Question

Using the molecular orbital model, write electron configurations for the following diatomic species and calculate the bond orders. Which ones are paramagnetic? Place the species in order of increasing bond length and bond energy. a. CO b. \(\mathrm{CO}^{+}\) c. \(\mathrm{CO}^{2+}\)

Short answer

Electron configurations: CO: \(\sigma_{1s}^2\sigma_{1s*}^2\sigma_{2s}^2\sigma_{2s*}^2\pi_{2p}^4\sigma_{2p}^2\); \(\mathrm{CO}^{+}\): \(\sigma_{1s}^2\sigma_{1s*}^2\sigma_{2s}^2\sigma_{2s*}^2\pi_{2p}^4\sigma_{2p}^1\); \(\mathrm{CO}^{2+}\): \(\sigma_{1s}^2\sigma_{1s*}^2\sigma_{2s}^2\sigma_{2s*}^2\pi_{2p}^4\). Bond orders: CO (3), \(\mathrm{CO}^{+}\) (2.5), and \(\mathrm{CO}^{2+}\) (2). The paramagnetic species is \(\mathrm{CO}^{+}\). The order by bond length is CO < \(\mathrm{CO}^{+}\) < \(\mathrm{CO}^{2+}\), and the order by bond energy is \(\mathrm{CO}^{2+}\) < \(\mathrm{CO}^{+}\) < CO.

Step-by-step solution

1. Write electron configurations using the molecular orbital model

To write the electron configurations for CO, \(\mathrm{CO}^{+}\), and \(\mathrm{CO}^{2+}\), first recall the atomic electron configurations for Carbon (C) and Oxygen (O): Carbon (C) has 6 electrons: 1s² 2s² 2p² Oxygen (O) has 8 electrons: 1s² 2s² 2p⁴ Now, combine the electron configurations for C and O, and distribute the electrons based on molecular orbital energy diagram for diatomic molecules: CO (14 electrons): \(\sigma_{1s}^2\sigma_{1s*}^2\sigma_{2s}^2\sigma_{2s*}^2\pi_{2p}^4\sigma_{2p}^2\). \(\mathrm{CO}^{+}\) (13 electrons): \(\sigma_{1s}^2\sigma_{1s*}^2\sigma_{2s}^2\sigma_{2s*}^2\pi_{2p}^4\sigma_{2p}^1\). \(\mathrm{CO}^{2+}\) (12 electrons): \(\sigma_{1s}^2\sigma_{1s*}^2\sigma_{2s}^2\sigma_{2s*}^2\pi_{2p}^4\).

2. Calculate the bond orders

To calculate the bond orders, use the formula: Bond order = (Number of electrons in bonding orbitals - Number of electrons in antibonding orbitals) / 2. CO: Bond order = (10 - 4) / 2 = 3. CO\(^{+}\): Bond order = (9 - 4) / 2 = 2.5. CO\(^{2+}\): Bond order = (8 - 4) / 2 = 2.

3. Identify paramagnetic species

A species is paramagnetic if it has unpaired electrons. From the electron configurations: CO: All electrons are paired; it is diamagnetic. CO\(^{+}\): One unpaired electron in \(\sigma_{2p}\); it is paramagnetic. CO\(^{2+}\): All electrons are paired; it is diamagnetic.

4. Order species by bond length and bond energy

Bond length is inversely proportional to the bond order and bond energy is directly proportional to the bond order. - Bond order: CO\(^{2+}\) < CO\(^{+}\) < CO - Bond length: CO < CO\(^{+}\) < CO\(^{2+}\) - Bond energy: CO\(^{2+}\) < CO\(^{+}\) < CO In summary: 1. Electron configurations: CO: \(\sigma_{1s}^2\sigma_{1s*}^2\sigma_{2s}^2\sigma_{2s*}^2\pi_{2p}^4\sigma_{2p}^2\). CO\(^{+}\): \(\sigma_{1s}^2\sigma_{1s*}^2\sigma_{2s}^2\sigma_{2s*}^2\pi_{2p}^4\sigma_{2p}^1\). CO\(^{2+}\): \(\sigma_{1s}^2\sigma_{1s*}^2\sigma_{2s}^2\sigma_{2s*}^2\pi_{2p}^4\). 2. Bond orders: CO (3), CO\(^{+}\) (2.5), CO\(^{2+}\) (2). 3. Paramagnetic species: CO\(^{+}\). 4. Order by bond length: CO < CO\(^{+}\) < CO\(^{2+}\). 5. Order by bond energy: CO\(^{2+}\) < CO\(^{+}\) < CO.

