The Nernst Equation

The effect of concentration on cell emf can be obtained from the effect of concentration on free-energy change, qpo(Section 19.7) Recall that the free-energy change for any chemical reaction, AG, is related to the standard free-energy change for the reaction, AG°  

The quantity Q is the reaction quotient, which has the form of the equihbrium-constant expression except that the concentrations are those that exist in the reaction mixture at a given moment, aao (Section 15.6) 

Solving this equation for E gives the Nernst equation  

This equation is customarily expressed in terms of the base-10 logarithm  

At T = 298 K, the quantity 2.303 RT/F equals 0.0592, with units of volts, and so the Nernst equation simplifies to 

Substituting AG = —tiFE (Equation 20.11) into Equation 20.15 gives -nFE = -tiFE + RT]n Q Solving this equation for Egives the Nernst equation  

The standard cell potential gq of a cell reaction is the equilibrium cell potential of the hypothetical galvanic cell in which each reactant and product of the cell reaction is in its standard state and there is no Uquid junction potential. The value of for a 

To derive a relation between ceii, eq and activities for a cell without liquid junction, or with a liquid junction of negligible liquid junction potential, we substitute expressions for AfG and for AjG° from Eqs. 14.3.13 and Eq. 14.3.15 into A G = ArG° + RTlnQ n 

Equation 14.4.1 is tbe Nernst equation for the cell reaction. Here Qrxn is the reaction quotient for the cell reaction defined by Eq. 11.8.6 Qrxn = Ot  

The rest of this section will assume that the cell reaction takes place in a cell without liquid junction, or in one in which Ej is negligible. 

At the same time, the Fe ions migrate through the moisture on the surface. The overall reaction is obtained hy halancing the electron transfer and adding the two half-reactions. 

The Fe ions can migrate from the anode through the solution toward the cathode region, where they comhine with OH ions to form ironflf) hydroxide. Iron is further oxidized hy O2 to the + 3 oxidation state. The material we call rust is a complex hydrated form of iron(ni) oxides and hydroxides with variable water composition it can be represented as Fc203 a-H20.The overall reaction for the rusting of iron is 

There are several methods for protecting metals against corrosion. The most widely used are 

Plating the metal with a thin layer of a less easily oxidized metal 

Connecting the metal directly to a sacrificial anode, a piece of another metal that is more active and therefore is preferentially oxidized 

For the generalized half-cell reaction, written as a reduction 

T = absolute temperature (Red) or a = activity of reduced form (Ox) or flox = activity of the oxidized form 

If numerical values are inserted for the constants and the temperature is 25 C, the Nemst equation becomes 

A change of one unit in the logarithmic term changes the value of the electrode potential by 59.16/n mV. If the copper electrode of cell 2.3 is dipped into a solution of copper ion at a = 0.001 M, the electrode potential is 

Since the activity of a solid-phase, such as the copper metal, is unity, 

Note these are written as reductions involving one electron. We therefore subtract to obtain a cell reaction  

All the species in Eq. 1.2 are present at unit activity, either as explicitly stated in the problem or implicitly in the case of copper since it is a pure solid and so must also be at unit activity. Note we assume that in the case of hydrogen at one atmosphere pressure that the gas will be sufficiently close to ideality that the effects of gas imperfections can be neglected. 

During the operation of a galvanic cell, electrons flow from the anode to the cathode, resulting in product formation and a decrease in reactant concentration. Thus, Q increases, which means that E decreases. Eventually, the cell reaches equilibrium. At equilibrium, there is no net transfer of electrons, so E = 0 and Q = K, where K is the equilibrium constant. 

The Nemst equation enables us to calculate as a function of reactant and product concentrations in a redox reaction. For example, for the cell pictured in Hgure 19.1, 

If the ratio [Zn ]/[Cu ] is less than 1, log ([Zn ]/[Cu l) is a negative number, making the second term on the right-hand side of the preceding equation a positive quantity. Under this condition, E is greater than the standard potential °. If the ratio is greater than 1. is smaller than °. Sample Problem 19.6 shows how to use the Nemst equation. 

Predict whether the following reaction will occur spontaneously as written at 298 K  

Strategy Use E° values from Table 19.1 to detamine E° for the reaction, and use Equation 19.7 to calculate . If is positive, the reaction will occur spontaneously. 

