The energy for vacancy formation[tex]E_{v}[/tex] can be calculated as:
[tex]N_{v}=Ne^{-\frac{E_{v}}{kT}}[/tex]
Here, [tex]N_{v}[/tex] is equilibrium number of vacancies, N is number of atomic sites per unit vacancies, k is Boltzmann constant, T is temperature.
Here, number of atomic sites per unit vacancies can be calculated as follows:
[tex]N=\frac{\rho N_{A}}{A}[/tex]
Here, ρ is density, [tex]N_{A}[/tex] is Avogadro's number and A is atomic weight.
Putting the values,
[tex]N=\frac{8.80 g/cm^{3}(6.023\times 10^{23} mol^{-1})}{58.69 g/mol}=9.03\times 10^{22} cm^{-3}[/tex]
Converting [tex]cm^{-3}[/tex] to [tex]m^{-3}[/tex]
Since, 1 [tex]cm^{-3}[/tex] =[tex]10^{-6} m^{3}[/tex]
Thus, [tex]9.03\times 10^{22} cm^{-3}=9.03\times 10^{28} m^{-3}[/tex]
Now, the energy for vacancy formation[tex]E_{v}[/tex] at 850 °C or 1123 K can be calculated using the following equation:
[tex]N_{v}=Ne^{-\frac{E_{v}}{kT}}[/tex]
Rearranging,
[tex]E_{v}=kTln\frac{N}{N_{v}}[/tex]
Putting the values,
[tex]E_{v}=(1.38\times 10^{-23} J/K)(1123 K)ln\frac{(9.03\times 10^{28} m^{-3})}{(4.7\times 10^{22}m^{-3})}=2.23\times 10^{-19}J[/tex]
Therefore, energy for vacancy formation in nickel is [tex]2.23\times 10^{-19}J[/tex]
