Problem 30 - General Chemistry: Principles and Modern Applications

Question

The paramagnetism of gaseous \(\mathrm{B}_{2}\) has been established. Explain how this observation confirms that the \(\pi_{2 p}\) orbitals are at a lower energy than the \(\sigma_{2 p}\) orbital for \(\mathrm{B}_{2}\)

Step-by-Step Solution

Answer

The paramagnetism of \(\mathrm{B}_{2}\) confirms that the \(\pi_{2 p}\) orbitals are at a lower energy level than the \(\sigma_{2 p}\) orbital because the unpaired electrons indicate that the \(\pi_{2 p}\) orbital filled before the \(\sigma_{2 p}\) orbital. If the \(\sigma_{2 p}\) had a lower energy level, it would fill before the \(\pi_{2 p}\), resulting in paired electrons and a diamagnetic molecule.

Step 1: Understanding the Electron Filling Order

The filling order of electron orbitals typically goes from lower energy levels to higher ones. Paired electrons in a specific orbital must have opposite spins because of a quantum mechanical principle called the Pauli Exclusion Principle.

Step 2: Constructing the Electron Orbital Diagram for the Molecule

For \(\mathrm{B}_{2}\) molecule, the electrons fill the lower energy \(\pi_{2 p\}}\) orbitals first. As a result, there are two unpaired electrons, one in each \(\pi_{2 p}\) orbital, which makes \(\mathrm{B}_{2}\) paramagnetic.

Step 3: Analyzing the Observation

The fact that \(\mathrm{B}_{2}\) is paramagnetic confirms that the \(\pi_{2 p}\) orbitals are at a lower energy than the \(\sigma_{2 p}\) orbital. If the \(\sigma_{2 p}\) orbital had a lower energy level, it would fill up before the \(\pi_{2 p}\) orbitals, resulting in paired electrons and a diamagnetic molecule.

Key Concepts

Electron Orbital Diagram

In chemistry, an electron orbital diagram helps us visualize where electrons are likely to be found around a nucleus in an atom or molecule. These diagrams represent the specific orbitals—defined regions in space around a nucleus—where electrons reside. For the molecule \(\text{B}_2\), the diagram illustrates how electrons fill these orbitals, starting from the lowest energy levels to the highest. When constructing an electron orbital diagram, you begin by noting the number of available electrons and pair them (or leave them unpaired) according to specific rules and principles.

An electron orbital diagram typically includes:

• Energy levels: These are shown vertically on the diagram. Electrons occupy higher levels as energy increases.
• Orbital types: Common types are \(s\), \(p\), \(d\), and \(f\), each with different shapes and orientations.
• Electron spins: Each orbital can hold two electrons, and they must have opposite spins (an upward arrow and a downward arrow).

In the case of \(\text{B}_2\), the electrons follow a particular filling order that aligns with the molecule’s paramagnetic nature. Here, the lower-energy \(\pi_{2p}\) orbitals are filled first, followed by the higher-energy \(\sigma_{2p}\) orbital. This reflects the unique arrangement crucial for understanding \(\text{B}_2\)'s magnetic properties.

Pauli Exclusion Principle

The Pauli Exclusion Principle is a fundamental rule in quantum mechanics that holds significant implications for electron arrangements in atoms and molecules. According to this principle, no two electrons can have the same set of quantum numbers within an atom. In simpler terms, it means that two electrons cannot occupy the same space and must be distinguishable by their spins.This principle manifests in electron orbital diagrams by requiring that paired electrons in the same orbital have opposite spins, represented by an up arrow and a down arrow. This visual rule helps predict an atom's or molecule's properties, such as its chemical reactivity and magnetic nature.

Importantly, the Pauli Exclusion Principle helps explain the paramagnetism observed in \(\text{B}_2\). As electrons fill the \(\pi_{2p}\) orbitals, they must do so in a way that obeys this principle. Thus, rather than two electrons occupying the same state, they remain unpaired as they fill, resulting in two unpaired electrons contributing to the molecule's paramagnetic behavior. The presence of these unpaired electrons is essential for \(\text{B}_2\)'s ability to be attracted to a magnetic field.

Molecular Orbital Theory

Molecular Orbital (MO) Theory is a key concept in understanding the electronic structure and properties of molecules. It provides a framework where atomic orbitals combine to create molecular orbitals, which are spread over the entire molecule. This theory differs from the simpler atomic orbital theory in that it explains the electronic structure of molecules more accurately by considering the interactions between all atomic orbitals involved.In MO theory, electrons fill the molecular orbitals based on their energy levels and availability, similar to atomic orbitals. However, molecular orbitals can be bonding, antibonding, or nonbonding, each influencing molecule stability and magnetism:

• Bonding Orbitals: Lower energy, stability-promoting orbitals.
• Antibonding Orbitals: Higher energy, destabilizing orbitals represented with an asterisk (*).
• Nonbonding Orbitals: Orbitals that have no net influence on bonding.

For \(\text{B}_2\), MO theory predicts that the \(\pi_{2p}\) orbitals are filled before the \(\sigma_{2p}\) orbital, reflecting their lower energy and contributing to the molecule's paramagnetic properties. This correct ordering and filling confirm the magnetic behavior of \(\text{B}_2\) and substantiate why the \(\pi_{2p}\) orbitals are energetically prioritized, leading to its distinguishable properties.