Question

Calculate the energy for vacancy formation in nickel (ni), given that the equilibrium number of vacancies at 850°c (1123 k) is 4.7 × 1022 m–3. the atomic weight and density (at 850°c) for ni are, respectively, 58.69 g/mol and 8.80 g/cm3.

Answer 1

The energy for vacancy formation$E_{v}$ can be calculated as:

$N_{v}=Ne^{-\frac{E_{v}}{kT}}$

Here, $N_{v}$ 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:

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

Here, ρ is density, $N_{A}$ is Avogadro's number and A is atomic weight.

Putting the values,

$N=\frac{8.80 g/cm^{3}(6.023\times 10^{23} mol^{-1})}{58.69 g/mol}=9.03\times 10^{22} cm^{-3}$

Converting $cm^{-3}$ to $m^{-3}$

Since, 1 $cm^{-3}$ =$10^{-6} m^{3}$

Thus, $9.03\times 10^{22} cm^{-3}=9.03\times 10^{28} m^{-3}$

Now, the energy for vacancy formation$E_{v}$ at 850 °C or 1123 K can be calculated using the following equation:

$N_{v}=Ne^{-\frac{E_{v}}{kT}}$

Rearranging,

$E_{v}=kTln\frac{N}{N_{v}}$

Putting the values,

$E_{v}=(1.38\times 10^{-23} J/K)(1123 K)ln\frac{(9.03\times 10^{28} m^{-3})}{(4.7\times 10^{22}m^{-3})}=2.23\times 10^{-19}J$

Therefore, energy for vacancy formation in nickel is $2.23\times 10^{-19}J$