Past exam of the chemistry course of the year 4 of the natural sciences course of the University of Cambridge / 2022 / Paper 1 / 13

One standard description of sodium-rich NaCl is removal of lattice chloride into a NaCl reservoir, leaving a positively charged chloride vacancy and an electron:This Kröger–Vink notation equation conserves atoms, chloride sites and effective charge. The electron can be trapped in the vacancy, forming an electrically neutral F center:Thus sodium excess introduces electronic defects and can give coloration. It does not imply that every added electron is a mobile conduction electron: electronic transport depends on trapping and thermal release. Chloride vacancies can also assist anion transport. An alternative microscopic incorporation route is a sodium interstitial plus an electron, but the vacancy–electron model describes the usual alkali-metal-colored halide.

The trivalent dopant is comparable in radius to Na${}^{+}$ and is naturally accommodated on a sodium site rather than a chloride site. One charge-balanced incorporation reaction isThree cation sites are filled by one Y ion and two vacancies; the three chloride ions occupy the corresponding anion sites. The effective $+2$ substitutional charge is balanced by two singly negative sodium vacancies.

YCl${}_3$ doping primarily increases sodium-vacancy-mediated ionic conductivity, provided the vacancies remain mobile rather than strongly bound to the dopant. In contrast, sodium-rich doping introduces electrons and anion vacancies or interstitial-related defects. Neither mechanism requires holes as its compensating carrier, and an increase in vacancy number need not translate proportionally into conductivity if defect association reduces mobility.

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.

A one-dimensional probability density has units $L^{-1}$, while $D$ has units $L^2{T}^{-1}$. With no other length scale, dimensional analysis requiresNormalization gives ${\int }_{-\infty }^{\infty }F(\xi )d\xi =1$ and unbiased diffusion gives an even $F$. Dimensional arguments alone do not specify $F$.

For ordinary Fickian diffusion, impose ${\partial }_{t}n=D{\partial }_x^{2}n$ with $n(x,0)=\delta (x)$. Substitution of the similarity form yieldsA decaying symmetric solution obeys $F^{\prime }=-\xi F/2$, so $F=C{e}^{-{\xi }^2/4}$. Normalization fixes $C=1/\sqrt{4\pi }$:This heat kernel integrates to one, has variance $2Dt$, and tends to the initial Dirac delta distribution as $t\downarrow 0$.