Chapter 19 Electrochemistry

Galvanic (Voltaic) Cells

To illustrate the basic principles of a galvanic cell, let’s consider the reaction of metallic zinc with cupric ion (Cu²⁺) to give copper metal and Zn²⁺ ion. The balanced chemical equation is as follows:

We can cause this reaction to occur by inserting a zinc rod into an aqueous solution of copper(II) sulfate. As the reaction proceeds, the zinc rod dissolves, and a mass of metallic copper forms (Figure 19.2 "The Reaction of Metallic Zinc with Aqueous Copper(II) Ions in a Single Compartment"). These changes occur spontaneously, but all the energy released is in the form of heat rather than in a form that can be used to do work.

Figure 19.2 The Reaction of Metallic Zinc with Aqueous Copper(II) Ions in a Single Compartment

When a zinc rod is inserted into a beaker that contains an aqueous solution of copper(II) sulfate, a spontaneous redox reaction occurs: the zinc electrode dissolves to give Zn²⁺(aq) ions, while Cu²⁺(aq) ions are simultaneously reduced to metallic copper. The reaction occurs so rapidly that the copper is deposited as very fine particles that appear black, rather than the usual reddish color of copper.

This same reaction can be carried out using the galvanic cell illustrated in part (a) in Figure 19.3 "The Reaction of Metallic Zinc with Aqueous Copper(II) Ions in a Galvanic Cell". To assemble the cell, a copper strip is inserted into a beaker that contains a 1 M solution of Cu²⁺ ions, and a zinc strip is inserted into a different beaker that contains a 1 M solution of Zn²⁺ ions. The two metal strips, which serve as electrodes, are connected by a wire, and the compartments are connected by a salt bridgeA U-shaped tube inserted into both solutions of a galvanic cell that contains a concentrated liquid or gelled electrolyte and completes the circuit between the anode and the cathode., a U-shaped tube inserted into both solutions that contains a concentrated liquid or gelled electrolyte. The ions in the salt bridge are selected so that they do not interfere with the electrochemical reaction by being oxidized or reduced themselves or by forming a precipitate or complex; commonly used cations and anions are Na⁺ or K⁺ and NO₃⁻ or SO₄²⁻, respectively. (The ions in the salt bridge do not have to be the same as those in the redox couple in either compartment.) When the circuit is closed, a spontaneous reaction occurs: zinc metal is oxidized to Zn²⁺ ions at the zinc electrode (the anode), and Cu²⁺ ions are reduced to Cu metal at the copper electrode (the cathode). As the reaction progresses, the zinc strip dissolves, and the concentration of Zn²⁺ ions in the Zn²⁺ solution increases; simultaneously, the copper strip gains mass, and the concentration of Cu²⁺ ions in the Cu²⁺ solution decreases (part (b) in Figure 19.3 "The Reaction of Metallic Zinc with Aqueous Copper(II) Ions in a Galvanic Cell"). Thus we have carried out the same reaction as we did using a single beaker, but this time the oxidative and reductive half-reactions are physically separated from each other. The electrons that are released at the anode flow through the wire, producing an electric current. Galvanic cells therefore transform chemical energy into electrical energy that can then be used to do work.

Figure 19.3 The Reaction of Metallic Zinc with Aqueous Copper(II) Ions in a Galvanic Cell

(a) A galvanic cell can be constructed by inserting a copper strip into a beaker that contains an aqueous 1 M solution of Cu²⁺ ions and a zinc strip into a different beaker that contains an aqueous 1 M solution of Zn²⁺ ions. The two metal strips are connected by a wire that allows electricity to flow, and the beakers are connected by a salt bridge. When the switch is closed to complete the circuit, the zinc electrode (the anode) is spontaneously oxidized to Zn²⁺ ions in the left compartment, while Cu²⁺ ions are simultaneously reduced to copper metal at the copper electrode (the cathode). (b) As the reaction progresses, the Zn anode loses mass as it dissolves to give Zn²⁺(aq) ions, while the Cu cathode gains mass as Cu²⁺(aq) ions are reduced to copper metal that is deposited on the cathode.

The electrolyte in the salt bridge serves two purposes: it completes the circuit by carrying electrical charge and maintains electrical neutrality in both solutions by allowing ions to migrate between them. The identity of the salt in a salt bridge is unimportant, as long as the component ions do not react or undergo a redox reaction under the operating conditions of the cell. Without such a connection, the total positive charge in the Zn²⁺ solution would increase as the zinc metal dissolves, and the total positive charge in the Cu²⁺ solution would decrease. The salt bridge allows charges to be neutralized by a flow of anions into the Zn²⁺ solution and a flow of cations into the Cu²⁺ solution. In the absence of a salt bridge or some other similar connection, the reaction would rapidly cease because electrical neutrality could not be maintained.

A galvanic cell. This galvanic cell illustrates the use of a salt bridge to connect two solutions.

A voltmeter can be used to measure the difference in electrical potential between the two compartments. Opening the switch that connects the wires to the anode and the cathode prevents a current from flowing, so no chemical reaction occurs. With the switch closed, however, the external circuit is closed, and an electric current can flow from the anode to the cathode. The potential (E_cell)Related to the energy needed to move a charged particle in an electric field, it is the difference in electrical potential beween two half-reactions. of the cell, measured in volts, is the difference in electrical potential between the two half-reactions and is related to the energy needed to move a charged particle in an electric field. In the cell we have described, the voltmeter indicates a potential of 1.10 V (part (a) in Figure 19.3 "The Reaction of Metallic Zinc with Aqueous Copper(II) Ions in a Galvanic Cell"). Because electrons from the oxidation half-reaction are released at the anode, the anode in a galvanic cell is negatively charged. The cathode, which attracts electrons, is positively charged.

