Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 4: Problem 4

Calculate the number of vacancies per cubic meter in gold (Au) at \(900^{\circ} \mathrm{C}\). The energy for vacancy formation is \(0.98 \mathrm{eV} /\) atom. Furthermore, the density and atomic weight for Au are \(18.63 \mathrm{~g} / \mathrm{cm}^{3}\) (at \(900^{\circ} \mathrm{C}\) ) and \(196.9 \mathrm{~g} /\) mol, respectively.

Short Answer

Expert verified
Answer: At $900^{\circ}\mathrm{C}$, the number of vacancies per cubic meter in gold is approximately $4.522 \times 10^{24}$.

Step by step solution

01

Convert the temperature to Kelvin

First, we need to convert the given temperature from Celsius to Kelvin. To do this, add 273.15 to the given temperature. \(T_{K} = 900^{\circ} \mathrm{C} + 273.15 = 1173.15\ \mathrm{K}\)
02

Calculate the equilibrium vacancy concentration

Now, we'll use the Arrhenius equation to find the equilibrium vacancy concentration, given by: \(C_v = C_0 \times e^{-E_v/(k_B T)}\) where \(C_v\) is the equilibrium vacancy concentration, \(C_0\) is the total concentration of lattice sites, \(E_v\) is the energy for vacancy formation, \(k_B\) is Boltzmann's constant (approximately \(8.617 \times 10^{-5} \mathrm{eV/K}\)), and \(T\) is the temperature in Kelvin. We have \(E_v = 0.98 \mathrm{eV/atom}\). Now we need to find \(C_0\). Since we are given the density and atomic weight of gold, we can find the number of moles of gold per cubic meter and then multiply by Avogadro's number to find the total concentration of lattice sites.
03

Calculate the total concentration of lattice sites (C0)

We are given the density as \(18.63 \mathrm{g/cm^3}\). To convert it to \(\mathrm{kg/m^3}\), we multiply by \(10^6\): \(\rho = 18.63 \mathrm{g/cm^3} \times 10^6 = 18.63 \times 10^6 \mathrm{g/m^3}\) Now, we can find the number of moles of gold per cubic meter: \(n = \frac{\rho}{M} = \frac{18.63 \times 10^6 \mathrm{\ g/m^3}}{196.9 \mathrm{g/mol}} = 9.458 \times 10^4 \mathrm{mol/m^3}\) Finally, we can find the total concentration of lattice sites by multiplying the number of moles by Avogadro's number (\(N_A \approx 6.022 \times 10^{23} \mathrm{mol^{-1}}\)): \(C_0 = n \times N_A = 9.458 \times 10^4 \mathrm{mol/m^3} \times 6.022 \times 10^{23} \mathrm{mol^{-1}} = 5.691 \times 10^{28} \mathrm{m^{-3}}\)
04

Compute the equilibrium vacancy concentration

Now that we have \(C_0\), we can find the equilibrium vacancy concentration using the Arrhenius equation: \(C_v = C_0 \times e^{-E_v/(k_B T)} = 5.691 \times 10^{28} \mathrm{m^{-3}} \times e^{-0.98 \mathrm{eV /atom}/(8.617 \times 10^{-5} \mathrm{eV/K} \times 1173.15\ \mathrm{K})}\) \(C_v \approx 5.691 \times 10^{28} \mathrm{m^{-3}} \times e^{-9.1849} \approx 4.522 \times 10^{24} \mathrm{vacancies/m^3}\) Thus, the number of vacancies per cubic meter in gold at \(900^{\circ}\mathrm{C}\) is approximately \(4.522 \times 10^{24}\).

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

For an FCC single crystal, would you expect th surface energy for a (100) plane to be greater less than that for a (111) plane? Why? (Note: Yo may want to consult the solution to Problem \(3.6\) at the end of Chapter 3.).

Determine the approximate density of a Ti-6Al-4V titanium (Ti) alloy that has a composition of \(90 \mathrm{wt} \%\) Ti, \(6 \mathrm{wt} \% \mathrm{Al}\), and \(4 \mathrm{wt} \% \mathrm{~V}\)

The concentration of \(\mathrm{Si}\) in an Fe-Si alloy is \(0.25 \mathrm{wt} \%\). What is the concentration in kilograms of Si per cubic meter of allov?

Cite the relative Burgers vector-dislocation line orientations for edge, screw, and mixed dislocations.

What is the composition, in atom percent, of an alloy that contains \(33 \mathrm{~g}\) of \(\mathrm{Cu}\) and \(47 \mathrm{~g}\) of \(\mathrm{Zn}\) ?
See all solutions

Recommended explanations on Physics Textbooks

Rotational Dynamics

Read Explanation

Engineering Physics

Read Explanation

Measurements

Read Explanation

Atoms and Radioactivity

Read Explanation

Solid State Physics

Read Explanation

Fundamentals of Physics

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