Walter Hermann Nemst (1864-1941). German chemist and physicist. Nemst s work was mainly on electrolyte solutions and thermodynamics. He also invented an electric piano. Nemst was awarded the Nobel Prize in Chemistry in 1920 for his contribution to thermodynamics. 

The Nernst equation relates the redox process and the potential giving zero current flow. In the solution of an electrolytic cell, it indicates the redox equilibrium potential V with zero DC current flow. 

Here Vq is the standard electrode potential of the redox system (with respect to the hydrogen reference electrode at 1 mol concentration), n is the number of electrons in the unit reaction, R is not resistance but the universal gas constant, and F is the Faraday constant (see Section 7.8). aox and area are activities, a = yc, where c is the concentration and y is the activity coefficient. Y = 1 for low concentrations (no ion interactions), but 1 at higher concentrations. The halfcell potentials are referred to standardized conditions, meaning that the other electrode is considered to be the standard hydrogen electrode (implying the condition pH = 0, hydrogen ion activity 1 mol/L). The Nernst concept is also used for semipermeable membranes with different concentration on each side of the membrane (see Section 7.6.4). 

the RT/nF factor can be substituted with —61 mV for room temperature and using the common instead of the natural logarithm. Then the Nernst equation becomes  

The Nernst equation presupposes a reversible reaction that the reaction is reasonably fast in both directions. This implies that the surface concentration of reactants and products are maintained close to their equilibrium values. If the electrode reaction rate is slow in any direction, the concentration at the electrode surface will not be equilibrium values, and the Nernst equation is not valid, the reactions are irreversible. 

For metal cathodes, the activity of the reduced forms ared is 1, and the equation, for example, for an AgCl electrode is  

As we mentioned, the cell voltage depends on the activities of all the ions and compounds in the cell reaction in this case it depends not only on the hydrogen gas pressure and but on acu ao,2+ as well. Standard conditions is defined as u = 1 for all products and reactants in the cell reaction, and so if the hydrogen electrode is operating under SHE conditions (uh2( ) = /njfe) = 1 bar), the copper electrode is pure Cu (acu(s) = 1) and the cupric ion concentration and activity coefficient are adjusted to give 00,2+ = 1, the ceU voltage will be the standard cell voltage, S°. 

The electrical work w required to move a charge of 5 coulombs through a potential difference S volts is 

Because some convention must be adopted to know whether the voltage is positive or negative, you may see Equation (12.8) [as well as (12.9) and (12.10), below] written without the minus sign in some references. The conventions we have adopted ( 12.6.1) require the minus sign. 

This electrical work is by definition (Chapter 4) the AG associated with the process, as long as the electrical work is the only non-PAV work done. Therefore for any process in which coulombs are moved through a potential difference , 

Recall from Chapter 9 that this activity term is referred to as Q rather than K because it refers to a metastable equilibrium. Substitution of Equations (12.12) and (12.13) gives 

For any chemical reaction the reaction Gibbs energy is written 

Equations (17.27) are various forms of the Nernst equation for the cell. The Nernst equation relates the cell potential to a standard value, °, and the activities of the species taking part in the cell reaction. Knowing the values of and the activities, we can calculate the cell potential. 

In working this problem it is incorrect to directly combine the F values they must first be converted into AG values. 

In view of this, the student may wonder why then it is legitimate to calculate AF values for overall cell reactions by simply combining the F values for the individual electrodes. Consider, for example, the following F values  

In preceding examples we combined two F values to obtain A values  

The fact that this is justified can be seen by writing down the AG values  

Thus far we have limited our discussion to standard electrode potentials, F, and to AF values for cells in which the active species are present at I m concentrations. The corresponding standard Gibbs energies have been written as AG . 

