Nernst equation

In electrochemistry, the Nernst equation is a chemical thermodynamical relationship that permits the calculation of the reduction potential of a reaction from the standard electrode potential, absolute temperature, the number of electrons involved in the redox reaction, and activities of the chemical species undergoing reduction and oxidation respectively. It was named after Walther Nernst, a German physical chemist who formulated the equation.

Expression

General form with chemical activities

When an oxidized species accepts a number z of electrons to be converted in its reduced form, the half-reaction is expressed as:
The reaction quotient, also often called the ion activity product, is the ratio between the chemical activities of the reduced form and the oxidized form. The chemical activity of a dissolved species corresponds to its true thermodynamic concentration taking into account the electrical interactions between all ions present in solution at elevated concentrations. For a given dissolved species, its chemical activity is the product of its activity coefficient by its molar, or molal, concentration : a = γ C. So, if the concentration of all the dissolved species of interest are sufficiently low and that their activity coefficients are close to unity, their chemical activities can be approximated by their concentrations as commonly done when simplifying, or idealizing, a reaction for didactic purposes:
At chemical equilibrium, the ratio ' of the activity of the reaction product by the reagent activity is equal to the equilibrium constant ' of the half-reaction:
The standard thermodynamics also says that the actual Gibbs free energy is related to the free energy change under standard state by the relationship:
where is the reaction quotient and R is the universal ideal gas constant.
The cell potential associated with the electrochemical reaction is defined as the decrease in Gibbs free energy per coulomb of charge transferred, which leads to the relationship The constant is a unit conversion factor, where is the Avogadro constant and is the fundamental electron charge. This immediately leads to the Nernst equation, which for an electrochemical half-cell is
For a complete electrochemical reaction, the equation can be written as
where:

is the half-cell reduction potential at the temperature of interest,
is the standard half-cell reduction potential,
is the cell potential at the temperature of interest,
is the standard cell potential in volts,
is the universal ideal gas constant:,
is the temperature in kelvins,
is the number of electrons transferred in the cell reaction or half-reaction,
is Faraday's constant, the magnitude of charge per mole of electrons:,
is the reaction quotient of the cell reaction, and,
is the chemical activity for the relevant species, where is the activity of the reduced form and is the activity of the oxidized form.
Thermal voltage

At room temperature, the thermal voltage is approximately 25.693 mV. The Nernst equation is frequently expressed in terms of base-10 logarithms rather than natural logarithms, in which case it is written:
where λ = ln ≈ 2.3026 and λV_T ≈ 0.05916 Volt.

Form with activity coefficients and concentrations

Similarly to equilibrium constants, activities are always measured with respect to the standard state. The chemical activity of a species,, is related to the measured concentration via the relationship, where is the activity coefficient of the species. Because activity coefficients tend to unity at low concentrations, or are unknown or difficult to determine at medium and high concentrations, activities in the Nernst equation are frequently replaced by simple concentrations and then, formal standard reduction potentials used.
Taking into account the activity coefficients the Nernst equation becomes:
Where the first term including the activity coefficients is denoted and called the formal standard reduction potential, so that can be directly expressed as a function of and the concentrations in the simplest form of the Nernst equation:

Formal standard reduction potential

When wishing to use simple concentrations in place of activities, but that the activity coefficients are far from unity and can no longer be neglected and are unknown or too difficult to determine, it can be convenient to introduce the notion of the "so-called" standard formal reduction potential which is related to the standard reduction potential as follows:
So that the Nernst equation for the half-cell reaction can be correctly formally written in terms of concentrations as:
and likewise for the full cell expression.
According to Wenzel, a formal reduction potential is the reduction potential that applies to a half reaction under a set of specified conditions such as, e.g., pH, ionic strength, or the concentration of complexing agents.
The formal reduction potential is often a more convenient, but conditional, form of the standard reduction potential, taking into account activity coefficients and specific conditions characteristics of the reaction medium. Therefore, its value is a conditional value, i.e., that it depends on the experimental conditions and because the ionic strength affects the activity coefficients, will vary from medium to medium. Several definitions of the formal reduction potential can be found in the literature, depending on the pursued objective and the experimental constraints imposed by the studied system. The general definition of refers to its value determined when. A more particular case is when is also determined at pH 7, as e.g. for redox reactions important in biochemistry or biological systems.

