15.1.1 Defining the terms of electrochemistry

Electrochemical reactions involve the transfer of free charge (electrons). Because of conservation of mass and charge, the transfer of electrons must always balance – the number of donated electrons must equal the number of accepted electrons. Therefore, an electrochemical reaction must always consist of both oxidation and reduction.

• Oxidation is the removal of electrons from a species (oxidation state increases). Oxidation occurs at the anode. D^x → D^{x + 1} + e⁻.
  
• Reduction is the addition of electrons to a species (oxidation state is reduced). Reduction occurs at the cathode. A^x + e⁻ → A^{x − 1}.
  
• A galvanic cell is an electrochemical cell in which a spontaneous reaction leads to the production of electricity. Chemical energy is converted to electrical energy.
  
• An electrolytic cell is an electrochemical cell in which a nonspontaneous reaction is driven by an external source of current. Electrical energy is converted to chemical energy.

An electrochemical cell, as depicted in Fig. 15.2, contains an anode and a cathode. These terms only apply to electrodes through which a net current flows. The anode is the electrode at which the predominating electrochemical reaction is oxidation. Anode oxidation. Electrons are produced spontaneously at the anode of a galvanic cell, and are extracted from the anode of an electrolytic cell. The cathode is the electrode at which the predominating electrochemical reaction is reduction. Cathode reduction. Electrons produced from the anode are consumed at the cathode.

The signs of the anode and the cathode change depending on whether the cell is galvanic or electrolytic, just as the reaction changes from spontaneous to forced. Electrons are negatively charged and flow from low potential to high potential. The spontaneous direction of flow is for electrons to move to the more positive electrode. In a galvanic cell electrons flow from the anode (the negative electrode) to the cathode (the positive electrode). In an electrolytic cell electrons are extracted from the anode (the positive electrode) and flow to the cathode (the negative electrode). The electrons are extracted from the anode by applying an appropriate voltage (bias) with an external power supply.

The electrode at which the reaction we are studying is occurring is called the working (or indicator) electrode. This is opposed to a reference electrode, which is an electrode that is constructed to have a constant potential and through which no current should flow. The electrodes are connected by an electrolyte that conducts ions. If the electrodes are in different compartments, they may be connected by a salt bridge (often concentrated KCl in agar jelly), which allows electrical contact but avoids mixing (in other words, ions can flow through the salt bridge but not the solvent).

The redox reaction can be expressed as the difference between two half-reactions. The oxidizing agent (oxidant) is an electron acceptor, while the reducing agent (reductant) is an electron donor. The reduced and oxidized species in a half-reaction form a redox couple. The couple is written Ox/Red and the corresponding reduction half-reaction is

(15.1)

The constant z is the electron number of the electrochemical reaction. It is also called the charge number or valence of the reaction.

By combining a series of half-reactions, we can write a balanced overall equation. Even if the reaction does not occur electrochemically, the electrochemical pathway may be useful for interpreting or calculating state functions because, of course, these are path-independent. As examples, consider the dissolution of AgCl(s), which can be modeled as the reduction of AgCl(s)

(15.2)

and the reverse of the reduction of Ag⁺(aq) (i.e., the oxidation of Ag(s))

(15.3)

to give the overall reaction

(15.4)

or the reduction of O₂(g) with protons

(15.5)

with the oxidation of H₂(g)

(15.6)

which, when added stoichiometrically to balance the number of electrons lost with the number of electrons gained, gives

(15.7)

Electrons appear as a reactant or product in a half-reaction. A reduction half-reaction is the addition of one or more electrons to the oxidant. No electrons appear as a reactant or product in a balanced overall electrochemical reaction. The charge on both sides of any reaction must be balanced. For an overall reaction, there can be no net gain or loss of electrons.

When current flows from the electrode to a solution-phase species, we say that a reduction current flows; this is also called a cathodic current. The flow of electrons from solution to electrode is an oxidation current; this is also called an anodic current. Which current is taken as positive is a matter of choice, and when reading the literature one should always check which convention the author has chosen. Here, we will take anodic currents as positive and cathodic currents as negative as is preferred by IUPAC.