Not all electrodes undergo a chemical transformation during a redox reaction. The electrode can be made from an inert, highly conducting metal such as platinum to prevent it from reacting during a redox process, where it does not appear in the overall electrochemical reaction. This phenomenon is illustrated in Example 1.

Constructing a Cell Diagram

Because it is somewhat cumbersome to describe any given galvanic cell in words, a more convenient notation has been developed. In this line notation, called a cell diagram, the identity of the electrodes and the chemical contents of the compartments are indicated by their chemical formulas, with the anode written on the far left and the cathode on the far right. Phase boundaries are shown by single vertical lines, and the salt bridge, which has two phase boundaries, by a double vertical line. Thus the cell diagram for the Zn/Cu cell shown in part (a) in Figure 19.3 "The Reaction of Metallic Zinc with Aqueous Copper(II) Ions in a Galvanic Cell" is written as follows:

Figure 19.4

A cell diagram includes solution concentrations when they are provided.

Galvanic cells can have arrangements other than the examples we have seen so far. For example, the voltage produced by a redox reaction can be measured more accurately using two electrodes immersed in a single beaker containing an electrolyte that completes the circuit. This arrangement reduces errors caused by resistance to the flow of charge at a boundary, called the junction potential. One example of this type of galvanic cell is as follows:

This cell diagram does not include a double vertical line representing a salt bridge because there is no salt bridge providing a junction between two dissimilar solutions. Moreover, solution concentrations have not been specified, so they are not included in the cell diagram. The half-reactions and the overall reaction for this cell are as follows:

A single-compartment galvanic cell will initially exhibit the same voltage as a galvanic cell constructed using separate compartments, but it will discharge rapidly because of the direct reaction of the reactant at the anode with the oxidized member of the cathodic redox couple. Consequently, cells of this type are not particularly useful for producing electricity.

Measuring Standard Electrode Potentials

It is physically impossible to measure the potential of a single electrode: only the difference between the potentials of two electrodes can be measured. (This is analogous to measuring absolute enthalpies or free energies. Recall from Chapter 18 "Chemical Thermodynamics" that only differences in enthalpy and free energy can be measured.) We can, however, compare the standard cell potentials for two different galvanic cells that have one kind of electrode in common. This allows us to measure the potential difference between two dissimilar electrodes. For example, the measured standard cell potential (E°) for the Zn/Cu system is 1.10 V, whereas E° for the corresponding Zn/Co system is 0.51 V. This implies that the potential difference between the Co and Cu electrodes is 1.10 V − 0.51 V = 0.59 V. In fact, that is exactly the potential measured under standard conditions if a cell is constructed with the following cell diagram:

This cell diagram corresponds to the oxidation of a cobalt anode and the reduction of Cu²⁺ in solution at the copper cathode.

All tabulated values of standard electrode potentials by convention are listed for a reaction written as a reduction, not as an oxidation, to be able to compare standard potentials for different substances. (Standard electrode potentials for various reduction reactions are given in Chapter 29 "Appendix E: Standard Reduction Potentials at 25°C".) The standard cell potential (E°_cell) is therefore the difference between the tabulated reduction potentials of the two half-reactions, not their sum:

In contrast, recall that half-reactions are written to show the reduction and oxidation reactions that actually occur in the cell, so the overall cell reaction is written as the sum of the two half-reactions. According to Equation 19.10, when we know the standard potential for any single half-reaction, we can obtain the value of the standard potential of many other half-reactions by measuring the standard potential of the corresponding cell.

Although it is impossible to measure the potential of any electrode directly, we can choose a reference electrode whose potential is defined as 0 V under standard conditions. The standard hydrogen electrode (SHE)The electrode chosen as the reference for all other electrodes, which has been assigned a standard potential of 0 V and consists of a Pt wire in contact with an aqueous solution that contains 1 M ${\text{H}}^{\text{ + }}$ in equilibrium with ${\text{H}}_2$ gas at a pressure of 1 atm at the Pt-solution interface. is universally used for this purpose and is assigned a standard potential of 0 V. It consists of a strip of platinum wire in contact with an aqueous solution containing 1 M H⁺. The [H⁺] in solution is in equilibrium with H₂ gas at a pressure of 1 atm at the Pt-solution interface (Figure 19.6 "The Standard Hydrogen Electrode"). Protons are reduced or hydrogen molecules are oxidized at the Pt surface according to the following equation:

One especially attractive feature of the SHE is that the Pt metal electrode is not consumed during the reaction.

Figure 19.6 The Standard Hydrogen Electrode

The SHE consists of platinum wire that is connected to a Pt surface in contact with an aqueous solution containing 1 M H⁺ in equilibrium with H₂ gas at a pressure of 1 atm. In the molecular view, the Pt surface catalyzes the oxidation of hydrogen molecules to protons or the reduction of protons to hydrogen gas. (Water is omitted for clarity.) The standard potential of the SHE is arbitrarily assigned a value of 0 V.

Figure 19.7 "Determining a Standard Electrode Potential Using a Standard Hydrogen Electrode" shows a galvanic cell that consists of a SHE in one beaker and a Zn strip in another beaker containing a solution of Zn²⁺ ions. When the circuit is closed, the voltmeter indicates a potential of 0.76 V. The zinc electrode begins to dissolve to form Zn²⁺, and H⁺ ions are reduced to H₂ in the other compartment. Thus the hydrogen electrode is the cathode, and the zinc electrode is the anode. The diagram for this galvanic cell is as follows:

The half-reactions that actually occur in the cell and their corresponding electrode potentials are as follows:

Figure 19.7 Determining a Standard Electrode Potential Using a Standard Hydrogen Electrode

The voltmeter shows that the standard cell potential of a galvanic cell consisting of a SHE and a Zn/Zn²⁺ couple is E°_cell = 0.76 V. Because the zinc electrode in this cell dissolves spontaneously to form Zn²⁺(aq) ions while H⁺(aq) ions are reduced to H₂ at the platinum surface, the standard electrode potential of the Zn²⁺/Zn couple is −0.76 V.

