Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 20: Problem 13

The chemical formula for copper ferrite may be written as \(\left(\mathrm{CuFe}_{2} \mathrm{O}_{4}\right)_{8}\) because there are eight formula units per unit cell. If this material has a saturation magnetization of \(1.35 \times\) \(10^{5} \mathrm{A} / \mathrm{m}\) and a density of \(5.40 \mathrm{g} / \mathrm{cm}^{3},\) estimate the number of Bohr magnetons associated with each \(\mathrm{Cu}^{2+}\) ion.

Short Answer

Expert verified
Question: Estimate the number of Bohr magnetons associated with each Cu²⁺ ion in copper ferrite. Answer: The number of Bohr magnetons associated with each Cu²⁺ ion in copper ferrite is approximately \(1.82\times10^{23}\).

Step by step solution

01

Calculate the number of Cu²⁺ ions per unit cell

For copper ferrite, the chemical formula is given as \((\mathrm{CuFe}_{2}\mathrm{O}_{4})_{8}\), which means there are 8 formula units per unit cell. Each formula unit contains 1 Cu²⁺ ion, thus there are 8 Cu²⁺ ions per unit cell.
02

Calculate the mass of one unit cell

To calculate the mass of one unit cell, we will use the density (\(\rho\)) of copper ferrite provided in the exercise: \(\rho = \frac{m}{V} \Rightarrow m = \rho\cdot V\) The density is \(\rho = 5.40\ \mathrm{g/cm^{3}}\). Next, we need to find the volume (V) of one unit cell: \(V = \frac{1}{\text{number of atoms per unit cell}} = \frac{1}{8} \ \mathrm{cm^3}\) Now we can calculate the mass of one unit cell: \(m = \rho \cdot V = 5.40\ \mathrm{g/cm^{3}} \cdot \frac{1}{8}\ \mathrm{cm^3} = 0.675\ \mathrm{g}\)
03

Calculate the saturation magnetization per Cu²⁺ ion

The saturation magnetization is given as \(1.35\times10^5\ \mathrm{A/m}\). Since there are 8 Cu²⁺ ions per unit cell, the saturation magnetization per Cu²⁺ ion is: \(\frac{1.35\times10^5\ \mathrm{A/m}}{8\ \mathrm{Cu^{2+}\ ions}} = 1.6875\times10^4\ \mathrm{A/m\ per\ Cu^{2+}\ ion}\)
04

Calculate the number of Bohr magnetons per Cu²⁺ ion

The Bohr magneton (\(\mu_B\)) can be defined as: \(\mu_B = \frac{e \hbar}{2m_e} = 9.27\times10^{-24}\ \mathrm{Am^2}\) Now we can calculate the number of Bohr magnetons per Cu²⁺ ion by dividing the saturation magnetization per Cu²⁺ ion by the Bohr magneton value: \(\text{Number of Bohr magnetons per Cu²⁺ ion} = \frac{1.6875\times10^4\ \mathrm{A/m}}{9.27\times10^{-24}\ \mathrm{Am^2}} \approx 1.82\times10^{23}\) The number of Bohr magnetons associated with each Cu²⁺ ion is approximately \(1.82\times10^{23}\).

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!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Cite the differences between hard and soft magnetic materials in terms of both hysteresis behavior and typical applications.

Cite the primary limitation of the new superconducting materials that have relatively high critical temperatures.

Briefly describe the phenomenon of magnetic hysteresis, and why it occurs for ferromagnetic and ferrimagnetic materials.

Compute (a) the saturation magnetization and (b) the saturation flux density for iron, which has a net magnetic moment per atom of 2.2 Bohr magnetons and a density of \(7.87 \mathrm{g} / \mathrm{cm}^{3}\).

The magnetic flux density within a bar of some material is 0.630 tesla at an \(H\) field of \(5 \times 10^{5} \mathrm{A} / \mathrm{m} .\) Compute the following for this material: (a) the magnetic permeability, and (b) the magnetic susceptibility. (c) What type(s) of magnetism would you suggest is(are) being displayed by this material? Why?
See all solutions

Recommended explanations on Physics Textbooks

Electricity

Read Explanation

Further Mechanics and Thermal Physics

Read Explanation

Famous Physicists

Read Explanation

Dynamics

Read Explanation

Turning Points in Physics

Read Explanation

Thermodynamics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free