15.1.2 Relating current flow to amount of substance reacted: Faraday’s law

We write a reaction quotient, which we denote Q_r, to distinguish it from the electrical charge Q, for the chemical species in a half-reaction just as we would for any other reaction. For instance, for the following half-reaction

(15.8)

and assuming O₂ to act as an ideal gas, the reaction quotient is

(15.9)

Michael Faraday’s law of electrolysis helps us to understand electrochemistry quantitatively. The mass of species produced at an electrode is proportional to the quantity of electricity Q passed through the electrolyte. Electricity is quantified in coulombs (C), and is defined as the current I in amperes (abbreviated A) multiplied by the time t in seconds (abbreviated s). The mass of a species liberated at an electrode is proportional to the product of the amount of substance (moles) of the species and its charge. The magnitude of the charge on a single electron is e = 1.602 × 10^–19 C. The proportionality factor is called the Faraday constant, which gives the charge on 1 mol of electrons,

(15.10)

The charge on a mole of ions with charge z is given by zF. Thus, the quantity of charge required to convert n moles of a species according to a half-reaction that has an electron number z is

(15.11)

15.1.2.1 Example

Electrolysis of an aqueous solution of Au(NO₃)₃ deposits 1.200 g of Au at the cathode after passing a current of 25.0 mA. The counter-reaction is the electrolysis of water. Calculate (a) the quantity of electricity passed (the total charge). (b) How long it took to carry out the electrolysis. (c) The volume of O₂ liberated at SATP (also called NSTP, 1 atm, 25 °C) at the anode.

1. Start with a balanced reaction. The half-reactions are

The overall balanced reaction is found by balancing the number of electrons. If 12 e^– are consumed in the reduction (that is, four times the first half-reaction), then an equal number of 12 e^– will be consumed in the oxidation (that is three times the second half-reaction).

Therefore 3 mol electrons lead to the deposition of 1 mol Au. Or 1 mol of electrons leads to the deposition of 1/3 mol of Au. The electron number in the electrolysis of Au³⁺ is z = 3. The total charge required to deposit 1.200 g Au is then

1. Note that 1 C = 1 A s.
   
2. 4 mol of electrons lead to the liberation of 1 mol of O₂, or 1 mol of electrons lead to the formation of ¼ mol O₂. z now equals 4.

The volume at NSTP is then

Reactions that involve charge transfer such as the reduction of Au³⁺ and the oxidation of H₂O shown in this example are called faradaic processes. Faradaic processes (electrochemical reactions) follow Faraday’s law.

In a faradic process, the amount of chemical reaction is directly proportional to the amount of current passed, Q = It = znF.

Current is the rate of charge flow

(15.12)

Therefore, Faraday’s law shows us that rate of reaction is directly proportional to the current

(15.13)

However, because electrode reactions are heterogeneous reactions, their rate depends on the surface area of the electrode A. Therefore, we introduce the current density j

(15.14)

and express the rate in units of mol m^–2 s^–1 as

(15.15)

There are also nonfaradaic processes – that is, processes in which current flows but no electrochemistry happens. These are processes in which the electrode/solution interface acts as a capacitor. The interfacial region can store charge without causing electrochemistry. To understand this we need to understand something about the structure of the electrode/solution interface.

15.1.3 The electrical double layer

Let us consider specifically the interaction of water and aqueous solutions with a metal surface. Water is a dipolar molecule with a negative charge that is preferentially displaced towards the O atom, and a corresponding amount of positive charge on the opposite side of the molecule. When a dipole approaches a metal surface the electrons in the surface rearrange to lower the energy of the system. Or, looked at from the perspective of the water molecule, if the surface has an excess of negative charge on it the positive H-side of the molecule will preferentially point towards the surface. If the surface has an excess of positive charge, the negative O-end of the molecule will preferentially point towards the surface. The orientation of the molecular water dipoles near the surface responds to the potential placed on the surface.

Anions and cations in an aqueous electrolyte also respond to the surface charge of the metal electrode. In addition, we need to consider the solvation shell of the solvated ions. The ions may interact with the surface with or without loss of their solvation shell. When a solvated ion or molecule approaches a surface and binds to it, but does not lose its solvation shell, it is nonspecifically adsorbed. When a solvated species binds and loses all (or part) of its solvation shell it is specifically adsorbed. If the surface is biased negative, it will attract cations, but if biased positive it will attract anions. At a certain potential, known as the potential of zero charge (pzc), it will attract both equally and the interface is neutral.

