Question

Use the MO model to determine which of the following has the smallest ionization energy: \(\mathrm{N}_{2}, \mathrm{O}_{2}, \mathrm{~N}_{2}{ }^{2-}, \mathrm{N}_{2}^{-}, \mathrm{O}_{2}^{+} .\) EX- plain your answer.

Short answer

The molecule with the smallest ionization energy is \(\mathrm{O}_{2}{ }^{+}\), as it has the highest energy level for its Highest Occupied Molecular Orbital (HOMO) at the \(3\sigma_{g}\) level, requiring the least amount of energy to remove an electron.

Step-by-step solution

Understand Molecular Orbital Theory

Molecular Orbital (MO) theory is a model that explains the electronic structure and bonding in molecules by considering the combination of atomic orbitals to create molecular orbitals. These molecular orbitals can be bonding, anti-bonding, or non-bonding, and they can be occupied by electrons.

Determine the MO Energy Diagrams and Electron Configurations

We will consider the energy diagrams for the molecules \(\mathrm{N}_{2}\), \(\mathrm{O}_{2}\), \(\mathrm{N}_{2}{ }^{2-}\), \(\mathrm{N}_{2}^{-}\), and \(\mathrm{O}_{2}{ }^{+}\). The electron configurations for N2, O2, N2^2-, N2^-, and O2^+ are as follows: 1. \(\mathrm{N}_{2}\): \([1\sigma_{g}]^{2}[1\sigma_{u}{*}]^{2}[2\sigma_{g}]^{2}[2\sigma_{u}{*}]^{2}[1\pi_{u}]^{4}[3\sigma_{g}]^{2}[1\pi_{g}{*}]^{2}\) 2. \(\mathrm{O}_{2}\): \([1\sigma_{g}]^{2}[1\sigma_{u}{*}]^{2}[2\sigma_{g}]^{2}[2\sigma_{u}{*}]^{2}[1\pi_{u}]^{4}[3\sigma_{g}]^{2}[1\pi_{g}{*}]^{2}\) 3. \(\mathrm{N}_{2}{ }^{2-}\): \([1\sigma_{g}]^{2}[1\sigma_{u}{*}]^{2}[2\sigma_{g}]^{2}[2\sigma_{u}{*}]^{2}[1\pi_{u}]^{4}[3\sigma_{g}]^{2}[1\pi_{g}{*}]^{4}\) 4. \(\mathrm{N}_{2}^{-}\): \([1\sigma_{g}]^{2}[1\sigma_{u}{*}]^{2}[2\sigma_{g}]^{2}[2\sigma_{u}{*}]^{2}[1\pi_{u}]^{4}[3\sigma_{g}]^{2}[1\pi_{g}{*}]^{3}\) 5. \(\mathrm{O}_{2}{ }^{+}\): \([1\sigma_{g}]^{2}[1\sigma_{u}{*}]^{2}[2\sigma_{g}]^{2}[2\sigma_{u}{*}]^{2}[1\pi_{u}]^{4}[3\sigma_{g}]^{2}[1\pi_{g}{*}]^{1}\)

Find the Highest Occupied Molecular Orbital (HOMO) for each molecule

Next, we need to determine the HOMO for each molecule, as the ionization energy is related to the energy of the highest occupied molecular orbital. 1. \(\mathrm{N}_{2}\): HOMO = \(1\pi_{g}{*}\) 2. \(\mathrm{O}_{2}\): HOMO = \(1\pi_{g}{*}\) 3. \(\mathrm{N}_{2}{ }^{2-}\): HOMO = \(1\pi_{g}{*}\) 4. \(\mathrm{N}_{2}^{-}\): HOMO = \(1\pi_{g}{*}\) 5. \(\mathrm{O}_{2}{ }^{+}\): HOMO = \(3\sigma_{g}\)

Compare the ionization energies

To determine which molecule has the smallest ionization energy, we need to compare the energy levels of the HOMO in each molecule. The molecule with the highest HOMO energy level is the one with the smallest ionization energy since it will require the least amount of energy to remove an electron from the highest occupied molecular orbital. Out of all five molecules, \(\mathrm{O}_{2}{ }^{+}\) has the HOMO at the \(3\sigma_{g}\) level, which is higher in energy compared to the other four molecules having their HOMO at the \(1\pi_{g}{*}\) level. Therefore, the molecule with the smallest ionization energy is \(\mathrm{O}_{2}{ }^{+}\).

Key concepts

Ionization Energy

Ionization energy refers to the amount of energy required to remove an electron from an atom or molecule in its gaseous state. Understanding ionization energy is crucial when investigating the behavior of molecules and their chemical reactions.

Electrons occupy specific energy levels, and the energy needed to remove the most loosely bound electron (generally in the highest energy level) is termed the first ionization energy. Successive ionization energies refer to the removal of additional electrons, with each requiring more energy due to the increasing attraction between the positively charged nucleus and the remaining electrons.

This concept is incredibly useful when comparing different elements or molecules, as it can provide insights into their stability, reactivity, and trends within the periodic table. For instance, elements with low ionization energy tend to form cations easily and are generally more reactive. This information helps in deducing the behavior of molecules under study, such as in the exercise, where the ionization energies of different molecules are compared.

Molecular Orbitals

Molecular Orbitals (MOs) are a fundamental concept in Molecular Orbital Theory, which describes the wave-like behavior of electrons in molecules. Unlike atomic orbitals which are associated with individual atoms, molecular orbitals extend over the entire molecule and are the result of the additive and subtractive overlap of atomic orbitals.

There are two primary types of MOs - bonding and antibonding orbitals. Bonding orbitals are lower in energy and result from the constructive interference of atomic orbital wave functions, leading to a higher electron density between the atoms and hence, stronger bonding. On the contrary, antibonding orbitals arise from destructive interference and are higher in energy, with a reduced electron density between the atoms leading to weaker or no bonding.

Understanding the arrangement of electrons within MOs is crucial for predicting molecular properties such as magnetic behavior, bond order, and yes, ionization energy too. The concept of bonding and antibonding orbitals allows us to determine the stability of the molecules and understand why some molecules have higher or lower ionization energies than others.

Electron Configurations

Electron configurations provide a way to describe the distribution of electrons among the available orbitals in an atom or molecule. This is essentially the 'address' of electrons, indicating in which orbitals they reside and in what numbers.

In atoms, electron configuration follows the Aufbau principle, Pauli exclusion principle, and Hund's rule to ensure the most stable arrangement of electrons. Similarly, molecular electron configurations follow a specific order filling the molecular orbitals from lowest to highest energy. The knowledge of electron configuration is especially valuable when predicting chemical properties and behaviors such as reactivity, bonding, and ionization energy.

For example, in the exercise solution, by looking at the electron configurations of different nitrogen and oxygen molecules and ions, we can predict which one would have the smallest ionization energy by identifying the orbital from which an electron is most easily removed. This forms an integral part of solving problems related to molecular reactivity and stability.