Nemstian diffusion

The general theoretical treatment of ion-selective membranes assumes a homogeneous membrane phase and thermodynamic equilibrium at the phase boundaries. Obvious deviations from a Nemstian behavior are explained by an additional diffusion potential inside the membrane. However, allowing stationary state conditions in which the thermodynamic equilibrium is not established some hitherto difficult to explain facts (e.g., super-Nemstian slope, dependence of the selectivity of ion-transport upon the availability of co-ions, etc.) can be understood more easily. 

Recall that Nemstian behavior of diffusing species yields a r1 /2 dependence, hi practice, the ideal behavior is approached for relatively slow scan rates, and for an adsorbed layer that shows no intermolecular interactions and fast electron transfers. 

Irreversibility versus reversibility inpolarography. Previously in this chapter we dealt only with reversible redox systems, i.e., with truly Nemstian behaviour and merely diffusion control. This also applies to combined processess of electron transfer and chemical reaction (e.g., complexation) provided that both take place instantly. For instance, in EC such as... 

As compared to the Nemstian case, the plateau is the same but the wave is shifted toward more negative potentials, the more so the slower the electrode electron transfer. An illustration is given in Figure 4.13 for a value of the kinetic parameter where the catalytic plateau is under mixed kinetic control, in between catalytic reaction and substrate diffusion control. For the kjet(E) function, rather than the classical Butler-Volmer law [equation (1.26)], we have chosen the nonlinear MHL law [equation (1.37)]. 

In the framework of Scheme 2.1, we start with the case where the electron transfer does not interfere kinetically. As compared to the simple Nemstian electron transfer case (Section 6.1.2), the main change occurs in die partial derivative equation pertaining to B, where a kinetic term is introduced in Fick s second law. A corresponding equation for C should also be taken into account, leading to the following system of partial derivative equations, accompanied by a series of initial and boundary conditions (assuming that the diffusion coefficients of A, B, and C are the same) ... 

First consider the system in which no diffusion potential is formed in the membrane. The membrane potential is then determined by the conditions at the membrane/aqueous electrolyte solution boundary. In the simplest situation, a salt of a monovalent ion-exchanger ion, anion A", with monovalent determinand cation J is dissolved in the membrane. In order for this system to be the basis for a usable ISE with Nemstian response to the determinand ion in a sufficiently broad activity interval, it is necessary that the distribution coefficient kj be... 

The diffusion of the electroactive ions is both physical and due to electron transfer reactions.45 The occurrence of either or both mechanisms is a function of the electroactive species present. It has been observed that the detailed electrochemical behaviour of the electroactive species often deviates from the ideal thin film behaviour. For example, for an ideal nemstian reaction under Langmuir isotherm conditions there should be no splitting between the anodic and cathodic peaks in the cyclic voltammogram further, for a one-electron charge at 25 °C the width at half peak height should be 90.6 mV.4 In practice a difference between anodic and cathodic potentials may be finite even at slow scan rates. This arises from kinetic effects of phase formation and of interconversion between different forms of the polymer-confined electroactive molecules with different standard potentials.46... 

The mass transport coefficient is, in general, a complex time and potential-dependent function through the linear diffusion layer thickness, <% ,. Only under certain conditions does this dependence disappear (as, for example, for nemstian... 

Under reversible conditions and electrode geometries different from the planar one when Do / DR, the mass transport coefficient is a complex function of both diffusion coefficients and, in general, the expressions of nii and mi oo do not coincide. The only case in which m-x =m (Xl when the diffusion coefficients of species O and R are assumed as different is that corresponding to nemstian processes under planar diffusion (macroelectrodes). 

Equation (2.11) refers to the flux conservation and Eq. (2.12) to the establishment of the nemstian equilibrium. Note that under these conditions, the original problem in terms of variables x and t has been transformed into a one-variable problem (s ), that is, c0 and cR can be expressed only as functions of the variables o and sR, respectively (which include distance and time variables), because they diffuse with different diffusion coefficients D0 and DR. This problem can now be solved by making = dci/dsf, and Eq. (2.9) becomes... 

It is important to note that Eq. (3.38) obtained with this treatment (based on the consideration that the diffusion layer corresponding to a non-nemstian electrode process coincides with that corresponding to a reversible one) can also be directly deduced from Eq. (3.17) if Fix) is replaced by its simplified form given by Eq. (E.10) (which is valid for x < 0.185 and x > 19.7 with an error smaller than 5 % see Appendix E) ... 

Note that the usual definition of the mass transfer coefficient is related to limiting diffusion conditions or nemstian conditions (mo, no = yDo/nt for a planar electrode see Sect. 1.8.4). The definition given in Eq. (3.43) is general for any reversibility degree of the electrode process at planar electrodes. 

Equation (5.62) for the current-potential response in CV has been deduced by assuming that the diffusion coefficients of species O and R fulfill the condition Do = >r = D. If this assumption cannot be fulfilled, this equation is not valid since in this case the surface concentrations are not constant and it has not been possible to obtain an explicit solution. Under these conditions, the CV curves corresponding to Nemstian processes have to be obtained by using numerical procedures to solve the diffusion differential equations (finite differences, Crank-Nicholson methods, etc. see Appendix I and ([28])3. 

For nonplanar electrodes there are no analytical expressions for the CV or SCV curves corresponding to non-reversible (or even totally irreversible) electrode processes, and numerical simulation methods are used routinely to solve diffusion differential equations. The difficulties in the analysis of the resulting responses are related to the fact that the reversibility degree for a given value of the charge transfer coefficient a depends on the rate constant, the scan rate (as in the case of Nemstian processes) and also on the electrode size. For example, for spherical electrodes the expression of the dimensionless rate constant is... 

Equation (A.2.5) indicates that the establishment of the Nemstian concentrations (0 at the electrode requires that mass transfer of analyte to and from the electrode be controlled and limited by diffusion due to a concentration difference only, called complete concentration polarization. Once the analyte reaches the electrode surface, the rate of electron transfer must be rapid (i.e., mass transfer not electron transfer limits the rate of... 

However, one should not forget that the potentiometric mode has several drawbacks. The fabrication of most potentiometric tips is much more involved than that of the conventional amperometric tips. It is necessary to have the electrode made by a very skilled technician. Despite this, even when following a proven recipe the success rate is relatively low. The response of potentiometric tips is not always Nemstian and a calibration is required before and after performing the experiment. The behavior typically varies from one microelectrode to another. Potentiometric tips cannot rely on positive and negative feedback diffusion, thus it is difficult to assess the tip-substrate distance from the tip response. Several approaches are available, but most are cumbersome. In the potentiometric mode the response... 

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