Determination of the formal standard reduction potential when 1

The formal standard reduction potential can be defined as the measured reduction potential of the half-reaction at unity concentration ratio of the oxidized and reduced species under given conditions.
Indeed:
as,, when,
because, and that the term is included in.
The formal reduction potential makes possible to more simply work with molar or molal concentrations in place of activities. Because molar and molal concentrations were once referred as formal concentrations, it could explain the origin of the adjective formal in the expression formal potential.
The formal potential is thus the reversible potential of an electrode at equilibrium immersed in a solution where reactants and products are at unit concentration. If any small incremental change of potential causes a change in the direction of the reaction, i.e. from reduction to oxidation or vice versa, the system is close to equilibrium, reversible and is at its formal potential. When the formal potential is measured under standard conditions it becomes de facto a standard potential.
According to Brown and Swift :

"A formal potential is defined as the potential of a half-cell, measured against the standard hydrogen electrode, when the total concentration of each oxidation state is one formal".

In this case, as for the standard reduction potentials, the concentrations of dissolved species remain equal to one molar or one molal, and so are said to be one formal. So, expressing the concentration in molarity :
The term formal concentration is now largely ignored in the current literature and can be commonly assimilated to molar concentration, or molality in case of thermodynamic calculations.
The formal potential is also found halfway between the two peaks in a cyclic voltammogram, where at this point the concentration of Ox and Red at the electrode surface are equal.
The activity coefficients and are included in the formal potential, and because they depend on experimental conditions such as temperature, ionic strength, and pH, cannot be referred as an immutable standard potential but needs to be systematically determined for each specific set of experimental conditions.
Formal reduction potentials are applied to simplify calculations of a considered system under given conditions and measurements interpretation. The experimental conditions in which they are determined and their relationship to the standard reduction potentials must be clearly described to avoid to confuse them with standard reduction potentials.

Formal standard reduction potential at pH 7

Formal standard reduction potentials are also commonly used in biochemistry and cell biology for referring to standard reduction potentials measured at pH 7, a value closer to the pH of most physiological and intracellular fluids than the standard state pH of 0. The advantage is to defining a more appropriate redox scale better corresponding to real conditions than the standard state. Formal standard reduction potentials allow to more easily estimate if a redox reaction supposed to occur in a metabolic process or to fuel microbial activity under some conditions is feasible or not.
While, standard reduction potentials always refer to the standard hydrogen electrode, with = 1 M corresponding to a pH 0, and fixed arbitrarily to zero by convention, it is no longer the case at a pH of 7. Then, the reduction potential of a hydrogen electrode operating at pH 7 is −0.413 V with respect to the standard hydrogen electrode.

Expression of the Nernst equation as a function of pH

The and pH of a solution are related by the Nernst equation as commonly represented by a Pourbaix diagram. explicitly denotes expressed versus the standard hydrogen electrode. For a half cell equation, conventionally written as a reduction reaction :
The half-cell standard reduction potential is given by
where is the standard Gibbs free energy change, is the number of electrons involved, and is the Faraday's constant. The Nernst equation relates pH and as follows:
where curly brackets indicate activities, and exponents are shown in the conventional manner. This equation is the equation of a straight line for as a function of pH with a slope of volt.
This equation predicts lower at higher pH values. This is observed for the reduction of O₂ into H₂O, or OH⁻, and for the reduction of H⁺ into H₂. is then often noted as to indicate that it refers to the standard hydrogen electrode whose = 0 by convention under standard conditions.