2.1 Electrochemical Reactions

An electrochemical reaction is a reaction where the transfer of electrons from a species being oxidized to a species undergoing reduction takes place through an electronic conductor. Typically, that conductor is a metal. Because the electron transfer takes place through a conductor rather than directly between the reacting species, we can separate the two electron-transfer reactions and use the flow of electrons (current) between them to do work. The oxidation or anodic reaction takes place at the anode, and the reduction reaction or cathodic reaction takes place at the cathode. An example of an anodic reaction is

where 1 mol of electrons is produced for every mole of Fe(II) that is oxidized. An example of a cathodic reaction is

Both an oxidation reaction and a reduction reaction are required to make an electrochemical cell. Therefore, an anodic or a cathodic reaction is referred to as a half-cell reaction as described in Chapter 1. The full-cell reaction is obtained by adding two half-cell reactions. For example, the two reactions above can be added to yield the following full-cell reaction:

Note that there are no net electrons in the full-cell reaction, and that the charge on both sides of the reaction must balance. Identifying and understanding these half-cell reactions is essential to the analysis of electrochemical systems. Common reactions are tabulated in Appendix A.

2.2 Cell Potential

At open circuit, no current flows between the electrodes. Furthermore, for our thermodynamic analysis, each electrode half-cell reaction is at equilibrium (no net anodic or cathodic reaction occurs at either electrode). Under these conditions, it is appropriate to write the half-cell reactions as reversible reactions. Since electrons are participants in each of the two reactions, the energy of the electrons in each conductor is determined by the reaction equilibrium at that electrode, and the electrical potential is different for each electrode. The difference in energy of the electrons at the two electrodes can be easily measured with a voltmeter as a voltage or potential difference. That potential difference at equilibrium is the principal topic of this chapter. This potential is also called the thermodynamic potential, U, since the half-cell reactions are at equilibrium.

Before proceeding with the analysis of electrochemical systems, it is productive to consider a simple, chemical-reacting system. A common exercise is to determine whether a reaction will occur spontaneously.

The criterion used is the sign of the change in Gibbs energy for the reaction as written, namely,

where ΔG_f is the Gibbs energy of formation of the compound. If the change in Gibbs energy of the reaction, ΔG_Rx, is less than zero, then the reaction is said to proceed spontaneously. The first thing to note is that the sign of the change in Gibbs energy for the reaction depends on how the reaction is written, more specifically what is considered a product and what is treated as a reactant. Thus, it is more precise to say that the reaction will occur spontaneously as written. If the reaction is spontaneous, then we can obtain work from the reaction. If it is not spontaneous as written, then we would need to add work to force the reaction to go in that direction.

You are undoubtedly familiar with the reaction of hydrogen and oxygen, which combust to form water.

Using values from Appendix C for the Gibbs energy of formation for the reactants and products at their standard states, a large negative ΔG_Rx is calculated. Thus, we conclude that the reaction as written will occur spontaneously, which is consistent with our expectations. Now of course, pure hydrogen and oxygen present together at near atmospheric pressure doesn't sound like the true thermodynamic equilibrium we seek. In fact, if these three species were together in equilibrium, we would find almost all water and only trace amounts of oxygen and hydrogen gas in our system. At this equilibrium state, the reactants and products are far from their standard states, and the change in Gibbs energy would be zero.

A hallmark of equilibrium of electrochemical systems is that there are two electrodes and at least one of the species is absent from each electrode. Furthermore, as noted earlier, electrons are involved in the half-cell reactions and, while the circuit is open, no electrons can be transferred between the two electrodes. Therefore, the half-cell reaction at each electrode will reach a dynamic equilibrium involving electrons where the rates of the forward and reverse reactions at that electrode are equal and there is no net current. The energy of the electrons for each of the half-cell reactions at equilibrium is different; this difference in electron energy, characterized by the potential difference, is what drives the full-cell reaction when the circuit is closed.

The same overall reaction of hydrogen and oxygen described above can occur electrochemically with two half-cell reactions. For instance,

and

Notably, hydrogen is absent from the oxygen electrode, and oxygen is missing from the hydrogen electrode. At open circuit (i.e., no external current flow, and allowing sufficient time for equilibrium to be established), we can use thermodynamics to develop a relationship between the equilibrium cell potential, U, and the thermodynamic state of the reactants and products. In contrast to the combustion situation, it is possible for equilibrium to exist with significant amounts of hydrogen, oxygen, and water all present in the same electrochemical cell. This is because the hydrogen and oxygen are separated, with the hydrogen reaction in equilibrium at one electrode and the oxygen reaction in equilibrium at the other. In fact, we can now vary the amount of hydrogen, oxygen, and water independent of each other. However, the equilibrium potential of the cell, U, changes, and it is precisely this change in potential that is of interest here.

The laws of thermodynamics for a closed reversible system at constant temperature and pressure tell us that the maximum work that can be done by the system on the surroundings is equal to the change in the Gibbs energy per mol, ΔG [J·mol⁻¹]. In the electrochemical system of interest, that work is electrical work. In other words,

(2.1)

where work done by the system on the surroundings is defined as positive.

The cell potential, U, has units of joules per coulomb (J·C⁻¹, which is also V) and represents the work per unit charge required to move charge from one electrode to the other. The amount of charge transferred in the full reaction is equal to nF, where n is the number of moles of electrons in the reaction as written, and F is Faraday's constant, the number of coulombs per mole of electrons. The electrical work is therefore

(2.2)

Combining (Equations 2.1) and (2.2) yields

(2.3)

Equation 2.3 applies to a full cell consisting of an anodic and cathodic reaction, and relates the change in Gibbs energy for the chemical species to the difference in the electron energy at the two electrodes. Generally, we treat the potential of a full cell, U, as the potential difference between the more positive electrode and the more negative electrode. When this is done, U is positive and ΔG is negative for a full electrochemical cell at equilibrium. As described previously, a negative change in Gibbs energy implies a spontaneous reaction. This is in fact correct; when the circuit is closed for any cell at equilibrium that has a nonzero potential, current flows spontaneously. In order to ascertain the reactions that are occurring and the direction of the current, we simply choose the direction of the reaction such that the change in Gibbs energy is negative. In our example with hydrogen and oxygen, writing the reaction as

resulted in a negative change in Gibbs energy, indicating that at one electrode hydrogen is oxidized (anode), and at the other electrode oxygen is reduced (cathode) to produce water when the circuit is closed. We now turn our attention to calculating the equilibrium potential of cells before the circuit is completed; i.e., at open circuit.

