Problem 78 - Chemistry: The Central Science

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: \(\mathrm{O}_{2}^{+}, \mathrm{N}_{2}{\underline{\phantom{xx}}}^{2-}, \mathrm{Li}_{2}^{+}, \mathrm{O}_{2}{\underline{\phantom{xx}}}^{2-}\) ? For those ions that are paramagnetic, determine the number of unpaired electrons.

Step-by-Step Solution

Answer

Paramagnetism refers to the property exhibited by certain materials when they become weakly attracted to an external magnetic field, due to the presence of unpaired electrons. Experimentally, one can determine if a substance is paramagnetic using the Gouy balance method or the Faraday method. Using molecular orbital theory, we find that \(\mathrm{O}_{2}^{+}\) and \(\mathrm{Li}_{2}^{+}\) ions are paramagnetic with one unpaired electron each, while \(\mathrm{N}_{2}{\underline{\phantom{xx}}}^{2-}\) and \(\mathrm{O}_{2}{\underline{\phantom{xx}}}^{2-}\) ions are diamagnetic with zero unpaired electrons.

Step 1: Definition of Paramagnetism

Paramagnetism refers to the property exhibited by certain materials when they become weakly attracted to an external magnetic field. This behavior is due to the presence of one or more unpaired electrons in their atomic or molecular orbital structure. These unpaired electrons have magnetic moments that align with the external magnetic field, resulting in a net magnetic attraction.

Step 2: Experimentally Determining Paramagnetism

One can experimentally determine if a substance is paramagnetic by placing it within an external magnetic field and observing if it is attracted to the field. Two common methods are: 1. The Gouy balance method: The sample is placed between two magnetic poles, and the weight change due to the magnetic attraction is measured. A paramagnetic substance will experience an increase in weight compared to its weight in the absence of a magnetic field. 2. The Faraday method: The sample is placed in a non-uniform magnetic field and is subjected to a force proportional to the product of the field gradient and the sample's magnetic susceptibility. If the substance is paramagnetic, it will be drawn towards the region of higher magnetic field intensity.

Step 3: Determining Unpaired Electrons for the Given Ions

To find the number of unpaired electrons and determine which ions are paramagnetic, we will use molecular orbital (MO) theory. MO theory explains the electronic structure of molecules and ions by considering the linear combination of atomic orbitals (LCAO) to form molecular orbitals. For the given ions, we can apply MO theory as follows: 1. \(\mathrm{O}_{2}^{+}\): The electronic configuration of neutral O2 is \(\sigma_{1s}^{2} \sigma_{1s*}^{2} \sigma_{2s}^{2} \sigma_{2s*}^{2} \sigma_{2p}^{2} \pi_{2p}^{4} \pi_{2p*}^{2}\). For the cation \(\mathrm{O}_{2}^{+}\), there will be one less electron, resulting in the configuration \(\sigma_{1s}^{2} \sigma_{1s*}^{2} \sigma_{2s}^{2} \sigma_{2s*}^{2} \sigma_{2p}^{2} \pi_{2p}^{4} \pi_{2p*}^{1}\). As it has one unpaired electron, this ion is paramagnetic. 2. \(\mathrm{N}_{2}{\underline{\phantom{xx}}}^{2-}\): The electronic configuration of neutral N2 is \(\sigma_{1s}^{2} \sigma_{1s*}^{2} \sigma_{2s}^{2} \sigma_{2s*}^{2} \sigma_{2p}^{2} \pi_{2p}^{4}\). For the anion \(\mathrm{N}_{2}{\underline{\phantom{xx}}}^{2-}\), there will be two additional electrons, resulting in the configuration \(\sigma_{1s}^{2} \sigma_{1s*}^{2} \sigma_{2s}^{2} \sigma_{2s*}^{2} \sigma_{2p }^{2} \pi_{2p}^{6}\). As there are no unpaired electrons, this ion is not paramagnetic (it is diamagnetic). 3. \(\mathrm{Li}_{2}^{+}\): The electronic configuration of neutral Li2 is \(\sigma_{1s}^{2} \sigma_{1s*}^{2} \sigma_{2s}^{2}\). For the cation \(\mathrm{Li}_{2}^{+}\), there will be one less electron, resulting in the configuration \(\sigma_{1s}^{2} \sigma_{1s*}^{2} \sigma_{2s}^{1}\). As it has one unpaired electron, this ion is paramagnetic. 4. \(\mathrm{O}_{2}{\underline{\phantom{xx}}}^{2-}\): As mentioned earlier, the electronic configuration of neutral O2 is \(\sigma_{1s}^{2} \sigma_{1s*}^{2} \sigma_{2s}^{2} \sigma_{2s*}^{2} \sigma_{2p}^{2} \pi_{2p}^{4} \pi_{2p*}^{2}\). For the anion \(\mathrm{O}_{2}{\underline{\phantom{xx}}}^{2-}\), there will be two additional electrons, resulting in the configuration \(\sigma_{1s}^{2} \sigma_{1s*}^{2} \sigma_{2s}^{2} \sigma_{2s*}^{2} \sigma_{2p}^{2} \pi_{2p}^{4} \pi_{2p*}^{4}\). As there are no unpaired electrons, this ion is not paramagnetic (it is diamagnetic). In summary, the \(\mathrm{O}_{2}^{+}\) and \(\mathrm{Li}_{2}^{+}\) ions are paramagnetic, with one unpaired electron each. The \(\mathrm{N}_{2}{\underline{\phantom{xx}}}^{2-}\) and \(\mathrm{O}_{2}{\underline{\phantom{xx}}}^{2-}\) ions are diamagnetic, with zero unpaired electrons.

