Thermodynamics is a scientific branch that describes the relationship between heat and all forms of energy. Thermodynamics represents a core template to just about every single scientific discipline. For an electrochemical cell that is outside of equilibrium, the difference in free energy drives the redox reaction. Thus, there is a quantitative link between the free energy as dictated by the properties of the cell (temperature, concentration-activities of the active redox species), and the cell potential, a measurable quantity.
A.3.1
Nernst Equation
Consider an electrochemical cell with an infinite impedance voltmeter connected across the cathode and anode; the voltmeter measures and displays the cell potential - also known as the open circuit potential (OCP). Most importantly, the cell potential, Erxn, is directly proportional to
the change in the redox free energy,∆G.
∆G = −nFErxn (A.3)
Furthermore, the redox process in standard state is shown as:
∆G0 = −nFE0 (A.4)
Equations A.3 and A.4 show that Erxnis proportional to∆G by a pre-factor n, which is the number
of electrons that transfer per unit (mol, atom) for a given reaction, and Faradays constant, F (charge per unit mol of 96,500 C/mol).
Figure S.3: An inert working electrode (Pt) against a NHE reference. The working electrode is being driven by an external source towards reduction.
We can derive an explicit relationship between the thermodynamic states of the redox species and the cell potential using the illustration shown in Figure S.3. Two wires, a platinum
(Pt) wire and an inert metal wire (also Pt), are both immersed in an aqueous electrolyte solution of hydrochloric acid (HCl). The solution also contains a small concentration of reversible redox couples, such as ferrocene, an organometallic compound, and its oxidized cation counterpart (fer- rocenium). Because the base chemical structure of the ferrocene/ferrocenium couple are essentially the same, the redox reaction is facile and reversible. For simplicity, the reactants will be designated as reduced ferrocene (R= ferrocene) and oxidized ferrocenium (O = ferrocenium). In the above illustration, the entire reaction of the cell is written as shown:
H2+ 2O → 2R + 2H+ (A.5)
When the reaction is broken up into their half-reactions at the cathode and anode, we see that the NHE (anode) oxidizes H2into H+protons.
H2 2H++ 2e −
(A.6) At the cathode, O is reduced to R.
O+ e− R (A.7)
The spontaneity of the reaction in Equation A.5 is governed by the partial molar change in Gibbs free energy,∆G. Under constant temperature and pressure, we express ∆G as the chemical poten- tial, µi, difference between the reactants and products. If we rewrite Reaction A.5 in its broadest
form, we get:
aH2+ bO → cR + dH+ (A.8)
Thus, the chemical potential of each active specie is:
∆G = dµH++ cµR− bµO− aµH2 (A.9)
The standard state free energy,∆G0, is similarly written as:
∆G0= dµ0 H+ + cµ 0 R− bµ 0 O− aµ 0 H2 (A.10)
We then write∆G as a function of activity:
∆G = ∆G0+ RTln a d H+a c R aa ab ! (A.11)
Assuming that the partial pressure of H2 and activity of H+ is fixed to their standard states (aH2,
aH+= 1), which is the set condition for an NHE reference, Equation A.11 is simplified to:
∆G = ∆G0+ RTln a c R ab O ! (A.12)
Because the NHE potential is zero, Equation A.12 shows that ∆G depends soley on the cathode half-reaction. Since the cell potential (Erxn) is proportional to∆G, Equation A.12 can be re-written
as: ∆Erxn = ∆E0+ RT nFln ab O ac R ! (A.13) This relationship is known as the Nernst Equation, which explicitly represents Erxn as a function
of the chemical states of the redox species. The Nernst equation is most often represented as a function of redox concentration (ex. mol/cm3).
∆Erxn= ∆E0 0 + RT nFln Cb O Cc R ! (A.14)
Equation A.14 now incoporates the term containing the activity coefficients, γ, with E0, rewritten as the formal potential, E00.
∆E00 = ∆E0+ RT nFln γb O γc R ! (A.15) To balance Reaction A.7, the powers b and c are both equal to 1, with n= 1. Therefore, the final equation for Erxnis:
∆Erxn= ∆E0 0 + RT F ln CO CR ! (A.16) Sign Conventions
The designated sign convention of the cell potential in relation to the free energy can often be a source of confusion. Many define the cell potential as a directionless quantity, whereby an infinitesimal change in the overall value dictates the direction of the redox reaction. This contrasts the definition of the Gibbs free energy, a directional quantity by which the sign corresponds to the direction of the reaction. Consequently, separate definitions now exist in order to reconcile the
direct relationship between the free energy and cell potential. For instance, Erxnis also known as
the electromotive force (emf), which is designated as a directional quantity tied to the cell reaction as opposed to the physical cell. For example, if we consider a convention of assigning the reduction half-reaction spontaneously taking place at the right electrode with respect to the oxidation half- reaction taking place at the left electrode, Erxnbecomes:
Erxn= ERight− ELe f t (A.17)
If the left electrode (oxidation) is at a lower potential than the right electrode (reduction), the reaction will spontaneously reduce, resulting in a positive emf quantity. Conversely, if the cell reaction reverses, the potential difference is now between the left electrode (reduction) and the right electrode (oxidation), resulting in a negative emf. This facilitates consistency in quantitatively connecting Erxnto∆G (Equation A.3), which are opposite in sign since a positive emf and negative
∆G defines a spontaneous reaction.
Reversibility
Reversibility is an important descriptor for characterizing the nature of various electro- chemical processes. In practice, however, there are different degrees of reversibility in electro- chemistry. Thermodynamic reversibility represents the absolute limit in which a system is always at equilibrium - the path from one state to another is changed infinitesimally, incurring no net change in entropy. Consequently, reals systems are not perfectly reversible in nature, but they can exhibit behavior that closely aligns with thermodynamic reversibility. In the context of electro- chemistry, a system can be characterized as being chemically and/or practically reversible. Chem- ical reversibility is defined by a chemical reaction that can proceed forwards or backwards. Many practical systems show chemical reversibility; one example is the following redox reaction of a silver/silverchloride (Ag/AgCl) (eq. 2.xx) is an example a chemically reversible reaction:
In forward direction, AgCl is reduced, dissociating to form metallic Ag. If the reaction is reversed, that same metallic Ag will re-oxidize to form AgCl. For irreversible reactions, the product, R, may decompose into a different byproduct in the presence of another reactant. Thus, the direction of the reaction can no longer be reversed.
O+ ne−→ R → T (A.19)
If a reaction is thermodynamically reversible, then it is by extension chemically reversible, though a chemically reversible reaction may not always be thermodynamically reversible. Common exam- ples include reactions that are extremely limited kinetically, resulting in a significant perturbation threshold to activate the reaction. Facile reactions that are close to thermodynamically reversible are often defined as practically reversible, or ”Nernstian”, which is notably found in simple reac- tions that exhibit rapid reaction kinetics within short time domains.