Question

4.14 (a) Calculate the diffusion coefficient for oxygen ions in a pure ZrO₂ electrolyte at T = 1000^∘C given ΔGact = 100 kJ∕mol, ?₀ = 1013 Hz. ZrO₂ has a cubic unit cell with a lattice constant a = 5 Å and contains four Zr atoms and eight O atoms. Assume that the oxygen–oxygen “jump”distance Δx = 1 2 a. (b) Calculate the intrinsic carrier concentration in the electrolyte givenΔh? = 1 eV. (Assume vacancies are the dominant carrier.) (c) From your answers in (a) and (b), calculate the intrinsic conductivity of this electrolyte at 1000^∘C.

Answer 1

Given: Temperature, T=1000⁰C=1273.15 K

Activation barrier for hoping process, G_act=100 kJ/mol

Jump attempt frequency, v₀=10¹³ Hz=10¹³ s^-1

Lattic constant for ZrO₂, a=5=5*10^-8 cm

Oxygen-oxygen jump distance, x, is

(a) Diffusion coefficine for oxygen atom, D, is obtained using the expression

where x is the Oxygen-oxygen jump distance

v₀ is the Jump attempt frequency

G_act is the Activation barrier for hoping process

R is the Gas constant, R=8.314 J/mol K

T is the Temperature

Thus the diffusion coefficient for oxygen in pure ZrO₂ electrolyte is D=2.465*10^-7 cm²/s

(b) For vacancies as the dominant carrier, the intrinsic carrier concentration, c_V, is

where x_V is the Vacancy fraction

c₀ is the Concentration of oxygen sites

Vacancy fraction, x_V, is

where h_v is the Enthalpy of vacancy. Given that h_v=1 eV=1.602*10^-9 J

k is the Boltzmann's constant, k=1.3807*10^-23 J/K

Concentration of oxygen sites, c₀, is

where n is the Number of oxygen atoms per unit cell. In this case, n=8 atoms

a is the Lattice constant

Convert atoms/cm³ to mol/cm³ by dividing it with Avogadro's number, N_A=6.022*10²³ atoms/mol

Thus the intrinsic carrier concentration is

Thus the intrinsic carrier concentration is c_V=1.115*10^-3 mol/cm³

(c) The intrinsic conductivity of the electrolyte, , is

where z is the Charge of the vacancy. Here z=2 which is the opposite charge of the ion O^2-.

F is the Faraday constant, F=96485 C/mol

c is the Intrinsic carrier concentration

D is the Diffusion coefficient

(Units: C is A s and J is kg m²/s²)

(Units: kg m²/A²s³ is ohm, )

Thus the conductivity of the electrolyte is =9.67*10^-4 ( cm)^-1