Definition of the Electrochemical Potential

Knowledge of the driving force is of the utmost importance for the understanding of any system. It determines the direction in which a chemical reaction can proceed spontaneously, as well as its position of equilibrium, at which the driving force is zero along all coordinates. 

The driving force in chemistry is the gradient of the chemical potential, (x. 

Let us turn our attention now to processes involving charged species, in particular charge-transfer processes. We recall that the chemical potential relates to the activity of the species  

The activity is related to the concentration via the activity coefficient y which is itself a function of concentration  

It would seem that pi, as given in Eq. (2.2), does not accoxmt for the effect of the electrical field or its gradient, unless we include that implicitly in the activity coefficient. 

---

Using the definition of the electrochemical potential this can be rewritten in the following forms ... 

As a side remark, the most consistent form of the thermodynamic definition of the electrochemical potential is... 

When we add an electron to the system at a given temperature and pressure, the electron is necessarily positioned in a level close to The increase in free energy of the electron system due to the addition of one electron is hence (Figure 5). Hence, the Fermi-Dirac occupation function is in accordance with the thermodynamic definition of the electrochemical potential. 

Eq. (8)] represents by definition the zero point of the electrochemical potential scale (standard hydrogen electrode, often denoted SHE). 

Equation (3.1.2) and the definition of the chemical potential yield the equation for the electrochemical potential of species i with activity a,(j3) and charge z, in phase )8 in the form... 

Electrochemical interfaces are sometimes referred to as electrified interfaces, meaning that potential differences, charge densities, dipole moments, and electric currents occur. It is obviously important to have a precise definition of the electrostatic potential of a phase. There are two different concepts. The outer or Volta potential ij)a of the phase a is the work required to bring a unit point charge from infinity to a point just outside the surface of the phase. By just outside we mean a position very close to the surface, but so fax away that the image interaction with the phase can be ignored in practice, that means a distance of about 10 5 — 10 3 cm from the surface. Obviously, the outer potential i/ a U a measurable quantity. 

For electrons in a metal the work function is defined as the minimum work required to take an electron from inside the metal to a place just outside (c.f. the preceding definition of the outer potential). In taking the electron across the metal surface, work is done against the surface dipole potential x So the work function contains a surface term, and it may hence be different for different surfaces of a single crystal. The work function is the negative of the Fermi level, provided the reference point for the latter is chosen just outside the metal surface. If the reference point for the Fermi level is taken to be the vacuum level instead, then Ep = —, since an extra work —eoV> is required to take the electron from the vacuum level to the surface of the metal. The relations of the electrochemical potential to the work function and the Fermi level are important because one may want to relate electrochemical and solid-state properties. 

Electrochemistry deals with charged particles that have both electrical and chemical properties. Since electrochemical interfaces are usually referred as electrified interfaces, it is clear that potential differences, charge densities, dipole moments, and electric currents occur at these interfaces. The electrical properties of systems containing charged species are very important for understanding how they behave at interfaces. Therefore, it is important to have a precise definition of the electrostatic potential of a phase [1-6]. Note that what really matters in electrochemical systems is not the value of the potential but its difference at a given interface, although it is illustrative to discuss its main properties. 

Any surface (typically a piece of metal) on which an electrochemical reaction takes place will produce an electrochemical potential when in contact with an electrolyte (typically water containing dissolved ions). The unit of the electrochemical potential is volt (TV = 1JC1 s 1 in SI units).The metal, or strictly speaking the metal-electrolyte interface, is called an electrode and the electrochemical reaction taking place is called the electrode reaction. The electrochemical potential of a metal in a solution, or the electrode potential, cannot be determined absolutely. It is referred to as a potential relative to a fixed and known electrode potential set up by a reference electrode in the same electrolyte. In other words, an electrode potential is the potential of an electrode measured against a reference electrode. The standard hydrogen electrode (SHE) is universally adopted as the primary standard reference electrode with which all other electrodes are compared. By definition, the SHE potential is OV, i.e. the zero-point on the electrochemical potential scale. Electrode potentials may be more positive or more negative than the SHE. 

