Problem 11 - An Introduction to Materials Engineering and Science for Chemical and Materials Engineers

Question

$$ \text { Which of the following molecules is (are) paramagnetic: } \mathrm{O}_{2}{\underline{\phantom{xx}}}^{2+} ; \mathrm{Be}_{2}{\underline{\phantom{xx}}}^{2+} ; \mathrm{F}_{2}{\underline{\phantom{xx}}}^{2+} \text { ? } $$

Step-by-Step Solution

Answer

\( \mathrm{F}_{2}^{2+} \) is paramagnetic.

Step 1: Understand Paramagnetism

Paramagnetism occurs when a substance has unpaired electrons in its molecular orbitals. Substances with paired electrons are typically diamagnetic.

Step 2: Apply Molecular Orbital Theory

To determine the magnetic properties of a molecule, we need to apply molecular orbital (MO) theory and fill in the electron configuration for each molecule using their total number of electrons.

Step 3: Analyze \( \mathrm{O}_{2}^{2+} \)

The molecule \( \mathrm{O}_{2} \) normally has 16 electrons. Removing 2 electrons to form \( \mathrm{O}_{2}^{2+} \) gives it 14 electrons. The MO configuration up to 14 electrons ends with: \( (\pi_{2p_x}^2)(\pi_{2p_y}^2) \). There are no unpaired electrons as the electrons fill up the orbitals symmetrically.

Step 4: Analyze \( \mathrm{Be}_{2}^{2+} \)

The molecule \( \mathrm{Be}_{2} \) typically has 8 electrons. Removing 2 electrons, \( \mathrm{Be}_{2}^{2+} \) has 6 electrons. The MO configuration is: \( (\sigma_{1s}^2)(\sigma^*_{1s}^2)(\sigma_{2s}^2) \). All electrons are paired, hence it is diamagnetic.

Step 5: Analyze \( \mathrm{F}_{2}^{2+} \)

The molecule \( \mathrm{F}_{2} \) normally has 18 electrons. Removing 2 electrons form \( \mathrm{F}_{2}^{2+} \) gives it 16 electrons. The MO configuration with 16 electrons is: \( (\sigma_{2s}^2)(\sigma^*_{2s}^2)(\sigma_{2p_z}^2)(\pi_{2p_x}^2)(\pi_{2p_y}^2)(\pi_{2p_x}^{*1})(\pi_{2p_y}^{*1}) \). The last two electrons are unpaired.

Key Concepts

Molecular Orbital Theory

Molecular Orbital (MO) Theory provides a detailed understanding of how electrons are distributed in a molecule. This theory combines atomic orbitals from each atom in a molecule to form molecular orbitals that are spread over the entire molecule, not just localized between two atoms. This concept helps us better understand which molecules are paramagnetic or diamagnetic.
In MO theory, electrons fill the molecular orbitals starting from the lowest energy level to the highest, similar to electron filling in atomic orbitals. The molecular orbitals can be bonding or antibonding. Bonding orbitals are lower in energy and lead to stability by helping hold atoms together. Antibonding orbitals are higher in energy and can destabilize the molecule.

• Bonding Orbitals: Electrons here increase molecule stability.
• Antibonding Orbitals: Electrons here can decrease molecule stability.

Understanding the filling of these orbitals is crucial to determining the electronic structure and properties of a molecule.

Electron Configuration

Electron configuration in molecular orbital theory involves placing electrons in the molecular orbitals according to their energy levels. This step-by-step filling based on available electrons in a molecule's orbitals helps describe its bonding and magnetic properties.
For example:

• In the case of \( ext{O}_2^{2+}\), which has 14 electrons, a specific sequence of MO filling helps determine that all orbitals are completely filled and paired, making it diamagnetic.
• Comparatively, in the case of \( ext{F}_2^{2+}\) with 16 electrons, electrons are placed in a way that results in unpaired electrons remaining. This configuration suggests a paramagnetic nature due to these unpaired electrons.

Proper electron configuration aligns with principles like the Aufbau principle, Hund's Rule, and Pauli’s exclusion principle, ensuring electrons fill in the most stable arrangement possible.

Unpaired Electrons

Unpaired electrons play a critical role in determining the magnetic properties of molecules. When electrons in a molecule's molecular orbitals are unpaired, the molecule becomes paramagnetic. Paramagnetic substances are attracted to magnetic fields due to these unpaired electrons.
In the case of \( ext{F}_2^{2+}\), the occurrence of unpaired electrons in its MO configuration makes the molecule paramagnetic. Conversely, when all electrons in a molecule are paired, it is diamagnetic, like \( ext{O}_2^{2+}\) and \( ext{Be}_2^{2+}\), which are not influenced by magnetic fields.
In summary:

• Unpaired Electrons: Lead to paramagnetic behavior.
• Paired Electrons: Result in diamagnetic properties.

Understanding how electrons are paired or unpaired within molecular orbitals is essential to predict the magnetic characteristics of molecules correctly.