2.3 Expression for Cell Potential

Note that Equation 2.3 relates the Gibbs energy to the cell potential, U, which is the difference between the potential of the two electrodes. From a practical standpoint, that potential will be positive if you measure it one way, and negative if you switch the wires and measure it again. The fact that the potential can be measured either way illustrates the need to establish a convention so that the connection between the half-cell reactions, the cell potential, and the direction of the full-cell reaction is clear. To do this, we use a diagram similar to that illustrated in Figure 2.1 for a Daniell cell consisting of a zinc electrode in a zinc sulfate solution on one side of a semipermeable separator, and a Cu electrode in a copper sulfate solution on the other. The physical device (left side) is represented as the cell (right side) for thermodynamic analysis.

The vertical lines in the cell diagram indicate phase boundaries. In this example, we have four phase boundaries and five phases. On the left is zinc metal. The zinc electrode is in contact with an aqueous solution of zinc sulfate, which does not conduct electrons but allows for ions to move between the two metal electrodes. For simplicity, here we assume that the zinc sulfate solution is separated from the copper sulfate solution by a semipermeable membrane. This membrane allows for the transport of sulfate ions, but excludes all other species. Recall that at least one species is absent from each electrode. At the interface between the Zn and the solution of zinc sulfate, copper ions are absent. On the other side, zinc ions are absent. On the right is the copper electrode, which is more positive at standard conditions as we'll see shortly. The two electrochemical half-cell reactions are

(2.4)

and

(2.5)

The bidirectional arrow is used to emphasize the fact that each half-cell reaction is at equilibrium. By convention, the equilibrium potential of the cell, U, is equal to the potential of the electrode on the right minus the potential of the electrode on the left, or

(2.6)

Since we generally define U so that it is positive, we put the more positive electrode on the right and the negative electrode on the left as shown in Figure 2.1. At this point, it is instructive to consider briefly what happens when the circuit is closed and a small amount of current is allowed to flow. Electrons naturally flow from low to high potential because of their negative charge. Therefore, if U is positive, the electrons would flow from the electrode on the left to the electrode on the right when the circuit is closed. Applying this knowledge to the cell considered above, we see that oxidation is occurring at the left electrode; that is, the left electrode is giving up electrons. Conversely, reduction is occurring at the right electrode:

(2.7)

and

(2.8)

Thus, the spontaneous path is for zinc to be oxidized at the electrode on the left, and copper to be reduced at the other electrode.

(2.9)

In other words, the direction of the spontaneous full-cell reaction that correctly corresponds to the cell potential is obtained by writing the reaction on the right as the cathodic reaction and that on the left as the anodic reaction. We now have a well-defined connection between the half-cell reactions, the overall or full reaction for the cell, and the equilibrium cell potential. In addition, Equation 2.3 provides a relationship between the Gibbs energy of the reaction and the equilibrium potential of the cell.

How can we use this information to determine the equilibrium potential of the cell? To answer this question, we next write the following expression for ΔG:

(2.10)

where is the Gibbs energy change for the reaction at standard conditions, a_i is the activity of species i, and s_i is the stoichiometric coefficient of species i in the reaction (positive for products and negative for reactants) at the conditions of the cell. Because it is infeasible to measure and list the Gibbs energy for every reaction under every set of conditions, we tabulate the Gibbs energy at an arbitrary standard condition (25 °C, 1 bar) for species in a reference state (more on this later), and then correct this value for the particular conditions of interest. Combining Equation 2.10 with Equation 2.3 yields

(2.11)

where is the standard potential for the full cell () at 25 °C and is defined at the same standard conditions as . Also, similar to , is not an absolute quantity, but a difference or relative value. One way to determine the standard potential for the full cell is to calculate it from the standard change in Gibbs energy for the reaction. However, it is often easier to use tabulated values of the standard potential for the reactions of interest as described in the next section.

2.4 Standard Potentials

For convenience, values of standard potentials have been determined and tabulated for a large number of electrochemical reactions, a subset of which is found in Appendix A. Although listed as half-cell reactions, the potentials represent the difference between the potential of the reaction of interest and a reference reaction. The universal reference is the standard hydrogen electrode (SHE). Hence, the standard potential for the hydrogen reaction is defined to be zero. Tables of standard potentials are typically written as cathodic (reduction) reactions. Be mindful that for our thermodynamic analysis these reactions are assumed to be in equilibrium. Reactions whose standard potentials are positive relative to hydrogen naturally act as cathodes when coupled with hydrogen. In contrast, reactions with negative potentials are more anodic than hydrogen. A table such as that found in Appendix A is also a good place to start when you are unsure of how a particular compound might react. For example, the number of electrons transferred and the products produced can be readily determined from the table for a variety of elements.

The standard potential for the overall reaction can be obtained from the standard potentials of the two half-cell reactions (each relative to SHE) as follows:

(2.12)

where the “right” and “left” refer to the cell diagram similar to that illustrated in Figure 2.1b. Since the values of for the two half-cell reactions are both relative to hydrogen, the influence of the reference subtracts out, and we are left with the desired quantity.

Note that in the second example shown in the illustration it was necessary to multiply the stoichiometry of the negative reaction by two in order to get the electrons to balance and yield the correct overall reaction. However, we did not multiply the standard potential by the same factor. This practice may seem counterintuitive, but is correct. The standard potential represents the energy per charge, which does not change when you multiply the equation by the factor needed to make the electrons balance. The Gibbs energy changes because more electrons are involved, but the value per charge remains the same. This invariability will persist as long as the ratio of the number of electrons per mole(s) of reacting species remains the same. In other words, it doesn't matter if you have 2 electrons per mole of zinc, 4 electrons per 2 mol of zinc, or 5000 electrons per 2500 mol of zinc; the energy per unit charge is the same. That ratio always remains constant for the situation that we have considered involving a balanced anodic and cathodic reaction. Reworking the problem above with use of helps illustrate the point:

As you can see, the total number of electrons for the anodic and cathodic reactions is the same, which results in equal to the difference between that of the two half-cell values (Equation 2.12).

