The electrochemical potential

The electrochemical potential of charged species i, jl is defined as the work done when this species is moved from charge-free infinity to the interior of a homogeneous phase a which carries no net charge [6]. The opposite process is known as the work function and is familiar for the case of removal of an electron from a metal. In fact, the work function for single ions in electrolyte solutions can also be measured experimentally, as described in detail in chapter 8. This means that the electrochemical potential is an experimentally determinable quantity. However, separation of the electrochemical potential into chemical and electrostatic contributions is arbitrary, even though it is conceptually very useful. 

the two parts of phase a are separated (fig. 6.4) and the process of introducing a charged species to each part repeated. The work done in the case of the homogeneous phase a is the chemical potential On the other hand, the work done in moving the charged particle through the surface shell into the empty interior is associated with the electrostatic characteristics of phase a and is written as (j) is called the inner potential of phase a, and cannot be measured for 

As a result of the thought experiment described above, the expression for the electrochemical potential of species i in phase a is 

the distinction between the electrochemical potential and the chemical potential is lost when the properties of the electrolyte as a whole are considered. The development of the thermodynamic properties of the electrolyte are those discussed earlier in section 3.6. 

The irmer potential is an important property of individual phases. Much more will be said about this property when the interface between two phases is discussed in chapter 8. For the moment, ) is regarded as a property of phase a which is the same throughout the phase with a value defined with respect to charge-free infinity. On the other hand, in the case of an electrolyte solution, the local electrostatic potential varies from point to point due to the presence of discrete charges on the ions. Thus the electrostatic potential is more positive at a cation and more negative at an anion. These fluctuations occur about the average value, ) . This can be seen more clearly by writing out the chemical potential of ion i in terms of its concentration c, and activity coefficient y,. Thus, from equation (6.6.1) 

In ordinary thermodynamics the chemical potential of a species i is defined as  

If the particles of species i in Eq. (2.1) are charged, one speaks of an electrochemical potential instead, and writes pi- The usual thermodynamic equilibrium conditions are now in terms of the pi. For example, if a species i is present both in a phase a and in a phase fj, and the interface between a and /3 is permeable to i, then pi,a = Pi,p at equilibrium. 

In adding a charged particle work is done against the inner potential f , and it may be useful to separate this out and write  

At zero temperature the electrons in a solid occupy the lowest energy levels compatible with the Pauli exclusion principle. The highest energy level occupied at T = 0 is the Fermi level, Ep. For metals the Fermi level and the electrochemical potential are identical at T = 0, since any electron that is added to the system must occupy the Fermi level. At finite temperatures Ep and the electrochemical potential p of the electrons differ by terms of the order of (kT)2, which are typically 

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 = — h — eoV , 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. 

---

Here, the only surface adsorption is taken to be that of the charge balancing the double-layer charge, and the electrochemical potential change is equated to a change in o- Integration then gives... 

The treatments that are concerned in more detail with the nature of the adsorbed layer make use of the general thermodynamic framework of the derivation of the Gibbs equation (Section III-5B) but differ in the handling of the electrochemical potential and the surface excess of the ionic species [114-117]. The derivation given here is after that of Grahame and Whitney [117]. Equation III-76 gives the combined first- and second-law statements for the surface excess quantities... 

The chemical potential pi, has been generalized to the electrochemical potential Hj since we will be dealing with phases whose charge may be varied. The problem that now arises is that one desires to deal with individual ionic species and that these are not independently variable. In the present treatment, the difficulty is handled by regarding the electrons of the metallic phase as the dependent component whose amount varies with the addition or removal of charged components in such a way that electroneutrality is preserved. One then writes, for the ith charged species. 

The electrochemical potentials pi, may now be expressed in terms of the chemical potentials pt, and the electrical potentials (see Section V-9) ... 

The electrochemical potential is defined as the total work of bringing a species i from vacuum into a phase a and is thus experimentally defined. It.may be divided into a chemical work p , the chemical potential, and the electrostatic work ZiC0 ... 

When two dissimilar metals are connected, as illustrated in Fig. V-16, ]here is a momentary flow of electrons from the metal with the smaller work function to the other so that the electrochemical potential of the electrons becomes the same. For the two metals a and /3... 

The chemical potential now includes any such effects, and one refers to the gmvochemicalpotential, the electrochemical potential, etc. For example, if the system consists of a gas extending over a substantial difference in height, it is the gravochemical potential (which includes a tenn m.gh) that is the same at all levels, not the pressure. The electrochemical potential will be considered later. 

In these equations the electrostatic potential i might be thought to be the potential at the actual electrodes, the platinum on the left and the silver on the right. However, electrons are not the hypothetical test particles of physics, and the electrostatic potential difference at a junction between two metals is nnmeasurable. Wliat is measurable is the difference in the electrochemical potential p of the electron, which at equilibrium must be the same in any two wires that are in electrical contact. One assumes that the electrochemical potential can be written as the combination of two tenns, a chemical potential minus the electrical potential (- / because of the negative charge on the electron). Wlien two copper wires are connected to the two electrodes, the... 

If two metals with different work functions are placed m contact there will be a flow of electrons from the metal with the lower work function to that with the higher work fimction. This will continue until the electrochemical potentials of the electrons in the two phases are equal. This change gives rise to a measurable potential difference between the two metals, temied the contact potential or Volta potential difference. Clearly... 

Figure Bl.28.9. Energetic sitiration for an n-type semiconductor (a) before and (b) after contact with an electrolyte solution. The electrochemical potentials of the two systems reach equilibrium by electron exchange at the interface. Transfer of electrons from the semiconductor to the electrolyte leads to a positive space charge layer, W. is the potential drop in the space-charge layer.

