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^{-(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=(\rho N_(A))/(A)`

Here, ρ is density,
`N_(A)` is Avogadro's number and A is atomic weight.

Putting the values,

`N=(8.80 g/cm^(3)(6.023* 10^(23) mol^(-1)))/(58.69 g/mol)=9.03* 10^(22) cm^(-3)`

Converting
`cm^(-3)` to
`m^(-3)`

Since, 1
`cm^(-3)` =
`10^(-6) m^(3)`

Thus,
`9.03* 10^(22) cm^(-3)=9.03* 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^{-(E_(v))/(kT)}`

Rearranging,

`E_(v)=kTln(N)/(N_(v))`

Putting the values,

`E_(v)=(1.38* 10^(-23) J/K)(1123 K)ln((9.03* 10^(28) m^(-3)))/((4.7* 10^(22)m^(-3)))=2.23* 10^(-19)J`

Therefore, energy for vacancy formation in nickel is
`2.23* 10^(-19)J`