 Solution

The diffusion coefficient relates particle flux to concentration gradient through Fick's first law, $\mathbf{j}=-D\nabla c$. For unbiased isotropic motion in $d$ dimensions it can equivalently be defined by the long-time mean squared displacement:An uncorrelated random walk with jump length $a$ and total jump rate $\Gamma$ has $⟨r^2⟩=\Gamma a^{2}t$, hence $D=\Gamma a^2/(2d)$; correlations introduce a multiplicative correlation factor. The factor $d$ counts the directions sharing the total displacement. It does not mean that the same material's single-coordinate diffusivity changes merely by choosing to observe fewer coordinates: $⟨x^2⟩=2Dt$ in every Cartesian direction of the same isotropic system.

At $500$ K, silver iodide is in its fast-ion-conducting $\alpha$ phase. For an order-of-magnitude estimate, take a jump length $a\sim 3$ Å and an effective successful jump rate $\Gamma \sim 10^{11}{\mathrm{s}}^{-1}$. ThenThus $D$ of order $10^{-9}{\mathrm{m}}^2{\mathrm{s}}^{-1}$ is a reasonable estimate for this superionic conductor. The length and rate are explicitly assumed material scales; the question supplies neither a measured conductivity nor jump data, so an exact numerical coefficient cannot be deduced from temperature alone. The estimate is not presented as an experimentally measured value.