The solution must always be neutral overall. Therefore, if the interface has acquired a charge (i.e., a surface charge layer) there must be a layer adjacent to it (i.e., the diffuse charge layer) that contains the opposite charge in order to maintain neutrality. This region (i.e., the combination of the surface charge layer and the diffuse charge layer) is known as the electrical double layer. A charged interfacial region is depicted in Fig. 15.3. The surface charge layer is also called the Stern layer and lies closest to the surface. The diffuse charge layer is also called the Gouy layer and is situated on top of the Stern layer. The Helmholtz layer is defined by the plane that contains a layer of fully or partially solvated ions. Whether ions are specifically or nonspecifically adsorbed depends on the electric field strength and the chemical identity of the ions. The outer Helmholtz layer is defined by the plane of fully solvated ions, whereas an inner Helmholtz plane is defined by the plane that passes through the center of specifically adsorbed ions. Depending on the extent to which the charge of the first layer is compensated by excess charge at the solid surface, the diffuse charge layer is composed either of like-charged ions (full compensation), counterions (no compensation), or a mixture of the two.

15.2.1 Why does an electron ‘hop’?

Any chemical system evolves to lower its Gibbs energy; that is, dG < 0 defines the way forward for a reaction as long as there is no kinetic barrier to progression. During electrochemistry, free charge is exchanged. The electron has a negative charge and therefore it is influenced by the presence of an electric field. We need to take this into account in our treatment of the reaction. We do so by introducing the electrochemical potential, which adds a component proportional to the electric field strength to the chemical potential. We will do that in the next section. Now, we want to think about why an electron ‘hops’ from one species to the next. The answer must be that it lowers the Gibbs energy of the system to move an electron from an occupied orbital on one species to an unoccupied (or partially occupied but not a fully occupied) orbital on another species. We shall learn shortly that electrons are transferred through a transition state in which the electron donor and the electron acceptor are degenerate (have the same energy) in terms of their Gibbs energy. We shall also see that we can change the relative energy of the states in an electrode compared to a species in solution by changing the electrical potential of that electrode. However, because an electron has a negative charge the sign of the electrical potential scale is the opposite to that of the energy scale. That is, electrons move spontaneously from high-energy states to lower-energy states. This corresponds to moving from states of some given electrical potential to states with more positive electrical potential.

This is illustrated in Fig. 15.4 for two different cases. In panel (a) a metal electrode is in contact with a liquid electrolyte solution. The metal has electronic energy levels filled up to the Fermi energy E_F. Below this energy, the electronic energy levels of the metal are full, but above it they are empty. The frontier orbitals of the solution-phase species line up with the Fermi energy of the metal in a manner consistent with the electronic structure of that species, the electronic structure of the metal, and the electrical potential applied to the metal electrode. A little care must be used in interpreting Fig. 15.4 in that we need to consider Gibbs energy and not just electronic energy, but don’t let this important subtlety distract us for the moment.

If a positive potential is applied to the metal, the electrochemical potential of the electrons in the metal moves down. If the electrical potential is made positive enough, an electron can be transferred from an occupied orbital in the solution-phase species to the electrode (anodic current). If the electrode is biased to a negative potential, then when this potential is negative enough, the electrochemical potential of the electrons in the metal will exceed the electrochemical potential of the unoccupied orbital in the solution species. An electron can then be transferred from the metal to the acceptor in solution (cathodic current). Note how the role of donor and acceptor can be changed based on the applied electrical potential.

The situation is somewhat more complicated if we introduce a semiconductor rather than a metal into the solution as our electrode. The presence of a band gap means that there are no electronic states at E_F in the semiconductor. Instead, there is a (nominally) full valence band below E_F that is filled up to the valence band maximum E_V. A (nominally) empty conduction band extends down in electronic energy to the conduction band minimum at E_C. An orbital on the solution species can exchange electrons with either the conduction band or the valence band. However, as always, electron transfer must occur from an occupied state to a less than fully occupied state. The presence of a band gap, as well as a phenomenon called ‘band bending’ adds considerable complexity to semiconductor electrochemistry. Therefore, we will always assume that the electrodes we are dealing with are metals, unless we explicitly specify otherwise. Nonetheless, semiconductor electrochemistry and photoelectrochemistry are increasingly important areas of research and technology. They represent important methods of nanomaterial production, as well as being relevant to solar cell (photovoltaic) and solar fuels technologies.

This discussion demonstrates that there are potentials at which we expect a reduction current to flow (working electrode to solution electron transfer), potentials at which an oxidation current flows (solution to electrode electron transfer), and potentials in between at which negligible net current flows. This is shown in the Fig. 15.5, which displays a current–potential curve.

