It is possible to express the magnetic susceptibility \(\chi_{m}\) in several different units. For the discussion of this chapter, \(\chi_{m}\) was used to designate the volume susceptibility in SI units, that is, the quantity that gives the magnetization per unit volume \(\left(\mathrm{m}^{3}\right)\) of material when multiplied by \(H\). The mass susceptibility \(\chi_{m}(\mathrm{kg})\) yields the magnetic moment (or magnetization) per kilogram of material when multiplied by \(H ;\) and, similarly, the atomic susceptibility \(\chi_{m}(\text { a })\) gives the magnetization per kilogram- mole. The latter two quantities are related to \(\chi_{m}\) through the relationships $$\begin{aligned}\chi_{m} &=\chi_{m}(\mathrm{kg}) \times \text { mass density }\left(\mathrm{in} \mathrm{kg} / \mathrm{m}^{3}\right) \\\\\chi_{m}(\mathrm{a}) &=\chi_{m}(\mathrm{kg}) \times \text { atomic weight }(\mathrm{in} \mathrm{kg})\end{aligned}$$ When using the cgs-emu system, comparable parameters exist, which may be designated by \(\chi_{m}^{\prime}, \chi_{m}^{\prime}(\mathrm{g}),\) and \(\chi_{m}^{\prime}(\mathrm{a})\); the \(\chi_{m}\) and \(\chi_{m}^{\prime}\) are related in accordance with Table 20.1 From Table \(20.2, \quad \chi_{m}\) for copper is \(-0.96 \times 10^{-5} ;\) convert this value into the other five susceptibilities.

Step by step solution

01

Gather known values and conversion factors

First, let's collect all the known values and conversion factors that we will need for this problem: 1. \(\chi_m\) for copper: \(-0.96 \times 10^{-5}\) 2. Mass density of copper: \(8960\ \mathrm{kg}/\mathrm{m}^3\) 3. Atomic weight of copper: \(63.55\ \mathrm{kg}/\mathrm{mol}\) 4. Conversion factor between SI and cgs-emu units: \(1\ \mathrm{SI} = 4\pi \times 10^{-3}\ \mathrm{cgs-emu}\)

02

Determine mass susceptibility in SI units (\(\chi_{m}(\mathrm{kg})\))

Find the mass susceptibility in SI units by rearranging the first given relationship and plugging in the known values: \(\chi_{m}(\mathrm{kg}) = \frac{\chi_{m}}{\text{mass density}}= \frac{-0.96 \times 10^{-5}}{8960} = -1.071 \times 10^{-9}\ \mathrm{SI}\)

03

Determine atomic susceptibility in SI units (\(\chi_{m}(\text {a})\))

Find the atomic susceptibility in SI units by rearranging the second given relationship and plugging in the known values: \(\chi_{m}(\text {a}) = \frac{\chi_{m}(\mathrm{kg})}{\text{atomic weight}} = \frac{-1.071 \times 10^{-9}}{63.55} = -1.68 \times 10^{-11}\ \mathrm{SI}\)

04

Determine volume susceptibility in cgs-emu units (\(\chi_{m}^{\prime}\))

Convert the volume susceptibility from SI units to cgs-emu units using the given conversion factor: \(\chi_{m}^{\prime} = \chi_{m} \times 4\pi \times 10^{-3} = -0.96 \times 10^{-5} \times 4\pi \times 10^{-3} = -1.21 \times 10^{-8}\ \mathrm{cgs-emu}\)

05

Determine mass susceptibility in cgs-emu units (\(\chi_{m}^{\prime}(\mathrm{g})\))

Convert the mass susceptibility from SI units to cgs-emu units using the given conversion factor: \(\chi_{m}^{\prime}(\mathrm{g}) = \chi_{m}(\mathrm{kg}) \times 4\pi \times 10^{-3} = -1.071 \times 10^{-9} \times 4\pi \times 10^{-3} = -1.35 \times 10^{-11}\ \mathrm{cgs-emu}\)

06

Determine atomic susceptibility in cgs-emu units (\(\chi_{m}^{\prime}(\mathrm{a})\))

