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?

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 $2.77\times 1{\text{0}}^{\text{24}}\text{vacancies}/{\text{m}}^{\text{3}}$.

Explanation of Solution

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

 $n_v=NC{e}^{-E_v/kT}$                                                                           (I)

Here, number of vacancies per cubic meter of metal is $n_v$, total number of atom sites per cubic meter of metal is $N$, activation energy to form a vacancy is $E_v\left(\text{eV}\right)$, absolute temperature is $T$, Boltzmann’s constant is $k$ and constant is $C$.

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

 $N=\frac{N_0{\rho }_{\text{Cu}}}{\text{at}\text{.}\text{mass}\text{Cu}}$                                                                          (II)

Here, Avogadro’s number is $N_0$ and density of copper is ${\rho }_{\text{Cu}}$.

Conclusion:

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

 $\begin{array}{l}{N}_0=6.02\times {10}^{23}\text{atoms}\\ {\rho }_{\text{Cu}}=8.96\text{Mg}/{\text{m}}^{\text{3}}\\ \text{at}\text{.}\text{mass}\text{Cu}=63.54\text{g}/\text{at}\text{.}\text{mass}\end{array}$

Write the value of Boltzmann’s constant.

 $k=8.62\times {10}^{-5}\text{eV}/\text{K}$

Substitute $6.02\times {10}^{23}\text{atoms}$ for $N_0$, $8.96\text{Mg}/{\text{m}}^{\text{3}}$ for ${\rho }_{\text{Cu}}$ and $63.54\text{g}/\text{at}\text{.}\text{mass}$ for $\text{at}\text{.}\text{mass}\text{Cu}$ in Equation (II).

 $\begin{array}{c}N=\frac{\left(6.02\times {10}^{23}\text{atoms}\right)\left[\left(8.96\text{Mg}/{\text{m}}^{\text{3}}\right)\frac{{10}^6\text{g}}{\text{Mg}}\right]}{63.54\text{g}/\text{at}\text{.}\text{mass}}\\ =\frac{\left(6.02\times {10}^{23}\text{atoms}\right)\left(8.96\times {10}^6\text{g}/{\text{m}}^{\text{3}}\right)}{63.54\text{g}/\text{at}\text{.}\text{mass}}\\ =8.49\times {10}^{28}\text{atoms}/{\text{m}}^{\text{3}}\end{array}$

Substitute $8.49\times {10}^{28}\text{atoms}/{\text{m}}^{\text{3}}$ for $N$, $8.62\times {10}^{-5}\text{eV}/\text{K}$ for $k$, $\text{850°C}$ for $T$ and $\text{1}\text{eV}$ for $E_v$ in Equation (I).

 $\begin{array}{c}{n}_v=\left(8.49\times {10}^{28}\text{atoms}/{\text{m}}^{\text{3}}\right)\left\{\exp \left[-\frac{\text{1}\text{eV}}{\left(8.62\times {10}^{-5}\text{eV}/\text{K}\right)\left(\text{850°C}\right)}\right]\right\}\\ =\left(8.49\times {10}^{28}\text{atoms}/{\text{m}}^{\text{3}}\right)\left\{\exp \left[-\frac{\text{1}\text{eV}}{\left(8.62\times {10}^{-5}\text{eV}/\text{K}\right)\left[\left(\text{850}+273\right)\text{K}\right]}\right]\right\}\\ =\left(8.49\times {10}^{28}\text{atoms}/{\text{m}}^{\text{3}}\right)\left\{\exp \left[-\frac{\text{1}\text{eV}}{\left(8.62\times {10}^{-5}\text{eV}/\text{K}\right)\left(1123\text{K}\right)}\right]\right\}\\ =2.77\times 1{\text{0}}^{\text{24}}\text{vacancies}/{\text{m}}^{\text{3}}\end{array}$

Hence, the equilibrium concentration of vacancies per cubic meter in pure copper is $2.77\times 1{\text{0}}^{\text{24}}\text{vacancies}/{\text{m}}^{\text{3}}$.

To determine

The vacancy fraction at 800°C.

Answer to Problem 11AAP

The vacancy fraction at 800°C is $2.02\times 1{\text{0}}^{-5}\text{vacancies}/\text{atom}$.

Explanation of Solution

Express the vacancy fraction at 800°C.

 $\frac{n_v}{N}=e^{-E_v/kT}$                                                                                         (III)

Conclusion:

Substitute $8.62\times {10}^{-5}\text{eV}/\text{K}$ for $k$, $\text{800°C}$ for $T$ and $\text{1}\text{eV}$ for $E_v$ in Equation (III).

 $\begin{array}{c}\frac{n_v}{N}=\exp \left[-\frac{\text{1}\text{eV}}{\left(8.62\times {10}^{-5}\text{eV}/\text{K}\right)\left(\text{800°C}\right)}\right]\\ =\exp \left[-\frac{\text{1}\text{eV}}{\left(8.62\times {10}^{-5}\text{eV}/\text{K}\right)\left(\text{1073}\text{K}\right)}\right]\\ =e^{-10.81}\\ =2.02\times 1{\text{0}}^{-5}\text{vacancies}/\text{atom}\end{array}$

Hence, the vacancy fraction at 800°C is $2.02\times 1{\text{0}}^{-5}\text{vacancies}/\text{atom}$.