3. Transport in electrolyte solutions

3.5. Transport equation in a non-stationary state

The transport equations presented above apply only in steady state where the flux AJi (mol s-1) is not a function of the position but constant throughout the entire system; the flux density Ji (mol cm2 s-1) can vary depending on the cell geometry which is known as primary current distribution. In a non-stationary or transient state, solution concentrations vary as a function of location, and the flux is at its highest on the surface of the electrode. The non-stationary transport equation is reached calculating the divergence of the Nernst-Planck equation:

\displaystyle\frac{\partial c_i}{\partial t}=-\nabla \cdot\vec{J}_i=D_i\nabla^2c_i+\frac{z_iF}{RT}D_i(\nabla c_i \cdot\nabla\phi+c_i\nabla^2\phi)-(c_i\nabla \cdot\vec{v}+\vec{v} \cdot\nabla c_i) (3.52)

It has been assumed above that the diffusion coefficient is independent of the concentration. In electroneutral systems,  \nabla^2\phi=0 is known as the Laplace equation and its solutions as harmonic functions. Furthermore, in incompressible solutions \nabla \cdot\vec{v} =0 Therefore

\displaystyle\frac{\partial c_i}{\partial t}=D_i\nabla^2c_i+\frac{z_iF}{RT}D_i\nabla c_i \cdot \nabla \phi-\vec{v}\nabla c_i (3.53)

In the supporting electrolyte case which is overwhelmingly the most common in electrochemistry, \nabla\phi \approx\frac{i}{\kappa}=0 . Convection only concerns in hydrodynamic methods that will be discussed later. In the absence of convection, Equation (3.53) reduces to Fick’s second law:


\displaystyle\frac{\partial c_i}{\partial t}=D_i\nabla^2c_i

(3.54)

An electrochemical problem constitutes the solution of the above equation with the boundary condition (3.48) as well as other pertinent initial and boundary conditions, depending on the experimental method. These issues are addressed in Chapter 7.

 



You can now test your conceptual knowledge by taking Quiz Chapter 3.