The Nemst Equation

The electrode potential of a redox couple varies with the activities of the reduced and oxidized forms of the couple in the sense that increasing activity of oxidant increases the value of the potential. Quantitatively, for the reversible half-reaction (12-9) we have the Nemst equation 

We may substitute in (12-10) for each activity the product of activity coefficient and concentration and thus obtain 

For more complicated electrode reactions the Nemst equation includes terms to represent all species of variable activity involved in the reaction, such as hydrogen or hydroxyl ion and complexing molecules or ions. Reactants at invariant unit activity, for instance the solvent, pure metals, and pure solids, are omitted in the Nemst equation. Examples are 

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For more complicated redox reactions, a general fonn of the Nemst equation may be derived by analogy with A2.4.113. If we consider a stoichiometric reaction of the following type ... 

The Nemst equation above for the dependence of the equilibrium potential of redox electrodes on the activity of solution species is also valid for uncharged species in the gas phase that take part in electron exchange reactions at the electrode-electrolyte interface. For the specific equilibrium process involved in the reduction of chlorine ... 

Once current passes tluough the interface, the Galvani potential difference will differ from that expected from the Nemst equation above the magnitude of the difference is temied the overpotential, which is defined heiiristically as... 

For a simple electron transfer reaction containing low concentrations of a redox couple in an excess of electrolyte, the potential established at an inert electrode under equilibrium conditions will be governed by the Nemst equation and the electrode will take up the equilibrium potential for the couple 0/R. In temis of... 

Cyclic voltammetry provides a simple method for investigating the reversibility of an electrode reaction (table Bl.28.1). The reversibility of a reaction closely depends upon the rate of electron transfer being sufficiently high to maintain the surface concentrations close to those demanded by the electrode potential through the Nemst equation. Therefore, when the scan rate is increased, a reversible reaction may be transfomied to an irreversible one if the rate of electron transfer is slow. For a reversible reaction at a planar electrode, the peak current density, fp, is given by... 

The expression for the mass-transport-limiting current density may be employed together with the Nemst equation to deduce the complete current-potential response in a solution containing only oxidized or reduced species... 

Gibbs values and the effective electrode potential follows the Nemst equation (see section C2.11). For the oxidation (anodic) reaction, the potential (E ) of the Nemst equation can be written as ... 

Combining equation 6 and 7 with the Van t Hoff isotherm the Nemst equation for electrochemicA reactions is obtained ... 

The doped Zr02 stmctures are used as electrochemical sensors, as, for example, when used to detect oxygen in automotive exhaust (see Exhaust CONTROL, automotive). The sensor voltage is governed by the Nemst equation (eq. 17) where the activities are replaced by oxygen partial pressures and the air inside the chamber is used as reference. 

Corrosion occurs even if the two reactants involved are not at standard conditions. In this case the nonstandard equiUbrium potential for each reaction, often referred to as the reversible potential, can be calculated from the Nemst equation. Additional information on thermodynamic aspects of corrosion can be found in Reference 10. 

Smaller values of necessitate the appHcation of voltages greater than those calculated from the Nemst equation to obtain a corresponding set of surface concentrations of electroactive species. These voltages are called overpotentials and iadicate chemically related difficulties with the electrolysis. In other words, electron exchange between the electrode and the electroactive species is impeded by the chemistry of the process itself. 

The chronoamperometric technique illustrates the principle that analytically useful current responses depend critically on the efficiency of analyte mass transport within the solution. The analyte mass transport in turn depends on the efficiency with which an appHed voltage can maintain the surface concentrations of oxidized and reduced species at values specified by the Nemst equation. It is generally the case in chronoamperometry that the bulk concentration of one of the species is zero whereas the surface concentration of the other species is forced to zero by the appHed potential, but this is not always so. 

Equation 14 is a form of the Nemst equation. The overall chemical reaction for the passage of 2 Earadays of charge in the external circuit from right to left... 

Each reactant and product appears in the Nemst equation raised to its stoichiometric power. Thermodynamic data for cell potentials have been compiled and graphed (3) as a function of pH. Such graphs are known as Pourbaix diagrams, and are valuable for the study of corrosion, electro deposition, and other phenomena in aqueous solutions.Erom the above thermodynamic analysis, the cell potential can be related to the Gibbs energy change... 

As the Nemst equation suggests, concentration variations in the electrolyte lead to potential differences between electrodes of the same kind. These potential differences are concentration polarizations or concentration overpotentials. Concentration polarizations can also affect the current distribution. Predicting these is considerably more difficult. If concentration gradients exist, equations 25 and 27 through 29 must generally be solved simultaneously. 

The actual potential of the couple is given by the Nemst equation,... 

The transmembrane potential derived from a concentration gradient is calculable by means of the Nemst equation. If K+ were the only permeable ion then the membrane potential would be given by Eq. 1. With an ion activity (concentration) gradient for K+ of 10 1 from one side to the other of the membrane at 20 °C, the membrane potential that develops on addition of Valinomycin approaches a limiting value of 58 mV87). This is what is calculated from Eq. 1 and indicates that cation over anion selectivity is essentially total. As the conformation of Valinomycin in nonpolar solvents in the absence of cation is similar to that of the cation complex 105), it is quite understandable that anions have no location for interaction. One could with the Valinomycin structure construct a conformation in which a polar core were formed with six peptide N—H moieties directed inward in place of the C—O moieties but... 

Otherwise it has been shown that the accumulation of electrolytes by many cells runs at the expense of cellular energy and is in no sense an equilibrium condition 113) and that the use of equilibrium thermodynamic equations (e.g., the Nemst-equation) is not allowed in systems with appreciable leaks which indicate a kinetic steady-state 114). In addition, a superposition of partial current-voltage curves was used to explain the excitability of biological membranes112 . In interdisciplinary research the adaptation of a successful theory developed in a neighboring discipline may be beneficial, thus an attempt will be made here, to use the mixed potential model for ion-selective membranes also in the context of biomembrane surfaces. 

In chemistry, the most important use of the Nemst equation lies in the experimental determination of the concentration of ions in solution. Suppose you measure the cell voltage and know the concentration of all but one species in the two half-cells. It should then be possible to calculate the concentration of that species by using the Nemst equation (Example 18.7). 

Strategy First (1) set up the Nemst equation for the reduction half-reaction and calculate red. Then (2) repeat the calculation for the oxidation half-reaction, finding Eox. Finally (3), add rcd + ra if the sum is positive, disproportionation should occur. 

Carry out electrochemical calculations involving E°, the Nemst equation, and/or electrolysis. (Examples 21.4,21.9 Problems 55-62) 56... 

Derive the Nemst equation from the Butler-Vohner equation. 

From the foregoing discussion it will be clear that the stoichiometry of the oxidation [n in Eq. (1)] has no thermodynamic significance. It should not be used in the Nemst equation to describe the potential dependence of the equilibrium shown in Eq. (1). It is therefore better to describe n as the degree of oxidation of the polymer (i.e., the average number of holes per monomer unit), n is a potential-dependent parameter,... 

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