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. \(\mathrm{CO}\) b. \(\mathrm{CO}^{+}\) c. \(\mathrm{CO}^{2+}\)

Short answer

The electron configurations are: a. \(\mathrm{CO}\): σ1s², σ*1s², σ2s², σ*2s², π2px², π2py², σ2pz², π*2px¹, π*2py¹ (paramagnetic) b. \(\mathrm{CO}^+\): σ1s², σ*1s², σ2s², σ*2s², π2px², π2py², σ2pz², π*2px¹ (paramagnetic) c. \(\mathrm{CO}^{2+}\): σ1s², σ*1s², σ2s², σ*2s², π2px², π2py², σ2pz² (diamagnetic) All three species have a bond order of 3, so their bond lengths and energies are approximately equal.

Step-by-step solution

Determine the number of electrons in each species

We need to find how many total electrons are present in each diatomic species. For CO, there are 6 valence electrons from C (Carbon) and 8 from O (Oxygen), totaling 14 electrons. For CO+, one electron is removed, resulting in 13 electrons. Lastly, for CO2+, two electrons are removed, leaving 12 electrons.

Fill the molecular orbitals using the aufbau principle

Using the aufbau principle, we will put the electrons into the molecular orbitals in the order: σ1s, σ*1s, σ2s, σ*2s, π2px, π2py, σ2pz, π*2px, π*2py, σ*2pz. - For \(\mathrm{CO}\) (14 electrons): σ1s², σ*1s², σ2s², σ*2s², π2px², π2py², σ2pz², π*2px¹, π*2py¹ - For \(\mathrm{CO}^+\) (13 electrons): σ1s², σ*1s², σ2s², σ*2s², π2px², π2py², σ2pz², π*2px¹ - For \(\mathrm{CO}^{2+}\) (12 electrons): σ1s², σ*1s², σ2s², σ*2s², π2px², π2py², σ2pz²

Calculate the bond order of each species

To find the bond order, use the formula: Bond Order = (Number of electrons in bonding orbitals - Number of electrons in anti-bonding orbitals) / 2. - Bond Order for \(\mathrm{CO}\): (10 - 4) / 2 = 3 - Bond Order for \(\mathrm{CO}^+\): (9 - 3) / 2 = 3 - Bond Order for \(\mathrm{CO}^{2+}\): (8 - 2) / 2 = 3

Determine which species are paramagnetic

A species is paramagnetic if it has one or more unpaired electrons. Examine the electron configurations: - \(\mathrm{CO}\) has two unpaired electrons (in π*2px¹ and π*2py¹ orbitals), so it is paramagnetic. - \(\mathrm{CO}^+\) has one unpaired electron (in π*2px¹ orbital), so it is also paramagnetic. - \(\mathrm{CO}^{2+}\) has no unpaired electrons, so it is diamagnetic.

Order the species based on bond length and bond energy

As the bond order increases, bond energy increases, and bond length decreases. All the species have the same bond order (3), so we can rank them as follows: - Bond Length: \(\mathrm{CO} \approx \mathrm{CO}^+ \approx \mathrm{CO}^{2+}\) - Bond Energy: \(\mathrm{CO} \approx \mathrm{CO}^+ \approx \mathrm{CO}^{2+}\) In conclusion, the electron configurations for each species are: a. \(\mathrm{CO}\): σ1s², σ*1s², σ2s², σ*2s², π2px², π2py², σ2pz², π*2px¹, π*2py¹ (paramagnetic) b. \(\mathrm{CO}^+\): σ1s², σ*1s², σ2s², σ*2s², π2px², π2py², σ2pz², π*2px¹ (paramagnetic) c. \(\mathrm{CO}^{2+}\): σ1s², σ*1s², σ2s², σ*2s², π2px², π2py², σ2pz² (diamagnetic) Bond lengths and energies are approximately equal for all three species since they all have a bond order of 3.

Key concepts

Bond Order

The concept of bond order provides insight into the stability and strength of a bond between two atoms in a molecule. Bond order is a numerical representation that predicts the bond strength and is derived from molecular orbital theory. It is calculated using the formula:\[\text{Bond Order} = \frac{\text{Number of bonding electrons} - \text{Number of anti-bonding electrons}}{2}\]A higher bond order indicates a stronger bond. For example, a bond order of 3, as observed in the given diatomic species \(\mathrm{CO}\), \(\mathrm{CO}^+\), and \(\mathrm{CO}^{2+}\), indicates a triple bond, which is strong and stable. Conversely, a bond order of 1 would correspond to a single bond, which is generally weaker.In summary:

• Bond order determines bond strength and stability.
• Higher bond order equals stronger and shorter bonds.
• Bond order can be whole or fractional.

Paramagnetic

Paramagnetism is a property of certain substances that have unpaired electrons. A substance is paramagnetic if its electrons align with an external magnetic field, allowing it to be attracted to the magnetic field. This occurs because unpaired electrons create magnetic dipoles.In the context of the molecular orbitals of diatomic species, we can determine paramagnetism by examining their electron configurations for unpaired electrons. For instance, \(\mathrm{CO}\) and \(\mathrm{CO}^+\) have unpaired electrons, making them paramagnetic. Conversely, \(\mathrm{CO}^{2+}\) has all its electrons paired, making it \emph{dia}\magnetic, which means it's not attracted to a magnetic field.Key points include:

• Paramagnetic species have unpaired electrons.
• Paramagnetism is detected by the attraction to external magnetic fields.
• Electron configuration reveals whether a species is paramagnetic or diamagnetic.

Electron Configuration

An electron configuration is a systematic arrangement of electrons within a molecule into different molecular orbitals, based on their energy levels. This distribution determines the chemical and physical properties of a molecule.In molecular orbital theory, each diatomic species, like \(\mathrm{CO}\), has a distinct electron configuration. For example:- \(\mathrm{CO}\) follows \(\sigma 1s^2, \sigma^* 1s^2, \sigma 2s^2, \sigma^* 2s^2, \pi 2p_x^2, \pi 2p_y^2, \sigma 2p_z^2, \pi^* 2p_x^1, \pi^* 2p_y^1\).The electron configuration helps in predicting:

• Magnetic properties (whether it is paramagnetic or diamagnetic).
• Bond order and molecular stability.
• The molecule's reactivity and interaction with other substances.

Understanding electron configuration is essential for explaining both the strong covalent bonds in diatomic molecules and any variations in their properties when electrons are added or removed.

Diatomic Species

Diatomic species are molecules composed of two atoms, which can either be the same element or different ones. In this context, we examine diatomic species such as carbon monoxide (\(\mathrm{CO}\)) and its ions, \(\mathrm{CO}^+\) and \(\mathrm{CO}^{2+}\).An important attribute of diatomic species is their ability to form simple molecular models which facilitate the study and understanding of bond formation using molecular orbitals. These species are typically used as examples to apply theoretical concepts such as:

• Molecular Orbital Theory: Helps to understand how atoms combine to form molecules.
• Bond Order: Provides an understanding of bond strength and stability.
• Electronic Structure: Useful in predicting properties such as bond length and energy.

Given these insights into diatomic species, concepts such as paramagnetism, electron configuration, and bond order can be thoroughly examined and applied in practice. This makes diatomic species ideal models for learning about molecular interactions and equilibrium.
Overall, understanding diatomic molecules enhances our comprehension of more complex molecular systems.