Although the reaction at the anode is an oxidation, by convention its tabulated E° value is reported as a reduction potential. The potential of a half-reaction measured against the SHE under standard conditions is called the standard electrode potentialThe potential of a half-reaction measured against the SHE under standard conditions. for that half-reaction.In this example, the standard reduction potential for Zn²⁺(aq) + 2e⁻ → Zn(s) is −0.76 V, which means that the standard electrode potential for the reaction that occurs at the anode, the oxidation of Zn to Zn²⁺, often called the Zn/Zn²⁺ redox couple, or the Zn/Zn²⁺ couple, is −(−0.76 V) = 0.76 V. We must therefore subtract E°_anode from E°_cathode to obtain E°_cell: 0 − (−0.76 V) = 0.76 V.

Because electrical potential is the energy needed to move a charged particle in an electric field, standard electrode potentials for half-reactions are intensive properties and do not depend on the amount of substance involved. Consequently, E° values are independent of the stoichiometric coefficients for the half-reaction, and, most important, the coefficients used to produce a balanced overall reaction do not affect the value of the cell potential.

To measure the potential of the Cu/Cu²⁺ couple, we can construct a galvanic cell analogous to the one shown in Figure 19.7 "Determining a Standard Electrode Potential Using a Standard Hydrogen Electrode" but containing a Cu/Cu²⁺ couple in the sample compartment instead of Zn/Zn²⁺. When we close the circuit this time, the measured potential for the cell is negative (−0.34 V) rather than positive. The negative value of E°_cell indicates that the direction of spontaneous electron flow is the opposite of that for the Zn/Zn²⁺ couple. Hence the reactions that occur spontaneously, indicated by a positive E°_cell, are the reduction of Cu²⁺ to Cu at the copper electrode. The copper electrode gains mass as the reaction proceeds, and H₂ is oxidized to H⁺ at the platinum electrode. In this cell, the copper strip is the cathode, and the hydrogen electrode is the anode. The cell diagram therefore is written with the SHE on the left and the Cu²⁺/Cu couple on the right:

The half-cell reactions and potentials of the spontaneous reaction are as follows:

Thus the standard electrode potential for the Cu²⁺/Cu couple is 0.34 V.

Balancing Redox Reactions Using the Half-Reaction Method

In Chapter 4 "Reactions in Aqueous Solution", we described a method for balancing redox reactions using oxidation numbers. Oxidation numbers were assigned to each atom in a redox reaction to identify any changes in the oxidation states. Here we present an alternative approach to balancing redox reactions, the half-reaction method, in which the overall redox reaction is divided into an oxidation half-reaction and a reduction half-reaction, each balanced for mass and charge. This method more closely reflects the events that take place in an electrochemical cell, where the two half-reactions may be physically separated from each other.

We can illustrate how to balance a redox reaction using half-reactions with the reaction that occurs when Drano, a commercial solid drain cleaner, is poured into a clogged drain. Drano contains a mixture of sodium hydroxide and powdered aluminum, which in solution reacts to produce hydrogen gas:

In this reaction, Al(s) is oxidized to Al³⁺, and H⁺ in water is reduced to H₂ gas, which bubbles through the solution, agitating it and breaking up the clogs.

The overall redox reaction is composed of a reduction half-reaction and an oxidation half-reaction. From the standard electrode potentials listed in Chapter 29 "Appendix E: Standard Reduction Potentials at 25°C", we find the corresponding half-reactions that describe the reduction of H⁺ ions in water to H₂ and the oxidation of Al to Al³⁺ in basic solution:

The half-reactions chosen must exactly reflect the reaction conditions, such as the basic conditions shown here. Moreover, the physical states of the reactants and the products must be identical to those given in the overall reaction, whether gaseous, liquid, solid, or in solution.

In Equation 19.21, two H⁺ ions gain one electron each in the reduction; in Equation 19.22, the aluminum atom loses three electrons in the oxidation. The charges are balanced by multiplying the reduction half-reaction (Equation 19.21) by 3 and the oxidation half-reaction (Equation 19.22) by 2 to give the same number of electrons in both half-reactions:

Adding the two half-reactions,

Simplifying by canceling substances that appear on both sides of the equation,

We have a −2 charge on the left side of the equation and a −2 charge on the right side. Thus the charges are balanced, but we must also check that atoms are balanced:

The atoms also balance, so Equation 19.26 is a balanced chemical equation for the redox reaction depicted in Equation 19.20.

We can also balance a redox reaction by first balancing the atoms in each half-reaction and then balancing the charges. With this alternative method, we do not need to use the half-reactions listed in Chapter 29 "Appendix E: Standard Reduction Potentials at 25°C" but instead focus on the atoms whose oxidation states change, as illustrated in the following steps:

Step 1: Write the reduction half-reaction and the oxidation half-reaction.

For the reaction shown in Equation 19.20, hydrogen is reduced from H⁺ in OH⁻ to H₂, and aluminum is oxidized from Al⁰ to Al³⁺:

Step 2: Balance the atoms by balancing elements other than O and H. Then balance O atoms by adding H₂O and balance H atoms by adding H⁺.

Elements other than O and H in the previous two equations are balanced as written, so we proceed with balancing the O atoms. We can do this by adding water to the appropriate side of each half-reaction:

Balancing H atoms by adding H⁺, we obtain the following:

We have now balanced the atoms in each half-reaction, but the charges are not balanced.

