text{C-O bond length})+(\text{C=O bond length})}{2} = \dfrac{143 + 123}{2} = 133 \, \mathrm{pm} \).

Comparing the Calculated Length to the Given Length

While the calculated average length (133 pm) is not exactly equal to the given identical bond lengths in CO₃²⁻ ion (136 pm), it is close enough to support the idea that the actual CO₃²⁻ ion structure is an average of the resonance structures. This also explains why the bond length in the CO₃²⁻ ion is different from all the given single, double, and triple bond lengths.

Key concepts

Lewis Structures

Lewis structures, named after Gilbert N. Lewis who introduced them, are diagrams that show the bonding between atoms of a molecule, as well as the lone pairs of electrons that may exist. These structures are pivotal in teaching students how to represent molecules visually, as they provide a basis for understanding molecular geometry, electron distribution, and the resulting chemical properties.

When drawing Lewis structures, one must account for the valence electrons of each atom, ensuring that they obey the octet rule, where applicable. This rule suggests that atoms are at their most stable when they have eight electrons in their valence shell. However, some atoms can have fewer or more electrons, an example being the hydrogen atom which is stable with two valence electrons. Creating a Lewis structure is typically the first step in predicting how atoms in a molecule bind together.

CO3(2-) Ion

The carbonate ion \(\mathrm{CO}_{3}^{2-}\) is a common polyatomic ion encountered in chemistry. It has a trigonal planar structure and carries a -2 charge. Understanding its configuration is a core principle that helps students become familiar with concepts of charge distribution and molecular geometry.

The carbonate ion consists of one carbon atom centrally bonded to three oxygen atoms. Each oxygen atom in \(\mathrm{CO}_{3}^{2-}\) has a valence of 6 electrons, while carbon has 4 valence electrons. In drawing the Lewis structure, it's essential to distribute electrons in such a way that carbon has four bonds while fulfilling the octet for the oxygen atoms. Since the ion carries a negative charge, this further implies that two extra electrons must be present. It leads to two oxygen atoms each carrying an additional negative charge, in resonance structures at least, to account for the overall charge of the ion.

Bond Length Averaging

Bond length averaging is a concept that arises when considering resonance structures in molecules such as \(\mathrm{CO}_{3}^{2-}\). Since multiple resonance structures can depict the bonding in a molecule, the actual bond lengths observed experimentally are often an average of these possible structures.

This happens because the molecule doesn't take on just one structure, but resonates between them, meaning that the actual bond is a hybrid, a kind of blending of all the possible resonance forms. Consequently, we can expect the bond lengths to be intermediate between the extremes of single and multiple bonds. This explains why, in the case of the carbonate ion, the measured bond lengths (136 pm) do not correspond precisely to the expected lengths of single (143 pm) or double (123 pm) C-O bonds, but instead lie somewhere in between.