In the above example, it appears that we used Equation 2.3 to describe of the half-cell reactions. However, as apparent from its derivation, Equation 2.3 only applies to full-cell reactions where the electrons are balanced; this equation relates the Gibbs energy change of the chemical species to the electrical work resulting from the difference in the electron energies that correspond to each of the half-cell reactions at equilibrium. Implicit in the use of Equation 2.3 for a “half-cell” is the fact that the other half is the standard hydrogen electrode, whose Gibbs energy is, by definition, equal to zero. The hydrogen portion simply subtracts out when we calculate as shown above. Subtracting off a balanced full-cell reaction does indeed reverse the sign of its as shown in the example. In contrast to the above, calculation of the Gibbs energy change for the half-cell alone must involve the chemical potential of the electrons as demonstrated in Section 2.12.

Half-Cell Potentials Not in Table

What do we do if the reaction that we need is not in the table of standard potentials? One possibility is to combine existing half-cell reactions to create the desired half-cell reaction. In doing so, we always go through the Gibbs energy route (just demonstrated above) in order to avoid errors. The creation of a new half-cell reaction will of necessity not balance electrons since we want electrons in the final expression.

To illustrate:

To get the desired reaction, we subtract Reaction 1 from Reaction 2:

As you can see, the standard potential for the new half-cell reaction is not just the difference between half-cell potentials of the reactions that were combined. The reason for this adjustment is that the Gibbs energies above actually correspond to full-cell values with hydrogen on the left as the anode. In cases where we have electrons left over, the hydrogen reaction does not cancel out as it does when the electrons balance, and we are left with the difference between the half-cell of interest and the hydrogen electrode, consistent with other standard potentials. Consequently, it is important to use Gibbs energies when combining half-cell reactions to form a new half-cell reaction.

Standard Potential from Thermodynamic Data

Standard potentials for the cell can also be calculated directly from Gibbs energy data using Equation 2.3. The standard Gibbs energy of formation is tabulated for compounds in different states (gas, liquid, aqueous, and solid) assuming they are formed from the elements taken at 25 °C, 1 bar pressure. For a cell reaction,

(2.13)

where s_i are the stoichiometric coefficients for the reactants and products. By convention, s_i is positive for products and negative for reactants. As we have seen above, reactants and products of electrochemical reactions usually include ions. Because of the definition of the standard state for ions in electrolyte systems, which includes only interactions between the solvent and the ion of interest, the standard state Gibbs energy does not depend on the counterion(s) in the system. Rather, the impact of the counterions is included in the activity term. Therefore, tabulated values of for the ions involved in the reaction should be used where available (see Appendix C), and are frequently tabulated for aqueous systems. When single-ion values are not available, the difference in the Gibbs energy of formation for ionic species can be calculated from for two neutral aqueous (aq) species with the same counterion. For the dissociation of a neutral species,

and from Equation 2.13

(2.14)

Therefore, the counterion portion will cancel out when the aqueous values are subtracted (see Illustration 2.2). Finally, a combination of and S^° data can be used to estimate . The following illustration demonstrates how to determine for each of the situations mentioned above.

2.5 Effect of Temperature on Standard Potential

The cell potential in the relationships shown previously was determined at 25 °C, where the standard values have been tabulated. However, it is often desirable to calculate the cell voltage at a different temperature. To do so, we need the standard potential at that temperature. This section describes how to correct the standard potential to the temperature of interest. Beginning with one of the fundamental relations from thermodynamics,

(2.15)

we find that

(2.16a)

If we further assume that does not change significantly over the temperature range of interest, it follows that

(2.17)

and

(2.18)

Tabulated values of are available in the literature. If the entropy change varies significantly with temperature over the range of interest, then integration of the temperature-dependent entropy term may be required as follows:

(2.19)

Once the standard potential is known at the new temperature, activity corrections can be made as usual. This process is illustrated in the example at the end of the next section.

The following alternative to Equation 2.16 is equally rigorous:

(2.16b)

This approach has an advantage when the temperature range is sufficiently large that the assumption of constant ΔS or ΔH is not valid. If constant pressure heat capacity data (C_p) are available, then the change in enthalpy can be calculated as a function of temperature,

(2.20)

See Problem 2.29 to explore further this method.

2.6 Simplified Activity Correction

Now that we have a way of obtaining for use in Equation 2.11, we must add the activity correction to get the desired expression for U. A discussion of activities and standard states for electrolytes is provided later in Section 2.14. Activity is a dimensionless quantity that depends on the standard state for each species. As a first approximation, we simply use the following:

(2.21a)

(2.21b)

(2.21c)

(2.21d)

The standard state for gaseous species is an ideal gas at 1 bar, and for ions in solution it is an ideal 1 molal solution. As a first approximation, we will use concentration as a proxy for molality. With these assumptions, Equation 2.11 becomes

(2.22)

where s_i is positive for products and negative for reactants. This equation is similar to the classical Nernst equation. The assumptions implicit therein are those most frequently used to approximate cell potential. Note that because of our assumption of unit activity for any solid reactants and the solvent, these species do not appear in Equation 2.22. The following illustration demonstrates the use of this equation and the process developed above to calculate the cell potential.

Our approach to calculating the cell potential in the above illustration, consistent with the discussion in the chapter to this point, has been the following:

1. Write down the two half-cell reactions of interest and the standard potential for each reaction.
2. Put the reaction with the most positive standard potential on right side of the electrochemical cell as the cathode. The other reaction is the anodic reaction.
3. Write the full-cell reaction that appropriately combines the cathodic and anodic reactions. Make sure that the electrons are balanced.
4. Take the difference between the two standard half-cell potentials to get the standard potential for the full cell (cathode potential – anode potential).
5. Use the simplified activity corrections to correct the cell potential with the products on the top and the reactants on the bottom. Use the full-cell reaction to get the stoichiometric coefficient for each species and the total number of electrons for the reaction.