Step 3: Balance the charges in each half-reaction by adding electrons.

Two electrons are gained in the reduction of H⁺ ions to H₂, and three electrons are lost during the oxidation of Al⁰ to Al³⁺:

Step 4: Multiply the reductive and oxidative half-reactions by appropriate integers to obtain the same number of electrons in both half-reactions.

In this case, we multiply Equation 19.34 (the reductive half-reaction) by 3 and Equation 19.35 (the oxidative half-reaction) by 2 to obtain the same number of electrons in both half-reactions:

Step 5: Add the two half-reactions and cancel substances that appear on both sides of the equation.

Adding and, in this case, canceling 8H⁺, 3H₂O, and 6e⁻,

We have three OH⁻ and one H⁺ on the left side. Neutralizing the H⁺ gives us a total of 5H₂O + H₂O = 6H₂O and leaves 2OH⁻ on the left side:

Step 6: Check to make sure that all atoms and charges are balanced.

Equation 19.39 is identical to Equation 19.26, obtained using the first method, so the charges and numbers of atoms on each side of the equation balance.

Figure 19.8 The Reaction of Dichromate with Iodide

The reaction of a yellow solution of sodium dichromate with a colorless solution of sodium iodide produces a deep amber solution that contains a green Cr³⁺(aq) complex and brown I₂(aq) ions.

Calculating Standard Cell Potentials

The standard cell potential for a redox reaction (E°_cell) is a measure of the tendency of reactants in their standard states to form products in their standard states; consequently, it is a measure of the driving force for the reaction, which earlier we called voltage. We can use the two standard electrode potentials we found earlier to calculate the standard potential for the Zn/Cu cell represented by the following cell diagram:

We know the values of E°_anode for the reduction of Zn²⁺ and E°_cathode for the reduction of Cu²⁺, so we can calculate E°_cell:

This is the same value that is observed experimentally. If the value of E°_cell is positive, the reaction will occur spontaneously as written. If the value of E°_cell is negative, then the reaction is not spontaneous, and it will not occur as written under standard conditions; it will, however, proceed spontaneously in the opposite direction.As we shall see in Section 19.7 "Electrolysis", this does not mean that the reaction cannot be made to occur at all under standard conditions. With a sufficient input of electrical energy, virtually any reaction can be forced to occur. Example 4 and its corresponding exercise illustrate how we can use measured cell potentials to calculate standard potentials for redox couples.

Reference Electrodes and Measuring Concentrations

When using a galvanic cell to measure the concentration of a substance, we are generally interested in the potential of only one of the electrodes of the cell, the so-called indicator electrodeThe electrode of a galvanic cell whose potential is related to the concentration of the substance being measured., whose potential is related to the concentration of the substance being measured. To ensure that any change in the measured potential of the cell is due to only the substance being analyzed, the potential of the other electrode, the reference electrodeAn electrode in an galvanic cell whose potential is unaffected by the properties of the solution., must be constant. You are already familiar with one example of a reference electrode: the SHE. The potential of a reference electrode must be unaffected by the properties of the solution, and if possible, it should be physically isolated from the solution of interest. To measure the potential of a solution, we select a reference electrode and an appropriate indicator electrode. Whether reduction or oxidation of the substance being analyzed occurs depends on the potential of the half-reaction for the substance of interest (the sample) and the potential of the reference electrode.

There are many possible choices of reference electrode other than the SHE. The SHE requires a constant flow of highly flammable hydrogen gas, which makes it inconvenient to use. Consequently, two other electrodes are commonly chosen as reference electrodes. One is the silver–silver chloride electrodeA reference electrode that consists of a silver wire coated with a very thin layer of AgCl and dipped into a chloride ion solution with a fixed concentration., which consists of a silver wire coated with a very thin layer of AgCl that is dipped into a chloride ion solution with a fixed concentration. The cell diagram and reduction half-reaction are as follows:

If a saturated solution of KCl is used as the chloride solution, the potential of the silver–silver chloride electrode is 0.197 V versus the SHE. That is, 0.197 V must be subtracted from the measured value to obtain the standard electrode potential measured against the SHE.

A second common reference electrode is the saturated calomel electrode (SCE)A reference electrode that consists of a platinum wire inserted into a moist paste of liquid mercury (calomel; ${\text{Hg}}_2{\text{Cl}}_2$) and KCl in an interior cell, which is surrounded by an aqueous KCl solution., which has the same general form as the silver–silver chloride electrode. The SCE consists of a platinum wire inserted into a moist paste of liquid mercury (Hg₂Cl₂; called calomel in the old chemical literature) and KCl. This interior cell is surrounded by an aqueous KCl solution, which acts as a salt bridge between the interior cell and the exterior solution (part (a) in Figure 19.9 "Three Common Types of Electrodes"). Although it sounds and looks complex, this cell is actually easy to prepare and maintain, and its potential is highly reproducible. The SCE cell diagram and corresponding half-reaction are as follows:

Figure 19.9 Three Common Types of Electrodes

(a) The SCE is a reference electrode that consists of a platinum wire inserted into a moist paste of liquid mercury (calomel; Hg₂Cl₂) and KCl. The interior cell is surrounded by an aqueous KCl solution, which acts as a salt bridge between the interior cell and the exterior solution. (b) In a glass electrode, an internal Ag/AgCl electrode is immersed in a 1 M HCl solution that is separated from the sample solution by a very thin glass membrane. The potential of the electrode depends on the H⁺ ion concentration of the sample. (c) The potential of an ion-selective electrode depends on the concentration of only a single ionic species in solution.