Figure 15.5 makes it clear that current flows in response to changes in the potential placed upon the electrodes. The reactions that occur also depend on the magnitude and sign of the potential. But where does the electron transfer occur? A clue is found when we change the Pt electrode for a Hg electrode in the system shown in Fig. 15.6. H₂ is produced at 0 V in an ideal hydrogen electrode constructed with Pt (assuming, of course, that it is attached to a counter-electrode, not shown). No H₂ production is found near 0 V in the presence of a Hg electrode. Instead, a much greater overpotential must be applied in order to overcome a substantial activation barrier for the reduction of H⁺. The overpotential is defined as the potential η in excess of the calculated reversible cell potential E_rev that would allow for reaction at equilibrium compared to the cell potential E that is required to actually make the reaction occur,

(15.16)

This behavior would not be expected if the reaction occurred in solution. In fact, solvated electrons have a very short lifetime in aqueous solutions. Charge exchange is occurring predominantly on the surface of the electrode. The general case of heterogeneous electron transfer follows the dynamics that can be described in the following manner. A solvated solution-phase species diffuses to a surface, and then either specifically or nonspecifically adsorbs onto the surface of the electrode. Specific absorption is when the solvated species loses part or all of its solvation shell. Nonspecific adsorption is when the adsorbed species maintains its solvation shell. Electron transfer occurs in the adsorbed phase. Before or after charge transfer there may be additional surface reactions that occur. After charge transfer (and perhaps after subsequent reactions) a reacted species desorbs from the surface; this species may be the final reaction product or further solution-phase chemistry may be required before the final reaction product is obtained.

15.2.2 Relating the electrochemical potential to the chemical potential

Work is done to move charge across a potential difference. The amount of work it takes to reversibly transfer dQ of charge (dQ = –zF dn) from a phase at potential φ₁ to a phase at φ₂ is

(15.17)

The work performed is reversible, nonexpansion work; therefore, it is equal to the change in Gibbs energy,

(15.18)

The change in Gibbs energy also represents the difference in the electrochemical potential of the charged particle in the two phases

(15.19)

The electrochemical potential, introduced by John A.V. Butler² and Edward A. Guggenheim,³ is the sum of the chemical potential and a term that includes the effect of the electric field. The electrochemical potential is defined as

(15.20)

Experimentally, we directly measure the difference in potentials rather than an absolute potential. Thus, if we set φ₁ to zero, we find that the electrochemical potential in phase 2 is related to the electrochemical potential in phase 1 by

(15.21)

Charged particles in otherwise identical phases that have different chemical potentials in the two phases are at different electrical potentials. This is the basis for understanding all of electrochemistry and its utility because we can easily change the electrical potential. The equilibrium condition can be written in terms of the electrochemical potential as

(15.22)

The convention is to choose the standard state of the solution (and all ions in solution) to be at φ = 0. With this choice of standard state, the electrochemical potential of any ion in solution is equal to its chemical potential,

(15.23)

We take the standard state chemical potential of an electron in a metal electrode at zero potential to be zero. Thus, the electron’s electrochemical potential at any potential φ is given by

(15.24)

The standard hydrogen electrode (SHE), depicted in Fig. 15.6, is used as a reference electrode against which all standard electrode potentials are measured. It is also called the normal hydrogen electrode (NHE) with a corresponding half-reaction

(15.25)

The equilibrium half-cell is described by

(15.26)

Half-cell reactions are conventionally written as reductions. Now expand Eq. (15.26) in terms of activities and fugacity

(15.27)

Now we recognize that the potential φ(H⁺/H₂) in Eq. (15.27) is what we call the cell potential, which according to convention is denoted E(H⁺/H₂). Solving for the cell potential, we obtain

(15.28)

If both the activity of H⁺ and the fugacity of H₂ are equal to 1, the cell potential has its standard-state potential. We denote this standard cell potential E°(H⁺/H₂). By definition, the standard chemical potential of H₂ is zero (it is an element in its standard state), , thus,

(15.29)

By convention, the Gibbs energy of formation for H⁺ set to zero,

(15.30)

Therefore, this convention combined with Eq. (15.29) means that the standard potential of the hydrogen electrode must also be zero,

(15.31)

15.4.1 The Nernst equation

We now relate the emf to composition. For the Daniell cell above, the cell reaction is

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