In using this approach, it is necessary to keep track of the location of the reactant and product species in order to make the required activity corrections, since those corrections, of necessity, use the environment adjacent to the electrode of interest. For example, in Illustration 2.4, the pH on the anodic side of the cell is approximately 4, and that on the cathodic side is about 14. Therefore, we have two different pH values that might be used to make the activity correction. In the example, we correctly selected a pH of 14 since the cathodic reaction depends on pH, whereas the anodic reaction does not. In other words, the activity corrections require us to use the concentration(s) that applies locally to each half cell. An alternative approach to finding the cell potential that many students find easier to use is to first determine the potential of each half cell relative to the SHE, including activity corrections, and then take the difference between the two half-cell potentials to get the full-cell potential. This approach is analogous to that used to calculate the standard potential, but has been expanded to include the needed activity corrections. The potential of any reaction versus SHE can be found from the following equation:

(2.23)

where all of the activity corrections belong to the half-cell reaction of interest since there are no activity corrections, by definition, for the SHE. Note also that and that the half-cell reaction was assumed to be expressed as

(1.10)

For half-cell reactions that are of the simple form

(2.24)

U can be written as

(2.25)

where we have kept the standard potential for hydrogen in the equation to emphasize the fact that this potential is relative to SHE. This equation assumes that both the oxidized and reduced species are ions in solution, where the oxidized form of A is on the same side as the electrons in Equation 2.24. The reference concentrations were not explicitly included for the purpose of simplification since they are equal to unity. The following illustration demonstrates the alternative approach to calculating the full-cell potential.

In these illustrations, the activity correction was small relative to the equilibrium voltage determined under standard conditions. While this is often the case, there are situations where the corrections make a significant difference and change the conclusions that would otherwise be made.

2.7 Use of the Cell Potential

Now that we have a way of determining the equilibrium potential, let's look at how we can use it. First, the convention we have adopted tells us that when the circuit is closed, a spontaneous reaction will occur. More specifically, a cathodic or reduction reaction takes place on the positive electrode, and an anodic or oxidation reaction occurs on the negative electrode. Electrochemical cells in which the reactions take place spontaneously to produce electrical current are called galvanic cells. A battery that is discharging is an example of such a cell. The maximum work that can be extracted from such a cell is equal to nFU (see Equation 2.3). However, to achieve this maximum, the cell would need to be discharged at an infinitely slow rate to avoid irreversible losses associated with the generation and movement of current in the cell. We will learn more about current and irreversible losses later in this book. In practice, as a consequence of irreversible processes, the operating voltage of a galvanic cell during discharge is less than the equilibrium voltage. But, as you can see, the equilibrium or thermodynamic potential serves as an essential reference point for the analysis of electrochemical cells.

In contrast to galvanic cells, an electrochemical cell in which energy must be added in order to drive the reactions in the desired direction is called an electrolytic cell. The magnitude of the applied potential, which is the operating voltage of the cell, must be greater than the equilibrium potential since energy must be added to overcome the equilibrium voltage in order to generate and move current through the cell. The chlor-alkali cell in Illustration 2.4 is an example of an electrolytic cell, where the full-cell reaction corresponding to the positive cell potential was opposite the desired direction. Thus, a potential greater than 2.164 V would need to be applied to such a cell in order to produce the desired products.

2.8 Equilibrium Constants

The equilibrium constant for a chemical reaction is defined as

(2.26)

Making use of Equation 2.11, the equilibrium constant can be related to the standard potential for the cell,

(2.27)

Also, since U for a reaction at equilibrium is equal to zero (as is ),

(2.28a)

and

(2.28b)

You are probably familiar with the solubility product for a solute dissolving in water. Now we want to see how to use standard cell potential data to calculate it. To do this, let's determine the solubility product for AgCl. The relevant reaction is the following equilibrium reaction:

This reaction can be viewed as the sum of the following two half-cell reactions:

and

Therefore, at 25 °C, . From Equations 2.23 and 2.28b,

Thus, the equilibrium constant can be readily calculated for any reaction that can be written as the sum of two or more known half-cell reactions. Note that must be known at the temperature of interest.

The ratio of activities and hence concentrations (if the activity coefficients are known) can also be determined for a half-cell at equilibrium if the cell potential at open circuit, U, is known and the other electrode reaction is fully specified. In this case, the full cell is not at equilibrium.

2.9 Pourbaix Diagrams

A knowledge of the equilibrium potential for reactions involving a specified set of elements allows us to determine the species that are thermodynamically stable under a particular set of conditions. A common way to present such data in aqueous media is with a Pourbaix diagram, which has been particularly useful for studying corrosion. For example, the Pourbaix diagram for Zn at 25 °C is shown in Figure 2.2. This diagram presents a regional map of stable species as a function of the potential (versus SHE) and the pH. The construction of this diagram is outlined in the paragraphs that follow.

First, each diagram contains two reference lines (dashed in the figure) that represent the reactions for hydrogen and oxygen. For convenience, we reference reactions and equations to the corresponding lines on the Pourbaix diagram:

and

We can express the potential of these two reactions using the methodology described above. Reaction (a) relative to a standard hydrogen electrode is

since is zero. If the hydrogen pressure is unchanged (at standard pressure), the cell potential varies only with the proton concentration. Assuming a reference state of ions as 1 M, this variation is most commonly expressed as a function of the pH as follows:

at 25 °C. Note that the switch from concentration to pH required us to change to a base 10 logarithm and then apply the simplified definition of pH (−log₁₀ c, where c is in mole per liter). U_a is represented as a line on the Pourbaix diagram as shown in Figure 2.2. At potentials greater than U_a, the anodic reaction is favored, and H⁺ is the stable species. Conversely, the cathodic reaction is favored below U_a, and the stable species is H₂. The line, of course, represents the equilibrium potential.

Similarly, the potential for reaction (b) is

where the anodic evolution of oxygen takes place at potentials above U_b.

Next, consider equilibrium for the dissolution of zinc:

The equilibrium potential for the dissolution of zinc relative to SHE is

Metallic Zn is stable at potentials below U_c. Since neither H⁺ nor OH⁻ is involved in the reaction, it makes sense that the potential does not depend on pH. Also, whereas the standard equilibrium potential for the dissolution of zinc is (Appendix A), this is not the value plotted on the graph. As seen by equation (c), if the concentration of the zinc ion changes, the equilibrium potential shifts. A series of lines could be plotted for the dissolution of Zn corresponding to different concentrations of ions in solution. By convention, a concentration of 10⁻⁶ M is usually assumed. At this concentration, the potential shifts negatively to −0.94 V, which is the value plotted in Figure 2.2.

As you may begin to realize, the Pourbaix diagram and associated calculations depend on the selection of species. This choice is not always clear. Up to 16 different species have been used in just the Pourbaix diagram for zinc. We will limit ourselves to just a few of these in order to get an idea of how these diagrams are constructed and how they can be used.

