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 12AAP

(a)

To determine

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

(a)

Expert Solution
Check Mark

Answer to Problem 12AAP

The equilibrium concentration of vacancies per cubic meter in pure silver is 2.24×1023vacancies/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=N0ρAgat.massAg                                                                          (II)

Here, Avogadro’s number is N0 and density of silver is ρAg.

Conclusion:

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

 N0=6.02×1023atomsρAg=10.5×106g/m3at.massAg=107.87g/at.mass

Write the value of Boltzmann’s constant.

 k=8.62×105eV/K

Substitute 6.02×1023atoms for N0, 10.5×106g/m3 for ρAg and 107.87g/at.mass for at.massAg in Equation (II).

 N=(6.02×1023atoms)(10.5×106g/m3)107.87g/at.mass=5.86×1028atoms/m3

Substitute 5.86×1028atoms/m3 for N, 8.62×105eV/K for k, 750°C for T and 1.10eV for Ev in Equation (I).

 nv=(5.86×1028atoms/m3){exp[1.10eV(8.62×105eV/K)(750°C)]}=(5.86×1028atoms/m3){exp[1.10eV(8.62×105eV/K)[(750+273)K]]}=(5.86×1028atoms/m3){exp[1.10eV(8.62×105eV/K)(1023K)]}=2.24×1023vacancies/m3

Hence, the equilibrium concentration of vacancies per cubic meter in pure silver is 2.24×1023vacancies/m3.

(b)

To determine

The vacancy fraction at 700°C.

(b)

Expert Solution
Check Mark

Answer to Problem 12AAP

The vacancy fraction at 8700°C is 2.01×106vacancies/atom.

Explanation of Solution

Express the vacancy fraction at 700°C.

 nvN=eEv/kT                                                                                         (III)

Conclusion:

Substitute 8.62×105eV/K for k, 700°C for T and 1.10eV for Ev in Equation (III).

 nvN=exp[1.10eV(8.62×105eV/K)(700°C)]=exp[1.10eV(8.62×105eV/K)(973K)]=e13.12=2.01×106vacancies/atom

Hence, the vacancy fraction at 700°C is 2.01×106vacancies/atom.

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