If the fraction of vacant sites at a temperature T1 is F1 and the fraction of vacant sites at a temperature T2 is F2, determine the relationship between T1, T2, F1, F2, Qv and k where Qv is the activation energy to form a vacancy and k is Boltzmann’s constant

F₁ / F₂ = exp (-Q_{v} / k)  exp [(T₂-T₁) / T₁T₂]

Q_{v} = -k ΔT / T₁T₂  ln (F₁ / F₂)

Explanation:

The Boltzmann equation for vacancies is

           n = n₀ exp (-Eₙ /kT)

Where n and n₀ are  the number of current and maximum vacancies, respectively, Eₙ is the activation energy (), K the Boltzmann constant and T the absolute temperature

The fraction is

         F = n / n₀ = exp (- Q_{v} / kT)

Let's apply this equation to our case

For temperature T = T₁

       F₁ = exp (-Qv / k T₁)

For temperature T = T₂

      F₂ = exp (-Qv / kT₂)

The ratio of the vacancy fraction is

     F₁ / F₂ = exp (-Q_{v} / k T₁) / exp (- Q_{v} / k T₂)

     F₁ / F₂ = exp (-Q_{v} / k)  exp [(T₂-T₁) / T₁T₂]

In general, the activation energy is sought by clearing this equation

     Q_{v} = -k ΔT / T₁T₂  ln (F₁ / F₂)

the bal will not change its velocity.