So far we have considered only standard cell potentials, that is, the electric potential difference developed by a chemical reaction that is at equilibrium in an electrochemical cell at normal atmospheric pressure and a temperature of 25 C, and when the chemical species are present in standard concentrations. We can derive an expression for the electric potential difference generated under nonequilibrium and nonstandard conditions (Fcdi) follows. If we write Eq. (2.41) in terms of concentrations and remove the requirement of molar concentrations, we get 

This last relationship is called the Nernst equation. Substituting the values of R and F into Eq. (6.25), and taking T=298K, 

Exercise 6.9. Calculate the initial electric potential difference generated in an electrochemical cell at 298K by the redox reaction 

Solution. In Exercise 6.8 we calculated that the electric potential difference generated by this reaction under standard conditions (i.e., eii) was 0.74 V we also saw that n = 6. We can use the Nernst equation to derive the electric potential difference generated by the reaction under the nonstandard concentrations specified in the present exercise if we know the value of Q. Applying Eq. (1.10) to the reaction (remembering that the concentrations of liquids and solids are equated to unity in this expression), we get 

We saw in Exercise 6.8 that when all the species in the specified reaction were present in concentrations of 1 M, the cell potential was 0.74 V and the reaction was spontaneous from left to right. Exercise 

We know that an electric charge q is associated with an electric field E. The work which must be done to bring a charge q from infinity to a given point P within an electric field is given by  

The integral term is called the electrostatic potential of the point in question  

In the description of an electrode/electrolyte interface, the electrochemist defines a so-called outer potential p. This is the potential in the vicinity ( 10 cm) of the interface, but still out of range of any image force interactions (in a thought experiment one imagines electrode and electrolyte separated and in a vacuum). The difference between the outer potentials of the electrode and electrolyte solution 

In order to arrive at the potential difference between the interiors of the two adjacent phases, the layer of partially oriented dipoles which develops at the phase boundary must not be overlooked. The work which must be done for a unit charge to cross a dipole layer can formally be assigned to a surface potential difference Ax- This voltage Ax must then be added to the Volta potential Ai//, which only includes charge separations at various locations in a vacuum, in order to get the total potential difference between the electrode interior and the interior of the adjacent electrolyte solution  

Calculation of the electrode potential at thermodynamic equilibrium is simple one can define a chemical potential p, analogous to the electrostatic potential 0, as the work which must be done or is set free if a mole of a given type of particle is to be brought from infinity into the interior of some uncharged phase free of any dipole layer coating. All chemical interactions can be determined with this property. 

Here s how you use Equation (22.4). Suppose that you are interested in the distribution of ions that are subject to a fixed spatial gradient of electrostatic potential. Consider two different locations in space, ri and r2, within a single solution. To eliminate some vector complexity, let s look at a one-dimensional problem. Suppose that ri = Xi and Tt = x- - At location Xi, some constellation of fixed charges or electrodes creates the electrostatic potential ip(xi), and at X2 you have ip(x ). Now consider a single species of mobile ions (so that you can drop the subscript i) that is free to distribute between locations Xi and x. The mobile ions will be at equilibrium when the electrochemical potentials are equal. 

By regarding as independent of x in this case, we are restricting our attention to situations where X and X) are in the same phase of matter. 

In earlier chapters we used mole fraction concentrations. Other concentration units are often useful, and just require a conversion factor that can be absorbed into q°. Here we express concentrations in a general way as c(x), in whatever units are most useful for the problem at hand. Substituting Equation (22.6) into (22.5) gives the Nernst equation, named after WH Nernst (1864-1941), a German chemist who was awarded the 1920 Nobel Prize in Chemistry for his work in chemical thermodynamics  

You should be aware of one important difference between the electrochemical potential and the chemical potential. The chemical potential describes the free energy of inserting one particle into a particular place or phase, subject to any appropriate constraint. Constraints are introduced explicitly. In contrast, the electrochemical potential always carries an implicit constraint with it overall electroneutrality must be obeyed. This is a very strong constraint. You can never insert a single ion in a volume of macroscopic dimensions because that would violate electroneutrality. You can insert only an electroneutral combination of ions. 

