POTENTIAL 1
One can obtain a quantitative relationship between the activity of the ions in solution and the potential that develops at the electrode/solution interface. At equilibrium, the change in free energy with respect to changes in reactants and products is zero (ΔGr×n=0). The free energy of formation for various species can often be found in the literature. These energies describe the system at standard state (a reference point for temperature, pressure, and activities of reactants and products). The standard conditions in most electrochem- ical work refer to 298.15 K, 1.00 atm pressure for gaseous species, and unit activities of each reactant and product. When the activities are expressed in molar quantities, the free energy of a reactant or product is also called the chemical potential,𝜇, for that species.
For example, the chemical potential for the Fe3+ion in an aqueous solution is given by the following:
𝜇Fe3+ =𝜇o
Fe3++RTln(aFe3+) (E.1)
In Eq. (2.2),Ris the universal gas constant, 8.314 J/(K mol);Tis the temperature in Kelvin. The activity,aFe3+, is the product of the concentration,c, and the activity coefficient, 𝛾Fe+3.
ai=𝛾ici (E.2)
When the system is at the standard state, the activity of the Fe3+is 1 M and the chemical potential,𝜇Fe3+, is equal to the chemical potential for the standard state,𝜇o
Fe3+. The term RTln(aFe+3)adjusts for conditions that are not at standard state. The activity is the effective reaction concentration of the species. The presence of other ions in solution effects the behavior or the Fe3+. The activity coefficient,𝛾, accounts for this effect. Other ions in the neighborhood tend to screen the electric field of a reactant species decreasing its ability to interact with other species that would, otherwise, exchange an electron or a proton or react in some other way had the interfering ions had not been there. Generally, the activity of an ion decreases as the concentration of the surrounding electrolyte increases. (A discussion of activity and activity coefficients appears in Appendix A at the end of this book.)
Consider the one-electron reduction of Fe+3to Fe+2at a Pt electrode. Because the Fe3+
ion carries a charge, a complete representation of the free energy associated with the ion
1Source: Bakker [1].
Electroanalytical Chemistry: Principles, Best Practices, and Case Studies, First Edition. Gary A. Mabbott.
© 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.
k k should also include the work done to bring this charge into the aqueous solution. The
electrochemical potential,𝜇, of an ion is the sum of the chemical potential and the electrical work required to bring into the medium the charge accompanying a mole of ions.
𝜇=electrochemical potential=chemical potential+electrical work done∕mole of ions (E.3) For Fe3+ions, the electrochemical potential is given here:
àFe3+ = àoFe3+ + RT ln(aFe3+) + 3 Fϕsoln
Charge on ion
Work done to bring a Coulomb of charge into solution Accounts for changes
from standard state
Chemical potential for formation of a mole of
Fe3+ ions at standard state (E.4)
The term𝜙solnis used to represent the work required to bring a coulomb of charge from outer space into the solution (in volts, or joules per coulomb). Because the other energy terms are expressed in joules per mole,𝜙solnmust be multiplied by Faraday’s constant,F (96 485 C/mol) to give units of J/mol. The same term is also multiplied by the charge on the ion,z, to give the work done for inserting a mole of ions with three charges/ion into the medium.
The strategy here is to write out the electrochemical potential for each reactant and product species. Then, because this is an equilibrium situation
(ΔGr×n=0= 𝜇products−𝜇reactants) (E.5)
the sum of the electrochemical potentials for the products must be equal to that of the reactants.
𝜇products=𝜇reactants (E.6)
The electrochemical potential for the electron can be written as follows:
Charge on electron
Work done to bring a coulomb of charge into Pt metal
àe(Pt) = àoe(Pt) + RT ln(ae(Pt)) – FϕPt (E.7)
For Fe2+ions, the electrochemical potential is 𝜇Fe2+=𝜇o
Fe2++RTln(aFe2+) +2F𝜙soln (E.8)
k k Now, one can set𝜇products=𝜇reactants
𝜇Fe2+ =𝜇Fe3++𝜇e(Pt) (E.9)
𝜇o
Fe2++RTln(aFe2+) +2F𝜙soln=𝜇oe(Pt)+RTln(ae(Pt)) −F𝜙Pt+𝜇o
Fe3++RTln(aFe3+) +3F𝜙soln (E.10) The objective is to obtain an expression for the electric potential energy difference 𝜙Pt− 𝜙soln, so, it is helpful to group all of the𝜙terms on the left and everything else on the right side.
