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

Short answer

The electron configurations for the given diatomic species are as follows: \\ a. \(\mathrm{CN}^{+}\): \(\sigma_{1s}^2\sigma^*_{1s}^2\sigma_{2s}^2\sigma^*_{2s}^2\pi_{2p}^4\), bond order 1, paramagnetic \\ b. CN: \(\sigma_{1s}^2\sigma^*_{1s}^2\sigma_{2s}^2\sigma^*_{2s}^2\pi_{2p}^4\sigma_{2p}^1\), bond order 1.5, paramagnetic \\ c. \(\mathrm{CN}^{-}\): \(\sigma_{1s}^2\sigma^*_{1s}^2\sigma_{2s}^2\sigma^*_{2s}^2\pi_{2p}^4\sigma_{2p}^2\), bond order 2, diamagnetic The order of increasing bond length is: \(\mathrm{CN}^{-} < \mathrm{CN} < \mathrm{CN}^{+}\) The order of increasing bond energy is: \(\mathrm{CN}^{+} < \mathrm{CN} < \mathrm{CN}^{-}\)

Step-by-step solution

Understand the molecular orbital model and bond order formula

The molecular orbital (MO) model describes the behavior of electrons in molecules. In this model, molecular orbitals are formed by combinations of atomic orbitals. For diatomic molecules, there are two types of molecular orbitals: bonding and anti-bonding. Bonding orbitals result from constructive overlap of atomic orbitals, while anti-bonding orbitals result from destructive overlap. Once we have the MO diagram, we can calculate the bond order using the following formula: \[Bond\,Order = \frac{1}{2}(Number\,of\,electrons\,in\,bonding\,orbitals - Number\,of\,electrons\,in\,anti-bonding\,orbitals)\] A species is paramagnetic if it contains unpaired electrons. We will determine this by looking at the electron configuration. Now, let's apply these concepts to each diatomic species.

Obtain electron configurations and calculate bond orders for the diatomic species

a. \(\mathrm{CN}^{+}\) The molecular orbital (MO) model for \(\mathrm{CN}^{+}\) is obtained by filling molecular orbitals according to the energy level and Hund's rule. C has 6 electrons and N has 7 electrons. In total, we have \(6+7 = 13\) electrons, but we need to subtract one electron because our species has a positive charge, resulting in \(13-1=12\) electrons. The electron configuration for \(\mathrm{CN}^{+}\) is: \(\sigma_{1s}^2\sigma^*_{1s}^2\sigma_{2s}^2\sigma^*_{2s}^2\pi_{2p}^4\) Now, we can calculate the bond order: \[Bond\,Order = \frac{1}{2}(6-4)=1\] Since there are unpaired electrons in the electron configuration, \(\mathrm{CN}^{+}\) is paramagnetic. b. CN For CN, we have a total of 13 electrons as \(6+7 = 13\). The electron configuration for CN is: \(\sigma_{1s}^2\sigma^*_{1s}^2\sigma_{2s}^2\sigma^*_{2s}^2\pi_{2p}^4\sigma_{2p}^1\) Now, we can calculate the bond order: \[Bond\,Order = \frac{1}{2}(7-4)=1.5\] Since there are unpaired electrons in the electron configuration, CN is paramagnetic. c. \(\mathrm{CN}^{-}\) For \(\mathrm{CN}^{-}\), we have a total of 14 electrons as \(6+7+1=14\). The electron configuration for \(\mathrm{CN}^{-}\) is: \(\sigma_{1s}^2\sigma^*_{1s}^2\sigma_{2s}^2\sigma^*_{2s}^2\pi_{2p}^4\sigma_{2p}^2\) Now, we can calculate the bond order: \[Bond\,Order=\frac{1}{2}(8-4)=2\] Since there are no unpaired electrons in the electron configuration, \(\mathrm{CN}^{-}\) is diamagnetic.

Order the species in terms of bond length and bond energy

Now, we can simply order the species based on their bond orders. Higher bond order indicates a stronger bond and shorter bond length. Bond energy is directly related to bond strength. So, the order for increasing bond length is: \(\mathrm{CN}^{-} < \mathrm{CN} < \mathrm{CN}^{+}\) And the order for increasing bond energy is: \(\mathrm{CN}^{+} < \mathrm{CN} < \mathrm{CN}^{-}\)

Key concepts

Bond Order Calculation

Bond order is a critical concept when exploring the molecular orbital model. It offers insight into the strength and stability of bonds between atoms in a molecule. Calculating bond order can seem tricky at first but becomes manageable with practice and understanding of the underlying principles.

Here's a simplified formula for bond order: \[Bond\,Order = \frac{1}{2}(Number\,of\,electrons\,in\,bonding\,orbitals - Number\,of\,electrons\,in\,anti-bonding\,orbitals)\]

To calculate the bond order, you need to know the distribution of electrons in both bonding and anti-bonding molecular orbitals. In a molecular orbital diagram, bonding orbitals have lower energy levels compared to anti-bonding orbitals. Electrons will fill the lowest available energy levels according to Hund's rule and the Pauli exclusion principle. Bond order correlates directly with bond energy and inversely with bond length, meaning higher bond orders signify stronger and shorter bonds.

Understanding this concept allows students to predict the stability of a molecule and how it might interact with other substances, providing a foundation for exploring chemical reactions and properties.

Paramagnetism in Molecules

Magnetism at the molecular level can be a fascinating concept to explore. Paramagnetism, in particular, refers to the magnetic properties of a molecule that arise from the presence of unpaired electrons. Molecules with unpaired electrons are attracted by a magnetic field, while those with paired electrons (diamagnetic) are not.

In the molecular orbital model, the presence or absence of unpaired electrons can be determined after assigning the electron configuration of the molecule. If a molecular orbital houses a single unpaired electron, the entire molecule is considered paramagnetic. This can significantly affect the behavior of a substance in the presence of a magnetic field. For example, oxygen (O2) is a classic example of a paramagnetic molecule due to its two unpaired electrons.

Understanding paramagnetism enriches the study of molecular interactions and can be essential in fields such as material science and chemistry, where magnetic properties are crucial. It's also a key factor when using techniques like electron paramagnetic resonance spectroscopy to study materials at the atomic level.

Electron Configurations for Diatomic Species

Understanding electron configurations within diatomic species is the backbone of grasping how these simple molecules bond and behave. Diatomic species, two atoms bonded together, are the simplest molecular systems and their electron configurations lay the groundwork for deeper exploration into chemical bonding.

To depict these configurations, we use molecular orbital theory and diagrams. These diagrams help visualize the bonding interactions between the atomic orbitals that combine to form molecular orbitals, which can be 'bonding' or 'anti-bonding'. The electron configuration follows a set order influenced by the energy level of each orbital—lower energy levels fill up first.

For diatomic species, we often consider the 's' and 'p' orbitals. Electrons pair up in the lower energy bonding orbitals before occupying higher energy anti-bonding orbitals. Hund's rule further refines this arrangement by ensuring that each orbital in a subshell is singly occupied before any is doubly occupied. This attention to detail is crucial in predicting molecular properties and behaviors. Consider the case of CN variants, where each form has a unique electron configuration, leading to different magnetic properties and levels of stability. This makes the concept of electron configurations an indispensable tool for predicting both the physical and chemical properties of diatomic species.