Consider again the generalized electrochemical reaction:
xM + mXx+↔ xMm++ mX (Eq 2.62)
One of the most significant equations derived from chemical thermody-namics permits calculation of the change in the GFE for this reaction at constant total pressure and temperature as a function of∆Gfoof the reac-tant and product species in their standard states and the concentrations of those species with concentrations that can be varied. The equation is:
∆G ∆G RT ln
In this equation,∆Greacto is the change in the GFE for the reaction as written for reactants and products in their standard states; it is calcu-lated from Eq 2.20. The a’s are the activities of the species indicated by the subscripts: each activity is raised to a power equal to the stoichiometric coefficient of the species as it appears in the reaction.
The activity is frequently called the effective concentration of the spe-cies because it naturally arises as a function of the concentration, that is necessary to satisfy the changes in the thermodynamic functions (here, the GFE). In electrochemical systems, the activity is usually related to the molality of the species (moles per 1000 g of solvent) by the follow-ing equation:
a =γ m (Eq 2.64)
whereγ is the activity coefficient and m, the molality. Although in prin-ciple the activity of a single ionic species has meaning, and theoreti-cally, expressions have been developed for it, direct experimental mea-surement is not possible. The reason for the latter limitation is not discussed in detail here; it is sufficient to state that the problem relates to the fact that writing a single activity for an ionic species implies that this species can be added to a solution independent of other species.
This is not possible because of the necessity of simultaneously adding or having present in the solution ions of opposite charge in amounts to satisfy electrical neutrality. Although Eq 2.62 is frequently written with the ions of opposite charge present, as in Eq 2.32 or 2.33, and Eq. 2.63 can be modified to include the activities of the actual species dissolved to give the solution (FeCl2, for example), this is not done in the present treatment. The primary reason for using individual ion activities in the
present treatment is that it allows focus of attention on the ions involved in the individual electrode reactions, the influence of which is important in controlling corrosion rates. In many corrosion calculations, it is suffi-cient to use estimates of the individual ion activities, or to use the molality directly. Some reasons and justifications for this often-neces-sary approach are as follows. Measurement or calculation of accurate activities in concentrated electrolytes and in electrolytes of complex mixtures is generally not possible. Also, a tenfold change in the concen-tration results in a change of less than 100 mV in the electrode potential, which is frequently small compared to the potentials involved in cell re-actions (i.e., Ecell values). And, finally, metal ion concentrations in many corrosive environments are usually small (<10–4), in which case the activity coefficient is essentially unity and, therefore, a ≈ m.
The standard state for reactants and products in reaction 2.62 is pure solid for solid species, one atmosphere pressure for gas species, and unit activity (approximately unit molality) for ionic species. The activ-ity is unactiv-ity in each of these standard states, and if these conditions are substituted into Eq 2.63,∆Greactwill equal∆Greacto ; this must follow if the derivation of this equation is examined. If, under the actual condi-tions of the reaction, one or more species are solids, or a gas exists at one atmosphere pressure, then unit activity for each of these species is substituted in Eq 2.63, which effectively removes these activities from the log term. Also, the activity of water can usually be set equal to unity because its concentration changes insignificantly in most reactions in aqueous solution. Thus, taking M and X as solids, Eq 2.63 reduces to:
∆G ∆G RT ln
From the convention relating the cell reaction to the cell representation (Table 2.2), the cell potentials are written as:
Ecell ERHE ELHE E E
X,Xx M,Mm +
= ′ − ′ = ′ + − ′ (Eq 2.69)
and
Equations 2.72 and 2.73 are Nernst half-cell equations. For example, with Eq 2.73, when aMm += 1, E′M,Mm + = EM,Mo m +. Hence, EM,Mo m + is the half-cell potential at unit activity of the ions (i.e., the standard electrode half-cell potential). Values of the standard potentials of many electrode reactions are available in the literature, some of which are given in Ta-ble 2.1 (Ref 2, 7, 8). All values are given in sign and magnitude relative to the standard hydrogen electrode as previously discussed.
Many half-cell reactions involve species on both sides of the reaction that have variable concentrations in solution. These circumstances are handled by using the half-cell equation in the following more general form:
In this equation, X, Y, and Z are symbolic representatives of the impor-tant species involved in the reaction; Π Ox
[ ]
i υi is the product of the ac-tivities of the species on the “oxidized side” of the reaction (the side showing electrons produced), each raised to its stoichiometric coeffi-cient (υi);Π Red[
i]
υi has similar meaning for the “reduced side” of the reaction; and n is the number of mols of electrons produced (or con-sumed) per unit of the half reaction. Application of Eq 2.74 is illustrated in the following examples:Example 2.
Five examples are given of the application of the Nernst equation to half-cell reactions. These examples illustrate the influence of ion con-centration, pH, precipitate phases, and complex-ion formation on the electrode potential. All of these variables have significance in aqueous corrosion:
The half-cell potential will be –763 mV (SHE) when the activity is unity. An increase in the activity causes the potential to become more positive. The change is shown graphically in Fig. 2.6.
Example 2: Hydrogen Electrode.
It is very convenient to introduce the pH as a measure of the hydrogen ion activity since the acidity of solutions is usually expressed in these