= Solution
For an oxide <ion>, or an <oxygen vacancy> measured relative to the ideal lattice, the charge magnitude is $|q|=2e$. The independent-carrier Nernst-Einstein inversion at $1250\,\mathrm K$ gives
$$
\boxed{n_{\rm eff}=\frac{\sigma k_BT}{(2e)^2D}
=6.569\times10^{27}\,\mathrm{m^{-3}}.}
$$
Here $D$ and $n_{\rm eff}$ must describe the same carrier convention. If the quoted diffusivity is a mobile-vacancy diffusivity, this is the vacancy <density>. If it is literally an oxygen-ion tracer diffusivity, it is an effective oxygen-ion <density> under an uncorrelated model. Vacancy <density> cannot be substituted for oxygen-ion <density> while retaining the same diffusivity. This distinction is consequential in the last part.
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