William M. White - Geochemistry

and

(3.107)

where z is the number of electrons per mole exchanged (e.g., 2 in the reduction of zinc) and is the Faraday constant ( = 96,485 coulombs; 1 joule = 1 volt-coulomb). The free energy of formation of a pure element is 0 (by convention). Thus, the Δ G in a reaction that is opposite one such as 3.105, such as:

is the free energy of formation of the ion from the pure element. From eqn. 3.106we can calculate the Δ G for the reduction of zinc as 147.24 kJ/mol. The free energy of formation of Zn 2+would be −147.24 kJ/mol. Given the free energy of formation of an ion, we can also use eqn. 3.105to calculate the hydrogen scale potential. Since

(3.108)

we can substitute eqns. 3.106and 3.107into 3.108and also write

(3.109)

Equation 3.109is known as the Nernst equation . †At 298 K and 0.1 MPa it reduces to:

(3.110)

We can deduce the meaning of this relationship from the relationship between ΔG and E in eqn. 3.106. At equilibrium Δ G is zero. Thus, in eqn. 3.108, activities will adjust themselves such that E is 0 .

We can calculate E Hvalues for reactions not listed in Table 3.3by algebraic combinations of the reactions and potentials that are listed. There is, however, a catch. Let's see how this works.

Calculate the E Hfor the reaction:

Answer: This reaction is the algebraic sum of two reactions listed in Table 3.3:

Since the reactions sum, we might assume that we can simply sum the E Hvalues to obtain the E Hof the net reaction. Doing so, we obtain an E Hof 0.33 V. However, the true E Hof this reaction is −0.037 V. What have we done wrong?

We have neglected to take into consideration the number of electrons exchanged. In the algebraic combination of E Hvalues, we need to multiply the E Hfor each component reaction by the number of electrons exchanged. We then divide the sum of these values by number of electrons exchanged in the net reaction to obtain the E Hof the net reaction, i.e.,

(3.111)

where the sum is over the component reactions i . Looking at eqn. 3.106, we can see why this is the case. By Hess's law, the Δ G of the net reaction must be the simple sum of the component reaction Δ G s, but E Hvalues are obtained by multiplying Δ G by z. Equation 3.111is derived by combining eqn. 3.106and Hess's law. Using eqn. 3.111, we obtain the correct E Hof −0.037 V.

Consider again the reaction:

(3.102)

If we were to express the equilibrium constant for this reaction, we would write:

Thus, we might find it convenient to define an activity for the electron. For this reason, chemists have defined an analogous parameter to pH, called pε , which is the negative log of the activity of electrons in solution:

(3.112)

The log of the equilibrium constant for eqn. 3.101may then be written as:

Upon rearranging we have:

(3.113)

When the activities of reactants and products are in their standard states (i.e., a = 1), then:

(3.114)

(where z again is the number of electrons exchanged: 1 in reaction 3.102). pε ° values are empirically determined and may be found in various tables. Table 3.3lists values for some of the more important reactions. For any state other than the standard state, pε is related to the standard state pε ° by:

(3.115)

pε and E Hare related by the following equation:

(3.116)

(the factor 2.303 arises from the switch from natural log units to base 10 log units).

In defining electron activity and representing it in log units, there is a clear analogy between pε and pH. However, the analogy is purely mathematical, and not physical. Natural waters do not contain significant concentrations of free electrons. Also, although a system at equilibrium can have only one value for pε , just as it will have only one value of pH, redox equilibrium is often not achieved in natural waters. The pε of a natural system is therefore often difficult to determine. Thus, pε is a hypothetical unit , defined for convenience of incorporating a representation of redox state that fits readily into established thermodynamic constructs such as the equilibrium constant. In this sense, eqn. 3.116provides a more accurate definition of pε than does eqn. 3.112.