The driving force for migration is established by the different electrochemical potentials (AU) that exist at the two interfaces of the oxide. In other words, the electrochemical potential at the outer interface is controlled by the dominant redox species present in the electrolyte (e.g. O2). 

The standard-state electrochemical potential, E°, provides an alternative way of expressing the equilibrium constant for a redox reaction. Since a reaction at equilibrium has a AG of zero, the electrochemical potential, E, also must be zero. Substituting into equation 6.24 and rearranging shows that... 

Ladder diagrams can also be used to evaluate equilibrium reactions in redox systems. Figure 6.9 shows a typical ladder diagram for two half-reactions in which the scale is the electrochemical potential, E. Areas of predominance are defined by the Nernst equation. Using the Fe +/Fe + half-reaction as an example, we write... 

In a redox reaction, one of the reactants is oxidized while another reactant is reduced. Equilibrium constants are rarely used when characterizing redox reactions. Instead, we use the electrochemical potential, positive values of which indicate a favorable reaction. The Nernst equation relates this potential to the concentrations of reactants and products. 

You will recall from Chapter 6 that the Nernst equation relates the electrochemical potential to the concentrations of reactants and products participating in a redox reaction. Consider, for example, a titration in which the analyte in a reduced state, Ared) is titrated with a titrant in an oxidized state, Tox- The titration reaction is... 

The electrochemical potential for the reaction is the difference between the reduction potentials for the reduction and oxidation half-reactions thus,... 

Derive a general equation for the electrochemical potential at the equivalence point for the titration of Fe + with Mn04 the reaction is... 

The most important class of redox indicators, however, are substances that do not participate in the redox titration, but whose oxidized and reduced forms differ in color. When added to a solution containing the analyte, the indicator imparts a color that depends on the solution s electrochemical potential. Since the indicator changes color in response to the electrochemical potential, and not to the presence or absence of a specific species, these compounds are called general redox indicators. 

Laboratory experiments have shown that IGSCC can be mitigated if the electrochemical potential (ECP) could be decreased to —0.230 V on the standard hydrogen electrode (SHE) scale in water with a conductivity of 0.3 ]lS/cm (22). This has also been demonstrated in operating plants. Equipment has been developed to monitor ECP in the recirculation line and in strategic places such as the core top and core bottom, in the reactor vessel during power operation. 

Practical developers must possess good image discrimination that is, rapid reaction with exposed silver haUde, but slow reaction with unexposed grains. This is possible because the silver of the latent image provides a conducting site where the developer can easily give up its electrons, but requires that the electrochemical potential of the developer be properly poised. For most systems, this means a developer overpotential of between —40 to +50 mV vs the normal hydrogen electrode. 

The two dashed lines in the upper left hand corner of the Evans diagram represent the electrochemical potential vs electrochemical reaction rate (expressed as current density) for the oxidation and the reduction form of the hydrogen reaction. At point A the two are equal, ie, at equiUbrium, and the potential is therefore the equiUbrium potential, for the specific conditions involved. Note that the reaction kinetics are linear on these axes. The change in potential for each decade of log current density is referred to as the Tafel slope (12). Electrochemical reactions often exhibit this behavior and a common Tafel slope for the analysis of corrosion problems is 100 millivolts per decade of log current (1). A more detailed treatment of Tafel slopes can be found elsewhere (4,13,14). 

Determining the cell potential requites knowledge of the thermodynamic and transport properties of the system. The analysis of the thermodynamics of electrochemical systems is analogous to that of neutral systems. Eor ionic species, however, the electrochemical potential replaces the chemical potential (1). 

The electrochemical potential, )T, of a species is a function of the electrical state as well as temperature, pressure, and composition is the absolute activity, which can be broken down into three parts as shown. Eor an electrolyte. A, which dissociates into cations and v anions, the chemical potential of the electrolyte can be expressed by... 

At open-circuit, the current in the cell is 2ero, and species in adjoining phases are in equilibrium. Eor example, the electrochemical potential of electrons in phases d and P are identical. Furthermore, the two electrochemical reactions are equilibrated. Thus,... 

The driving force for the transport of all particles is a change in the electrochemical potential /i, which is related to the partial molar free enthalpy /i, and the electric potential 0 as follows ... 

For saturated hydrocarbons, exchange is too slow and reference points are so uncertain that direct determination of pAT values by exchange measurements is not feasible. The most useful proach to obtain pK data for such hydrocarbons involves making a measurement of the electrochemical potential for the reaction... 

Consider a phospholipid vesicle containing 10 mMNa ions. The vesicle is bathed in a solution that contains 52 mMNa ions, and the electrical potential difference across the vesicle membrane Ai/t = i/toutskie / inside = —30 mV. What is the electrochemical potential at 25°C for Na ions ... 

As shown in Figure 3.6-1, GC and Pt exhibit anodic and cathodic potential limits that differ by several tenths of volts. However, somewhat fortuitously, the electrochemical potential windows for both electrodes in this ionic liquid come out to be 4.7 V. What is also apparent from Figure 3.6-1 is that the GC electrode exhibits no significant background currents until the anodic and cathodic potential limits are reached, while the Pt working electrode shows several significant electrochemical processes prior to the potential limits. This observed difference is most probably due to trace amounts of water in the ionic liquid, which is electrochemically active on Pt but not on GC (vide supra). 

In aqueous solutions it becomes somewhat more feasible to modify the entry of hydrogen into the steel. This can be achieved by the addition of inhibitors to the solution, by control of the electrochemical potential of the metal and by coatings. 

---