Consider a different type of reaction:

How might this reaction be represented on the diagram? Since this is not an electron-transfer reaction, the reaction is not a function of potential and is therefore represented as a vertical line on the diagram. For our purposes, we use this equation to define the stability boundary between Zn²⁺ and Zn(OH)₂. Specifically, we are looking for the pH where Zn(OH)₂ is in equilibrium with 10⁻⁶ M Zn²⁺. Using the methods described in the previous section, we find

With the concentration of zinc ions specified, the pH can be determined. The calculated pH is 8.74 (see Figure 2.2). Note that the choice of zinc concentration is arbitrary and represents the concentration above which Zn²⁺ is considered to be the stable species. The specified value of 10⁻⁶ M is frequently used for the analysis of corrosion systems.

One more reaction is considered:

This equilibrium is described by

and the line is labeled e on the diagram.

Now is a good time to review the meaning of these lines. These Pourbaix diagrams indicate the regions of stability of different phases and ionic species in equilibrium with the solid phases. Referring to line c, if the potential is below −0.94 V, Zn is stable. When the potential rises above −0.94 V, dissolution of Zn occurs. At higher potentials, when the pH of the solution is increased above 8.74, zinc hydroxide precipitates. Finally, at pH values above 8.74 and potential values higher than those indicated by line e, zinc can react directly with water to form zinc hydroxide.

2.10 Cells with a Liquid Junction

The Daniell cell that was examined previously presupposed a selectively permeable membrane separator. In practice, simple porous media are often used. In this case, bulk mixing is avoided, but ions are able to move between electrodes. In fact, we often have electrochemical cells where the two electrodes are in solutions of different concentration and/or composition. The thermodynamic analysis described above accounts for the effect of the local solution composition on the equilibrium potential. However, there is also a small potential difference at open circuit associated with the junction between the two liquids of different composition. This potential difference is sometimes called the liquid junction potential, and is the topic of this section.

The liquid junction of interest is the region of varying composition between the two different electrolyte solutions. In practice, it is often a porous membrane that inhibits mixing of the two solutions, although several other physical possibilities exist such as use of a capillary tube to form a stable liquid junction. Ions must be able to move through the junction in order for current to flow in the electrochemical cell since ions are the current carriers. However, under the open-circuit conditions discussed in this chapter, the current is zero. Under such conditions, diffusion can still take place, but there can be no net transfer of charge. This situation violates two tenets of our thermodynamic analysis. The first is that the system is at true equilibrium—with concentration gradients and transport this condition is not strictly met. The second assumption is that at least one species is absent from each electrode. This condition too can no longer be guaranteed.

With this brief background, we can now describe the origin of the liquid junction potential. For illustration purposes, we can consider the situation where we have different concentrations of the same 1 : 1 binary electrolyte on opposite sides of a porous membrane as shown in Figure 2.3. At open circuit, there will be a diffusion driving force for ions to move from the high concentration side to the low concentration side. The problem is that the cation and anion typically have different diffusion coefficients. The ion with the largest diffusivity will initially move faster than the counterion. This difference in velocity causes a slight imbalance of charge, which results in a potential difference across the junction. The associated potential field in the junction serves to slow down the faster ion and speed up the slower ion so they move at the same rate. This potential difference is the liquid junction potential.

As you may have noticed from the above description, the liquid junction potential is the result of transport and is not thermodynamic in origin. The magnitude of this potential ranges from less than a millivolt to a few tens of millivolts (e.g., 20–30 mV), so that it is not a large correction. Hence, for the most part we will ignore it. Several methods exist for estimating the liquid junction potential. In general, the potential of a cell with a liquid junction can only be calculated with detailed knowledge of the concentration profile across the junction region. However, simplified methods such as the Henderson equation provide a common and straightforward way to estimate this potential. Awareness of potential errors from the junction potential may be important in selecting a reference electrode and in correcting measurements. Please refer to references in the “Further Reading” section for additional information.

There is one other aspect that is worth noting. Since the liquid junction potential is the result of the different transport rates of the anions and cations, the magnitude of this potential can be minimized by choosing anions and cations with similar diffusivities. Consequently, KCl is frequently used to minimize the junction potential. Finally, while we have described the situation for a binary electrolyte, the same physics and principles apply to multicomponent junctions.

2.11 Reference Electrodes

The potential scale in Appendix A is based on the SHE. This scale is arbitrary, and by convention is taken to be zero at standard state as mentioned previously. For experimental work it is generally desirable to have a reference electrode in the system. The purpose of the reference electrode is to provide a known, stable potential against which other potentials can be measured. In principle, no current is passed through the reference electrode; therefore, it remains at its equilibrium potential—a potential that is known and well defined. In practice, a very, very small current is passed through the reference electrode to allow measurement of the potential; however, this current is not sufficient to move the electrode from its equilibrium potential. The desired characteristics for a reference electrode include the following:

• Reversible reactions
• Stable and well-defined potential
• Ion(s) that participates in the reference electrode reaction is present in the solution
• No liquid junctions that cause an offset in potential

The purpose of this section is to provide several examples of reference electrodes and to demonstrate calculation of the potential for some of these electrodes. The book by Ives and Janz should be consulted for more details.

Hydrogen Electrode

The hydrogen electrode (Figure 2.4) can be used in aqueous solutions over a wide range of pH values. It consists of a metal, such as platinum, on which hydrogen reacts rapidly and reversibly. The electrode is immersed in an aqueous solution, and hydrogen gas is bubbled around it. The hydrogen ion concentration (pH) is known in the aqueous solution that surrounds the electrode. The pressure above the solution is a combination of the hydrogen gas pressure and the water pressure. For example, the vapor pressure of water is 5 kPa at room temperature, so that a total pressure of 100 kPa corresponds to a hydrogen pressure of 95 kPa. Hydrogen is vented through a trap, which prevents air from diffusing into the cell. The reference electrode is connected to the point of interest through a capillary. The hydrogen electrode is appropriate for most aqueous solutions, but is not practical for many situations where hydrogen gas must be avoided. It is more difficult to use in unbuffered neutral solutions because of the challenge of maintaining a constant solution composition under such conditions. The hydrogen reactions under acidic and basic conditions are

and

The traditional hydrogen reference electrode is a normal hydrogen electrode (NHE), which has a hydrogen ion concentration in solution of one molar and a hydrogen gas pressure of 1 bar, and operates at a temperature of 25 °C. The NHE potential is very close to the SHE potential, but differs slightly due to non-ideal effects that are not present in the theoretical SHE.