Convert the atomic susceptibility from SI units to cgs-emu units using the given conversion factor: \(\chi_{m}^{\prime}(\mathrm{a}) = \chi_{m}(\text{a}) \times 4\pi \times 10^{-3} = -1.68 \times 10^{-11} \times 4\pi \times 10^{-3} = -2.12 \times 10^{-14}\ \mathrm{cgs-emu}\) The other five susceptibilities for copper are: 1. Mass susceptibility in SI units: \(\chi_{m}(\mathrm{kg}) = -1.071 \times 10^{-9}\ \mathrm{SI}\) 2. Atomic susceptibility in SI units: \(\chi_{m}(\text {a}) = -1.68 \times 10^{-11}\ \mathrm{SI}\) 3. Volume susceptibility in cgs-emu units: \(\chi_{m}^{\prime} = -1.21 \times 10^{-8}\ \mathrm{cgs-emu}\) 4. Mass susceptibility in cgs-emu units: \(\chi_{m}^{\prime}(\mathrm{g}) = -1.35 \times 10^{-11}\ \mathrm{cgs-emu}\) 5. Atomic susceptibility in cgs-emu units: \(\chi_{m}^{\prime}(\mathrm{a}) = -2.12 \times 10^{-14}\ \mathrm{cgs-emu}\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Magnetization

Magnetization is a fundamental concept in the study of magnetism and refers to the magnetic moment per unit volume of a material. It's a measure of the extent to which a substance becomes magnetized when exposed to an external magnetic field. In the context of magnetic susceptibility, which is symbolized as \( \chi_m \), magnetization \( M \) can be defined using the equation \( M = \chi_m H \) where \( H \) represents the magnetic field strength.

Magnetizing a material aligns the magnetic moments (tiny magnetic fields due to electrons) within it, thus increasing its overall magnetic effect. Different materials respond to magnetization in varying degrees—some can be strongly magnetized, while others may show minimal magnetic response. This property is crucial in applications like magnetic data storage, electric motors, and transformers.

Informed by the exercise provided, we see that magnetization also ties into mass and atomic susceptibilities as they provide insights into how the magnetic properties of a material relate to its physical quantities like mass and mole.

SI and cgs-emu Conversion

Scientific measurements often require conversion between different unit systems, and magnetism is no exception. The System International (SI) and the centimetre-gram-second electro-magnetic unit (cgs-emu) are two systems used for measuring magnetic properties. Although SI is the modern standard for most scientific work, cgs-emu remains in use for certain magnetic measurements.

The conversion factor between these two units for magnetic susceptibility is particularly important. As exemplified in the textbook exercise, the conversion factor is \( 1 \text{ SI unit} = 4\pi \times 10^{-3} \text{ cgs-emu} \). Understanding how to convert between these systems is essential for interpreting magnetic data because mixing units without proper conversion can lead to significant errors in calculations and a misunderstanding of a material's magnetic properties.

Mass Density

Mass density is the mass of a substance divided by its volume, commonly expressed in units like kilograms per cubic meter (\(\mathrm{kg}/\mathrm{m}^3\)). This physical property plays a critical role in determining the mass susceptibility of a material, as it accounts for how much matter is contained within a given space.

In the context of magnetic susceptibility, the mass density allows us to translate the volume susceptibility (inherently volumetric) into a form that relates to the material's mass (mass susceptibility). This is particularly useful when dealing with pure substances or when comparing materials that might have very different densities. By using the relationship provided in the exercise, \(\chi_{m} = \chi_{m}(\mathrm{kg}) \times \text{mass density}\), students can connect the conceptual understanding of magnetism to a tangible physical property, enhancing their comprehension of the material's response to magnetism.

Atomic Weight

Atomic weight, often referred to as the relative atomic mass, represents the mass of one mole of an element and is measured in kilograms per mole (\(\mathrm{kg}/\mathrm{mol}\)). The atomic weight is fundamental for converting mass susceptibility to atomic susceptibility as it links the magnetization per kilogram to the magnetization per mole.

By utilizing the formula mentioned in the exercise, students learn how to tie the microscopic atomic properties of a substance to its macroscopic magnetic response. The equation \(\chi_{m}(\mathrm{a}) = \chi_{m}(\mathrm{kg}) \times \text{atomic weight}\) transforms the unit of analysis from mass-based magnetism to the number of atoms involved. This connection is especially important for chemists and physicists who often think in moles when considering large numbers of atoms or molecules.