Question

Which of these species has a longer bond, \(\mathrm{B}_{2}\) or \(\mathrm{B}_{2}^{+} ?\) Explain in terms of molecular orbital theory.

Short answer

\(\mathrm{B}_2\) has a longer bond than \(\mathrm{B}_2^{+}\).

Step-by-step solution

Understand Molecular Orbital Theory

Molecular Orbital (MO) Theory explains the electronic structure of molecules by combining atomic orbitals into molecular orbitals that can house the molecule's electrons. These are classified into bonding and antibonding orbitals. Bonding orbitals lower energy and stabilize the molecule, while antibonding orbitals have higher energy and destabilize the molecule.

MO Diagrams for B extsubscript{2} and B extsubscript{2} extsuperscript{+}

For \(\mathrm{B}_2\), we configure its molecular orbital diagram using 10 valence electrons (5 from each boron atom). Electrons fill \(\sigma_{2s}, \sigma_{2s}^{*}, \pi_{2p}, \sigma_{2p}\) in increasing energy order. For \(\mathrm{B}_2^{+}\), one electron is removed, leaving us with 9 electrons.

Calculate Bond Order

The bond order is calculated as:\[\text{Bond Order} = \frac{(\text{Number of Electrons in Bonding Orbitals}) - (\text{Number of Electrons in Antibonding Orbitals})}{2}\]For \(\mathrm{B}_2\), the bond order is:\[\frac{(4 \text{ bonding } - 2 \text{ antibonding})}{2} = 1\] For \(\mathrm{B}_2^{+}\), the bond order is:\[\frac{(5 \text{ bonding } - 2 \text{ antibonding})}{2} = 1.5\]

Compare Bond Orders and Inferences

A higher bond order indicates a stronger and shorter bond. Since \(\mathrm{B}_2^{+}\) has a bond order of 1.5, which is higher than \(\mathrm{B}_2\)'s bond order of 1, the bond in \(\mathrm{B}_2^{+}\) is stronger and shorter.

Key concepts

Bond Order

Bond order is a crucial concept in molecular orbital theory that helps to understand the stability and strength of a chemical bond. Essentially, bond order is calculated using a formula:

• Bond Order = \( \frac{(\text{Number of Electrons in Bonding Orbitals}) - (\text{Number of Electrons in Antibonding Orbitals})}{2} \)

This simple formula helps determine how many chemical bonds exist between two atoms in a molecule. The bond order indicates the molecular stability:

• A higher bond order usually translates to a more stable and shorter bond.
• A bond order of zero implies no bond between the atoms.

In our example with the \( \text{B}_2 \) and \( \text{B}_2^+ \) molecules, the bond orders are 1 and 1.5, respectively. Thus, \( \text{B}_2^+ \)'s bond is stronger and shorter than that of \( \text{B}_2 \).

Bonding Orbitals

Bonding orbitals are where the magic of molecular formation happens! These orbitals are formed when atomic orbitals combine constructively, leading to a lower energy system. This construction makes the molecule more stable.Key characteristics of bonding orbitals include:

• They are lower in energy compared to the contributing atomic orbitals.
• They help hold the atoms together, forming a molecule.

In molecular orbital theory, the presence of electrons in bonding orbitals is essential, as they contribute positively to the bond order. In the \( \text{B}_2 \) molecule, for example, the electrons in the \( \pi_{2p} \) and \( \sigma_{2p} \) orbitals act as bonding electrons, contributing to the molecule's bond order.

Antibonding Orbitals

Contrary to bonding orbitals, antibonding orbitals can disrupt the stability of a molecule. These orbitals result when atomic orbitals combine in a way that increases the energy of the system, thus having a destabilizing effect.Important aspects of antibonding orbitals are:

• They are higher in energy than the original atomic orbitals.
• They often contain a nodal plane where electron density is zero, which weakens the bond.

Electrons in antibonding orbitals decrease the bond order. In the context of \( \text{B}_2 \), the \( \sigma_{2s}^* \) orbital houses electrons that counteract the effects of bonding electrons.

B2 Molecule

The \( \text{B}_2 \) molecule, consisting of two boron atoms, is a fascinating example in molecular orbital theory. With 10 valence electrons to organize, the molecule's bonding characteristics are determined by how these electrons fill the molecular orbitals.- In the \( \text{B}_2 \) molecule: - Electrons fill both bonding and antibonding orbitals. - The bond order is calculated as 1, indicating a stable molecule with one bond.Molecular orbitals such as \( \sigma_{2s} \), \( \pi_{2p} \), and their antibonding counterparts play a crucial role in this bond structure. The interaction between these orbitals gives rise to bond formation in the \( \text{B}_2 \) molecule.

Chemical Bonding

Chemical bonding is the process that holds atoms together in molecules, and molecular orbital theory provides an elegant explanation for this process.Key insights into chemical bonding include:

• Bonding orbitals are the result of constructive overlap, stabilizing the molecule.
• Antibonding orbitals arise from destructive overlap, typically destabilizing or weakening bonds.

In the case of \( \text{B}_2 \) and \( \text{B}_2^+ \), understanding the layout of bonding and antibonding orbitals allows us to predict bond stability and strength. Molecular orbital theory elegantly illustrates how these quantum mechanical concepts translate into observable molecular properties.