It is possible to measure the chemical potentials for salts, but not for the individual ions that compose them because electroneutrality must be maintained. So when you are dealing with simple systems, for example systems in which salts are distributed uniformly throughout a macroscopic volume, you can work with the chemical potentials of neutral molecules such as NaCl. In contrast, w hen you are interested in microscopic nonuniformities, such as the counterion distribution near a DNA molecule, or the electrostatic potential microscopically close to an electrode surface, then you can use the electrochemical potential. Charge neutrality need not hold in a microscopic region of 

The Gibbs free energy of a dissolved species varies with the activity, a, and in the case of a gas, with the partial pressure, p. Consider the equation for the reaction isotherm (16) for the general chemical reaction Equation (9)  

Equation (17) is valid if interfacial charge-transfer equilibrium between the electrodes and both the reactants (A,B) and products (C,D) has been established. For illustration, consider two relevant special conditions If the underlying chemical reaction is at equilibrium, as characterized by the activity quotient of Equation (10), the driving force for the chemical, and thus for the electrochemical reaction, is zero - that is, AG (reaction) = -nFE - 0. On the other hand, if standard conditions prevail, then AG (reaction) = AG°(reaction) (i.e. E - E°). Equation (17) is valid for any combination of two electrodes making up a complete cell. 

As indicated previously, it is desirable to consider the individual electrode reactions independently. One might suppose that this could be achieved by characterizing the individual electrodes as described in Section 3.1.3. However, for reasons of sound thermodynamics, another method has been established. It was decided to relate all electrode reactions to one common reference electrode. Electrochemists have chosen the H+/H2 reaction under standard conditions (ct 1+ = 1M p 12 = 1 bar) as such a general reference electrode. It is termed the normal hydrogen electrode or the standard hydrogen electrode (SHE). Thus, whenever E and E° values are presented for individual electrode reactions (half cells), it is understood that these values pertain to a complete cell in which the SHE constitutes the second electrode. 

As shown in Equation (19), it is conventional for Nernst equations to represent the sum of E° and the logarithmic term, rather than the difference. Consequently, in the activity quotient, the activity of the oxidized form of the electroactive species has to be written as the numerator. Standard potentials of individual electrode reactions are conveniently available in textbooks of physical chemistry and electrochemistry, and in relevant handbooks. A small selection is presented in Table 3.1.3. 

The standard equilibrium cell voltage resulting from a combination of any two electrodes is the difference between the two standard potentials, E°(2) - E°( 1). For instance, the standard cell equilibrium voltage of the combination F2/F with the Li+/Li electrode would be 5,911 V. Correspondingly, the standard free energy change of the underlying chemical reaction, 1/2 F2 + Li — F + Li+, is AG° = -570 KJ (g-equivalent)-1. 

To write the cell reaction corresponding to the electrochemical cell diagram, we first write the half-reactions at both electrodes as reductions and then subtract the equation for the left-hand electrode from the equation for the right-hand electrode. Thus, we saw in Example 5.2 that for the electrochemical cell used to study the reaction between NADH and Oj, 

In other cases, it may be necessary to match the numbers of electrons in the two half-reactions by multiplying one of the equations through by a numerical factor there should be no spare electrons showing in the overall equation. 

The concentrations of electroactive species in biological systems do not normally have their standard values, so we need to be able to relate the potential difference of a cell to the actual concentrations. 

A galvanic cell does electrical work as the reaction drives electrons through an external circuit. The work done by a given transfer of electrons depends on the potential difference between the two electrodes. When the potential difference is large (for instance, 2 V), a given number of electrons travehng between the electrodes can do a lot of electrical work. When the potential difference is small (such as 2 mV), the same number of electrons can do only a httle work. An electrochemical cell in which the reaction is at equilibrium can do no work and the potential difference between its electrodes is zero. 

According to the discussion in Section 2.8, we know that the maximum nonexpansion work that a system (in this context, the electrochemical cell performing electrical work, w ) can do is given by the value of AG and in particular that 

In the examples used so far, we have only considered cases where the concentration of the solutions in each half-cell was 1.0 M. However, the concentration of the reactants and products in the half-cells does affect the emf. We just finished showing how free energy and emf are related. In the last chapter, we saw how the concentrations of reactants and products affect the free energy in Equation 18.9  

By combining this equation with Equation 18.3, AG = nFE, we come up with a new expression  

If this equation is rearranged to solve for E, it becomes the Nernst equation (Equation 18.4)  

This expression can also be written using the base 10 logarithm as opposed to the natural logarithm. In this form, Equation 18.5 reads  

Because R and Fare constants, the expression can be simplified if the reaction takes place at standard temperature (remember, you have to use the absolute temperature in this expression, so standard temperature is 298 K). The simplification is shown below  