Key Concepts

Unpaired Electrons

The concept of unpaired electrons is pivotal when discussing magnetic properties of substances. To understand this, one should know that electrons have a property called 'spin', which can generate a tiny magnetic field. In atoms or molecules where electrons are paired, their spins oppose each other, effectively cancelling out their magnetic fields. However, when electrons are unpaired, their spins do not cancel out, and the magnetic fields produce a net magnetic moment.

This magnetic moment is what imparts paramagnetic substances their characteristic attraction to external magnetic fields. The number of unpaired electrons is directly related to how strong this paramagnetic effect will be. Substances with more unpaired electrons will show a more considerable attraction to magnetic fields. For instance, in our textbook exercise, the substances \(\mathrm{O}_{2}^{+}\) and \(\mathrm{Li}_{2}^{+}\) both have one unpaired electron and thus are paramagnetic.

Magnetic Susceptibility

Magnetic susceptibility is a measure of how much a substance will become magnetized in an applied magnetic field. It is an intrinsic property that indicates whether the substance is diamagnetic or paramagnetic. Paramagnetic substances have positive susceptibility due to the aligned magnetic moments of unpaired electrons enhancing the external field. On the contrary, diamagnetic materials have negative susceptibility, as their paired electrons generate small, opposing fields to the applied magnetic field.

In experimental contexts, magnetic susceptibility can be measured using methods such as the Gouy balance or the Faraday technique. The Gouy balance detects weight changes in the presence of a magnetic field, while the Faraday method tracks movement due to magnetic forces. Both methods depend on the fundamental relationship between a substance's magnetic susceptibility and its behavior in a magnetic field.

Molecular Orbital Theory

Molecular Orbital Theory (MO theory) is a sophisticated way to describe how electrons are arranged in molecules. According to MO theory, individual atomic orbitals (the regions in atoms where electrons are most likely to be found) combine to form molecular orbitals when atoms bond together. These molecular orbitals can be occupied by electrons from both atoms, belonging to the whole molecule rather than any single atom.

In the context of our exercise, MO theory helps us determine the number of unpaired electrons in molecules or ions by examining their electronic configurations. For instance, it allows us to analyze diatomic molecules like \(\mathrm{O}_{2}^{+}\) and \(\mathrm{Li}_{2}^{+}\), explaining their paramagnetic nature due to unpaired electrons in their molecular orbitals. Conversely, the theory also explains why \(\mathrm{N}_{2}{\underline{\phantom{xx}}}^{2-}\) and \(\mathrm{O}_{2}{\underline{\phantom{xx}}}^{2-}\) have no unpaired electrons and are diamagnetic.