# Why can we use the arrhenius equation to measure the number of vacancies in a material?

In a materials science video I was watching today the prof. said we could use the arrhenius equation to estimate the number of vacancy point defects in a material. And gave the following equation:

Nv = N exp(-Qv / KBT)

Where Nv = number of vacancies

N= # of atom sites

Qv = energy of vacancy formation

KB = Boltzmann constant

T = Temp

I don't understand how we got to that equation from the Arrhenius equation. How do the two relate? How is the above equation derived from the Arrhenius equation?

• I'll try a heuristic undetailed explanation. The formation of vacacies has a cost in enthalpy (because atoms are no longer tightly bonded), although it naturally increases entropy (because there are more possible microstates). It follows that there is a certain Gibbs energy (G = H - TS) that induces an equlibrium, where G is minimized. In chemical equlibrium, G is associated with a given K (equilibrium constant), which is temperature-dependent according the Van't Hoff equation (which can be derived from basic thermodynamics), that is similar to the Arrhenius equation. Sep 8, 2022 at 23:33
• This looks like the Boltzmann equation, perhaps Arrhenius was a slip of the tongue. Sep 9, 2022 at 14:38