$9\mathrm{.}8$ An oxide compound, AO$,$ undergoes nonstoichiometry at certain oxygen partial pressures

by the reaction:

$2A_A+O_O=\frac{1}{2}{O}_2(g)+V_O^{\text{@}{?}^{}}+2A_A^{\text{'}}$;$G_R=4\mathrm{.}0eV$

Of course, the $A_A^{\text{'}}$ ion is an $A^{2+}$ ion reduced to an $A^{+}$ion in Kr$$ger$-$Vink notation. Also, for a

Schottky defect in AO:

$0=V_A^{\text{'}\text{'}}+V_O$;$G_S=2\mathrm{.}1eV.$

Because of three possible defects, the total electrical neutrality condition is

$2{}_{v_O^{**}}=2{}_{v_A^n}+{}_{A_A^{\text{'}}}$

But because ${}_A={}_O,$ so $2[V_O^{**}]=2[V_A^{\text{'}\text{'}}]+[A_A^{\text{'}}].$ This implies that two concentration ranges are

possible, the stoichiometric range where $[V_O^{**}]$~$=[V_A^{\text{'}\text{'}}]$ and the $A_A^{\text{'}}$ can be neglected and a

nonstoichiometric range where $2[V_O^{**}]$~$=[A_A^{\text{'}}]$ and $V_A^{\text{'}\text{'}}$ can be neglected.

a$.$ Calculate the vacancy concentration in the stoichiometry range where $[V_O^{**}]$~$=[V_A^{\text{'}\text{'}}]$ at

$1000C.$

b$.$ Calculate $A_A^{\text{'}}$ in the stoichiometric range when $p_{O_2}={10}^4$ bar at $1000C.$

c$.$ From parts $($a$)$ and $($b$),$ calculate the nonstoichiometry at $p_{O_2}={10}^4$ bar at $1000C$; i$.$e$.x$

in $A{O}_{1-x}.$

d$.$ In the nonstoichiometry range, $2[V_O^{**}]$~$=[A_A^{\text{'}}],$ calculate $V_O^{**}$ and $x$ at $p(O_2)={10}^{-14}$ bar

and $1000C.$

e$.$ Calculate all of the vacancy concentrations, $x,$ and the oxygen pressure when $2[V_A^{\text{'}\text{'}}]$~$=[A_A^{\text{'}}].$