Calomel Electrode

Another common reference electrode is the calomel electrode (Figure 2.5). Calomel refers to mercury(I) chloride, a sparingly soluble salt. This electrode is based on the reaction between Hg and Hg₂Cl₂:

For a saturated calomel electrode (SCE), the solution is often kept saturated by the addition of crystals of KCl to maintain a constant concentration of Cl⁻. Note that the standard potential given above corresponds to an ideal 1 molal solution, rather than the Cl⁻ concentration of a saturated solution. For the saturated solution, the potential is about 0.242 V (SHE). The electrode consisting of the Hg(ℓ), Hg₂Cl₂(s) and saturated KCl is connected to the electrolyte solution of interest through a porous frit; this porous frit is equivalent to a salt bridge or junction region. These electrodes will generally have a liquid junction (see Section 2.10), although the correction in potential is not accounted for in most measurements.

Calomel electrodes are best suited to electrolytes that contain Cl⁻ and, conversely, should not be used in situations where low levels of chloride contamination are not acceptable. Additional mercury salt electrodes such as Hg/Hg₂SO₄ (second in popularity to the calomel electrode) and Hg/HgO (alkaline solutions) are available for use with other electrolytes.

Silver–Silver Chloride

Another popular reference electrode is the Ag/AgCl electrode, which is based on the following reaction:

The Ag/AgCl electrode could consist of a simple silver wire upon which a silver chloride layer has been formed. Alternatively, a base metal such as Pt can be used for the deposition of both silver and the silver halide. These electrodes are small and compact and can be used in any orientation. They can be inserted directly into the electrolyte solution of interest with no significant contamination. Ag/AgCl electrodes can be formed by either electrolytic or thermal methods. Bromide and iodide electrodes can likewise be formed. The thermodynamic properties of these electrodes do depend slightly upon the method of preparation. Commercial Ag/AgCl electrodes have a controlled solution concentration, which further increases their stability and reproducibility.

A related reference electrode is the silver sulfate electrode, which is suitable for use in a lead–acid battery since it shares the sulfate ion with the acid.

2.12 Equilibrium at Electrode Interface

In this chapter, we have shown how to calculate the thermodynamic potential and the relationship between this equilibrium value and the change in the Gibbs energy for the full-cell reaction. We have also referred to this potential as an equilibrium potential. Clearly, however, the full-cell reaction is not in equilibrium, which would mean that ΔG_Rx = 0. Why, then, do we refer to this potential as the equilibrium voltage? What exactly is in equilibrium?

When there is no flow through an external circuit, then the half-cell reaction at each electrode approaches equilibrium. At equilibrium, the energy of the electrons in the metal is characteristic of the reaction, and related to the standard potential—more on this in a minute. First, what does it mean to approach equilibrium, and what exactly is in equilibrium?

An example in the form of a thought experiment may be helpful at this point. Let's imagine that we have a copper metal electrode in an acidic sulfate electrolyte. If the Gibbs energy for dissolution of the metal is negative, then some of the copper atoms will give up their electrons, leave the copper metal lattice, and move into solution as cations. This leaves excess charge in the copper metal, resulting in an increase in the electron energy. The addition of cations to the solution increases the copper ion activity. Since the half-cell reaction of interest is

both the increase in electron energy and the increase in copper ion activity increase the rate of the reduction reaction relative to that of the oxidation reaction. The process continues until equilibrium is reached and the rates of the forward and reverse reactions are equal. While the above is a simplification of the actual physics, it provides a conceptual context for understanding the equilibrium of half-cell reactions. Thus, the half-cell voltage really does represent the equilibrium point for the reaction of interest on the hydrogen scale.

The Gibbs energy change, is equal to zero for a half-cell reaction at equilibrium, which is approached under open-circuit conditions. At constant T and p, the Gibbs energy change for the reaction can be expressed in terms of the electrochemical potentials of the species that participate in the reaction:

(2.29)

where is the stoichiometric coefficient for species i as we have defined and used it previously in this book. The electrochemical potential, , is defined as

(2.30)

which includes a chemical portion () and an electrical portion (). The chemical portion is the chemical potential about which you may have learned in a course on chemical thermodynamics. The electrical portion represents the work required to bring a charge from infinity to a location inside the solution, and is known as the Galvani potential or inner potential. It is equal to zero for uncharged species. It is also zero for charged species in a bulk solution with no electric field or without surfaces where charges align to create fields. Thus, the columbic interactions of ions in a neutral bulk solution are included in the chemical term. In contrast, the impact of the electric field on ions in solution due to, for example, charge on the electrodes, is included in the electrical portion of the electrochemical potential.

Application of Equations 2.29 and 2.30 to the copper reaction yields, at equilibrium:

(2.31a)

where

(2.31b)

(2.31c)

(2.31d)

since the copper metal and the electrons are assumed to be in their standard state. Substituting Equations 2.31b–2.31d into Equation 2.31a and rearranging yields

(2.32)

(2.33)

where is the potential difference between the metal and the solution under standard conditions. The quantity is the potential difference across the interface at equilibrium. This interfacial potential plays an important role in electrochemical systems. For example, a potential difference across the interface that is greater than the value at equilibrium will cause the reaction to take place in the anodic direction. Conversely, the cathodic reaction will take place if the potential across the interface is less than the equilibrium value. We will examine this again in Chapter 3 when we discuss reaction kinetics. The purpose of this section was to illustrate the role of the potential in establishing interfacial equilibrium for single electrochemical reactions. This, of course, is unique to electrochemical reactions.

Unfortunately, is not accessible experimentally, since there is no way to measure the solution potential without introducing another interface, and therefore another interfacial potential drop, into the system. Consequently, how do we measure and quantitatively characterize the equilibrium potential for a half-cell reaction? We have already seen the results of the answer to this question. Since we can measure the potential between two electrodes, and we know that reversible half-cell reactions are at equilibrium at open circuit, we simply define a half-cell as our reference point and measure all other potentials relative to that reference. As long as the reference half-cell reaction remains at equilibrium, the process of defining a reference electrode is equivalent to adding a constant to the potential across the interface of the electrode of interest. The universal reference is the SHE, whose potential has been defined as zero. The hydrogen reaction is often highly reversible and reproducible. The potential of an electrode relative to a hydrogen electrode is measurable. Also, by making appropriate corrections, the potential of an electrode measured in a practical system relative to any other electrode at equilibrium can be quantitatively related to the potential of that electrode versus SHE. Thus, we have a well-defined way of determining the equilibrium potential of a half-cell reaction.