At 25°C, the potential of the SCE is 0.2415 V versus the SHE, which means that 0.2415 V must be subtracted from the potential versus an SCE to obtain the standard electrode potential.

One of the most common uses of electrochemistry is to measure the H⁺ ion concentration of a solution. A glass electrodeAn electrode used to measure the ${\text{H}}^{+}$ ion concentration of a solution and consisting of an internal Ag/AgCl electrode immersed in a 1 M HCl solution that is separated from the solution by a very thin glass membrane. is generally used for this purpose, in which an internal Ag/AgCl electrode is immersed in a 0.10 M HCl solution that is separated from the solution by a very thin glass membrane (part (b) in Figure 19.9 "Three Common Types of Electrodes"). The glass membrane absorbs protons, which affects the measured potential. The extent of the adsorption on the inner side is fixed because [H⁺] is fixed inside the electrode, but the adsorption of protons on the outer surface depends on the pH of the solution. The potential of the glass electrode depends on [H⁺] as follows (recall that pH = −log[H⁺]:

The voltage E′ is a constant that depends on the exact construction of the electrode. Although it can be measured, in practice, a glass electrode is calibrated; that is, it is inserted into a solution of known pH, and the display on the pH meter is adjusted to the known value. Once the electrode is properly calibrated, it can be placed in a solution and used to determine an unknown pH.

Ion-selective electrodesAn electrode whose potential depends on only the concentration of a particular species in solution. are used to measure the concentration of a particular species in solution; they are designed so that their potential depends on only the concentration of the desired species (part (c) in Figure 19.9 "Three Common Types of Electrodes"). These electrodes usually contain an internal reference electrode that is connected by a solution of an electrolyte to a crystalline inorganic material or a membrane, which acts as the sensor. For example, one type of ion-selective electrode uses a single crystal of Eu-doped LaF₃ as the inorganic material. When fluoride ions in solution diffuse to the surface of the solid, the potential of the electrode changes, resulting in a so-called fluoride electrode. Similar electrodes are used to measure the concentrations of other species in solution. Some of the species whose concentrations can be determined in aqueous solution using ion-selective electrodes and similar devices are listed in Table 19.1 "Some Species Whose Aqueous Concentrations Can Be Measured Using Electrochemical Methods".

The Relationship between Cell Potential and Free Energy

Electrochemical cells convert chemical energy to electrical energy and vice versa. The total amount of energy produced by an electrochemical cell, and thus the amount of energy available to do electrical work, depends on both the cell potential and the total number of electrons that are transferred from the reductant to the oxidant during the course of a reaction. The resulting electric current is measured in coulombs (C)The SI unit of measure for the number of electrons that pass a given point in 1 second; it is defined as $6.25\times {10}^{18}{\text{e}}^{-}\text{/s}$ and relates electron potential (in volts) to energy (in joules): 1 J/1 V = 1 C., an SI unit that measures the number of electrons passing a given point in 1 s. A coulomb relates energy (in joules) to electrical potential (in volts). Electric current is measured in amperes (A)The fundamental SI unit of electric current; it is defined as the flow of 1 C/s past a given point: 1A = 1 C/s.; 1 A is defined as the flow of 1 C/s past a given point (1 C = 1 A·s):

In chemical reactions, however, we need to relate the coulomb to the charge on a mole of electrons. Multiplying the charge on the electron by Avogadro’s number gives us the charge on 1 mol of electrons, which is called the faraday (F)The charge on 1 mol of electrons; it is obtained by multiplying the charge on the electron by Avogadro’s number., named after the English physicist and chemist Michael Faraday (1791–1867):

The total charge transferred from the reductant to the oxidant is therefore nF, where n is the number of moles of electrons.

The maximum amount of work that can be produced by an electrochemical cell (w_max) is equal to the product of the cell potential (E_cell) and the total charge transferred during the reaction (nF):

Work is expressed as a negative number because work is being done by a system (an electrochemical cell with a positive potential) on its surroundings.

As you learned in Chapter 18 "Chemical Thermodynamics", the change in free energy (ΔG) is also a measure of the maximum amount of work that can be performed during a chemical process (ΔG = w_max). Consequently, there must be a relationship between the potential of an electrochemical cell and ΔG, the most important thermodynamic quantity discussed in Chapter 18 "Chemical Thermodynamics". This relationship is as follows:

A spontaneous redox reaction is therefore characterized by a negative value of ΔG and a positive value of E_cell, consistent with our earlier discussions. When both reactants and products are in their standard states, the relationship between ΔG° and E°_cell is as follows:

Potentials for the Sums of Half-Reactions

Although Table 19.2 "Standard Potentials for Selected Reduction Half-Reactions at 25°C" and Chapter 29 "Appendix E: Standard Reduction Potentials at 25°C" list several half-reactions, many more are known. When the standard potential for a half-reaction is not available, we can use relationships between standard potentials and free energy to obtain the potential of any other half-reaction that can be written as the sum of two or more half-reactions whose standard potentials are available. For example, the potential for the reduction of Fe³⁺(aq) to Fe(s) is not listed in the table, but two related reductions are given:

Although the sum of these two half-reactions gives the desired half-reaction, we cannot simply add the potentials of two reductive half-reactions to obtain the potential of a third reductive half-reaction because E° is not a state function. However, because ΔG° is a state function, the sum of the ΔG° values for the individual reactions gives us ΔG° for the overall reaction, which is proportional to both the potential and the number of electrons (n) transferred. To obtain the value of E° for the overall half-reaction, we first must add the values of ΔG° (= −nFE°) for each individual half-reaction to obtain ΔG° for the overall half-reaction:

Solving the last expression for ΔG° for the overall half-reaction,

Three electrons (n = 3) are transferred in the overall reaction (Equation 19.55), so substituting into Equation 19.52 and solving for E° gives the following:

This value of E° is very different from the value that is obtained by simply adding the potentials for the two half-reactions (0.32 V) and even has the opposite sign.