We have seen how to use standard reduction potentials to calculate for cells. Real cells are usually not constructed at standard state conditions. In fact, it is almost impossible to make measurements at standard conditions because it is not reasonable to adjust concentrations and ionic strengths to give unit activity for solutes. We need to relate standard potentials to those measured for real cells. It has been found experimentally that certain variables affect the measured cell potential. These variables include the temperature, concentrations of the species in solution, and the number of electrons transferred. The relationship between these variables and the measured cell emf can be derived from simple thermodynamics (see any introductory general chemistry text). The relationship between the potential of an electrochemical cell and the concentration of reactants and products in a general redox reaction 

The logarithmic term has the same form as the equilibrium constant for the reaction. The term is called Q, the reaction quotient, when the concentrations (rigorously, the activities) are not the equilibrium values for the reaction. As in any equilibrium constant expression, pure liquids and pure solids have activities equal to 1, so they are omitted from the expression. If the values of R, T (25°C = 298 K), and F are inserted into the equation and the natural logarithm is converted to log to the base 10, the Nemst equation reduces to 

It should be noted that the square brackets literally mean the molar concentration of . For example, [Fe ] means the molar concentration of ferrous ion or moles of ferrous ion per liter of solution. 

Concentrations in molarity should really be the activities of the species, but we often do not know the activities. For that reason, we define the formal potential, fP, as the measured potential of the cell when the species being oxidized and reduced are present at concentrations such that the ratio of the concentrations of oxidized to reduced species is unity and other components of the cell are present at designated concentrations. The use of the formal potential allows us to avoid activity coefficients, which are often unknown. This gives us  

The Nernst equation is also used to calculate the electrode potential for a given half-cell at nonstandard conditions. For example, for the half-cell Fe - - e Fe which has an = 0.77 V and n= the Nemst equation would be  

Thus far we have confined our discussion to cells and half-reactions at standard state, that is, 25°C, 1 atm pressure, and unit activity for all species. To determine the effect of reactant and product concentrqtion, we need to draw further on our analogy between free energy change and electrode potential or cell emf. Using the free energy equation as developed in Chapter 3, 

The application of the Nernst equation to the reactions that take place in an electrochemical cell and in solution is illustrated in Examples 7-6 and 7-7, respectively. 

A Daniell cell consists of a zinc electrode in a zinc chloride solution connected to a copper electrode in a cupric chloride solution. 

What is the equilibrium constant of the cell reaction at 25°C From Table 7-1, Ezn2+,Zn,5, = -0.76 volt and Ecu +.cuis, = +0.34 volt. 

Assuming that Zn is oxidized and Cu is reduced in the cell, the cell reaction 

Le Chatelier s principle tells us that, if we increase reactant concentrations, we drive a reaction to the right. Increasing the product concentrations drives a reaction to the left. The net driving force for a redox reaction is expressed by the Nernst equation, whose two terms include the driving force under standard conditions ( °, which applies when concentrations are 1 M or 1 bar) and a term that shows the dependence on concentrations. 

The logarithmic term in the Nernst equation is the reaction quotient, Q (= [B] / [A] ). Q has the same form as the equilibrium constant, but the concentrations need not be at their equilibrium values. Concentrations of solutes are expressed as moles per liter and concentrations of gases are expressed as pressures in bars. Pure solids, pure liquids, and solvents are omitted from Q. When all concentrations are 1 M and all pressures are 1 bar, Q = 1 and log (2 = 0, and the Nernst equation reduces toE = E°. 

Let s write the Nernst equation for the reduction of white phosphorus to phosphine gas  

SOLUTION We omit solids from the reaction quotient, and the concentration of phosphine gas is expressed as its pressure in bars (Pph,)  

Test Yourself Write the Nernst equation for a hydrogen electrode (Reaction 14-5). (Answer E = 0 - (0.059 16/1) log(PHj/[H ])) 

In Experiment 5.2, we saw that the mathematical model describing electronic transitions in solution is Beer s law (A = sbc). Using visible spectroscopy we were able to determine s, the molar absorptivity, which gives us information about the probability of electronic transitions occurring in coordination complexes. Similarly, E° from the Nernst equation (A.2.1) gives us information about redox activity of species in solution, where n is the number of electrons transferred. 

