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3. Transport in electrolyte solutions

3.3. Transport equation in a steady state

Transport processes can be approached in various theoretical ways. The most traditional theory is thermodynamics of irreversible processes. In this book we use the most practical means, the Nernst-Planck equation. We bypass the simplifying approximation with which the Nernst-Planck equation can be derived from the theory of irreversible thermodynamics. The Nernst-Planck equation expresses an ionic flux, Jk (mol cm2 s-1), as the sum of three contributions:

\( \displaystyle\vec{J}_k=-D_k\nabla c_k-\frac{z_kF}{RT}D_kc_k\nabla \phi+\vec{v}c_k \) (3.33)


The first term of Equation (3.33) relates to diffusion, the second one to migration and the third one to convection. The first term alone is equal to Fick’s first law stating that concentration differences tend to level out due to Brownian movement. The second term describes the movement of a charged species in an electric field, in other words electrophoresis. The velocity \( \vec{v} \) found in the convection term is practically equal to the solvent velocity. Convection comes from stirring the solution (forced convection) or density gradients formed in the solution (natural convection).

In the bulk solution no concentration gradients are usually found and, therefore, transport takes place only as a result of migration and convection or, in the absence of stirring, plain migration. In the polarization layer, diffusion plays the main role because, depending on the viscosity of the solvent, an unstirred stagnant hydrodynamic layer is formed on the surface of the electrode. Inert electrolytes that do not react on the electrode are usually added in an electrochemical cell. Such an electrolyte is commonly called as a supporting electrolyte because it carries most of electric current. As a consequence, the flux of an electroactive species is in practice solely diffusion; more detailed evidence is provided later on. In the absence of a supporting electrolyte, the mathematical analysis of transport becomes rather complex and the ohmic losses in the cell may grow in an uncontrollable way.

Equation (3.34) can be written in a compact form with the gradient of electrochemical potential:

\( \vec{J}_k=-L\nabla \tilde{\mu}_k \), (3.34)

where L is known as the phenomenological coefficient. Inserting the expression of \( \tilde{\mu}_k \), Equation (2.7) here, the following is obtained
\(\displaystyle \vec{J}_k=-L(RT\nabla \text{ln}c_k+RT\nabla \text{ln} \gamma_k+z_kF \nabla \phi) \) (3.35)

Comparing the above equation with the Nernst-Planck equation it is seen that [1]

\( \displaystyle L=\frac{D_kc_k}{RT} \) (3.36)

\(\displaystyle \vec{J}_k=-D_k \nabla c_k(1+\frac{\partial \text{ln} \gamma_k}{\partial \text{ln} c_k})-\frac{z_kF}{RT}D_kc_k\nabla \phi \) (3.37)

If the dependence of the activity coefficient on concentration is omitted, the first two terms of the Nernst-Planck equation are thus obtained. The convection term comes from the notion that flux can always be written in the form \( \vec{J}_k=c_k \vec{v}_k \) and interpreting \( \vec{v} \) as the solvent velocity.

 

 


[1] Equation (3.36) does not apply in general. See, e.g. Kontturi et al., Ionic Transport Processes, Oxford University Press, 2nd Ed. 2015.


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