Question

Predict the ordering, from shortest to longest, of the bond lengths in \(\mathrm{CO}, \mathrm{CO}_{2},\) and \(\mathrm{CO}_{3}^{2-}\) .

Short answer

The bond lengths from shortest to longest are: CO < CO2 < CO3^2-. This is based on their bond orders: CO has a triple bond, CO2 has two double bonds, and CO3^2- has an average of single and double bonds due to resonance.

Step-by-step solution

Draw the Lewis Structures

Before comparing bond lengths, we need to draw the Lewis structures for each molecule. For CO: 1. Count the total number of valence electrons: C has 4 and O has 6, thus a total of 10 electrons. 2. Arrange the atoms by placing the least electronegative atom (C) in the center, surrounded by O. 3. Place a pair of electrons between C and O to form a bond. 4. Fill the remaining octets for both atoms, adding up to 10 electrons. For CO2: 1. Count the total number of valence electrons: C has 4 and each O has 6, thus a total of 16 electrons. 2. Arrange the atoms with C in the center, surrounded by the two O atoms. 3. Place pairs of electrons between C and each O atom to form bonds. 4. Fill the remaining octets for all atoms, adding up to 16 electrons. For CO3^2-: 1. Count the total number of valence electrons: C has 4, each O has 6, and we have an additional 2 electrons due to the charge, for a total of 24 electrons. 2. Arrange the atoms with C in the center, surrounded by the three O atoms. 3. Place pairs of electrons between C and each O atom to form bonds. 4. Fill the remaining octets for all atoms, considering the added 2 electrons due to the charge, adding up to 24 electrons.

Determine Bond Order

The bond order can be calculated for each molecule using the following formula: Bond Order = (number of bonding electrons - number of antibonding electrons) / 2 For CO: - Bond Order: (10-0) / 2 = 5 (a triple bond) For CO2: - Two double bonds between C and the O atoms. For CO3^2-: - Resonance structure presents a single bond and two double bonds between C and the O atoms, meaning that the bonds are an average of all three structures (1/3 of single bond and 2/3 of double bonds)

Predict Bond Length Ordering

Since bond length is inversely proportional to bond order, we can now predict the bond length ordering by comparing bond orders: 1. CO: Triple bond (highest bond order - shortest bond length) 2. CO2: Two double bonds (middle bond order - medium bond length) 3. CO3^2-: An average of single and double bonds (lowest bond order - longest bond length) So the predicted bond lengths for the molecules, from shortest to longest, are: CO < CO2 < CO3^2-.

Key concepts

Lewis Structures

Understanding Lewis structures is fundamental in predicting and explaining the properties and behaviors of molecules, including bond lengths. Lewis structures use simple notations to represent the valence electrons of atoms within molecules and show how they are shared or transferred to form chemical bonds.

When drawing a Lewis structure, start by determining the number of valence electrons for each atom, which are the electrons in the outer shell. For example, carbon (C) has four valence electrons, while oxygen (O) has six. The goal is to arrange the electrons around the atoms such that each atom achieves a stable electron configuration, often resembling the nearest noble gas.

Steps to Draw Lewis Structures:

• Count the valence electrons for each atom.
• Determine the central atom, typically the least electronegative.
• Place electron pairs to form chemical bonds between the central atom and surrounding atoms.
• Complete octets for each atom by distributing the remaining valence electrons, modifying for ions with charges when necessary.

Considering the CO molecule, it is formed by a triple bond between C and O, which is clearly illustrated in its Lewis structure. CO₂ has two double bonds with C in the center, and CO₃²⁻ has a resonance structure with an average of single and double bonds shared among three O atoms. The correct drawing of these structures is critical in predicting molecule behaviors, such as bond lengths.

Bond Order Calculation

The bond order of a molecule represents the number of chemical bonds between a pair of atoms. In essence, a larger bond order correlates with a shorter and stronger bond since more bonding electrons imply more robust interactions between the atoms.

To calculate bond order, use the formula: \[\begin{equation}Bond Order = \frac{(\text{number of bonding electrons} - \text{number of antibonding electrons})}{2}\end{equation}\]Bonding electrons are those directly involved in bond formation, whereas antibonding electrons are found in antibonding orbitals and counteract bond formation. In most cases you encounter through high school chemistry or introductory college courses, antibonding electrons are not part of the bond order calculation.

In our example, the CO bond order was determined to be 5, indicative of a triple bond. CO₂ displays a bond order reflecting two double bonds, while CO₃²⁻, with a charge of 2-, has resonance structures, which effectively result in an overall bond order that is an average of single and double bonds.

Valence Electrons

Valence electrons are the electrons present in the outermost shell of an atom. These electrons play a key role in chemical reactivity and bond formation since they can be transferred or shared to achieve a stable electronic configuration. The number of valence electrons is reflected in an element's position on the periodic table; for example, elements in group 1 have one valence electron, and elements in group 14, such as carbon, have four.

In the case of CO, CO₂, and CO₃²⁻, we carefully counted the valence electrons to construct their respective Lewis structures: 10 for CO, 16 for CO₂, and 24 for CO₃²⁻, considering the extra negative charge which contributes additional electrons. The valence electrons determine how atoms bond and ultimately influence molecular geometry and bond lengths.

By paying attention to valence electrons and invoking octet rule concepts, students can accurately predict how atoms combine to form molecules and polyatomic ions. Correctly identifying valence electrons lead us not only to the proper Lewis structures but also to more advanced concepts such as molecular shape and bond strength, essential in the discussion of chemical bond lengths.