Using the molecular orbital model to describe the bonding in \(\mathrm{F}_{2}{ }^{+}, \mathrm{F}_{2}\), and \(\mathrm{F}_{2}^{-}\), predict the bond orders and the relative bond lengths for these three species. How many unpaired electrons are present in each species?

Bond order represents the number of chemical bonds between a pair of atoms, providing insight into the stability and strength of the bond. In the molecular orbital model, bond order is calculated by taking the difference between the number of electrons in bonding molecular orbitals and those in antibonding molecular orbitals, then dividing by two. The formula is:
\[ \text{Bond order} = \frac{(\text{Number of electrons in bonding orbitals} - \text{Number of electrons in antibonding orbitals})}{2} \].

influences molecular stability and determines the length of the bond. A higher bond order indicates a stronger and shorter bond, while a lower bond order suggests a weaker and longer bond. For example, a single bond has a bond order of 1, a double bond, 2, and a triple bond, 3. In the fluorine molecules from the exercise:

• \(F_2\) has a bond order of 1, representing a single bond.
• \(F_2^{+}\) has a bond order of 0.5, indicating a bond weaker than a single bond.
• \(F_2^{-}\) has a bond order of 1.5, suggesting a bond stronger than a single bond but weaker than a double bond.

Thus, \(F_2\) has the longest bond length, being the weakest, followed by \(F_2^{+}\) , and \(F_2^{-}\) with the shortest bond length due to its highest bond order.

Electron configuration describes the distribution of electrons among the various atomic or molecular orbitals. For atoms, it follows a specific order filling subshells such as 1s, 2s, 2p, etc. In molecular systems, electrons occupy molecular orbitals, which arise from the combination of atomic orbitals. When several atomic orbitals combine, they form the same number of molecular orbitals, with some being of lower energy (bonding) and some of higher energy (antibonding).

Following Hund's rule and the Pauli exclusion principle, each electron is placed into the lowest energy orbital available, with electrons in the same orbital spinning in opposite directions to comply with Pauli's rule. In the exercise, the electron configurations for \(F_2\), \(F_2^{+}\), and \(F_2^{-}\) are filled out according to their respective number of electrons with the specific aim to minimize energy. Each species has a distinct configuration affecting the bond order:

• \(F_2\) has a full set of electrons in both bonding and antibonding orbitals leading to a bond order of 1.
• \(F_2^{+}\) has one less electron, causing a decrease in bond order to 0.5.
• \(F_2^{-}\) has an additional electron, which increases the bond order to 1.5.

The electron configuration is crucial in understanding the reactivity, color, magnetism, and other properties of molecules.

Molecular orbitals (MOs) form the foundation of the molecular orbital theory, which explains the electron structure of molecules. MOs result from the linear combination of atomic orbitals (AOs), and they extend over the entire molecule. According to this theory, electrons in a molecule are not confined to individual AOs of atoms but are instead delocalized over MOs that belong to the molecule as a whole.

Formation of Molecular Orbitals

When two atomic orbitals overlap, they can form two kinds of molecular orbitals: bonding orbitals, which are lower in energy and encourage electron sharing that leads to bond formation, and antibonding orbitals, which are higher in energy and can discourage bond formation if occupied. The MO diagram for \(F_2\), \(F_2^{+}\), and \(F_2^{-}\) includes \(\sigma\) and \(\pi\) orbitals, with the \(\sigma\) orbitals resulting from end-to-end overlap and the \(\pi\) orbitals from side-by-side overlap of p atomic orbitals. Understanding the construction and filling of these molecular orbitals is crucial in predicting physical and chemical properties of molecules, such as bond orders, magnetic properties, and the potential for chemical reactions.