Question

Calculate the number of vacancies per cubic meter at 1000°C for a metal that has an energy for vacancy formation of 1.22 eV/atom, a density of 6.25 g/cm3, and an atomic weight of 37.4 g/mol.

Answer 1

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,

$N = \frac{N_A\rho}{A}$

Where,

$N_A = 6.02210^{23}atoms/mol \rightarrow$ Avogadro's number

$\rho$ density

A = Atomic weight

Replacing with our values we have

$N = \frac{(6.02210^{23}atoms/mol)(6.25g/cm^3)(10^6cm^3/m^3)}{37.4g/mol}$

$N = 1*10^{29}atoms/m^3$

At the same time we know that the number of vacancies $N_v$ is defined as,

$[tex]N_v = Ne^{-\frac{Q_v}{KT}}$[/tex]

Where,

$Q_v =$ Energy of vacancy

$K = 8.62*10^{-5} ev/Atom.K \rightarrow$ Boltzman constant

T = Temperature

Replacing with the values given we have

$N_v = (1*10^{29}atoms/m^3)e^{-\frac{1.22eV/atom}{(8.62*10^{-5}eV/atom.K)(1273K)}}$

$N_v = 1.49*10^{24}m^{-3}$

Therefore the number of vacancies per cubit meter at 1000°C is $N_v = 1.49*10^{24}m^{-3}$