Question

Calculate the equilibrium number of vacancies per cubic meter in gold(Au) at 25(celcius). For gold, the energy for vacancy formation is 0.98 eV/atom. Furthermore, the atomic mass and the theoretical density for gold are 196.9 g/mol and 19.32 g/cm^3(at 25 celcius), respectively.

Answer 1

Given, = 19.32 g /cm³ , atomic weight = 196.9 g/mol

Now, N = N_A / atomic weight

=> N = 6.022 x 10²³ atoms /mol * 19.32 g/cm³ / 196.9 g/mol

=> N = (116.34504 x 10²³/196.9) atom/cm³

=> N = 5.91 x 10^-1 x 10²³ atom / cm³

=> N = 5.91 x 10²² atom/cm³

Also, given that Q_v = 0.98 ev / atom , T = 25⁰C =( 25 +273 ) K = 298 K

We know, K= 8.62 x 10^-5 ev / atom K

Now, Number of vacancies, N_v = N exp (-Q_v / KT)

=> N_v = 5.91 x 10²² atom/cm³ * exp ( -0.98 ev/atom / 8.62 x 10^-5 ev/atom.K * 298 K )

=> N_v = 5.91 x 10²² * exp ( -0.98 / 0.0256876) vacancies / cm³

=> N_v = 5.91 x 10²² * exp ( - 38.15070306)

=> N_v = 5.91 x 10²² x 2.69 x 10^-17

=> N_v = 15.89 x 10⁵ vacancies/ cm³

=> N_v = 15.89 x 10⁵ x 10⁶ vacancies /m³

=> N_V = 15.89 x 10¹¹ vacancies /m³