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. \(\mathrm{CN}^{+}\) b. \(\mathrm{CN}\) c. \(\mathrm{CN}^{-}\)

Short answer

The electron configurations for \(\mathrm{CN}^{+}\), \(\mathrm{CN}\), and \(\mathrm{CN}^{-}\) are 1σ² 2σ² 1π⁴ 3σ¹, 1σ² 2σ² 1π⁴ 3σ², and 1σ² 2σ² 1π⁴ 3σ² 2π¹, respectively. The bond orders are 1.5, 1, and 0.5 respectively, with \(\mathrm{CN}^{+}\) and \(\mathrm{CN}^{-}\) being paramagnetic. The order of increasing bond length is \(\mathrm{CN}^{-}\) < \(\mathrm{CN}^{+}\) < \(\mathrm{CN}\), and the order of increasing bond energy is \(\mathrm{CN}\) < \(\mathrm{CN}^{+}\) < \(\mathrm{CN}^{-}\).

Step-by-step solution

Determine Electron Configurations

Firstly, write down the electron configurations of each element and count the total number of valence electrons in the molecule: Carbon (C): 1s² 2s² 2p² Nitrogen (N): 1s² 2s² 2p³ a. \(\mathrm{CN}^{+}\) (C and N have 9 valence electrons because of +1 charge): The molecular orbital model predicts the following electron arrangements: 1σ: 2 electrons 2σ: 2 electrons 1π: 4 electrons 3σ: 1 electron b. \(\mathrm{CN}\) (C and N have 10 valence electrons): The molecular orbital model predicts the following electron arrangements: 1σ: 2 electrons 2σ: 2 electrons 1π: 4 electrons 3σ: 2 electrons c. \(\mathrm{CN}^{-}\) (C and N have 11 valence electrons because of -1 charge): The molecular orbital model predicts the following electron arrangements: 1σ: 2 electrons 2σ: 2 electrons 1π: 4 electrons 3σ: 2 electrons 2π: 1 electron

Calculate Bond Orders and Identify Paramagnetic Species

The bond order is given by the formula: (number of electrons in bonding orbitals - number of electrons in anti-bonding orbitals)/2. We will use this formula to calculate bond orders for each diatomic species: a. \(\mathrm{CN}^{+}\): Bond Order (BO) = (6 - 3)/2 = 1.5 - Paramagnetic due to unpaired electron in 3σ orbital b. \(\mathrm{CN}\): BO = (6 - 4)/2 = 1 - Diamagnetic no unpaired electrons c. \(\mathrm{CN}^{-}\): BO = (6 - 5)/2 = 0.5 - Paramagnetic due to unpaired electron in 2π orbital

Place the Species in Order of Increasing Bond Length and Bond Energy

Higher bond order corresponds to shorter bond length and higher bond energy. So, \<bond length and energy of \( BO \) can be arranged as follows: Increasing Bond Length: \(\mathrm{CN}^{-}\) < \(\mathrm{CN}^{+}\) < \(\mathrm{CN}\) Increasing Bond Energy: \(\mathrm{CN}\) < \(\mathrm{CN}^{+}\) < \(\mathrm{CN}^{-}\) In conclusion, the electron configurations and bond orders have been determined for \(\mathrm{CN}^{+}\), \(\mathrm{CN}\) and \(\mathrm{CN}^{-}\). \(\mathrm{CN}^{+}\) and \(\mathrm{CN}^{-}\) were found to be paramagnetic species. The order of increasing bond length is \(\mathrm{CN}^{-}\) < \(\mathrm{CN}^{+}\) < \(\mathrm{CN}\), while the order of increasing bond energy is \(\mathrm{CN}\) < \(\mathrm{CN}^{+}\) < \(\mathrm{CN}^{-}\).