The Relationship between Cell Potential and the Equilibrium Constant

We can use the relationship between ΔG° and the equilibrium constant K, defined in Chapter 18 "Chemical Thermodynamics", to obtain a relationship between E°_cell and K. Recall that for a general reaction of the type aA + bB → cC + dD, the standard free-energy change and the equilibrium constant are related by the following equation:

Given the relationship between the standard free-energy change and the standard cell potential (Equation 19.52), we can write

Rearranging this equation,

For T = 298 K, Equation 19.59 can be simplified as follows:

Thus E°_cell is directly proportional to the logarithm of the equilibrium constant. This means that large equilibrium constants correspond to large positive values of E°_cell and vice versa.

Figure 19.10 "The Relationships among Criteria for Thermodynamic Spontaneity" summarizes the relationships that we have developed based on properties of the system—that is, based on the equilibrium constant, standard free-energy change, and standard cell potential—and the criteria for spontaneity (ΔG° < 0). Unfortunately, these criteria apply only to systems in which all reactants and products are present in their standard states, a situation that is seldom encountered in the real world. A more generally useful relationship between cell potential and reactant and product concentrations, as we are about to see, uses the relationship between ΔG and the reaction quotient Q developed in Chapter 18 "Chemical Thermodynamics".

Figure 19.10 The Relationships among Criteria for Thermodynamic Spontaneity

The three properties of a system that can be used to predict the spontaneity of a redox reaction under standard conditions are K, ΔG°, and E°_cell. If we know the value of one of these quantities, then these relationships enable us to calculate the value of the other two. The signs of ΔG° and E°_cell and the magnitude of K determine the direction of spontaneous reaction under standard conditions.

The Effect of Concentration on Cell Potential: The Nernst Equation

Recall from Chapter 18 "Chemical Thermodynamics" that the actual free-energy change for a reaction under nonstandard conditions, ΔG, is given as follows:

We also know that ΔG = −nFE_cell and ΔG° = −nFE°_cell. Substituting these expressions into Equation 19.61, we obtain

Dividing both sides of this equation by −nF,

Equation 19.44 is called the Nernst equationAn equation for calculating cell potentials $\left(E_{\text{cell}}\right)$ under nonstandard conditions; it can be used to determine the direction of spontaneous reaction for any redox reaction under an conditions: $E_{\text{cell}}=E_{\text{cell}}^{°}-\left(RT/nF\right)\text{ln }Q\text{.}$, after the German physicist and chemist Walter Nernst (1864–1941), who first derived it. The Nernst equation is arguably the most important relationship in electrochemistry. When a redox reaction is at equilibrium (ΔG = 0), Equation 19.44 reduces to Equation 19.59 because Q = K, and there is no net transfer of electrons (i.e., E_cell = 0).

Substituting the values of the constants into Equation 19.44 with T = 298 K and converting to base-10 logarithms give the relationship of the actual cell potential (E_cell), the standard cell potential (E°_cell), and the reactant and product concentrations at room temperature (contained in Q):

Equation 19.64 allows us to calculate the potential associated with any electrochemical cell at 298 K for any combination of reactant and product concentrations under any conditions. We can therefore determine the spontaneous direction of any redox reaction under any conditions, as long as we have tabulated values for the relevant standard electrode potentials. Notice in Equation 19.64 that the cell potential changes by 0.0591/n V for each 10-fold change in the value of Q because log 10 = 1.

Applying the Nernst equation to a simple electrochemical cell such as the Zn/Cu cell discussed in Section 19.2 "Standard Potentials" allows us to see how the cell voltage varies as the reaction progresses and the concentrations of the dissolved ions change. Recall that the overall reaction for this cell is as follows:

The reaction quotient is therefore Q = [Zn²⁺]/[Cu²⁺]. Suppose that the cell initially contains 1.0 M Cu²⁺ and 1.0 × 10⁻⁶ M Zn²⁺. The initial voltage measured when the cell is connected can then be calculated from Equation 19.64:

Thus the initial voltage is greater than E° because Q < 1. As the reaction proceeds, [Zn²⁺] in the anode compartment increases as the zinc electrode dissolves, while [Cu²⁺] in the cathode compartment decreases as metallic copper is deposited on the electrode. During this process, the ratio Q = [Zn²⁺]/[Cu²⁺] steadily increases, and the cell voltage therefore steadily decreases. Eventually, [Zn²⁺] = [Cu²⁺], so Q = 1 and E_cell = E°_cell. Beyond this point, [Zn²⁺] will continue to increase in the anode compartment, and [Cu²⁺] will continue to decrease in the cathode compartment. Thus the value of Q will increase further, leading to a further decrease in E_cell. When the concentrations in the two compartments are the opposite of the initial concentrations (i.e., 1.0 M Zn²⁺ and 1.0 × 10⁻⁶ M Cu²⁺), Q = 1.0 × 10⁶, and the cell potential will be reduced to 0.92 V.

The variation of E_cell with log Q over this range is linear with a slope of −0.0591/n, as illustrated in Figure 19.11 "The Variation of ". As the reaction proceeds still further, Q continues to increase, and E_cell continues to decrease. If neither of the electrodes dissolves completely, thereby breaking the electrical circuit, the cell voltage will eventually reach zero. This is the situation that occurs when a battery is “dead.” The value of Q when E_cell = 0 is calculated as follows:

Figure 19.11 The Variation of E_cell with Log Q for a Zn/Cu Cell

Initially, log Q < 0, and the voltage of the cell is greater than E°_cell. As the reaction progresses, log Q increases, and E_cell decreases. When [Zn²⁺] = [Cu²⁺], log Q = 0 and E_cell = E°_cell = 1.10 V. As long as the electrical circuit remains intact, the reaction will continue, and log Q will increase until Q = K and the cell voltage reaches zero. At this point, the system will have reached equilibrium.

