Chapter 5.13, Problem 1QP

To determine

The total number of lattice site in sliver $\left(Ag\right)$ at ${700}^{\circ }\text{C}$.

Answer to Problem 1QP

The total number of lattice site in sliver $\left(Ag\right)$ at ${700}^{\circ }\text{C}$ is $\underset{\_}{1.156\times {10}^{23}{\text{vacancies/m}}^{\text{3}}}$.

Explanation of Solution

Given:

The atomic weight of silver, $A_{\text{Ag}}=107.87\text{g/mol}$.

The theoretical density of silver, ${\rho }_{\text{Ag}}=10.35{\text{g/cm}}^{\text{3}}$.

The absolute temperature of sliver is ${700}^{\circ }\text{C}$.

The number of vacancies per unit volume, $N_{\nu }=2\times {10}^{-6}$.

Explanation:

Write the expression of atomic site per cubic meter.

$N_{\text{Ag}}=\frac{N_{\text{A}}{\rho }_{\text{Ag}}}{A_{\text{Ag}}}$ (I)

Here, Avogadro’s number $\left(6.022\times {10}^{23}\text{atoms/mol}\right)$ is $N_{\text{A}}$.

Write the expression of vacancies per meter cubic.

$N_{\nu }=\left(2\times {10}^{-6}\right)N_{\text{Ag}}$ (II)

Conclusion:

Substitute $107.87\text{g/mol}$ for $A_{\text{Ag}}$, $10.35g/c{m}^3$ for ${\rho }_{\text{Ag}}$, and $\left(6.022\times {10}^{23}\text{atoms/mol}\right)$ for $N_{\text{A}}$ in Equation (I).

$\begin{array}{c}{N}_{\text{Ag}}=\frac{\left(6.022\times {10}^{23}\text{atoms/mol}\right)\left(10.35g/c{m}^3\right)\left({10}^6{\text{cm}}^{\text{3}}{\text{/m}}^{\text{3}}\right)}{\left(107.87\text{g/mol}\right)}\\ =\frac{\left(6.23277\times {10}^{24}\right)\left({10}^6{\text{cm}}^{\text{3}}{\text{/m}}^{\text{3}}\right)}{\left(107.87\text{g/mol}\right)}\\ =5.78\times {10}^{28}{\text{atoms/m}}^{\text{3}}\end{array}$

Substitute $5.78\times {10}^{28}{\text{atoms/m}}^{\text{3}}$ for $N_{\text{Ag}}$ in Equation (II).

$\begin{array}{c}{N}_{\nu }=\left(2\times {10}^{-6}\right)\left(5.78\times {10}^{28}{\text{atoms/m}}^{\text{3}}\right)\\ =1.156\times {10}^{23}{\text{vacancies/m}}^{\text{3}}\end{array}$

Thus, the total number of lattice site in sliver $\left(Ag\right)$ at ${700}^{\circ }\text{C}$ is $\underset{\_}{1.156\times {10}^{23}{\text{vacancies/m}}^{\text{3}}}$.