Nernst equation example

In chemistry, the most important use of the Nernst equation Hes in the experimental determination of the concentrations of ions in solution. Suppose you measure the cell voltage E and know the concentration of all but one species in the two half-cells. It should then be possible to calculate the concentrations of that species by using the Nernst equation (Example 17.7). 

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. 

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... 

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

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). 

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... 

Traditionally concepts of ion selective permeation of biological membranes have centered on differences in the effective radii of hydrated nuclei. An example of that perspective derives from consideration of the resting membrane potential, E, which in the squid axon is approximated by the Nernst equation... 

The Nernst equation can also be used to determine the effect of changes in concentration on the voltage of an individual half-cell, E or Consider, for example, the half-reaction... 

Standard potentials Ee are evaluated with full regard to activity effects and with all ions present in simple form they are really limiting or ideal values and are rarely observed in a potentiometric measurement. In practice, the solutions may be quite concentrated and frequently contain other electrolytes under these conditions the activities of the pertinent species are much smaller than the concentrations, and consequently the use of the latter may lead to unreliable conclusions. Also, the actual active species present (see example below) may differ from those to which the ideal standard potentials apply. For these reasons formal potentials have been proposed to supplement standard potentials. The formal potential is the potential observed experimentally in a solution containing one mole each of the oxidised and reduced substances together with other specified substances at specified concentrations. It is found that formal potentials vary appreciably, for example, with the nature and concentration of the acid that is present. The formal potential incorporates in one value the effects resulting from variation of activity coefficients with ionic strength, acid-base dissociation, complexation, liquid-junction potentials, etc., and thus has a real practical value. Formal potentials do not have the theoretical significance of standard potentials, but they are observed values in actual potentiometric measurements. In dilute solutions they usually obey the Nernst equation fairly closely in the form ... 

An important application of the Nernst equation is the measurement of concentration. In a concentration cell, the two electrodes are identical except for their concentrations. For such a cell, E° = 0 and at 25°C the potential corresponding to the cell reaction is related to Q by E = —(0.025693 V//z) In Q. For example, a concentration cell having two Ag+/Ag electrodes is... 

Let us first take the example of titrating Fe2+ with Ce4+ and vice versa. There are two redox couples, Fe3+ /Fe2+ (1) and Ce4+ /Ce3+ (2), for each of which the Nernst equation... 

A description of an electrolytic cell has already been given under cell features (Section 1.3.2, Fig. 1.1c). Another example is the cell with static inert electrodes (Pt) shown in Fig. 3.1 where an applied voltage (Eappl) allows a current to pass that causes the evolution of Cl2 gas at the anode and the precipitation of Zn metal on the cathode. As a consequence, a galvanic cell, (Pt)Zn 2 ZnCl2 Cl2 iPt+, occurs whose emf counteracts the voltage applied this counter- or back-emf can be calculated with the Nernst equation to be... 

It is very often necessary to characterize the redox properties of a given system with unknown activity coefficients in a state far from standard conditions. For this purpose, formal (solution with unit concentrations of all the species appearing in the Nernst equation its value depends on the overall composition of the solution. If the solution also contains additional species that do not appear in the Nernst equation (indifferent electrolyte, buffer components, etc.), their concentrations must be precisely specified in the formal potential data. The formal potential, denoted as E0, is best characterized by an expression in parentheses, giving both the half-cell reaction and the composition of the medium, for example E0,(Zn2+ + 2e = Zn, 10-3M H2S04). 

EXAMPLE 14.7. Write the Nernst equation for the Cu/Ag cell of Example 14.6. 

Ans. The Daniell cell has a standard potential of 1.10V, as calculated in Example 14.5. The Nernst equation is used to calculate the actual potential. 

The reaction could also be written as the reverse of this one—the way we write it is arbitrary since the sign of E tells us what direction the reaction proceeds. Using the equation above, the Nernst equation is the following. Notice that there are two electrons involved, as determined in step 3 in the solution to Example 14.1. 