2.13 Potential in Solution Due to Charge: Debye–Hückel Theory

The previous section described a potential difference between a surface and the adjacent solution. This potential difference is due to unbalanced charge in solution as a result of charge on a surface or on an ion. Both of these situations (surface and ion) have been described in the literature using very similar approaches, and the key parameter that results is the same in both cases. Here we present the Debye–Hückel solution for the potential field surrounding a single charged central ion as shown in Figure 2.6. Our goal in presenting this material is to help you understand the basic physics of the problem and to introduce the Debye length. Later in this chapter we will use the solution developed here to provide a first approximation to the activity coefficient of an ion.

As shown in Figure 2.6, we shall evaluate the potential field surrounding a positively charged ion in solution. Ions near this central ion are affected by its charge, with negative ions drawn toward the central ion and positive ions pushed away. Ions in solution also experience random thermal motion, which tends to make the concentration more uniform. These two counteracting effects, the potential field, which tends to separate ions, and random thermal motion, which tends to make concentrations more uniform, can be expressed quantitatively by the Boltzmann factor:

(2.34)

The quantity z_iFϕ represents the work required to move a mole of ions to a different energy state as characterized by the local potential. This expression correctly yields a concentration of negative ions near the central ion that is greater than the bulk concentration, and a concentration of positive ions near the central ion that is less than the bulk. Also, the expression is quite sensitive to the potential. For example, a potential difference of only 10 mV leads to an increase in concentration of the negative ions of almost 50%. Note that the treatment presented in this section ignores the discrete nature of the ions and treats the concentration distributions as continuous.

Far away from the central ion, the concentration is that of the bulk; here the potential, ϕ, is arbitrarily set to zero. The potential distribution is given by Poisson's equation:

(2.35)

where ɛ is the permittivity with units [C·V⁻¹·m⁻¹]. Assuming spherical symmetry and substituting the Boltzmann distribution above (Equation 2.34) for the concentrations, Equation 2.35 becomes

(2.36)

where the summation is over each type of ion in solution. The boundary conditions are

The second boundary condition accounts for the fact that the charge density integrated throughout the volume surrounding the central ion must be equal to the charge of the central ion (z_c). Consequently, according to Gauss's law,

(2.37)

where the potential gradient and the surface area of the ion are constants and can be removed from the integral. In order to simplify Equation 2.36, the exponential term is approximated by the first two terms of a Maclaurin series, which is accurate for small values of the term in the exponent.

Substituting this expression back into the original differential equation (Equation 2.36) results in

(2.38)

Here, the Debye length, λ, has been introduced. This parameter is critical in the study of electrochemical systems, and is defined as

(2.39)

The Debye length is important in describing the potential distribution; more specifically, it is the characteristic length over which the charge density in solution varies as a result of the central ion. The solution of (2.38) is

(2.40)

which represents the variation of the solution potential with position beginning at the surface of the central ion and moving outward. The Debye length characterizes the distance over which the potential changes as shown in Figure 2.7, where the potential in the less concentrated solution takes significantly longer to decay. The potential is the result of the field from the central ion and the shielding effects of other ions in solution. The Debye length decreases with increasing concentration (see Equation 2.39), as more ions are available to shield the central ion. Typical λ values in aqueous solutions are on the order of 1 nm. While a nanometer may seem quite small, this characteristic distance is large relative to other types of interaction forces between molecules. The variation of the charge density in solution as a result of charge on an electrode surface also scales with the Debye length as discussed later in Chapter 3.

2.14 Activities and Activity Coefficients

The accuracy of our expression for cell potential can be increased by including the full activity corrections, rather than the approximations used earlier. The complexity of the calculations, however, increases significantly. The treatment below assumes that the reader has been exposed to the concepts of activity and fugacity. If these are new to you, you may want to learn more about them from a book on physical chemistry or chemical equilibrium (see Further Reading section at the end of the chapter). The fugacity is related to the chemical potential and is used as a surrogate for the chemical potential in phase-equilibrium calculations. The activity of a species is defined as the ratio of the fugacity of species i to the fugacity of pure i at the standard state:

(2.41)

Note that the activity is dimensionless.

For our analysis of electrochemical systems, we need the activity of solid, gas, solvent, and solute species. For solid species, the standard state is typically taken as the pure species. In this text, we will consider only pure solid species. Therefore, the fugacity is equal to the pure component fugacity, f_i, and

(2.42)

since the solid is in its standard state.

For the gas-phase species, the standard state fugacity is an ideal gas at a pressure of 1 bar. The fugacity of species i in a mixture is defined as

(2.43)

where is the fugacity coefficient of species i in a mixture, y_i is the mole fraction of the component in the gas phase, and p is the total pressure. For the purposes of this book, we assume that , which is equivalent to assuming ideal gas behavior. With this assumption, the activity of the gas is

(2.44)

Remember, is dimensionless, and the pressure units in the numerator cancel out with those of the standard state fugacity, p^o. However, since the numerical value of is 1 bar, sometimes it is left off the standard state when writing the activity. Do so with great care.

Electrochemical systems include an electrolyte, which is a material in which current flows due to the movement of ions. A common liquid electrolyte consists of a solvent (e.g., water) into which one or more salts are dissolved to provide the ionic species. For electrolyte solutions, molality (m_i = moles solute i per kg solvent) is the most commonly used form of expressing the composition when dealing with nonideal solutions and activities, and hence will be used here. Molality is convenient from an experimental perspective because it depends only upon the masses of the components in the electrolyte solution, and does not require a separate determination of density. The temperature dependence of the density may also introduce error when dealing with concentration rather than molality. The relationship between molality and concentration is

(2.45)

where the subscript “0” refers to the solvent and M is the molecular weight. The summation in the denominator is simply the total mass of solute species per volume of solution.

