- #1

Dario56

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Stefan-Maxwell equations model diffusion in gases while Onsanger equations model mass transfer in electrolytes where ions can both diffuse and move in the electric field (migration). Both equations can be expressed in the same form, but I will concentrate on the Onsanger type as I'm more interested in it: $$c_i \nabla \bar \mu_i = \sum_j K_{ij} (v_j - v_i) \tag {1}$$

where ##\bar \mu_i## is electrochemical potential of ion ##i##, ##K_{ij}## is interaction coefficient between ions ##i## and ##j##, ##v## is a velocity of the corresponding ion and ##c_i$## is the concentration.

Electrochemical potential gradient takes into account both chemical and electric potential gradient and is therefore sufficient to explain mass transfer in electrolytes.

In the textbook Electrochemical Systems by Newman and Alyea, chapter 12.6: Multicomponent Transport, this equation is written in a bit of a different form on the grounds that in the system of ##n## ions, there are ##n-1## independent velocity differences ##v_j - v_i## or electrochemical potential gradients ## \nabla\bar \mu _i##.

While I understand that there are ##n-1## independent variables previously mentioned, I don't see why does that lead to the equation of this form: $$ c_i \nabla \bar \mu_i = \sum_j M_{ij}(v_j - v_0) \tag {2}$$

where ##v_0## is the velocity of any ion in the solution and ##M_{ij}## is a matrix connected to the interaction coeffiecient matrix ##K_{ij}## and defined as: $$M_{ij} = \begin{cases} K_{ij}, & i \neq j \\ K_{ij} - \sum_k K_{ik}, & i=j \end{cases}$$

Equation (2) is written without a derivation or much explanation and therefore I don't understand how did we get equation (2) from equation (1).