Key concepts

Electron Configurations

Electron configurations describe how electrons are distributed among the molecular orbitals of a molecule. In molecular orbital theory, these configurations are key to understanding the chemical and physical behavior of molecules.
For diatomic molecules like CO and its ions ( CO^{+} , CO^{2+} ), we combine atomic orbitals of individual atoms into molecular orbitals. Carbon (C) provides 6 electrons, and Oxygen (O) supplies 8 electrons, which makes a total of 14 electrons for the carbon monoxide (CO) molecule.
Using the molecular orbital energy diagram, we distribute these 14 electrons among various orbitals:

• σ_{1s}^{2} , σ_{1s^{*}}^{2} (these represent the inner core electrons and have little effect on bonding)
• σ_{2s}^{2} , σ_{2s^{*}}^{2}
• π_{2p}^{4}
• σ_{2p}^{2}

Thus, the electron configurations are:

• CO: σ_{1s}^{2}σ_{1s^{*}}^{2}σ_{2s}^{2}σ_{2s^{*}}^{2}π_{2p}^{4}σ_{2p}^{2}
• CO⁺: σ_{1s}^{2}σ_{1s^{*}}^{2}σ_{2s}^{2}σ_{2s^{*}}^{2}π_{2p}^{4}σ_{2p}^{1}
• CO²⁺: σ_{1s}^{2}σ_{1s^{*}}^{2}σ_{2s}^{2}σ_{2s^{*}}^{2}π_{2p}^{4}

This approach helps in explaining different properties such as bond order and magnetic behavior.

Bond Order

Bond order indicates the strength and stability of a bond. It is calculated by taking the difference between the number of electrons in bonding orbitals and antibonding orbitals, divided by two.
The formula is:

• Bond Order = (Number of Bonding Electrons - Number of Antibonding Electrons) / 2

For CO, CO⁺, and CO²⁺:

• CO: Bond order = (10 - 4) / 2 = 3
• CO⁺: Bond order = (9 - 4) / 2 = 2.5
• CO²⁺: Bond order = (8 - 4) / 2 = 2

A higher bond order suggests a stronger, shorter, and more stable bond. Therefore, the bond order helps predict the molecular stability and bond characteristics. CO, with the highest bond order, is the most stable and has the strongest bond. As we remove electrons to go to CO⁺ and CO²⁺, the bonds become weaker and longer.

Paramagnetism

Paramagnetism arises in molecules with unpaired electrons. A molecule is paramagnetic if it can be attracted to an external magnetic field, which is a direct result of its unpaired electrons.
In the context of molecular orbitals:

• CO: All electrons are paired, making it diamagnetic (not attracted to magnetic fields).
• CO⁺: Has one unpaired electron in the σ_{2p} orbital, which makes it paramagnetic.
• CO²⁺: All electrons paired again, rendering it diamagnetic.

To identify paramagnetic species, look for unfilled molecular orbitals, which typically indicates unpaired electrons. This characteristic impacts a molecule's magnetic properties, demonstrating an intriguing aspect of chemistry where electronic configurations determine physical attributes.

Diatomic Molecules

Diatomic molecules consist of two atoms, which may either be the same element or different (like CO). Understanding these simple molecules is crucial because they form the basis for more complex structures in chemical theory.
Studying diatomic molecules using molecular orbital theory involves:

• Identifying the number of electrons involved by considering each atom.
• Filling these electrons into molecular orbitals, starting from the lowest energy level and following exclusion and filling rules.
• Using molecular orbital diagrams to infer properties such as bond order, bond length, and magnetic behavior.

In exercises like the one involving CO, CO⁺, and CO²⁺, the relative bond order allows us to predict properties like bond energy and bond length. Diatomic molecules are straightforward yet immensely valuable in providing insight into molecular interactions and structure-function relationships.