We first consider the activity of the solvent. Since the concentration of the solvent is usually much higher than that of the dissolved solute species, the standard state fugacity is that of the pure solvent at the same pressure and phase of the system (e.g., liquid water).

(2.46)

In most cases, the vapor pressures are sufficiently low that the fugacity can be approximated by the pressure, as shown. An osmotic coefficient is usually used to express the activity of the solvent as follows:

(2.47)

where the summation is over ionic species and does not include the solvent. M_solvent is the molecular weight of the solvent and is the osmotic coefficient. For a single salt (binary electrolyte), Equation 2.47 becomes

(2.48)

where m is the molality of the salt and ν is defined by Equation 2.54. When the ion concentration is zero, the right sides of Equations 2.47 and 2.48 both go to zero, and the solvent activity is equal to unity as expected. For our purposes in this text, we will assume unit activity for the solvent, unless otherwise specified.

We now turn our attention to the activity of the solute, which is the activity correction that is most frequently of concern in electrolyte solutions. The activity of a single ion is defined as

(2.49)

where is the single-ion activity coefficient (unitless). Again, we have left off the standard state in the final expression, since it has a value of one. Remember, however, that the activity is dimensionless.

Where does the standard state come from, and why is it used? What is meant by an ideal solution in this context? A thought experiment is useful for answering these questions. Think about an ion in solution. That ion will interact with the solvent (e.g., water molecules) and with other ions of all types in the solution. However, as the ion concentration in the electrolyte approaches zero, only the ion–solvent interactions are important. Under such conditions, the fugacity of the ion is equal to . In other words, the fugacity of the ion depends only on the amount present. This is because the ions interact only with the solvent, and the nature of the interactions does not change with molality as long as ion–ion interactions remain insignificant (i.e., as long as changing interactions do not contribute to the fugacity). We define an ideal solution as a solution in which only ion–solvent interactions are important. Such behavior is approximated in real systems as the concentration approaches small values, and is seen as a linear asymptote in a plot of the activity as a function of composition.

The standard state defined in Equation 2.49 represents the fugacity of an ideal solution where only ion–solvent interactions are important at a concentration of one molal. This state is clearly hypothetical because ion-ion interactions are important in real electrolytes at this concentration. However, it is a convenient reference state and is widely used. The activity coefficient is used to account for deviations from this ideal state due to ion–ion interactions, including complex formation in solution. From the above, it follows that

(2.50)

There is, however, a practical problem with the single-ion activity and activity coefficient. Electrolyte solutions are electrically neutral and solutions containing just a single ion don't exist. Therefore, single-ion activities cannot be measured, although they can be approximated analytically under some conditions (see Section 2.15).

To address this issue, we define a measureable activity that is related to the single-ion activities defined above. A single salt containing ν₊ positive ions and ν₋ negative ions dissociates as follows to form a binary electrolyte:

(2.51)

The activity of the salt in solution is

(2.52)

where the + and − subscripts are introduced for convenience to represent the positive and negative ions of the salt, respectively. We now define a mean ionic activity in terms of the single-ion activities:

(2.53)

where

(2.54)

Similarly, we define the mean ionic activity coefficient in terms of the single-ion activity coefficients

(2.55)

Assuming complete dissociation, the molality of the neutral salt in solution (moles of salt per kg solvent) is related to the molality of the individual ions according to

(2.56)

Combining Equations 2.52−2.56 with the definition of the single-ion activity coefficient (Equation 2.49) yields the following for the salt in solution:

(2.57)

Consistent with the above, the following limits are reached at low concentrations:

(2.58)

Equations 2.51–2.57 apply to a single salt in solution or binary electrolyte and permit activity corrections without requiring single-ion activity coefficients. Several systems of practical importance, such as Li-ion batteries and lead–acid batteries, have either binary electrolytes or electrolytes that can be approximated as binary. The mean ionic activity coefficient, , has been measured for a number of binary solutions, and the results have been correlated and can be found in the literature.

The activity relationships above provide a practical, measurable way to include activity corrections for binary electrolytes. Measurements for binary systems are typically fit to models in order to provide the needed activity coefficients in an accessible, usable way. Under some conditions, prediction of activity coefficients is possible. These issues are discussed briefly in the section that follows.

2.15 Estimation of Activity Coefficients

In dilute solutions, long-range electrical interactions between ions dominate, and the activity coefficient can be estimated from a determination of the electrical contribution alone. The chemical potential is equal to the reversible work of transferring 1 mol of the species to a large volume of the solution at constant temperature and pressure. The nonideal portion for dilute electrolytes is the electrical portion, which is the work required to charge 1 mol of ions in a solution where all of the other ions are already charged. This work can be calculated directly from the Debye–Hückel potential distribution that we determined earlier in Section 2.13. The result is

(2.59)

where the variables were defined in Section 2.13. Since , it follows directly that

(2.60)

The expression on the right arises after introducing the following definitions:

(2.61)

(2.62)

(2.63)

Finally, we can express the mean activity coefficient as

(2.64)

where a is the mean ionic radius of the two hydrated ions. This equation utilizes the relationship shown in Equation 2.55 and takes advantage of the fact that . Ionic strength is the key solution property, which depends on the amount of solute. The only other two parameters are the solvent density and the permittivity. Figure 2.8 shows the activity coefficient as a function of ionic strength for two salts in water. Thus, we can see that the activity coefficient is always less than one, and for dilute solutions, the logarithm of the activity coefficient is proportional to the square root of ionic strength. Also note that at the same ionic strength the 1 : 2 electrolyte has a lower activity coefficient (i.e., is more nonideal). As the ionic strength approaches zero, the limiting form of Equation 2.60, known as Debye–Hückel limiting law, is

(2.65)

In spite of the approximations made in the derivation, the Debye–Hückel model represents the activity coefficients reasonably well for dilute solutions. In fact, the limiting law is usually within 10% of measured values for concentrations less than 0.1 M. Equation 2.64 further increases accuracy. The size parameter is usually treated as a fitting parameter, although it approximates the expected sizes of ions with typical values from 0.3 to 0.5 nm.

Closure

The focus of this chapter has been on the potential of electrochemical cells at equilibrium. Specifically, we have shown how to relate this potential to the environmental conditions at the electrode: temperature, pressure, and composition. Principles of classical thermodynamics have provided the framework for these calculations. The reference electrode has also been introduced. The potential measured with a reference electrode will be a critical variable for subsequent chapters.