It is possible to express the magnetic susceptibility \(\chi_{m}\) in several different units. For the discussion in this chapter, \(\chi_{m}\) is 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 last two quantities are related to \(\chi_{m}\) through the following relationships: $$ \begin{aligned} &\chi_{m}=\chi_{m}(\mathrm{~kg}) \times \text { mass density }\left(\text { in } \mathrm{kg} / \mathrm{m}^{3}\right) \\ &\chi_{m}(a)=\chi_{m}(\mathrm{~kg}) \times \text { atomic weight }(\text { in } \mathrm{kg}) \end{aligned} $$ When using the cgs-emu system, comparable parameters exist that may be designated by \(\chi_{m}^{\prime}\), \(\chi_{m}^{\prime}(\mathrm{g})\), and \(\chi_{m}^{\prime}(a) ;\) the \(\chi_{m}\) and \(\chi_{m}^{\prime}\) are related in accordance with Table 20.1. From Table 20.2, \(\chi_{m}\) for copper is \(-0.96 \times 10^{-5}\); convert this value into the other five susceptibilities.

Short Answer

Expert verified
Question: Express the magnetic susceptibility for silver in 5 different units. Answer: The magnetic susceptibility for silver can be expressed as: 1. Volume susceptibility in SI units: \(\chi_{m} \approx -2.38 \times 10^{-5}\) kg m³ A⁻¹ s² 2. Mass susceptibility in SI units: \(\chi_{m}(\mathrm{kg}) \approx -2.27 \times 10^{-9} \,\text{kg}\) 3. Atomic susceptibility in SI units: \(\chi_{m}(\mathrm{a}) \approx - 2.44 \times 10^{-10}\, \text{kg-mol}\) 4. Volume susceptibility in cgs-emu system: \(\chi_{m}^{\prime}\approx -2.99 \times 10^{-4} \, \text{cm}^{3}\, \text{g}^{-1}\, \text{Oe}^{-1}\) 5. Mass susceptibility in cgs-emu system: \(\chi_{m}^{\prime}(\mathrm{g}) \approx - 2.27 \times 10^{-6} \, \text{g}\, \text{Oe}^{-1}\)

Step by step solution

01

Data

For silver, we have the following properties: Mass density: 10.5 g/cm³ or 10500 kg/m³ Atomic weight: 107.87 g/mol or 0.10787 kg/mol Volume susceptibility in SI units: \(\chi_{m} = -2.38 \times 10^{-5}\) Note: To convert SI to cgs-emu, use Table 20.1
02

Conversion to Mass Susceptibility in SI units

Using the first equation: $$\chi_{m}(\mathrm{kg}) = \frac{\chi_{m}}{\text{mass density}}$$ $$\chi_{m}(\mathrm{kg}) = \frac{-2.38 \times 10^{-5}}{10500}$$ $$\chi_{m}(\mathrm{kg}) \approx -2.27 \times 10^{-9} \,\text{kg}$$
03

Conversion to Atomic Susceptibility in SI units

Using the second equation: $$\chi_{m}(\mathrm{a}) = \chi_{m}(\mathrm{kg}) \times \text{atomic weight}$$ $$\chi_{m}(\mathrm{a}) = -2.27 \times 10^{-9} \times 0.10787$$ $$\chi_{m}(\mathrm{a}) \approx - 2.44 \times 10^{-10}\, \text{kg-mol}$$
04

Conversion to Volume Susceptibility in cgs-emu system

Using conversion factors in Table 20.1, we get: $$\chi_{m}^{\prime} = \chi_{m} \times 4 \pi = -2.38 \times 10^{-5} \times 4\pi$$ $$\chi_{m}^{\prime}\approx -2.99 \times 10^{-4} \, \text{cm}^{3}\, \text{g}^{-1}\, \text{Oe}^{-1}$$
05

Conversion to Mass Susceptibility in cgs-emu system

Using conversion factors in Table 20.1, we get: $$\chi_{m}^{\prime}(\mathrm{g}) = \chi_{m}(\mathrm{kg}) \times 10^3 = -2.27 \times 10^{-9} \times 10^3$$ $$\chi_{m}^{\prime}(\mathrm{g}) \approx - 2.27 \times 10^{-6} \, \text{g}\, \text{Oe}^{-1}$$
06

Conversion to Atomic Susceptibility in cgs-emu system

Using conversion factors in Table 20.1, we get: $$\chi_{m}^{\prime}(\mathrm{a}) = \chi_{m}(\mathrm{a}) \times 10^3 = - 2.44 \times 10^{-10} \times 10^3$$ $$\chi_{m}^{\prime}(\mathrm{a}) \approx - 2.44 \times 10^{-7}\, \text{g-mol}\, \text{Oe}^{-1}$$ The magnetic susceptibility of silver is now successfully expressed in all 5 required units.

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