Chapter 4, Problem 4.15P

To determine

(a)

The fraction of atom sites those are vacant in gold at the given temperature.

Answer to Problem 4.15P

The fraction of atom sites that are vacant in gold at the $1000\text{K}$ is $1.84\times {10}^{-5}\text{vacancies}/\text{atom}$.

Explanation of Solution

Given:

The enthalpy of formation of one vacancy in pure gold is $0.94\text{eV}$.

The temperature is $1000\text{K}$.

Formula used:

The expression to calculate the fraction of atom sites that are vacant is gold is give by,

$\frac{n_{\text{v}}}{N}=\exp \left(-\frac{\Delta H_{\text{v}}}{kT}\right)$   ......... (I)

Here, $\frac{n_{\text{v}}}{N}$ is the fraction of atom sites that are vacant, $k$ is the Boltzmann’s constant, $T$ is the temperature at which the vacancy sites are formed and $\Delta H_{\text{v}}$ is the enthalpy of vacancy formation.

Calculation:

The fraction of atom sites that are vacant is calculated as,

Substitute $8.62\times {10}^{-5}\text{eV}/\text{atom}\cdot \text{K}$ for $k$, $0.94\text{eV}$ for $\Delta H_{\text{v}}$, and $1000\text{K}$ for $T$ in equation (I).

$\begin{array}{c}\frac{n_{\text{v}}}{N}=\exp \left(-\frac{0.94\text{eV}}{\left(8.62\times {10}^{-5}\text{eV}/\text{atom}\cdot \text{K}\right)\left(1000\text{K}\right)}\right)\\ =\exp \left(-10.9048\right)\\ =1.84\times {10}^{-5}\text{vacancies}/\text{atom}\end{array}$

Conclusion:

Therefore, the fraction of atoms that are vacant in gold at the $1000\text{K}$ is $1.84\times {10}^{-5}\text{vacancies}/\text{atom}$.

To determine

(b)

The contribution to the volume coefficient of expansion for gold due to the formation of vacancies.

Answer to Problem 4.15P

The contribution to the volume coefficient of expansion for gold due to the formation of vacancies is $1.84\times {10}^{-8}{\text{K}}^{-1}$.

Explanation of Solution

Formula used:

The relation between the fraction of atom sites that are vacant and the volume change is given by,

$\frac{n_{\text{v}}}{N}=\frac{\Delta V}{V_0}$   ......... (II)

Here, $\Delta V$ is the change in volume, and $V_0$ is the initial volume.

The expression for the volume coefficient of expansion of gold is given by,

${\alpha }_{\text{V}}=\frac{1}{V_0}\frac{dV}{dT}$   ....... (III)

Here, ${\alpha }_{\text{V}}$ is the volume coefficient of expansion, $dT$ is the change in temperature and $dV$ is the change in volume.

Substitute $\frac{n_{\text{v}}}{N}$ for $\frac{dV}{V_0}$ and $\left(T_2-T_1\right)$ for $dT$ in equation (III).

${\alpha }_{\text{V}}=\frac{n_{\text{v}}}{N}\left(\frac{1}{T_2-T_1}\right)$   ....... (IV)

Here, $T_1$ is the initial temperature and $T_2$ is the final temperature.

Calculation:

The volume coefficient of expansion for gold is calculated as,

Substitute $1.84\times {10}^{-5}\text{vacancies}/\text{atom}$ for $\frac{n_{\text{v}}}{N}$, $1000\text{K}$ for $T_2$ and $0\text{K}$ for $T_1$ in equation (IV).

$\begin{array}{c}{\alpha }_{\text{V}}=\left(1.84\times {10}^{-5}\right)\left(\frac{1}{1000\text{K}-0\text{K}}\right)\\ =\frac{1.84\times {10}^{-5}}{1000\text{K}}\\ =1.84\times {10}^{-8}{\text{K}}^{-1}\end{array}$

Conclusion:

Therefore, the contribution to volume coefficient of expansion for gold is $1.84\times {10}^{-8}{\text{K}}^{-1}$.

To determine

(c)

The fraction of volume coefficient of expansion for gold.

Answer to Problem 4.15P

The fraction of volume coefficient of expansion for gold is $0.04$.

Explanation of Solution

Formula used:

The fraction of volume of coefficient of expansion of gold is given by,

${\left({\alpha }_{\text{V}}\right)}_{\text{F}}=\frac{{\alpha }_{\text{V}}}{\alpha }$   ....... (V)

Here, ${\left({\alpha }_{\text{V}}\right)}_{\text{F}}$ is the fraction of volume of coefficient of expansion of gold and $\alpha$ is the actual coefficient of expansion of gold.

Calculation:

The fraction of volume of coefficient of expansion of gold is calculated as,

Substitute $1.84\times {10}^{-8}{\text{K}}^{-1}$ for ${\alpha }_{\text{V}}$, and $42.6\times {10}^{-8}{\text{K}}^{-1}$ for $\alpha$ in equation (V).

$\begin{array}{c}{\left({\alpha }_{\text{V}}\right)}_{\text{F}}=\frac{1.84\times {10}^{-8}{\text{K}}^{-1}}{42.6\times {10}^{-8}{\text{K}}^{-1}}\\ =0.04\end{array}$

Conclusion:

Therefore, fraction of volume coefficient of expansion for gold is $0.04$.