## Principles and Fundamental Relations

Potentiometric measurements are based on thermody-namic relationships and, more particularly, the Nernst equation which relates potential to the concentration of electroactive species. For our purposes, it is most convenient to consider the redox process that occurs at a single electrode, although two electrodes are always essential for an electrochemical cell. However, by considering each electrode individually, the two electrode processes are easily combined to obtain the entire cell process. Furthermore, confusion can be minimized if the half reactions for electrode processes are written in a consistent manner. Here, these are always reduction processes with the oxidized species reduced by n electrons to give a reduced species, ox + ne rei.

For such a half reaction the free energy is given by the relation

where -AG indicates the tendency for the reaction to go to the right; R is the gas constant and in the units appropriate for electrochemistry has a value of 8.317 J mol-1 K-1; T is the temperature of the system in K; and the logarithmic terms in the bracketed expression represent the activities (effective concentrations) of the electroactive pair at the electrode surface. The free energy of this half reaction is related to the electrode potential, E, by the expression

The quantity AG° is the free energy of the half reaction when the activities of the reactant and product have values of unity and is directly proportional to the standard half-cell potential for the reaction as written. It also is a measure of the equilibrium constant for the half reaction, assuming the activity of electrons is unity,

An extensive summary of E° values is presented in the compilation by Bard, Parsons, and Jordan (1985); the most important are tabulated in Sawyer, Sobkowiak, and Roberts (1995). Standard potentials are thermodynamic quantities that usually are evaluated via caloriometry for a cell reaction (e.g., 2 H2 + O ^ 2H2O; Ec°ell = [EO2 - EH+/H2]) and the relationship of Eq. (30).

When Eqs. (29) and (30) are combined, the resulting Nernst expression relates the half-cell potential to the effective concentrations (activities) of the redox couple,

The activity of a species is indicated as the symbol of the species enclosed in a bracket. This quantity is equal to the concentration of the species times a mean activity coefficient,

Although there is no straightforward and convenient method for evaluating activity coefficients for individual ions, the Debye-Huckel relationship permits an evaluation of the mean activity coefficient (y±) for ions at low concentrations (usually below 0.01 M), log y± = -0.509z2

1 + Vm where z is the charge on the ion and m is the ionic strength

More rigorous methods for the calculation of aqueous activity coefficients are available.

The reaction of an electrochemical cell always involves a combination of two redox half reactions such that one species oxidizes a second species to give the respective redox products. Thus, the overall cell reaction can be expressed by a balanced chemical equation a ox1 + b red2—*~c red1 + dox2, Kt equil •

However, electrochemical cells are most conveniently considered as two individual half reactions, whereby each is written as a reduction in the form indicated by Eqs. (28)-(32). When this is done and values of the appropriate quantities are inserted, a potential can be calculated for each half-cell electrode system. Then that half-cell reaction with the more positive potential will be the positive terminal in a galvanic cell, and the electromotive force of that cell will be represented by the algebraic difference between the potential of the more positive half-cell and the potential of the less positive half-cell,

Ecell — E(more positive) E(less positive) — E1 E2 • (37)

Insertion of the appropriate forms of Eq. (32) into Eq. (37) gives an overall expression for the cell potential,

The equilibrium constant for the chemical reaction expressed by Eq. (36) is related to the difference of the standard half-cell potentials by the relation ln Kequil = (nF/RT)(Ef - E2°).

To apply potentiometric measurements to the determination of the concentration of electroactive species, a number of conditions have to be met. The basic measurement system must include an indicator electrode, which is capable of monitoring the activity of the species of interest, and a reference electrode, which gives a constant, known half-cell potential to which the indicator electrode potential can be referred. The voltage resulting from the combination of these two electrodes must be measured in a manner that minimizes the amount of current drawn by the measuring system. For low-impedance electrode systems, a conventional potentiometer is satisfactory. However, electrochemical measurements with high-impedance electrode systems, and in particular the glass-membrane electrode, require the use of an exceedingly high-input-impedance measuring instrument (usually an electrometer amplifier with a current drain of less than 10-12 A). Because of the logarithmic nature of the Nernst equation, the measuring instrumentation must have considerable sensitivity. For example, a one-electron half reaction of 25°C gives a voltage change of 59.1 mV for a 10-fold change in the concentration of the electroactive species. Another important point is that the potential response is directly dependent on the temperature of the measuring system. Thus, if the correct temperature is not used in the Nernst expression, large absolute errors can be introduced in the measurement of the activity for an electroactive species.

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