Past exam of the chemistry course of the year 4 of the natural sciences course of the University of Cambridge / 2022 / Paper 3 / Section D / 43 / b / i / Solution

Here $\mathbf{j}$ is electric current density, $n$ the number density of independently mobile ions, $Q$ their charge, $T$ absolute temperature and $k_B$ the Boltzmann constant. The sum runs over all nearest-neighbour destinations $j$ of site $i$, and ${\mathbf{r}}_{ij}$ is the displacement from $i$ to $j$. The applied field is $\mathcal{E}$, ${\omega }_0$ is an attempt frequency, and $\Delta E_{ij}$ is the zero-field reaction energy barrier for the indicated jump. With the normalization of the supplied equation,It has units s${}^{-1}$; it is not an equilibrium occupation probability. The current formula is the linear-response result of a small field-induced asymmetry in forward and reverse rates, requiring $|Q{\mathbf{r}}_{ij}\cdot \mathcal{E}|\ll k_{B}T$. Use its explicit factor $1/2$ consistently rather than absorbing it into a redefined jump rate.