Consider the transport of species $i$ in a material, and assume that there are no

other forces applied $($e$.$g$.,$ gradients in temperature or electrical potential$).$ Derive

the Nernst$-$Einstein equation that relates the mobility $M_i$ of $i$ to its diffusivity $D_i$ :

$D_i=M_{i}kT$

where $k$ is the Boltzmann constant and $T$ is temperature.

Hints: Invoke Fick's first law. Assume Raoultian behavior. It is convenient in

this case to use the definition of chemical potential per atom,

${}_i-=G_i^0+kTln{}_i{x}_i$

which would imply that the reference state $(G_i^0)$ is also on a per$-$atom basis. The

mole fraction $x_i$ above is related to concentration as

$x_i=($:$$:$)c_i$

where $($:$$:$)$ is the average atomic volume $($see also Eq$.$ A$.12$ in BAC$).$