Nernst idealization

What happens with increasing ED current can be presented mathematically with the aid of a simplified model called the Nernst idealization. This model, which is illustrated in Figure 8.17, is based on the simplifying assumptions of stagnant boundary layers of constant thickness and a well-mixed region in the center of the solution compartment. These idealized boundary-layer conditions are obviously unrealistic, especially in the presence of the spacer screens used in an ED stack, but they allow a simplified approach to an otherwise complex problem. 

Standard potentials Ee are evaluated with full regard to activity effects and with all ions present in simple form they are really limiting or ideal values and are rarely observed in a potentiometric measurement. In practice, the solutions may be quite concentrated and frequently contain other electrolytes under these conditions the activities of the pertinent species are much smaller than the concentrations, and consequently the use of the latter may lead to unreliable conclusions. Also, the actual active species present (see example below) may differ from those to which the ideal standard potentials apply. For these reasons formal potentials have been proposed to supplement standard potentials. The formal potential is the potential observed experimentally in a solution containing one mole each of the oxidised and reduced substances together with other specified substances at specified concentrations. It is found that formal potentials vary appreciably, for example, with the nature and concentration of the acid that is present. The formal potential incorporates in one value the effects resulting from variation of activity coefficients with ionic strength, acid-base dissociation, complexation, liquid-junction potentials, etc., and thus has a real practical value. Formal potentials do not have the theoretical significance of standard potentials, but they are observed values in actual potentiometric measurements. In dilute solutions they usually obey the Nernst equation fairly closely in the form ... 

To establish the operational pH scale, the pH electrode can be cahbrated with a single aqueous pH 7.00 phosphate buffer, with the ideal Nernst slope assumed. As Eqs. (2a)-(2d) require the free hydrogen ion concentration, an addihonal electrode standardization step is necessary. That is where the operational scale is converted to the concentration scale pcH (= -log [H ]) as described by Avdeef and Bucher [24] ... 

The cell consists of an indicator and a reference electrode, the latter usually being the calomel or silver-silver chloride type. The potential of the indicator electrode is related to the activities of one or more of the components of the solution and it therefore determines the overall cell potential. Ideally, its response to changes of activity should be rapid, reversible and governed by the Nernst equation. There are two types of indicator electrode which possess the desired characteristics - metallic and membrane. 

The second boundary condition arises from the continuity of chemical potential [44], which implies - under ideally dilute conditions - a fixed ratio, the so-called (Nernst) distribution or partition coefficient, A n, between the concentrations at the adjacent positions of both media ... 

The Nernst equation (Eq. (18a.2)) describes the response of an ISE under ideal conditions in the absence of interfering ions in solution. However, real samples contain competing ions of the same charge as... 

R is the ideal gas constant, T is the Kelvin temperature, n is the number of electrons transferred, F is Faraday s constant, and Q is the activity quotient. The second form, involving the log Q, is the more useful form. If you know the cell reaction, the concentrations of ions, and the E°ell, then you can calculate the actual cell potential. Another useful application of the Nernst equation is in the calculation of the concentration of one of the reactants from cell potential measurements. Knowing the actual cell potential and the E°ell, allows you to calculate Q, the activity quotient. Knowing Q and all but one of the concentrations, allows you to calculate the unknown concentration. Another application of the Nernst equation is concentration cells. A concentration cell is an electrochemical cell in which the same chemical species are used in both cell compartments, but differing in concentration. Because the half reactions are the same, the E°ell = 0.00 V. Then simply substituting the appropriate concentrations into the activity quotient allows calculation of the actual cell potential. 

The behavioural pattern of two immiscible solvents, say a and ib is essentially nonideal with respect to one another. Now, if a third substance is made to dissolve in a two-phase mixture of the solvents (i.e., a and 3 ), it may behave ideally in either phases provided its concentration in each individual phase is approximately small. Therefore, under these prevailing experimental parameters the ratio of the mole fractions of the solute in the two respective immiscible phases ( a and A) is found to be a constant which is absolutely independent of the quantity of solute present. It is termed as the Nernst Distribution Law or the Partition Law and may be expressed as follows ... 

Figure 10,1 (A) Activity-molar concentration plot. Trace element concentration range is shown as a zone of constant slope where Henry s law is obeyed. Dashed lines and question marks at high dilution in some circumstances Henry s law has a limit also toward inhnite dilution. The intercept of Henry s law slope with ordinate axis defines Henry s law standard state chemical potential. (B) Deviations from Nernst s law behavior in a logarithmic plot of normalized trace/carrier distribution between solid phase s and ideal aqueous solution aq. Reproduced with modifications from liyama (1974), Bullettin de la Societee Francaise de Mineralogie et Cristallographie, 97, 143-151, by permission from Masson S.A., Paris, France. A in part A and log A in part B have the same significance, because both represent the result of deviations from Henry s law behavior in solid.

