Question

(a) What does the term paramagnetism mean? (b) How can one determine experimentally whether a substance is paramagnetic? (c) Which of the following ions would you expect to be paramagnetic: \(\mathrm{O}_{2}^{+}, \mathrm{N}_{2}^{2-}, \mathrm{Li}_{2}^{+}, \mathrm{O}_{2}^{2-} ?\) For those ions that are paramagnetic, determine the number of unpaired electrons.

Short answer

Paramagnetism is a type of magnetism exhibited by materials with unpaired electrons in their atomic or molecular orbitals. Experimentally, paramagnetism can be determined using the Gouy balance method. Among the given ions, \(\mathrm{O}_{2}^{+}\), \(\mathrm{Li}_{2}^{+}\), and \(\mathrm{O}_{2}^{2-}\) are paramagnetic, with 1, 1, and 2 unpaired electrons, respectively. The ion \(\mathrm{N}_{2}^{2-}\) is not paramagnetic as it has no unpaired electrons.

Step-by-step solution

1. Understanding Paramagnetism

Paramagnetism is a type of magnetism exhibited by materials that have unpaired electrons in their atomic or molecular orbitals. These unpaired electrons give rise to a magnetic moment that interacts with an external magnetic field, causing the material to be attracted to the magnetic field.

2. Experimental Determination of Paramagnetism

One can determine if a substance is paramagnetic through an experiment called the Gouy balance method. In this method, a sample of a substance is placed between the poles of a strong electromagnet, and its mass change is measured upon applying the magnetic field. If the mass of the sample decreases, it means that the substance is attracted to the magnetic field and, therefore, is paramagnetic. If the mass remains the same or increases, the substance is not paramagnetic.

3. Checking for Paramagnetic Ions

We will now check if each of the given ions is paramagnetic, and if so, calculate the number of unpaired electrons: (a) \(\mathrm{O}_{2}^{+}\): Molecular oxygen has 12 valence electrons. The \(\mathrm{O}_{2}^{+}\) ion has one less electron, so it has 11 valence electrons. According to Molecular Orbital (MO) theory, these electrons fill the molecular orbitals in the following order: \(1\sigma_g, 1\sigma_u^*, 2\sigma_g, 2\sigma_u^*, 1\pi_u(2), 3\sigma_g, 1\pi_g(3)\). Since the \(\mathrm{O}_{2}^{+}\) ion has one unpaired electron in the \(1\pi_g\) molecular orbital, it is paramagnetic. (b) \(\mathrm{N}_{2}^{2-}\): Molecular nitrogen has 10 valence electrons. The \(\mathrm{N}_{2}^{2-}\) ion has two additional electrons, so it has 12 valence electrons. These electrons fill the following molecular orbitals, similar to molecular oxygen: \(1\sigma_g, 1\sigma_u^*, 2\sigma_g, 2\sigma_u^*, 1\pi_u(2), 3\sigma_g, 1\pi_g(2)\). Since the electrons completely fill the molecular orbitals, there are no unpaired electrons, so \(\mathrm{N}_{2}^{2-}\) is not paramagnetic. (c) \(\mathrm{Li}_{2}^{+}\): A lithium molecule (\(\mathrm{Li}_{2}\)) has 4 valence electrons. It loses one electron to form \(\mathrm{Li}_{2}^{+}\) which has 3 valence electrons. According to MO theory, these electrons fill the following molecular orbitals: \(1\sigma_g, 1\sigma_u^*\). Since \(\mathrm{Li}_{2}^{+}\) has an unpaired electron in the \(1\sigma_u^*\) molecular orbital, it is paramagnetic. (d) \(\mathrm{O}_{2}^{2-}\): Molecular oxygen has 12 valence electrons. The \(\mathrm{O}_{2}^{2-}\) ion has two additional electrons, so it has 14 valence electrons. Similar to molecular oxygen, these electrons fill the following molecular orbitals: \(1\sigma_g, 1\sigma_u^*, 2\sigma_g, 2\sigma_u^*, 1\pi_u(2), 3\sigma_g, 1\pi_g(4)\). Since the \(\mathrm{O}_{2}^{2-}\) ion has two unpaired electrons in the \(1\pi_g\) molecular orbital, it is paramagnetic.

4. Number of Unpaired Electrons

For the paramagnetic ions, the number of unpaired electrons are: - \(\mathrm{O}_{2}^{+}\): 1 unpaired electron - \(\mathrm{Li}_{2}^{+}\): 1 unpaired electron - \(\mathrm{O}_{2}^{2-}\): 2 unpaired electrons

Key concepts

Molecular Orbital Theory

Molecular Orbital (MO) Theory is a fundamental concept in chemistry that describes the behavior of electrons within a molecule. Unlike the Atomic Orbital Theory, which focuses on single atoms, MO theory shines light on how electrons are distributed across a molecule, rather than isolated atoms.

In MO theory, atomic orbitals combine to form molecular orbitals. These are spread out over the entire molecule. The combining process varies in complexity depending on the molecule being studied.

Some key points about MO theory include the following:

• Bonding molecular orbitals form when atomic orbitals combine constructively, resulting in lower energy states that help hold a molecule together.
  
• Antibonding molecular orbitals, on the other hand, arise when atomic orbitals combine destructively, leading to higher energy states that can destabilize the molecule if occupied.
  
• Electrons fill these molecular orbitals from lowest to highest energy, similar to how they fill atomic orbitals based on the Aufbau principle.
  

By examining how electrons populate these molecular orbitals, chemists can predict and explain reactivity, bond order, and magnetic properties of molecules, including paramagnetism.

Gouy Balance Method

The Gouy Balance Method is a practical technique used to determine the magnetic properties of materials, such as paramagnetism. This method provides a straightforward way to measure a substance's magnetic susceptibility.

Here's a simple breakdown of how it works:

• The sample of interest is positioned between the poles of a strong electromagnet.
  
• As the magnetic field is applied, the balance measures the change in the sample's weight.
  
• If the sample's weight decreases, it indicates that the sample is attracted to the magnetic field, classifying it as paramagnetic.
  

If there is no change or if the sample's weight increases, it suggests diamagnetism – meaning it's not attracted by the magnetic field.

This method is sensitive and allows for quick determination of a material's magnetic character. It gives valuable insights, especially when studying transition metals, complex ions, and materials with unpaired electrons.

Unpaired Electrons

Unpaired electrons are electrons that occupy an orbital without a partner of opposite spin. One of their most important roles is in determining whether a substance is magnetic.

Let's consider some key aspects about unpaired electrons:

• In a magnetic field, unpaired electrons generate a magnetic moment, which causes the material to interact with the magnetic field.
  
• Materials with unpaired electrons usually exhibit paramagnetism, being drawn into the magnetic field.
  
• The number of unpaired electrons can directly influence the magnetic strength of a substance.
  

Take the example of the ions \( ext{O}_2^+, ext{Li}_2^+, ext{and } ext{O}_2^{2-}\). These ions display paramagnetism as they have unpaired electrons. Specifically:

• \( ext{O}_2^+\) and \( ext{Li}_2^+\) each carry one unpaired electron.
  
• \( ext{O}_2^{2-}\) supports two unpaired electrons.
  

These unpaired electrons significantly contribute to the magnetic properties observed during analysis with methods like the Gouy balance. Their presence or absence can give deep insights into chemical behaviors and properties.