Recall that at equilibrium, Q = K. Thus the equilibrium constant for the reaction of Zn metal with Cu²⁺ to give Cu metal and Zn²⁺ is 1.7 × 10³⁷ at 25°C.

Concentration Cells

A voltage can also be generated by constructing an electrochemical cell in which each compartment contains the same redox active solution but at different concentrations. The voltage is produced as the concentrations equilibrate. Suppose, for example, we have a cell with 0.010 M AgNO₃ in one compartment and 1.0 M AgNO₃ in the other. The cell diagram and corresponding half-reactions are as follows:

As the reaction progresses, the concentration of Ag⁺ will increase in the left (oxidation) compartment as the silver electrode dissolves, while the Ag⁺ concentration in the right (reduction) compartment decreases as the electrode in that compartment gains mass. The total mass of Ag(s) in the cell will remain constant, however. We can calculate the potential of the cell using the Nernst equation, inserting 0 for E°_cell because E°_cathode = −E°_anode:

An electrochemical cell of this type, in which the anode and cathode compartments are identical except for the concentration of a reactant, is called a concentration cellAn electrochemical cell in which the anode and the cathode compartments are identical except for the concentration of a reactant.. As the reaction proceeds, the difference between the concentrations of Ag⁺ in the two compartments will decrease, as will E_cell. Finally, when the concentration of Ag⁺ is the same in both compartments, equilibrium will have been reached, and the measured potential difference between the two compartments will be zero (E_cell = 0).

Using Cell Potentials to Measure Solubility Products

Because voltages are relatively easy to measure accurately using a voltmeter, electrochemical methods provide a convenient way to determine the concentrations of very dilute solutions and the solubility products (K_sp) of sparingly soluble substances. As you learned in Chapter 17 "Solubility and Complexation Equilibriums", solubility products can be very small, with values of less than or equal to 10⁻³⁰. Equilibrium constants of this magnitude are virtually impossible to measure accurately by direct methods, so we must use alternative methods that are more sensitive, such as electrochemical methods.

To understand how an electrochemical cell is used to measure a solubility product, consider the cell shown in Figure 19.12 "A Galvanic Cell for Measuring the Solubility Product of AgCl", which is designed to measure the solubility product of silver chloride: K_sp = [Ag⁺][Cl⁻]. In one compartment, the cell contains a silver wire inserted into a 1.0 M solution of Ag⁺; the other compartment contains a silver wire inserted into a 1.0 M Cl⁻ solution saturated with AgCl. In this system, the Ag⁺ ion concentration in the first compartment equals K_sp. We can see this by dividing both sides of the equation for K_sp by [Cl⁻] and substituting: [Ag⁺] = K_sp/[Cl⁻] = K_sp/1.0 = K_sp. The overall cell reaction is as follows:

Thus the voltage of the concentration cell due to the difference in [Ag⁺] between the two cells is as follows:

Figure 19.12 A Galvanic Cell for Measuring the Solubility Product of AgCl

One compartment contains a silver wire inserted into a 1.0 M solution of Ag⁺, and the other compartment contains a silver wire inserted into a 1.0 M Cl⁻ solution saturated with AgCl. The potential due to the difference in [Ag⁺] between the two cells can be used to determine K_sp.

By closing the circuit, we can measure the potential caused by the difference in [Ag⁺] in the two cells. In this case, the experimentally measured voltage of the concentration cell at 25°C is 0.580 V. Solving Equation 19.72 for K_sp,

Thus a single potential measurement can provide the information we need to determine the value of the solubility product of a sparingly soluble salt.

Using Cell Potentials to Measure Concentrations

Another use for the Nernst equation is to calculate the concentration of a species given a measured potential and the concentrations of all the other species. We saw an example of this in Example 11, in which the experimental conditions were defined in such a way that the concentration of the metal ion was equal to K_sp. Potential measurements can be used to obtain the concentrations of dissolved species under other conditions as well, which explains the widespread use of electrochemical cells in many analytical devices. Perhaps the most common application is in the determination of [H⁺] using a pH meter, as illustrated in Example 12.

Batteries

There are two basic kinds of batteries: disposable, or primary, batteries, in which the electrode reactions are effectively irreversible and which cannot be recharged; and rechargeable, or secondary, batteries, which form an insoluble product that adheres to the electrodes. These batteries can be recharged by applying an electrical potential in the reverse direction. The recharging process temporarily converts a rechargeable battery from a galvanic cell to an electrolytic cell.

Batteries are cleverly engineered devices that are based on the same fundamental laws as galvanic cells. The major difference between batteries and the galvanic cells we have previously described is that commercial batteries use solids or pastes rather than solutions as reactants to maximize the electrical output per unit mass. The use of highly concentrated or solid reactants has another beneficial effect: the concentrations of the reactants and the products do not change greatly as the battery is discharged; consequently, the output voltage remains remarkably constant during the discharge process. This behavior is in contrast to that of the Zn/Cu cell, whose output decreases logarithmically as the reaction proceeds (Figure 19.11 "The Variation of "). When a battery consists of more than one galvanic cell, the cells are usually connected in series—that is, with the positive (+) terminal of one cell connected to the negative (−) terminal of the next, and so forth. The overall voltage of the battery is therefore the sum of the voltages of the individual cells.