F𝜙Pt−F𝜙soln=𝜇e(Pt)o +RTln(ae(Pt)) +𝜇o
Fe3+−𝜇o
Fe2++RTln
{(aFe3+) (aFe2+)
}
(E.11) àoe(Pt) + RT ln(ae(Pt)) + àoFe3+ – àoFe2+
This is a constant = EoFe3+/Fe2+
RTF
ϕPt–ϕsoln= F In (aFe2+)
(aFe3+) –
(E.12) It can be argued that the number of electrons that are transferred to or from the plat- inum will be a negligible fraction of the total electrons available in the conduction band of the metal. Consequently, the activity of the electron in the platinum is essentially constant, and it can be grouped together with other terms that are constant by definition to create a new constant, Eo
Fe3+∕Fe2+. This new constant is known as the standard electrode potential (or the standard reduction potential) for the Fe3+/Fe2+ electron transfer reaction. (Reac- tion equations in which electrons appear as a reactant or product are incomplete, since free electrons are not available as reagents, and the source of the electrons is not given. Nev- ertheless, the practice of conceptualizing the process in this form is very useful. Reaction equations in which electrons appear as a reactant or product are called “half-reactions.”) The term on the left-hand side of Eq. (E.12) is the difference in the electric potential between the solution and the platinum wire. It represents the voltage drop created by the separation of charge that appears at the interface (the electrical double layer). It is also apparent that this voltage is a characteristic of the type of electroactive species in solution. The electrode voltage is known as the redox potential of the solution,Esoln=𝜙Pt−𝜙soln, and is dependent on the activities of the Fe3+and Fe2+ions in solution. The equation becomes
Esoln=Eo
Fe2+∕Fe3+−RT F ln
{(aFe2+) (aFe3+)
}
(E.13) This format for the relationship between the redox potential and activities of redox components is known as the Nernst equation after Walther Nernst who introduced it in 1887. In writing Eq. (E.13) from the expression in (E.12), the ratio of ion activities was inverted, and the sign in front of the logarithm term was changed in order to compensate.
Of course, the two versions are equivalent, but it may be a little easier to remember the latter form. Equation (E.13) uses the ratio of activities for the products over the reactants as is the convention for defining equilibrium constants. Because the convention is to
k k write a half-reaction as a reduction process (electrons appear on the reactant side of the
arrows), one can also write the ratio of activities for products (the reduced form) over reactants (the oxidized form) in the Nernst equation, if a negative sign is used in front of the logarithm term. In other words, each species on the reduced side of the equation appears in the numerator, and a negative sign appears before the coefficient. A general procedure for formulating the equation for the redox potential for any balanced electron transfer reaction can be stated as follows. Consider the reduction ofamoles of an oxidized species, Ox, bynelectrons to formbmoles of the reduced product, Red
aOx+ne ⇄bRed (E.14)
The Nernst equation for the redox potential of a solution containing a mixture of the reactants and products can be written as:
Esoln=EoOx∕Red−RT
nFln(aRed)b
(aOx)a (E.15)
If one can write the Nernst equation easily using a direct procedure such as just demon- strated, then why spend time discussing electrochemical potentials? One reason is that working with electrochemical potentials is a more general approach. It covers all pro- cesses that develop a voltage at the interface between two media. The shortened rubric for writing the Nernst equation would appear to work only for electron transfer reactions.
Oxidation–reduction reactions are merely a subset of charge transfer processes that are important to analytical chemistry and other fields. Ion transfer reactions are another. For example the surface of the special glass in a pH electrode develops a potential because hydrogen ions are adsorbed onto the glass from the sample solution; electrons are not a part of that reaction equation. However, starting with the electrochemical potentials for the participating ions leads to the Nernst equation, if certain assumptions are made. Fur- thermore, it is prudent to know what those assumptions are that lead to the simplified expressions that one might be using.
REFERENCE
1. Bakker, E. (2014)Fundamentals of Electroanalysis: 1. Potentials and Transport. Apple iBook Store.
https://itunes.apple.com/us/book/fundamentals-electroanalysis-1-potentials-transport/
id933624613?mt=11(accessed 9 June 2019).
k k