E° represents the relative tendency for this reaction to occur. This can also be thought of as the driving force for the reaction. The half-cell free energy change, AG0 is a related quantity that gives the free energy change for the given reaction at standard conditions relative to the reduction for the standard hydrogen electrode, for which the half reaction is
The half-cell potential and the half-cell free energy change are related by the following relationship for reversible conditions:
where F is the Faraday constant (=94490 C/ mol or 94490 J/V/mol) and 2 is the number of electrons appearing in the balanced reduction half cell. A negative value for AG0 (or a positive value for E°) means that the given reduction has a greater tendency to occur than does the reduction of H+ to H2. AG0 values are easily obtained from tables of thermodynamic data. As mentioned above, AG0 is zero by convention for elements in their standard states and for H+.
As an alternative, the tendency for a reduction to occur may also be expressed in terms of a hypothetical electron activity based on the standard hydrogen electrode. Activity was functionally defined in Equation (9). The free energy of an electron is related to chemical activity of the electron by
where ae is electron activity. The free energy change for a process in which z electrons move from a standard hydrogen electrode to some other electrode is therefore
AG = [G° + RTln(«e)] - [G° + RT]n(asm)] = zRT\n(ae/oshe) (23)
where «she is the standard hydrogen electrode electron activity.
We connected our earlier definition of activity to a standard state of 1.0 bar or 1.0 M or a mole fraction of unity. None of these make much sense for electrons, but we may define electron activity in terms of the standard hydrogen electrode. We define «she to be unity, and we define a term related to electron activity, pe:
Hostettler (1984) discusses issues involved in associating pe with electron activity.
One of the conditions of spontaneity is that AG < 0 at constant T and P. A new statement of this condition is that in a spontaneous process electrons (or any other substance) move from a state of higher activity to a state of lower activity. Because of the definition of pe as a negative logarithm of activity, the condition of spontaneity for pe is that electrons move from a more negative to a more positive pe.
We can combine the definition of pe with the Nernst equation to obtain a relationship between pe and concentrations:
5.3.3 pe/pH Stability Diagrams
We can write pe expressions for the reduction of 02 to H20 and the reduction of H+ to H2. For the reaction
If we use the relationship between pe° and £° implied by Equation (22) and assume an oxygen pressure of 1.0 bar, then pe = +20.77 - pH
For the reduction of H+ at a hydrogen pressure of 1.0 bar, pe = -pH.
We draw two lines representing these two expressions for pe as a function of pH in Fig. 51. We discuss five cases for this system:
Case 1. Along the lower line H2 and H+ are in equilibrium when pe = —pH. Case 2. If pe < -pH (perhaps as the result of
an applied voltage or the presence of another electrode), the electrons will have a greater activity than those in equilibrium with a standard hydrogen electrode, and H+ will be reduced to H2. This is the stability region for H2.
Case 3. Along the upper line 02 (at 1.0 bar) and H20 are in equilibrium.
Case 4. Above the upper line, pe > +20.77 - pH and 02 is stable.
Case 5. If pe lies between the two lines, H20 is stable.
Consider for a moment what happens if we provide a path for electron flow between the 02/H20 electrode and the H+/H20 electrode. The activity of electrons in equilibrium with the 02/H20 electrode is 10^ that of electrons associated with H+/H20. Electrons will flow from the H+/H20 electrode to the 02/H20 electrode.
As a more complex example, we examine the stability of oxidation states of aqueous sulfur as a function of pH. This exercise will bring out the treatment of thermodynamically unstable species and the change of sulfur speciation with pH.
The sulfur species include those in the +6 oxidation state (or S(VI)) and also S(IV), S(0), and S(—II). The necessary reduction reactions are
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