Diffusion in ionic crystals.

MX is an ionic crystal that forms defects by creating Schottky pairs $($the cation $M$ and the anion $x$ are

univalent, as in NaCl $).$ The diffusivities of the anion vacancies and cation vacancies are given by:

$D_{av}=1{10}^{-8}$exp$(-\frac{E_1}{R}T)\frac{m^2}{s}$ and $D_{cv}=2{10}^{-9}$exp$(-\frac{E_2}{R}T)\frac{m^2}{s}$

respectively, with $E_1=200k\frac{J}{m}ol$ and $E_2=150k\frac{J}{m}ol.$

$($a$)$ The equilibrium cation and anion vacancy molar fractions, $x_{cv}$ and $x_{av}$ respectively, are $x_{cv}=x_{av}$

$=1{10}^{-5}at1500K.$ Calculate the diffusion coefficients of $M$ and $x$ at this temperature.

$($b$)$ From the equilibrium vacancy concentrations given in $($a$),$ determine the Gibbs free energy for

the formation of Schottky pairs $($hint: write a simple "chemical reaction" equation for the formation of a

Schottky pair, and write the equilibrium rate constant for that reaction$).$

$($c$)Mx$ is doped with $2at.\%Ca{x}_2,$ with the goal of increasing its ionic conductivity. Using the

constraint of charge neutrality and the data given in the previous questions, calculate $x_{av}$ in this doped

MX ionic crystal at $1500K.$

$($d$)$ Calculate the ratio of the cation conductivity for the doped and undoped crystals, assuming that

this conductivity is proportional to the cation vacancy concentration.