Answer:
the energy vacancies for formation in silver is ![\mathbf{Q_v = 3.069*10^{-4} \ J/atom}](https://tex.z-dn.net/?f=%5Cmathbf%7BQ_v%20%3D%203.069%2A10%5E%7B-4%7D%20%5C%20J%2Fatom%7D)
Explanation:
Given that:
the equilibrium number of vacancies at 800 °C
i.e T = 800°C is 3.6 x 10¹⁷ cm3
Atomic weight of sliver = 107.9 g/mol
Density of silver = 9.5 g/cm³
Let's first determine the number of atoms in silver
Let silver be represented by N
SO;
![N = \dfrac{N_A* \rho _{Ag}}{A_{Ag}}](https://tex.z-dn.net/?f=N%20%3D%20%20%5Cdfrac%7BN_A%2A%20%5Crho%20_%7BAg%7D%7D%7BA_%7BAg%7D%7D)
where ;
avogadro's number = ![6.023*10^{23} \ atoms/mol](https://tex.z-dn.net/?f=6.023%2A10%5E%7B23%7D%20%5C%20atoms%2Fmol)
= Density of silver = 9.5 g/cm³
= Atomic weight of sliver = 107.9 g/mol
![N = \dfrac{(6.023*10^{23} \ atoms/mol)*( 9.5 \ g/cm^3)}{(107.9 \ g/mol)}](https://tex.z-dn.net/?f=N%20%3D%20%20%5Cdfrac%7B%286.023%2A10%5E%7B23%7D%20%5C%20atoms%2Fmol%29%2A%28%209.5%20%5C%20g%2Fcm%5E3%29%7D%7B%28107.9%20%5C%20g%2Fmol%29%7D)
N = 5.30 × 10²⁸ atoms/m³
However;
The equation for equilibrium number of vacancies can be represented by the equation:
![N_v = N \ e^{^{-\dfrac{Q_v}{KT}}](https://tex.z-dn.net/?f=N_v%20%3D%20N%20%5C%20e%5E%7B%5E%7B-%5Cdfrac%7BQ_v%7D%7BKT%7D%7D)
From above; Considering the natural logarithm on both sides; we have:
![In \ N_v =In N - \dfrac{Q_v}{KT}](https://tex.z-dn.net/?f=In%20%5C%20N_v%20%3DIn%20N%20-%20%5Cdfrac%7BQ_v%7D%7BKT%7D)
Making
the subject of the formula; we have:
![{Q_v = - {KT} In( \dfrac{ \ N_v }{ N})](https://tex.z-dn.net/?f=%7BQ_v%20%3D%20%20-%20%7BKT%7D%20%20%20In%28%20%5Cdfrac%7B%20%5C%20N_v%20%7D%7B%20N%7D%29)
where;
K = Boltzmann constant = 8.62 × 10⁻⁵ eV/atom .K
Temperature T = 800 °C = (800+ 273) K = 1073 K
![Q _v =-( 8.62*10^{-5} \ eV/atom.K * 1073 \ K) \ In( \dfrac{3.6*10^{17}}{5.3 0*10^{28}})](https://tex.z-dn.net/?f=Q%20_v%20%3D-%28%208.62%2A10%5E%7B-5%7D%20%5C%20eV%2Fatom.K%20%2A%201073%20%5C%20K%29%20%5C%20In%28%20%5Cdfrac%7B3.6%2A10%5E%7B17%7D%7D%7B5.3%200%2A10%5E%7B28%7D%7D%29)
![\mathbf{Q_v = 2.38 \ eV/atom}](https://tex.z-dn.net/?f=%5Cmathbf%7BQ_v%20%3D%202.38%20%5C%20eV%2Fatom%7D)
Where;
1 eV = 1.602176565 × 10⁻¹⁹ J
Then
![Q_v = (2.38 \ * 1.602176565 * 10^{-19} ) J/atom }](https://tex.z-dn.net/?f=Q_v%20%3D%20%20%282.38%20%5C%20%2A%201.602176565%20%2A%2010%5E%7B-19%7D%20%29%20J%2Fatom%20%20%7D)
![\mathbf{Q_v = 3.069*10^{-4} \ J/atom}](https://tex.z-dn.net/?f=%5Cmathbf%7BQ_v%20%3D%203.069%2A10%5E%7B-4%7D%20%5C%20J%2Fatom%7D)
Thus, the energy vacancies for formation in silver is ![\mathbf{Q_v = 3.069*10^{-4} \ J/atom}](https://tex.z-dn.net/?f=%5Cmathbf%7BQ_v%20%3D%203.069%2A10%5E%7B-4%7D%20%5C%20J%2Fatom%7D)