Although the calculation of Ece in the previous section appears satisfactory, it is not very rigorous. In this section we show how a rigorous thermodynamic argument4 leads to the same result. For this we need the concept of the electrochemical potential /i, that obeys the same criteria at equilibrium as the chemical potential ju. Its definition for component i in phase a is... 

Both of these quantities contain an arbitrary constant, the zero from which the potentials are measured, but differences of either the electrostatic potential or of the electrochemical potential, between two phases, are definite. The thermionic work function, x, the work required to extract electrons from the highest energy level within the phase, to a state of rest just outside the phase, is also definite and the relation between the three definite quantities fa, V, and x is given by (3.1), where is the electrochemical potential of electrons very widely separated from all other charges. The internal electric potential , and other expressions relating to the electrical part of the potential inside a phase containing dense matter, are undefined, and so are the differences of these quantities between two phases of different composition. This indefiniteness arises from the impossibility of separating the electrostatic part of the forces between particles, from the chemical, or more complex interactions between electrons and atomic nuclei, when both types of force are present. 

The term electromotive force is misleading and undesirable. However, it is so firmly entrenched that we will continue to use it here. Note that the definition involves the electrochemical potential of the electrons on the product side of the equation as written, minus that for the electrons on the reagent side for the reaction as written. 

The thermodynamics of solutions and solid-liquid interfaces can be well described in terms of the chemical and electrochemical potentials of the system. The basic definition of the chemical potential [6] is... 

The over potential plays a central role in electrochemistry as it controls the electrochemical reactions. By convention it is generally measured as a positive value for reactions where electrons are transferred to the electrode. The associated current is also counted positively. In this case the electrode is called an anode. If electrons are transferred from, the electrode to the ions of the electrolyte, the over potential and the associated current are measured as negative values. The electrode is termed cathode. Using the definition of the over potential, the terminal voltage for an electrochemical cell is given by (see Fig. 3.2(b)) ... 

AG, 2 is calculated from electrochemical data using the definition of the standard potential (Equation 1.67), where n is the charge number of the reaction, that is, the number of electrons involved, and F is the Faraday constant. In convenient units, F = 23.06 kcal/(molV). 

The universal definition of the standard potential of a redox couple Red/Ox is as follows the standard potential is the value of emf of an electrochemical cell, in which diffusion potential and thermo-emf are eliminated. This cell consists of an electrode, on which the Red/Ox equilibria establish under standard conditions, and a SHE. 

With the general definition in Equation 12.14, it becomes possible to extend the use of the electrochemical potential in kinetics, which is not possible with the classical definition limited to simple coupling situations in fact. This point will be detailed in case study J2. 

In these three cases, new definitions of the mentioned efforts are proposed, which in the case of the electrochemical potential allows one to use it out of equilibrium (in kinetics notably) and in the case of the two latter, outlines the dependence of the classical definition of the force-field relationship on the linearity of the capacitive relationship in each domain. 

The notation cfef is not indicated in the expression of the electrochemical potential because this classical definition is not recommended. It is preferable to define the electrochemical potential as an apparent effort, that is, as the partial derivative of the energy versus the variation of a basic quantity, the substance amount, as is going to be demonstrated. 

One can also define, on the same scheme as the definition of an electrochemical potential (see case study J2), a resulting force, which is an apparent force given by the sum... 

For a clearer definition of the physical meaning of the relationship between A and we should take into account certain thermodynamic relations describing a reaction on an individual electrode. From the equality of the sum of the electrochemical potentials of the initial compounds and products (X/If = we find the equilibrium potential drop... 

The protection potential plays an important role in engineering because it defines the electrochemical conditions that protect a metal against corrosion. If the potential of the metal is at or below the protection potential, the rate of corrosion can be considered negligible. If required, the definition of the protection potential can be adapted to the precise conditions of a given situation by using another value for the surface concentration that defines the maximum rate of corrosion that can be tolerated. 

---