Recall that the Nernst equation is the mathematical model describing the relationship between cell potential and concentrations and is readily derived from the fact that cell potential shows a concentration dependence due to its relationship to free energy, equations (A.2.2) and (A.2.3), where Q is the concentration ratio of oxidized (e.g., [Fe3+L ]) to reduced (e.g., [Fe2+L ]) species. In our system, E° is the reduction potential for the one electron transfer half reaction [equation (5.4.4)]. 

Both half- and overall reaction tendencies change with temperature, pressure (if gases are involved), and concentrations of the ions involved. Thus far, we have only been concerned with standard conditions. Standard conditions, as stated previously, are 25°C, 1 atm pressure, and 1 M ion concentrations. An equation has been derived to calculate the cell potential when conditions other than standard conditions are present. This equation is called the Nernst equation and is used to calculate the true E (cell potential) 

For the general overall reaction, in which a moles of chemical A react (with the transfer of electrons) with b moles of chemical B to give c moles of chemical C and d moles of chemical D, 

If any species involved is a gas, the partial pressure of the gas is substituted for the concentration. If the temperature is different from 25°C, the constant 0.0592 changes. The number of electrons, n, in Equation (14.4) is the total number of electrons transferred, as discovered after equalizing the electrons in the two half-reactions in step 3 of the scheme for determining E, in Section 14.3. 

As seen in Equations (14.2) and (14.4), the potential of cells and half-cells is dependent on the concentrations of the dissolved species involved. Clearly, the measurement of a potential can lead to the determination of the concentration of an analyte. This, therefore, is the basis for all quantitative poten-tiometric techniques and measurements to be discussed in this chapter. 

Once again, to be thermodynamically correct, activity should be used rather than concentration. Use of concentration is an approximation. 

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The change in the redox potential is given quantitatively by the Nernst equation ... 

The redox (electrode) potential for ion-ion redox systems at any concentration and temperature is given by the Nernst equation in the form... 

Thus under standard conditions chloride ions are not oxidised to chlorine by dichromate(Vr) ions. However, it is necessary to emphasise that changes in the concentration of the dichromate(VI) and chloride ions alters their redox potentials as indicated by the Nernst equation. Hence, when concentrated hydrochloric acid is added to solid potassium dichromate and the mixture warmed, chlorine is liberated. 

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

Although this treatment of buffers was based on acid-base chemistry, the idea of a buffer is general and can be extended to equilibria involving complexation or redox reactions. For example, the Nernst equation for a solution containing Fe + and Fe + is similar in form to the Henderson-Hasselbalch equation. 

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

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

Before the equivalence point the titration mixture consists of appreciable quantities of both the oxidized and reduced forms of the analyte, but very little unreacted titrant. The potential, therefore, is best calculated using the Nernst equation for the analyte s half-reaction... 

Although EXo /ATcd is standard-state potential for the analyte s half-reaction, a matrix-dependent formal potential is used in its place. After the equivalence point, the potential is easiest to calculate using the Nernst equation for the titrant s half-reaction, since significant quantities of its oxidized and reduced forms are present. 

At the equivalence point, the moles of Fe + initially present and the moles of Ce + added are equal. Because the equilibrium constant for reaction 9.16 is large, the concentrations of Fe and Ce + are exceedingly small and difficult to calculate without resorting to a complex equilibrium problem. Consequently, we cannot calculate the potential at the equivalence point, E q, using just the Nernst equation for the analyte s half-reaction or the titrant s half-reaction. We can, however, calculate... 

In potentiometry the potential of an electrochemical cell is measured under static conditions. Because no current, or only a negligible current, flows while measuring a solution s potential, its composition remains unchanged. For this reason, potentiometry is a useful quantitative method. The first quantitative potentiometric applications appeared soon after the formulation, in 1889, of the Nernst equation relating an electrochemical cell s potential to the concentration of electroactive species in the cell. ... 

Potential and Concentration—The Nernst Equation The potential of a potentio-metric electrochemical cell is given as... 

Note, again, that the Nernst equations for both E and Ta are written for reduction reactions. The cell potential, therefore, is... 

Making appropriate substitutions into the Nernst equation for the electrochemical cell (see Example 11.2)... 

Despite the apparent ease of determining an analyte s concentration using the Nernst equation, several problems make this approach impractical. One problem is that standard-state potentials are temperature-dependent, and most values listed in reference tables are for a temperature of 25 °C. This difficulty can be overcome by maintaining the electrochemical cell at a temperature of 25 °C or by measuring the standard-state potential at the desired temperature. 

