The equilibrium number of vacancies (N_v) for a volume of material is given by t

Question

The equilibrium number of vacancies (N_v) for a volume of material is given by the following, N_v = N e^(-Q_v/KT). Where N is the number of atomic sites, Q_v is the energy required for vacancy formation, T is absolute temperature and k is Boltzmann's constant (8.62 Times 10^-5 eV/atom K). What is the ratio of the number of vacancies for copper at 1000 degree C over copper at 0 degree C (N_vcu @1ooo degree c/N_vcu @o degree c)? Energy required for vacancy formation is 0.9 eV/atom; atomic weight is 63.5 g/mol; p_cu is 8.4 g/cm^3; Avogadro's number is 6.023 Times 10^23atoms/mole (you can use density, atomic weight and Avogadro's number to get N). Remember that for temp K = 273 + degree C.

Explanation / Answer

Given

Nv=N exp(-Qv/kT)

N is number of atomic sites ,this is constant and doesnt vary with temperature

So

Nv@1000C/Nv@0C=(exp(-Qv1/kT1))/(exp(-Qv2/kT2))

Qv1 and Qv2 are energy required for vacancy formations which are constant with temperature

k is boltzmann constant

so

Nv@1000C/Nv@0C=(exp(-0.9/8.62*10^-5*1273))/(exp(-0.9/8.62*10^-5*273))

Nv@1000C/Nv@0C=1.32*10-4/8.2*10-19=1.6*10^14

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