Question

The highest occupied molecular orbital of a molecule is abbreviated as the HOMO. The lowest unoccupied molecular orbital in a molecule is called the LUMO. Experimentally, one can measure the difference in energy between the HOMO and LUMO by taking the electronic absorption (UV-visible) spectrum of the molecule. Peaks in the electronic absorption spectrum can be labeled as \(\pi_{2 p}-\pi_{2 p}^{\star}\) ,\(\sigma_{25}-\sigma_{25}^{*},\) and so on, corresponding to electrons being promoted from one orbital to another. The HOMO-LUMO transition corresponds to molecules going from their ground state to their first excited state. (a) Write out the molecular orbital valence electron configurations for the ground state and first excited state for \(N_{2} .\) (b) Is \(N_{2}\) paramagnetic or diamagnetic in its first excited state? (c) The electronic absorption spectrum of the \(N_{2}\) molecule has the lowest energy peak at 170 nm. To what orbital transition does this corre- spond? (a) Calculate the energy of the HOMO-LUMO transition in part (a) in terms of kJ/mol. (e) Is the N-N bondin the first excited state stronger or weaker compared to that in the ground state?

Short answer

The ground state valence electron configuration of \(N_{2}\) is \(1\sigma_{g}^{2}(1s)2\sigma_{u}^{2}(1s)1\sigma_{g}^{2}(2s)2\sigma_{u}^{2}(2s)1\pi_{u}^{4}(2p)1\pi_{g}^{2}(2p)\). The first excited state has the configuration \(1\sigma_{g}^{2}(1s)2\sigma_{u}^{2}(1s)1\sigma_{g}^{2}(2s)2\sigma_{u}^{2}(2s)1\pi_{u}^{4}(2p)1\pi_{g}^{1}(2p)3\sigma_{g}^{1}(2p)\). In its first excited state, \(N_{2}\) is paramagnetic. The low-energy peak in the electronic absorption spectrum corresponds to the \(\pi_{2p} \rightarrow \pi_{2p}^*\) transition. The energy of the HOMO-LUMO transition is approximately 693 kJ/mol. The N-N bond in the first excited state is weaker than in the ground state.

Step-by-step solution

Ground state valence electron configuration of \(N_{2}\)

Nitrogen has 7 electrons, so \(N_{2}\) has a total of 14 electrons. The valence electrons are in 2s and 2p orbitals. After applying the aufbau principle, and the molecular orbital diagram, we will find the ground state configuration: \(1\sigma_{g}^{2}(1s)2\sigma_{u}^{2}(1s)1\sigma_{g}^{2}(2s)2\sigma_{u}^{2}(2s)1\pi_{u}^{4}(2p)1\pi_{g}^{2}(2p)\).

First excited state valence electron configuration of \(N_{2}\)

The first excited state corresponds to promoting one electron from the HOMO (highest occupied molecular orbital) to the LUMO (lowest unoccupied molecular orbital). In the ground state configuration, the HOMO is the \(1\pi_{g}^{2}(2p)\) orbital and the LUMO is the \(3\sigma_{g}^{1}(2p)\) orbital. So, the first excited state configuration is: \(1\sigma_{g}^{2}(1s)2\sigma_{u}^{2}(1s)1\sigma_{g}^{2}(2s)2\sigma_{u}^{2}(2s)1\pi_{u}^{4}(2p)1\pi_{g}^{1}(2p)3\sigma_{g}^{1}(2p)\).

Determine if \(N_{2}\) is paramagnetic or diamagnetic

A species is paramagnetic if it has unpaired electrons, and diamagnetic if all electrons are paired. In the first excited state configuration, there are two unpaired electrons, one in the \(1\pi_{g}\) and one in the \(3\sigma_{g}\) orbitals. Therefore, \(N_{2}\) is paramagnetic in its first excited state.

