# Origin of terms in the Nernst-Planck equation

We know the Nernst Planck equation is $$\frac{\partial c}{\partial t} = - \nabla \cdot J \quad | \quad J = -\left[ D \nabla c - u c + \frac{Dze}{k_\mathrm{B} T}c\left(\nabla \phi+\frac{\partial \mathbf A}{\partial t}\right) \right]$$ $$\iff\frac{\partial c}{\partial t} = \nabla \cdot \left[ D \nabla c - u c + \frac{Dze}{k_\mathrm{B} T}c\left(\nabla \phi+\frac{\partial \mathbf A}{\partial t}\right) \right]$$ We know this is basically a continuity equation where a partial of a conserved density quantity is equal to the divergence of its flux. The first term is derived from Ficks Law, the second term is simply a convection term flux=velocity*density and the final term is due to Einstein's Relation stating $$D = \mu \, k_\text{B} T$$ Where μ is the "mobility", or the ratio of the particle's terminal drift velocity to an applied force, μ = vd/F and we know F=EQ and V=ED so the third term is derived from these equations. I am having trouble with the modified Nernst Planck equation for porous media where we have $$\nabla (\kappa_{eff}\:\ln C)=\mbox{Current Density}$$ where $$\kappa_{eff}=\frac{2RT\kappa}{F}\left(t_+-1\right)\left(1+\frac{d \ln f}{d \ln C}\right)=D\left(1+\frac{d \ln f}{d \ln C}\right)$$ Where does this term come from? Logarithm of the activity would imply a chemical potential but how is this related to diffusion?

I see now that for Chemical Systems of non-ideal solutions/mixtures Ficks First Law is: $$J_i = - \frac{D c_i}{RT} \frac{\partial \mu_i}{\partial x}$$ This is a restatement of the First Law as $$J_i = - \frac{D c_i}{RT} \frac{\partial \mu_i}{\partial x}= - \frac{D c_i}{RT} \frac{RT\partial \ln C}{\partial x} \implies d \ln C =\frac{dC}{C}\implies J_i = - D \frac{\partial C}{\partial x}.$$
The Nernst-Planck equation assumes an ideal thermodynamic behavior of ions, i.e., unit activity coefficients. Additionally, it uses self diffusivities, $$D$$, to describe the response of ions to an applied electric field.
A generalized treatment of ion motion employs Stefan-Maxwell diffusivities, $$\mathfrak{D}_{ij}$$, and non-ideal energies of the ions, i.e., $$f \left(c\right) \neq 1$$. This leads to concentrated solution theory (https://onlinelibrary.wiley.com/doi/abs/10.1002/bbpc.19650690712). In this description, the total ionic current density is defined as $$$$i = - \kappa \left( \nabla \phi - \frac{2RT}{F} \left(1-t_+^0\right) \left(1 + \frac{\mathrm{d} \ln f}{\mathrm{d} \ln c}\right) \nabla \ln c \right)$$$$. Note that this equation is only valid for an electrolyte with a monovalent cation, monovalent anion, and a solvent. One can derive equivalent expressions for other electrolytes.