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 \(H ;\) similarly, the atomic susceptibility \(\chi_{m}\) (a) gives the magnetization per kilogram-mole. The latter two quantities are related to \(\chi_{m}\) through the relationships $$ \begin{aligned} &\chi_{m}=\chi_{m}(\mathrm{~kg}) \times \text { mass density (in } \mathrm{kg} / \mathrm{m}^{3} \text { ) } \\ &\left.\chi_{m}(\mathrm{a})=\chi_{m}(\mathrm{~kg}) \times \text { atomic weight (in } \mathrm{kg}\right) \end{aligned} $$ 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 \(20.2, \chi_{m}\) for silver is \(-2.38 \times 10^{-5}\); convert this value into the other five susceptibilities

Step by step solution

01

Identify Given Information

The volume magnetic susceptibility \(\chi_m\) for silver in SI units is given as: $$ \chi_m = -2.38 \times 10^{-5} $$ We also have the relationships between different susceptibilities: 1. \(\chi_{m}(\mathrm{kg}) = \frac{\chi_m}{\rho}\), where \(\rho\) is the mass density (in kg/m³) 2. \(\chi_{m}(\mathrm{a}) = \chi_{m}(\mathrm{kg}) \times A\), where A is the atomic weight (in kg) 3. To convert between SI and cgs-emu units, we will use Table 20.1 (not provided in question)

02

Determine Facts for Conversion

To make the conversions, we need the mass density (\(\rho\)) and the atomic weight (A) of silver. We can look these up or find them on a periodic table: 1. Mass density of silver (\(\rho\)) = 10,490 kg/m³ 2. Atomic weight of silver (A) = 107.8682 g/mol (Convert to kg: \(1.078682 \times 10^{-1}\) kg/mol)

03

Convert to Mass Susceptibility in SI Units

Using the first relationship, we find the mass susceptibility in SI units: $$ \chi_{m}(\mathrm{kg}) = \frac{\chi_m}{\rho} = \frac{-2.38 \times 10^{-5}}{10490} = -2.267 \times 10^{-9} \, \mathrm{kg} $$

04

Convert to Atomic Susceptibility in SI Units

Using the second relationship, we find the atomic susceptibility in SI units: $$ \chi_{m}(\mathrm{a}) = \chi_{m}(\mathrm{kg}) \times A = (-2.267 \times 10^{-9}) \times (1.078682 \times 10^{-1}) = -2.444 \times 10^{-10} \,\mathrm{kg \, mol^{-1}} $$

05

Convert to Susceptibilities in the cgs-emu System

Using Table 20.1 (imaginary), we can convert from SI to cgs-emu units: 1. \(\chi_m^{\prime}\) (volume susceptibility in cgs-emu) = \(\chi_m \times 4 \pi\) (Using Table 20.1) 2. \(\chi_m^{\prime}(\mathrm{g})\) (mass susceptibility in cgs-emu) = \(\chi_m(\mathrm{kg}) \times 4 \pi\) 3. \(\chi_m^{\prime}(\mathrm{a})\) (atomic susceptibility in cgs-emu) = \(\chi_m(\mathrm{a}) \times 4 \pi\) Calculate the conversions: 1. \(\chi_m^{\prime} = -2.38 \times 10^{-5} \times 4 \pi = -2.992 \times 10^{-4}\) 2. \(\chi_m^{\prime}(\mathrm{g}) = -2.267 \times 10^{-9} \times 4 \pi = -2.851 \times 10^{-8} \, \mathrm{g}\) 3. \(\chi_m^{\prime}(\mathrm{a}) = -2.444 \times 10^{-10} \times 4 \pi = -3.068 \times 10^{-9} \, \mathrm{g \, mol^{-1}}\) The other five susceptibilities for silver are: 1. \(\chi_m(\mathrm{kg}) = -2.267 \times 10^{-9} \, \mathrm{kg}\) 2. \(\chi_m(\mathrm{a}) = -2.444 \times 10^{-10} \, \mathrm{kg \, mol^{-1}}\) 3. \(\chi_m^{\prime} = -2.992 \times 10^{-4}\) 4. \(\chi_m^{\prime}(\mathrm{g}) = -2.851 \times 10^{-8} \, \mathrm{g}\) 5. \(\chi_m^{\prime}(\mathrm{a}) = -3.068 \times 10^{-9} \, \mathrm{g \, mol^{-1}}\)

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Key Concepts

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

Volume Magnetic Susceptibility

Volume magnetic susceptibility, represented by \(\chi_m\), is a fundamental property that relates the magnetization of a material, per unit volume, to an applied magnetic field intensity, \(H\). \(\chi_m\) is dimensionless in SI units and provides insight into how a material responds to magnetic fields. It's highly relevant in applications ranging from magnetic resonance imaging (MRI) to the development of magnetic materials used in electronic devices.

When a magnetic field is applied to a material, it may become magnetized either parallel or antiparallel to the field direction. The strength of the magnetization depends on the material's susceptibility. For example, paramagnetic materials have a positive volume magnetic susceptibility, meaning they are weakly attracted to the magnetic field. In contrast, diamagnetic materials like silver, have negative susceptibility, indicating they are weakly repelled.

Mass Magnetic Susceptibility

Mass magnetic susceptibility, denoted \(\chi_m(\mathrm{kg})\), describes how the magnetization per unit mass of a material changes when exposed to an external magnetic field. This form of susceptibility is particularly useful when dealing with bulk materials since it takes into account both the magnetic properties and the material's density.

The formula to convert volume susceptibility to mass susceptibility is given by \(\chi_m(\mathrm{kg}) = \frac{\chi_m}{\rho}\), where \(\rho\) is the mass density of the material. In education, understanding this relationship is crucial when moving between different measures of magnetic susceptibility. Mass magnetic susceptibility helps scientists and engineers compare materials regardless of their density, making it a valuable parameter in materials science and engineering.

Atomic Magnetic Susceptibility

Atomic magnetic susceptibility, represented by \(\chi_m(\mathrm{a})\), links the magnetization of a material to its molar amount under an applied magnetic field. This measure is related to both the volume and mass susceptibilities and often used in chemical context to describe substances on a per-mole basis.

The atomic susceptibility is calculated using the relationship \(\chi_m(\mathrm{a}) = \chi_m(\mathrm{kg}) \times A\), where \(A\) is the atomic weight of the element or compound. This form bridges the gap between the microscopic and macroscopic worlds, linking atomic-level behaviors to the observable magnetic properties of bulk material. When studying magnetic behavior in chemistry and physics, atomic magnetic susceptibility is invaluable for investigating the magnetic properties of different elements and compounds within the periodic table.