Key concepts

Electron Configurations

Electron configuration describes the distribution of electrons in an atom or molecule's orbitals.
For diatomic molecules such as Molecular Orbital Theory helps us understand how these electrons are configured.
The theory is an extension of Quantum Mechanics and explains how atomic orbitals combine to form molecular orbitals.
When considering diatomic species like the cyanide ion (\( \mathrm{CN}^{+} \)), electrons fill molecular orbitals in a sequence starting from the lowest energy orbital.
It's important to consider the charge of the species since it affects the total count of valence electrons.
With molecular orbitals, electrons typically fill bonding orbitals first before filling anti-bonding orbitals.For instance, in \( \mathrm{CN}^{+} \), the \( 1\sigma, 2\sigma, and 1\pi \) orbitals are where the electrons primarily reside.

• A bonding orbital, such as \( \sigma \), has lower energy and contributes to the stability of a molecule.
• An anti-bonding orbital is denoted with an "*" (like \( \sigma^{*} \)) and has higher energy.

Determining electron configurations for diatomic molecules is essential for calculating other molecular properties like bond order.

Bond Order

Bond order is a concept used to determine the strength and stability of a bond in molecules.
In Molecular Orbital Theory, bond order is derived from the electron configurations within molecular orbitals.
The formula for bond order is:\[ \text{Bond Order} = \frac{\text{(Number of Electrons in Bonding Orbitals)} - \text{(Number of Electrons in Anti-bonding Orbitals)}}{2}\]A higher bond order indicates a stronger bond and consequently, a shorter bond length. Consider \( \mathrm{CN} \) and its molecular ions:

• For \( \mathrm{CN}^{+} \), the bond order is \( 1.5 \), due to 6 electrons in bonding and 3 in anti-bonding orbitals.
• For \( \mathrm{CN} \), the bond order is \( 1 \), leading to a comparatively weaker bond.
• For \( \mathrm{CN}^{-} \), the bond order is \( 0.5 \), implying the weakest bond among them.

When ordering molecules by bond length, those with lower bond order (\( \mathrm{CN}^{-} \)) will have the longest bonds, while higher bond orders imply shorter bonds. Understanding bond order helps explain molecular properties like length and energy.

Paramagnetism

Paramagnetism refers to the presence of unpaired electrons in a molecule.
When a molecule has unpaired electrons, it can be attracted to a magnetic field, displaying paramagnetic behavior.
This property is deduced from the electron configuration derived from Molecular Orbital Theory.For example, in the diatomic species discussed:

• \( \mathrm{CN}^{+} \) is paramagnetic as it has an unpaired electron in the \( 3\sigma \) orbital.
• \( \mathrm{CN} \)n is diamagnetic since all its electrons are paired.
• \( \mathrm{CN}^{-} \) exhibits paramagnetism due to an unpaired electron in the \( 2\pi \) orbital.

A quick check for paramagnetism is to look at the electron fill in molecular orbitals.
Unpaired electrons commonly indicate paramagnetic qualities, and this can influence the magnetic properties and reactivity of the species.

Diatomic Molecules

Diatomic molecules consist of only two atoms, which may or may not be identical.
Many elements naturally form diatomic covalent bonds, like hydrogen (\( H_2 \)), oxygen (\( O_2 \)), and, in this context, the varying forms of CN.
In molecular orbital theory, diatomic molecules are studied to understand electron distribution and molecular stability.
These molecules are fundamental in chemistry because they often form simpler models to demonstrate complex principles like bonding and molecular behavior.The cyanide variations (\( \mathrm{CN}^{+} \), \( \mathrm{CN} \), and \( \mathrm{CN}^{-} \)) illustrate how adding or removing electrons, thereby altering the molecular charge, changes properties such as bond order, length, and magnetic behavior.
Diatomic molecules like these are vital in showcasing how subtle changes can significantly influence molecular properties, offering insights into chemical reactions and stability.

• They are a baseline for comparing molecular stability and reactivity.
• They help explain theoretical concepts in a more tangible way.

Studying diatomic molecules also aids in constructing fundamental bonding models applicable across various chemical contexts.