Another problem is that the Nernst equation is a function of activities, not concentrations. As a result, cell potentials may show significant matrix effects. This problem is compounded when the analyte participates in additional equilibria. For example, the standard-state potential for the Fe "/Fe " redox couple is +0.767 V in 1 M 1TC104, H-0.70 V in 1 M ITCl, and -H0.53 in 10 M ITCl. The shift toward more negative potentials with an increasing concentration of ITCl is due to chloride s ability to form stronger complexes with Fe " than with Fe ". This problem can be minimized by replacing the standard-state potential with a matrix-dependent formal potential. Most tables of standard-state potentials also include a list of selected formal potentials (see Appendix 3D). 

Since the junction potential is usually of unknown value, it is normally impossible to directly calculate the analyte s concentration using the Nernst equation. Quantitative analytical work is possible, however, using the standardization methods discussed in Chapter 5. 

Activity Versus Concentration In describing metallic and membrane indicator electrodes, the Nernst equation relates the measured cell potential to the concentration of analyte. In writing the Nernst equation, we often ignore an important detail—the... 

Quantitative Analysis Using External Standards To determine the concentration of analyte in a sample, it is necessary to standardize the electrode. If the electrode s response obeys the Nernst equation. 

Sensitivity The sensitivity of a potentiometric analysis is determined by the term RT/nF or RT/zF in the Nernst equation. Sensitivity is best for smaller values of n or z. 

The difference between the potential actually required to initiate an oxidation or reduction reaction, and the potential predicted by the Nernst equation. 

Influence of Applied Potential on the Faradaic Current As an example, let s consider the faradaic current when a solution of Fe(CN)6 is reduced to Fe(CN)6 at the working electrode. The relationship between the concentrations of Fe(CN)6 , Fe(CN)6 A and the potential of the working electrode is given by the Nernst equation thus... 

Influence of the Kinetics of Electron Transfer on the Faradaic Current The rate of mass transport is one factor influencing the current in a voltammetric experiment. The ease with which electrons are transferred between the electrode and the reactants and products in solution also affects the current. When electron transfer kinetics are fast, the redox reaction is at equilibrium, and the concentrations of reactants and products at the electrode are those specified by the Nernst equation. Such systems are considered electrochemically reversible. In other systems, when electron transfer kinetics are sufficiently slow, the concentration of reactants and products at the electrode surface, and thus the current, differ from that predicted by the Nernst equation. In this case the system is electrochemically irreversible. 

Determining the Standard-State Potential To extract the standard-state potential, or formal potential, for reaction 11.34 from a voltammogram, it is necessary to rewrite the Nernst equation... 

The shift in the voltammogram for a metal ion in the presence of a ligand may be used to determine both the metal-ligand complex s stoichiometry and its formation constant. To derive a relationship between the relevant variables we begin with two equations the Nernst equation for the reduction of O... 

In the absence of ligand the half-wave potential occurs when [R]x=o and [O] c=o are equal thus, from the Nernst equation we have... 

When ligand is present we must account for its effect on the concentration of O. Solving equation 11.42 for [O] c=o and substituting into the Nernst equation gives... 

Electrochemical methods covered in this chapter include poten-tiometry, coulometry, and voltammetry. Potentiometric methods are based on the measurement of an electrochemical cell s potential when only a negligible current is allowed to flow, fn principle the Nernst equation can be used to calculate the concentration of species in the electrochemical cell by measuring its potential and solving the Nernst equation the presence of liquid junction potentials, however, necessitates the use of an external standardization or the use of standard additions. 

The free energy changes of the outer shell upon reduction, AG° , are important, because the Nernst equation relates the redox potential to AG. Eree energy simulation methods are discussed in Chapter 9. Here, the free energy change of interest is for the reaction... 

Because the ionic transference number for zirconia material is taken as being unity, then this equation reduces to the Nernst equation " ... 

Air is normally the reference gas used in the exhaust gas sensor. If the oxygen partial pressure in the engine exhaust gas is known as a function of the engine air/fuel ratio, the theoretical galvanic potential of the sensor is easily determined by the Nernst equation. 

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