Question

Consider the molecules \(\mathrm{CO}^{+}\) and \(\mathrm{CN}^{-}\) and use molecular orbital theory to answer the following: (a) Write the molecular orbital configuration of each ion (ignore the 1 s electrons). (b) Predict the bond order of each ion. (c) Which of these ions is paramagnetic? Which is diamagnetic? (d) Which of these ions do you think has the greater bond length? Explain.

Short answer

The molecular orbital configurations are \(\mathrm{CO}^{+}\): \(σ_{2s}^{2} σ_{2s^*}^{0} π_{2p}^{4} σ_{2p}^{2} π_{2p^*}^{1} σ_{2p^*}^{0}\) and \(\mathrm{CN}^{-}\): \(σ_{2s}^{2} σ_{2s^*}^{0} π_{2p}^{4} σ_{2p}^{2} π_{2p^*}^{2} σ_{2p^*}^{0}\). The bond order for \(\mathrm{CO}^{+}\) is 3.5 and for \(\mathrm{CN}^{-}\) is 3. \(\mathrm{CO}^{+}\) is paramagnetic with one unpaired electron while, \(\mathrm{CN}^{-}\) is diamagnetic. From the bond orders, it can be inferred that \(\mathrm{CN}^{-}\) has a greater bond length than \(\mathrm{CO}^{+}\)

Step-by-step solution

Determine the Molecular Orbital Configurations

Molecular orbital configurations can be determined by the total number of valence electrons in the ion. Carbon has 4, Oxygen has 6 and Nitrogen has 5 valence electrons. For \(\mathrm{CO}^{+}\), we have 4 valence electrons of Carbon + 6 of Oxygen - 1 since it is a positive ion = 9 valence electrons. For \(\mathrm{CN}^{-}\), we have 4 valence electrons of Carbon + 5 of Nitrogen + 1 since it is a negative ion = 10 valence electrons. The molecular orbital configurations can be written as:For \(\mathrm{CO}^{+}\): \(σ_{2s}^{2} σ_{2s^*}^{0} π_{2p}^{4} σ_{2p}^{2} π_{2p^*}^{1} σ_{2p^*}^{0}\)For \(\mathrm{CN}^{-}\): \(σ_{2s}^{2} σ_{2s^*}^{0} π_{2p}^{4} σ_{2p}^{2} π_{2p^*}^{2} σ_{2p^*}^{0}\)

Predict the Bond Order

Bond order is calculated by the formula: BO = 0.5 * (no. of electrons in bonding orbitals - no. of electrons in anti-bonding orbitals). For \(\mathrm{CO}^{+}\), BO = 0.5 * (8-1) = 3.5. For \(\mathrm{CN}^{-}\), BO = 0.5 * (8-2) = 3.

Calculate the Magnetic Behaviour

A molecule is paramagnetic if it has unpaired electrons and diamagnetic if all its electrons are paired. From the molecular orbital configurations, we see that \(\mathrm{CO}^{+}\) has 1 unpaired electron (is paramagnetic) while \(\mathrm{CN}^{-}\) has all paired electrons (is diamagnetic).

Determine the Bond Length

Typically, a higher bond order leads to a shorter bond length. So, since the bond order of \(\mathrm{CO}^{+}\) is greater than \(\mathrm{CN}^{-}\), we would expect the bond length of \(\mathrm{CO}^{+}\) to be shorter, meaning \(\mathrm{CN}^{-}\) seems to have the greater bond length.

Key concepts

Bond Order

Bond order is an important concept in molecular orbital theory because it provides insight into the stability and strength of a bond between two atoms. It can be calculated using the formula: \[\text{Bond Order} = 0.5 \times (\text{Number of bonding electrons} - \text{Number of antibonding electrons})\]A higher bond order indicates a stronger bond. For example, a bond order of 3 suggests a triple bond, typically resulting in a stronger and shorter bond compared to a single bond, which has a bond order of 1. In the given exercise, for \(\text{CO}^+\), the bond order is 3.5, while for \(\text{CN}^-\), it is 3. This suggests that \(\text{CO}^+\) has a stronger and shorter bond.A few important points about bond order include:

• Bond order can be a fractional value, showing bonds that might not be typical integers like 1 or 2.
• Higher bond order generally corresponds to greater bond energy and decreased bond length.
• Bonds with higher bond orders are typically more resistant to breaking apart or destroying.

Paramagnetism

Paramagnetism is a form of magnetism that occurs in materials which have unpaired electrons. These unpaired electrons react to external magnetic fields and cause the material to be attracted to the magnetic field.
In terms of molecular orbital theory, if a molecule has one or more unpaired electrons in its molecular orbital configuration, it will exhibit paramagnetic behavior.In the exercise, \(\text{CO}^+\) exhibits paramagnetism because its molecular orbital configuration reveals one unpaired electron. This means that \(\text{CO}^+\) is attracted to magnetic fields. Paramagnetic substances are often used in various applications where magnetic properties are required. Additionally:

• Paramagnetism is generally not as strong as ferromagnetism (such as the strongest magnets).
• Unpaired electrons are usually responsible for causing the magnetic properties.
• Transition metals commonly exhibit paramagnetic behavior due to unpaired d electrons.

Diamagnetic

Diamagnetic substances are those that are not attracted to a magnetic field; in fact, they may be weakly repelled by one. This occurs when all electrons in a molecule are paired. In molecular orbital theory, this implies that there are no unpaired electrons in the molecular orbitals configuration.In the exercise given, \(\text{CN}^-\) is diamagnetic because its electrons are all paired up. Therefore, it won't be attracted or moved by a magnetic field, making it quite stable in environments with magnetic influences.Essentially, key points about diamagnetic substances include:

• They generally show a very weak reaction to magnetic fields, often slightly repelling them.
• Most molecules with all paired electrons are diamagnetic.
• Diamagnetism is a universal property of all materials, as it is associated with the response of electron pairs to external magnetic fields.
• It is most noticeable only when there are no other magnetic behaviors present (e.g., paramagnetism).