Lithium–Iodine Battery

None of the batteries described above is actually “dry.” They all contain small amounts of liquid water, which adds significant mass and causes potential corrosion problems. Consequently, substantial effort has been expended to develop water-free batteries.

One of the few commercially successful water-free batteries is the lithium–iodine batteryA battery that consists of an anode of lithium metal and a cathode containing a solid complex of I2, with a layer of solid LiI in between that allows the diffusion of Li+ ions.. The anode is lithium metal, and the cathode is a solid complex of I₂. Separating them is a layer of solid LiI, which acts as the electrolyte by allowing the diffusion of Li⁺ ions. The electrode reactions are as follows:

Equation 19.84

cathode: I₂(s) + 2e⁻ → 2I⁻(LiI)

Equation 19.85

anode: 2Li(s) → 2Li⁺(LiI) + 2e⁻

Equation 19.86

overall: 2Li(s) + I 2 (s) → 2 LiI(s) E cell = 3.5 V

Cardiac pacemaker. An x-ray of a patient showing the location and size of a pacemaker powered by a lithium–iodine battery.

As shown in part (c) in Figure 19.13 "Three Kinds of Primary (Nonrechargeable) Batteries", a typical lithium–iodine battery consists of two cells separated by a nickel metal mesh that collects charge from the anode. Because of the high internal resistance caused by the solid electrolyte, only a low current can be drawn. Nonetheless, such batteries have proven to be long-lived (up to 10 yr) and reliable. They are therefore used in applications where frequent replacement is difficult or undesirable, such as in cardiac pacemakers and other medical implants and in computers for memory protection. These batteries are also used in security transmitters and smoke alarms. Other batteries based on lithium anodes and solid electrolytes are under development, using TiS₂, for example, for the cathode.

Figure 19.13 Three Kinds of Primary (Nonrechargeable) Batteries

(a) A Leclanché dry cell is actually a “wet cell,” in which the electrolyte is an acidic water-based paste containing MnO₂, NH₄Cl, ZnCl₂, graphite, and starch. Though inexpensive to manufacture, the cell is not very efficient in producing electrical energy and has a limited shelf life. (b) In a button battery, the anode is a zinc–mercury amalgam, and the cathode can be either HgO (shown here) or Ag₂O as the oxidant. Button batteries are reliable and have a high output-to-mass ratio, which allows them to be used in applications such as calculators and watches, where their small size is crucial. (c) A lithium–iodine battery consists of two cells separated by a metallic nickel mesh that collects charge from the anodes. The anode is lithium metal, and the cathode is a solid complex of I₂. The electrolyte is a layer of solid LiI that allows Li⁺ ions to diffuse from the cathode to the anode. Although this type of battery produces only a relatively small current, it is highly reliable and long-lived.

Dry cells, button batteries, and lithium–iodine batteries are disposable and cannot be recharged once they are discharged. Rechargeable batteries, in contrast, offer significant economic and environmental advantages because they can be recharged and discharged numerous times. As a result, manufacturing and disposal costs drop dramatically for a given number of hours of battery usage. Two common rechargeable batteries are the nickel–cadmium battery and the lead–acid battery, which we describe next.

Nickel–Cadmium (NiCad) Battery

The nickel–cadmiumA type of battery that consists of a water-based cell with a cadmium anode and a highly oxidized nickel cathode., or NiCad, battery is used in small electrical appliances and devices like drills, portable vacuum cleaners, and AM/FM digital tuners. It is a water-based cell with a cadmium anode and a highly oxidized nickel cathode that is usually described as the nickel(III) oxo-hydroxide, NiO(OH). As shown in Figure 19.14 "The Nickel–Cadmium (NiCad) Battery, a Rechargeable Battery", the design maximizes the surface area of the electrodes and minimizes the distance between them, which decreases internal resistance and makes a rather high discharge current possible.

Figure 19.14 The Nickel–Cadmium (NiCad) Battery, a Rechargeable Battery

NiCad batteries contain a cadmium anode and a highly oxidized nickel cathode. This design maximizes the surface area of the electrodes and minimizes the distance between them, which gives the battery both a high discharge current and a high capacity.

The electrode reactions during the discharge of a NiCad battery are as follows:

Equation 19.87

cathode: 2NiO(OH)(s) + 2H₂O(l) + 2e⁻ → 2Ni(OH)₂(s) + 2OH⁻(aq)

Equation 19.88

anode: Cd(s) + 2OH⁻(aq) → Cd(OH)₂(s) + 2e⁻

Equation 19.89

overall: Cd(s) + 2 NiO(OH)(s) + 2 H 2 O(l) → Cd(OH) 2 (s) + 2 Ni ( OH ) 2 (s) E cell = 1.4 V

Because the products of the discharge half-reactions are solids that adhere to the electrodes [Cd(OH)₂ and 2Ni(OH)₂], the overall reaction is readily reversed when the cell is recharged. Although NiCad cells are lightweight, rechargeable, and high capacity, they have certain disadvantages. For example, they tend to lose capacity quickly if not allowed to discharge fully before recharging, they do not store well for long periods when fully charged, and they present significant environmental and disposal problems because of the toxicity of cadmium.

A variation on the NiCad battery is the nickel–metal hydride battery (NiMH) used in hybrid automobiles, wireless communication devices, and mobile computing. The overall chemical equation for this type of battery is as follows:

NiO(OH)(s) + MH → Ni(OH)₂(s) + M(s)

The NiMH battery has a 30%–40% improvement in capacity over the NiCad battery; it is more environmentally friendly so storage, transportation, and disposal are not subject to environmental control; and it is not as sensitive to recharging memory. It is, however, subject to a 50% greater self-discharge rate, a limited service life, and higher maintenance, and it is more expensive than the NiCad battery.