To solve this problem, it is necessary to take into account the concepts of the number of vacations N_v at a given temperature as well as the calculation of the number of atom sites per cubic meter.
For definition the number of atomic sites per cubic meter is given as,
[tex]N = \frac{N_A\rho}{A}[/tex]
Where,
[tex]N_A = 6.02210^{23}atoms/mol \rightarrow[/tex] Avogadro's number
[tex]\rho[/tex] density
A = Atomic weight
Replacing with our values we have
[tex]N = \frac{(6.02210^{23}atoms/mol)(6.25g/cm^3)(10^6cm^3/m^3)}{37.4g/mol}[/tex]
[tex]N = 1*10^{29}atoms/m^3[/tex]
At the same time we know that the number of vacancies [tex]N_v[/tex] is defined as,
[tex][tex]N_v = Ne^{-\frac{Q_v}{KT}}[/tex][/tex]
Where,
[tex]Q_v =[/tex] Energy of vacancy
[tex]K = 8.62*10^{-5} ev/Atom.K \rightarrow[/tex] Boltzman constant
T = Temperature
Replacing with the values given we have
[tex]N_v = (1*10^{29}atoms/m^3)e^{-\frac{1.22eV/atom}{(8.62*10^{-5}eV/atom.K)(1273K)}}[/tex]
[tex]N_v = 1.49*10^{24}m^{-3}[/tex]
Therefore the number of vacancies per cubit meter at 1000°C is [tex]N_v = 1.49*10^{24}m^{-3}[/tex]
