Question

Write the ground-state electron configuration for \(\mathrm{B}_{2}\). Is the molecule diamagnetic or paramagnetic?

Short answer

The ground-state electron configuration is \( \sigma_{1s}^2 \sigma^*_{1s}^2 \sigma_{2s}^2 \sigma^*_{2s}^2 \pi_{2p}^2 \); \( \mathrm{B}_2 \) is diamagnetic.

Step-by-step solution

Determine the Number of Electrons

The molecule \( \mathrm{B}_2 \) consists of two boron atoms. Each boron atom has an atomic number of 5, meaning each contributes 5 electrons. Therefore, \( \mathrm{B}_2 \) has a total of \( 2 \times 5 = 10 \) electrons.

Configuration in Molecular Orbitals

For diatomic molecules, we use molecular orbital theory to write the electron configuration. According to the energy level diagram for homonuclear diatomic molecules like \( \mathrm{B}_2 \), the order of orbitals from lowest to highest energy is: \( \sigma_{1s}, \sigma^*_{1s}, \sigma_{2s}, \sigma^*_{2s}, \pi_{2p}, \sigma_{2p} \).

Fill the Molecular Orbitals

Fill the molecular orbitals in order according to the Aufbau principle. The electron configuration for \( \mathrm{B}_2 \) with 10 electrons is: \[ \sigma_{1s}^2 \sigma^*_{1s}^2 \sigma_{2s}^2 \sigma^*_{2s}^2 \pi_{2p}^2 \]

Determine Magnetic Properties

To identify if \( \mathrm{B}_2 \) is diamagnetic or paramagnetic, we check for unpaired electrons in the configuration. In this configuration, the \( \pi_{2p} \) orbitals are filled with 2 electrons, and both are paired. No unpaired electrons are present.

Key concepts

Homonuclear Diatomic Molecules

Homonuclear diatomic molecules consist of two identical atoms bonded together. Hence the term 'homonuclear,' indicating that the nuclei of the two atoms are the same. Examples of these molecules include

• the hydrogen molecule (\( \mathrm{H}_2 \))
• the nitrogen molecule (\( \mathrm{N}_2 \))
• and the oxygen molecule (\( \mathrm{O}_2 \)).

For the molecule \( \mathrm{B}_2 \), it consists of two boron atoms. Molecular orbital (MO) theory helps us conceptualize the bonding process within these molecules by mixing the atomic orbitals from individual atoms to form molecular orbitals. These molecular orbitals can spread over the entire molecule, unlike atomic orbitals, which are localized to individual atoms.In the case of \( \mathrm{B}_2 \), the MO theory provides insights into the arrangement and energy levels of orbitals. As such, understanding where electrons are most likely to reside in homonuclear diatomic molecules involves examining their orbitals created by the bonding process.

Ground-State Electron Configuration

The ground-state electron configuration of any molecule refers to the arrangement of electrons in its lowest energy state. This configuration gives us a glimpse into such properties as the reactivity and stability of the molecule. The electron configuration for homonuclear diatomic molecules like \( \mathrm{B}_2 \) is determined using molecular orbital diagrams which help in understanding the various energy levels and how electrons fill these levels.In the given scenario, \( \mathrm{B}_2 \) has a total of 10 electrons, contributed equally by each boron atom. The molecular orbital configuration following the energy sequence is: \[ \sigma_{1s}^2 \sigma^*_{1s}^2 \sigma_{2s}^2 \sigma^*_{2s}^2 \pi_{2p}^2 \] Here:

• The subscript numbers (1s, 2s, etc.) refer to the atomic orbitals.
• The asterisks in some terms (\( \sigma^* \)) indicate that these are anti-bonding orbitals, which are higher in energy and can potentially weaken the bond between atoms if occupied.
• The numbers written as superscripts indicate the number of electrons in those particular orbitals.

Filled from lowest to higher energy, this configuration informs us about the chemical behavior and bonding properties of the molecule.

Magnetic Properties

The magnetic properties of molecules give us insight into whether they will be attracted or repelled by a magnetic field. A key term here is "diamagnetic" and "paramagnetic". A molecule is considered diamagnetic if all its electrons are paired; it will not be attracted to a magnetic field and can even be slightly repelled. On the other hand, a molecule is paramagnetic if it contains unpaired electrons, making it attracted to magnetic fields.For \( \mathrm{B}_2 \), examining the full molecular electron configuration, we focus on the electrons in the \( \pi_{2p} \) orbital. The \( \pi_{2p} \) orbitals in \( \mathrm{B}_2 \) are fully paired, as seen with the configuration:

• \[ \pi_{2p}^2 \]

With no unpaired electrons, \( \mathrm{B}_2 \) is diamagnetic, indicating it won't have a strong interaction with magnetic fields.Thus, whether a diatomic molecule is paramagnetic or diamagnetic depends primarily on its complete electron configuration and the presence of unpaired electrons.

Aufbau Principle

The Aufbau principle is a fundamental concept in understanding how electrons populate orbitals within an atom or molecule. "Aufbau" means "building up" in German, and this principle starts with electrons filling the lowest energy orbitals before moving to higher energy ones. This process ensures that the total energy of the atom or molecule is minimized. In the specific case of \( \mathrm{B}_2 \), the Aufbau principle dictates how the 10 electrons are distributed across molecular orbitals: The electrons start filling from the \( \sigma_{1s} \) orbital, followed by \( \sigma^*_{1s} \), \( \sigma_{2s} \), \( \sigma^*_{2s} \), and finally the \( \pi_{2p} \) orbital. Each orbital is filled according to its energy level, in line with the Aufbau principle's rules. To remember:

• Fill lower energy orbitals first.
• Recognize that some orbitals (e.g., \( \sigma^* \)) are higher energy and fill after lower energy ones are full.
• Remember Hund's rule which complements the Aufbau principle, stating that electrons will fill unoccupied orbitals singly before pairing in order to maximize the total spin.

By following these principles, we can predict electron configurations accurately and understand the behavior of atoms and molecules.