Foundations of Materials Science and Engineering
Foundations of Materials Science and Engineering
6th Edition
ISBN: 9781259696558
Author: SMITH
Publisher: MCG
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Chapter 5.7, Problem 11AAP

(a) Calculate the equilibrium concentration of vacancies per cubic meter in pure copper at 850°C. Assume that the energy of formation of a vacancy in pure copper is 1.0 eV. (b) What is the vacancy fraction at 800°C?

(a)

Expert Solution
Check Mark
To determine

The equilibrium concentration of vacancies per cubic meter in pure copper.

Answer to Problem 11AAP

The equilibrium concentration of vacancies per cubic meter in pure copper is 2.77×1024vacancies/m3.

Explanation of Solution

Express the equilibrium concentration of vacancies per cubic meter in pure copper.

 nv=NCeEv/kT                                                                           (I)

Here, number of vacancies per cubic meter of metal is nv, total number of atom sites per cubic meter of metal is N, activation energy to form a vacancy is Ev(eV), absolute temperature is T, Boltzmann’s constant is k and constant is C.

Express total number of atom sites per cubic meter of metal.

 N=N0ρCuat.massCu                                                                          (II)

Here, Avogadro’s number is N0 and density of copper is ρCu.

Conclusion:

Write the value of Avogadro’s number, atomic mass and density of copper.

 N0=6.02×1023atomsρCu=8.96Mg/m3at.massCu=63.54g/at.mass

Write the value of Boltzmann’s constant.

 k=8.62×105eV/K

Substitute 6.02×1023atoms for N0, 8.96Mg/m3 for ρCu and 63.54g/at.mass for at.massCu in Equation (II).

 N=(6.02×1023atoms)[(8.96Mg/m3)106gMg]63.54g/at.mass=(6.02×1023atoms)(8.96×106g/m3)63.54g/at.mass=8.49×1028atoms/m3

Substitute 8.49×1028atoms/m3 for N, 8.62×105eV/K for k, 850°C for T and 1eV for Ev in Equation (I).

 nv=(8.49×1028atoms/m3){exp[1eV(8.62×105eV/K)(850°C)]}=(8.49×1028atoms/m3){exp[1eV(8.62×105eV/K)[(850+273)K]]}=(8.49×1028atoms/m3){exp[1eV(8.62×105eV/K)(1123K)]}=2.77×1024vacancies/m3

Hence, the equilibrium concentration of vacancies per cubic meter in pure copper is 2.77×1024vacancies/m3.

(b)

Expert Solution
Check Mark
To determine

The vacancy fraction at 800°C.

Answer to Problem 11AAP

The vacancy fraction at 800°C is 2.02×105vacancies/atom.

Explanation of Solution

Express the vacancy fraction at 800°C.

 nvN=eEv/kT                                                                                         (III)

Conclusion:

Substitute 8.62×105eV/K for k, 800°C for T and 1eV for Ev in Equation (III).

 nvN=exp[1eV(8.62×105eV/K)(800°C)]=exp[1eV(8.62×105eV/K)(1073K)]=e10.81=2.02×105vacancies/atom

Hence, the vacancy fraction at 800°C is 2.02×105vacancies/atom.

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Students have asked these similar questions
At 660°C the fractional concentration of vacancies in aluminum is 9 × 10-5. The energy for vacancy formation is therefore: a. 0.72 eV b. 1.44 eV c. 0.79 eV d. 0.75 eV e. 0.68 eV
1. For some hypothetical metal the equilibrium number of vacancies at 750°C is 2.8×1024?−3. If the density and atomic weight of this metal are 5.60 ? ??3⁄ and 65.6 ?/???, respectively, calculate the fraction of vacancies for this metal at 750°C.  2. Calculate the number of vacancies per cubic meter in iron at 850°C. The energy for vacancy formation is 1.08 eV/atom. The density and atomic weight for Fe are 7.65 g/cm3 and 55.85 g/mol, respectively.    Answer both please
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