The chemical formula for manganese ferrite may be written as \(\left(\mathrm{MnFe}_{2} \mathrm{O}_{4}\right)_{8}\) because there are eight formula units per unit cell. If this material has a saturation magnetization of \(5.6 \times 10^{5} \mathrm{~A} / \mathrm{m}\) and a density of \(5.00 \mathrm{~g} / \mathrm{cm}^{3}\) estimate the number of Bohr magnetons associated with each \(\mathrm{Mn}^{2+}\) ion.

Understanding the concept of saturation magnetization is crucial when diving into the realm of magnetic materials. Saturation magnetization, denoted as \(M_s\), is the maximum level of magnetization a material can achieve under an external magnetic field. When this field is removed, some materials retain a degree of magnetization known as remanence, while others, such as soft magnetic materials, may return to an unmagnetized state.
Manganese ferrite, featured in our textbook exercise, is an example of a ferrimagnetic material, which possesses a high saturation magnetization. To comprehend why the calculated value of \(5.6 \times 10^{5} \mathrm{~A} / \mathrm{m}\) is significant, we must first recognize that this figure is indicative of the magnetic moments within the unit cell aligning to their maximum extent. This alignment is critical in applications like magnetic storage devices, where greater saturation magnetization can translate to higher data storage capabilities.

Exercise Connection

In our example, the concept of saturation magnetization is employed to determine the number of Bohr magnetons per \(\mathrm{Mn}^{2+}\) ion. By knowing the saturation magnetization, we can deduce the magnetic moment per manganese ion, and thereby gauge the magnetic contribution of \(\mathrm{Mn}^{2+}\) ions in manganese ferrite. Since this is a tangible measure of magnetization, it enables us to link the atomic-scale magnetic properties to the macroscopic magnetic behavior of the material.

Bohr magnetons are a fundamental metric in magnetism, providing a quantum mechanical standard unit of magnetic moment. One Bohr magneton, expressed as \(\mu_B\), is approximately equal to \(9.27 \times 10^{-24} \mathrm{A m^2}\). This value is derived from the electron's orbital and spin magnetic moments – a reflection of the quantum scale of magnetism. The concept is named after the Danish physicist Niels Bohr, recognizing his contributions to our understanding of atomic structure and quantum theory.
In relation to magnetic materials, calculating the number of Bohr magnetons gives us insight into the magnetic strength of an ion or a molecule. For instance, in high spin \(\mathrm{Mn}^{2+}\) ions, we typically expect five Bohr magnetons based on the electron configuration, but in some cases, due to interactions within the lattice, this number can deviate.

Exercise Connection

In the given exercise, the aim is to estimate the number of Bohr magnetons associated with each \(\mathrm{Mn}^{2+}\) ion in manganese ferrite by utilizing the magnetic properties provided. This not only illustrates practical application of quantum mechanics in material science, but it also bridges the gap between theoretical physics and hands-on problem solving.

Each crystalline solid has a defining structure which determines its physical and chemical properties. For manganese ferrite, with the chemical representation \(\left(\mathrm{MnFe}_{2} \mathrm{O}_{4}\right)_{8}\), the subscript '8' indicates that there are eight formula units per unit cell. A formula unit is the lowest whole number ratio of ions represented in an ionic compound. In the case of manganese ferrite, the unit consists of one manganese ion, two iron ions, and four oxygen ions, essentially reflecting the compound's stoichiometry.
Understanding formula units is helpful in calculating various properties of a material, such as its density, molar mass, and as seen in our exercise, the magnetic properties. For magnetic calculations, it's the amount of magnetic ions per formula unit that primarily contributes to the overall magnetic behavior of a compound.

Exercise Connection

Linking back to the textbook problem, by knowing we have eight manganese ferrite formula units in the unit cell, we determine the number of Mn²⁺ ions and further calculate the unit cell's mass, volume, magnetization, and ultimately the magnetic contribution of these ions. Addressing the exercise improvement advice, it's clear that each formula unit's contribution is critical to answering the problem. This example showcases the importance of chemical formula units in comprehending the macroscopic properties that stem from microscopic structures.