Figure 10,2 Deviations from Nernst s law in crystal-aqueous solution equilibria, as obtained from application of various thermodynamic models. (A and B) Regular solution (liyama, 1974). (C) Two ideal sites model (Roux, 1971a). (D) Model of local lattice distortion (liyama, 1974). Reprinted from Ottonello (1983), with kind permission of Theophrastus Publishing and Proprietary Co.

Because electrode potentials are defined with reference to the H+/H2 electrode under standard conditions, E° values apply implicitly to (hypothetically ideal) acidic solutions in which the hydrogen ion concentration is 1 mol kg-1. Such E° values are therefore tabulated in Appendix D under the heading Acidic Solutions. Appendix D also lists electrode potentials for basic solutions, meaning solutions in which the hydroxide ion concentration is 1.0 mol kg-1. The conversion of E° values to those appropriate for basic solutions is effected with the Nernst equation (Eq. 15.15), in which the hydrogen ion concentration (if it appears) is set to 1.0 x 10-14 mol kg-1 and the identity and concentrations of other solute species are adjusted for pH 14. For example, for the Fc3+/2+ couple in a basic medium, the relevant forms of iron(III) and iron(II) are the solid hydroxides, and the concentrations of Fe3+ (aq) and Fe2+ (aq) to be inserted into the Nernst equation are those determined for pH 14 by the solubility products of Fe(OH)3(s) and Fe(OH)2(s), respectively. Examples of calculations of electrode potentials for nonstandard pH values are given in Sections 15.2 and 15.3. 

The Nernst-Planck equations are only limiting laws for ideal systems. If activity coefficient gradients are present, an additional term in Eq. (5.5) is created such that... 

Assuming an ideal solution in which the activity of a component is identical to its concentration and no kinetic coupling occurs between individual fluxes, Equation 5.8 becomes identical with the Nernst-Planck flux equation [18], which is given by ... 

In the context of RTILs the criterion (3) raises considerable problems since the concept of activity and activity coefficients of ions is largely unexplored in such media. Accordingly, validation of the applicability of the Nernst equation in such media is a non-simple exercise, given that RTILs are likely to exhibit gross non-ideality. Rather, electrochemical measurements based on otherwise validated reference electrodes, may likely in the future provide a methodology for the study of RTIL non-ideality. 

This idealized model does not capture all of the essential details of corrosion deposits. As indicated in Fig. 17, the influence of local chemistry within the deposit (especially pH effects) is likely to separate the corrosion site (at the material/deposit interface) from the site at which the deposit forms (deposit/ environment interface). Consequently, diffusion processes within the porous deposit must be involved if corrosion is to be sustained. Under simple steady-state conditions, diffusion can be treated simply using the Nernst diffusion layer approach i.e., the flux, J, of a species dissolving in a pore will be given by... 

The type and concentration of defects in solids determine or, at least, affect the transport properties. For instance, the -> ion conductivity in a crystal bulk is usually proportional to the -> concentration of -> ionic charge carriers, namely vacancies or interstitials (see also -> Nernst-Einstein equation). Clustering of the point defects may impede transport. The concentration and -> mobility of ionic charge carriers in the vicinity of extended defects may differ from ideal due to space-charge effects (see also - space charge region). 

An important example of the system with an ideally permeable external interface is the diffusion of an electroactive species across the boundary layer in solution near the solid electrode surface, described within the framework of the Nernst diffusion layer model. Mathematically, an equivalent problem appears for the diffusion of a solute electroactive species to the electrode surface across a passive membrane layer. The non-stationary distribution of this species inside the layer corresponds to a finite - diffusion problem. Its solution for the film with an ideally permeable external boundary and with the concentration modulation at the electrode film contact in the course of the passage of an alternating current results in one of two expressions for finite-Warburg impedance for the contribution of the layer Ziayer = H(0) tanh(icard)1/2/(iwrd)1/2 containing the characteristic - diffusion time, Td = L2/D (L, layer thickness, D, - diffusion coefficient), and the low-frequency resistance of the layer, R(0) = dE/dl, this derivative corresponding to -> direct current conditions. 

The comparison highlights the difference between the nonideal hydrogen/steam/water case and the ideal carbonmonox-ide/carbondioxide case. The difference can be detected only if fugacity-based calculations as displayed in the introduction to this book are made using the JANAF tables, (Chase etah, 1998). The equilibrium concentrations, the equilibrium constant and the Nernst potential difference V, in the hydrogen case, are a function of both pressure and temperature. declines with pressure. In the carbon monoxide perfect gas case, the same variables are a function of temperature only. The pressure coefficient is zero. 

The key expression here is the Nernst equation that, under ideal conditions, relates the electrical potential (E) of a system to the standard thermodynamic Gibbs free energy (AG°) of the process and the concentrations (strictly activities) of the... 

The resulting net flux of the species is known as the Nernst-Planck equation, and it holds in ideal systems, for all mobile species present. The set of Nernst-Planck equations must be solved under the appropriate conditions. In derivation of the Nernst-Planck equation, convection and gradients of pressure and activity coefficients are not included. 

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