Identify the orbital transition corresponding to the electronic absorption spectrum

We are given that the lowest energy peak in the electronic absorption spectrum of \(N_{2}\) is at 170 nm. This peak corresponds to the HOMO-LUMO transition, which is: \(\pi_{2p} \rightarrow \pi_{2p}^*\)

Calculate the energy of the HOMO-LUMO transition

The energy of the HOMO-LUMO transition can be calculated using the formula: Energy (kJ/mol) = \(\dfrac{6.626 \times 10^{-34} \; Js \times 3.00 \times 10^8 \; m/s}{170 \times 10^{-9} \; m \times 1.60 \times 10^{-19} \; J/eV} \times (10^{3} \; eV/mol)\) After performing the calculation, the energy is approximately 693 kJ/mol.

Determine if the N-N bond is stronger or weaker in the first excited state

Comparing the ground state and first excited state electron configurations of \(N_{2}\), we can observe that one electron was removed from the \(\pi_{g}\) (bonding) orbital and placed into the \(\sigma_{g}\) (antibonding) orbital. The presence of an additional electron in the antibonding orbital weakens the N-N bond. Hence, the N-N bond in the first excited state is weaker compared to the ground state.

Key concepts

Molecular Orbital Theory

Molecular Orbital Theory (MOT) is a fundamental concept in quantum chemistry that explains the electron structure of molecules. It is based on the idea that atomic orbitals combine to form new orbitals called molecular orbitals, which extend over the entire molecule.

In MOT, electrons are filled into these molecular orbitals following the Pauli exclusion principle and Hund's rule of maximum multiplicity. The lower energy orbitals, known as bonding orbitals, generally lead to the stability of the molecule as they hold electrons in areas of space between the nuclei. On the other hand, higher energy orbitals called antibonding orbitals can weaken and potentially destabilize the molecule because they house electrons with increased energy levels and keep them away from the region between the nuclei.

The highest occupied molecular orbital (HOMO) is the most energy-rich orbital containing electrons under normal circumstances, while the lowest unoccupied molecular orbital (LUMO) is the nearest orbital in energy terms that is vacant. Electrons can be excited from the HOMO to the LUMO, leading to what is known as an electronic transition. This principle serves as the underpinning for understanding electronic absorption spectra and is also central to explaining paramagnetism and diamagnetism in molecules.

Electronic Absorption Spectrum

The electronic absorption spectrum of a molecule displays the different frequencies of electromagnetic radiation absorbed by the molecule. This spectrum results from transitions between electron energy levels within the molecular orbitals of a molecule when it absorbs ultraviolet (UV) or visible light.

Peaks within this spectrum correspond to different electron transitions, such as the HOMO to LUMO transition, which is often the most important for determining molecules' color and photochemistry. The wavelengths of light absorbed are related to the energy gap between the involved molecular orbitals, and this can be calculated using the formula \( E = \frac{h \cdot c}{\lambda} \) where \( E \) is the energy of the transition, \( h \) is Planck's constant, \( c \) is the speed of light, and \( \lambda \) is the wavelength of the absorbed light.

In the solution provided, we see this calculation applied to determine the energy associated with the HOMO-LUMO transition for the nitrogen molecule (\(N_2\)) at a given peak wavelength of 170 nm in the absorption spectrum.

Paramagnetism and Diamagnetism

Paramagnetism and diamagnetism are properties that describe how a substance interacts with magnetic fields based on its electronic structure. Diamagnetic materials are those that do not contain any unpaired electrons and are repelled by a magnetic field. The electronic pairing leads to the cancellation of the magnetic moments, causing the repulsion.

On the other hand, paramagnetic materials possess at least one unpaired electron, giving them a net magnetic moment that aligns with magnetic fields, thus they are attracted to it. An everyday example of a paramagnetic material is oxygen, which has two unpaired electrons in its molecular form.

In the context of the nitrogen molecule (\(N_2\)) exercise, initially, in its ground state, nitrogen is diamagnetic because all of its electrons are paired. However, when an electron is promoted to an excited state, resulting in unpaired electrons, the molecule becomes paramagnetic. This transition to paramagnetism is key to understanding the magnetic properties of molecules and materials and can be explained using molecular orbital theory.