Equilibrium: no electric current, no spatial nor temporal changes in concentrations. Nernst equation applies Transport equations not needed, c ≠ c(x,y,z,t), Stationary state: electric current is constant, no temporal changes in concentrations.  2 c = 0, I ≠ I(t), c ≠ c(t), c = c(x,y,z) At the electrode, either Nernst equation or a kinetic equation (Butler-Volmer) as the boundary condition of the transport equation. Transient: concentrations change as a function of both time and position. Fick’s 2.law, I = I(t), c = c(x,y,z,t), In addition to boundary conditions, also initial conditions, c(x,y,z,t=0) = c 0. Electrochemical problems Obs.: Electric current is continuous everywhere, I ≠ I(x,y,z)

Transport equations Phenomenological equation, L k = phenomenological coefficient Nernst-Planck equation diffusion + migration + convection The driving force of transport is the gradient of the electrochemical potential. X non-physical!

Convection term in Nernst-Planck equation means a change from Hittorf reference system to a fixed laboratory coordinates; velocity is due to mechanical forces and has to be calculated using Navier-Stokes equation. Hittorf: solvent (water) velocity Laboratory coordinates: All the transport quantities (diffusion coefficient, transport number, conductivity) are defined only in Hittorf reference, but experiments are done in fixed laboratory coordinates. Strictly speaking, phenomenological equations should be written in the center-of-mass coordinates (barycentric reference), but in practice it is the same as the Hittorf reference. Solution flow is due to recycling, viz pumping, (electro)osmosis or stirring. Calculating it can be very difficult, because Navier-Stokes equation can be solved in closed form only in some special cases.

Do activity coefficients matter? As can be seen from the figure above, in a binary system an error can be as high as 10 %. Since in multicomponent systems the estimation of activity coefficients is, however, very difficult, they are usually ignored. Yet, in the derivation of Nernst-Planck equation a number of simplifying assumptions have been made, such as ignoring the mutual coupling of ionic fluxes; hence, ignoring the activity coefficient term is not that serious. Also, Nernst-Planck equation has proved to be very useful and workable. More rigorous approaches are hopelessly complicated and include several parameters the values of which are unknown. Figure: Mean acitivity coefficient of NaCl in the concentration range of 5 mM - 1 M. A single ionic diffusion coefficient cannot be measured, but it must be estimated with a non-thermodynamic procedure.

Analysis of transport equations Each species has its own Nernst-Planck equation. Constraints: electroneutrality electric current density Binary system = salt + water: c + =  + c ja c  =   c  z +  + + z    = 0 +

D ± = salt diffusion coefficient Nernst-Hartley equation D ±, t + ja t − are concentration independent constants, which apples only in binary systems! They can be measured, contrary to ionic diff. coefficients. Transport number: Q.E.D. t + + t − = 1 or in general

1:1 electrolyte (e.g. NaCl, c + = c − = c): Ionic diffusion coefficient can be calculated by measuring the salt diffusion coefficient and the transport number in the binary system: ja 1:1 electrolyte 2:1 electrolyte (CaCl 2 ) 1:2-electrolyte (Na 2 SO 4 )

Potential gradient in the solution in a general case: Nernst-Planck equation: Multiply with z i and sum: concentration dependent transport number

ohmic loss, irreversible, dissipates into heat diffusion potential, reversible, usually very low, as all ionic mobilities are of the same order of magnitude (except H + and OH − ) In electrochemical experiments, an inert electrolyte that does not react on the electrode is usually added in excess for two reasons: conductivity increases, making ohmic loss insignificant transport number of the electroactive ion reduces practically to zero, reducing N-P equation to Fick’s law, see below. A transport number can formally be written also as (no convection) if, again, ionic coupling is ignored. But since the transport number depends on concentrations and changes, therefore, as a function of position, the equation above cannot be solved. Since for a trace-ion t k ≈ 0, its N-P equation becomes Fick’s law:

} = 0 Time dependent transport equation: Poisson equation:  2  = −  e /  ;  e = charge density,  = permittivity =  r  0. In an electro- neutral system  e = 0. In an incompressible fluid  ∙v = 0. If it is arranged so that  ≈ 0 ja v = 0 Fick’s 2. law For a trace-ion, the above equation can be solved using the boudary condition Additionally, on the electrode, either Nernst equation (reversible reaction) or a kinetic equation is needed.