When a metal is heated, its volume expands while its mass remains constant, resulting in a decrease in density. This phenomenon can be attributed to two main factors: $1)$ the average bond length increases with temperature $($change in energy$),$ and $2)$ the number of vacancies within the material rises as temperature increases $($change in entropy$).$

Consider copper, with a density of $8\mathrm{.}92\frac{(g)}{c{m}^3}$ at $0C,$ heated to $1000C.$ Given that the vacancy formation energy for copper is $0\mathrm{.}91\frac{eV}{\text{a}\text{t}\text{o}\text{m}}$ and that copper will be treated as an isotropic material $($use$...$

$$ for the linear coefficient of thermal expansion$),$ complete the following:

Calculate the change in density of copper between $0C$ and $1000C$ due to thermal expansion.

Calculate the change in density of copper between $0C$ and $1000C$ due to the increase in the number of vacancies.

Which of the two phenomena $($thermal expansion or increase in vacancies$)$ is the primary driving force for the volume change of copper as it is heated? Justify your answer based on your calculations and understanding of the physical processes involved.

Remember the Volumetric Thermal Expansion equation is:

$V=V_{0}T$

Where:

$V-=$ change $in$ volume

$V_0^{}-=$ initial volume

$-=$ volumetric coefficient $of$ thermal expansion

$T-=$ change $in$ temperature

And, the Vacancy Concentration Equation is:

$N_v=N{e}^{(-\frac{Q_v}{kT})}$

Where:

$N_v-=$ number of vacancies

N $$ total number of available atomic sites

$Q_v$ vacancy formation energy

k $$ Boltzmann's constant

T $$ absolute temperature

Hint: the total number of atomic sites can be found using the materials density, molar mass, and Avogadro's number.