These observations are in accord with a scheme involving a reversible electron transfer, followed by a reaction that depletes the concentration of the initially formed reduced species, R. They are also reminiscent of the observations made earlier in regard to the electrohydrocyclization process. The greater the rate of the follow-up process, the more significant its effect on the concentration of R in a given time period, that associated with the CV scan rate, for example. From a moments consideration of the Nernst equation, it is clear that this event should manifest itself in terms of a shift in the peak potential to a more positive value, as observed for 255 and 257b [4]. In the present instance, it is suggested that a rapid or concerted loss of the mesylate anion in the reductive cyclization is likely to be associated with this so-called kinetic shift of the peak potentials [69]. 

Let us, for example, determine the of a [4Fe-4S] cluster in hydrogenase. The cluster has an EPR signal in its reduced form so we must rewrite the Nernst equation as... 

A redox couple that is wholly in solution can be analysed without recourse to a redox electrode - indeed, in the example given here, analysis with an iron rod would complicate the situation since the Fe " ", Fe " " system itself obeys the Nernst equation (equation (3.8)). 

Compton, R. G. and Sanders, G. H. W., Electrode Potentials, Oxford University Press, Oxford, 1996. This book is another in the Oxford Primer Series, and thus represents good value for money. The treatment of the Nernst equation, in particular, is thorough and straightforward. This book contains copious examples and exercises in the form of self-assessment questions (SAQs). Note, however, that it does not cover sensors. 

Eq. (8.25) indicates that as the pH is raised, oxidation becomes less favourable. The Nernst equation can also be used to predict which species will predominate in a solution at a particular redox potential. For the Fe /Fe couple (E = 0.77 V), for example, in an aqueous solution with Eh = 0.2 V, we have... 

Because electrode potentials are defined with reference to the H+/H2 electrode under standard conditions, E° values apply implicitly to (hypothetically ideal) acidic solutions in which the hydrogen ion concentration is 1 mol kg-1. Such E° values are therefore tabulated in Appendix D under the heading Acidic Solutions. Appendix D also lists electrode potentials for basic solutions, meaning solutions in which the hydroxide ion concentration is 1.0 mol kg-1. The conversion of E° values to those appropriate for basic solutions is effected with the Nernst equation (Eq. 15.15), in which the hydrogen ion concentration (if it appears) is set to 1.0 x 10-14 mol kg-1 and the identity and concentrations of other solute species are adjusted for pH 14. For example, for the Fc3+/2+ couple in a basic medium, the relevant forms of iron(III) and iron(II) are the solid hydroxides, and the concentrations of Fe3+ (aq) and Fe2+ (aq) to be inserted into the Nernst equation are those determined for pH 14 by the solubility products of Fe(OH)3(s) and Fe(OH)2(s), respectively. Examples of calculations of electrode potentials for nonstandard pH values are given in Sections 15.2 and 15.3. 

According to reaction 15.47, for which the Nernst equation gives Eh = +0.401 V at pH 14.0, the oxidation of water is clearly favored by alkaline conditions (see Exercise 15.3). At the same time, however, many oxidation half-reactions also have lower Eh values in basic media. For example, Eh for the manganate/manganese dioxide couple... 

The ranges of Eh and pH over which a particular chemical species is thermodynamically expected to be dominant in a given aqueous system can be displayed graphically as stability fields in a Pourbaix diagram,10-14 These are constructed with the aid of the Nernst equation, together with the solubility products of any solid phases involved, for certain specified activities of the reactants. For example, the stability field of liquid water under standard conditions (partial pressures of H2 and 02 of 1 bar, at 25 °C) is delineated in Fig. 15.2 by... 

Example Writing the Nernst Equation for a Half-Reaction... 

Even though this Nernst equation does not look like the one in the preceding example, Box 14-2 shows that the numerical value of E is unchanged. The squared term in the reaction quotient cancels the doubled value of n in front of the log term. 

Problem 14-20 gives an example of the use of the Nernst equation to find E... 

In the Figure 3.18 example, after imposing Ej across the electrode-solution interface, the potential is scanned negatively toward the standard redox potential of the O/R couple. The ratios of O and R that must exist at the electrode surface (Cr/Cq) at several potentials during the scan are given in each figure. These values are dictated by the Nernst equation for a reversible system (see Table 3.1). Since the solution initially contained only O, the R required to satisfy the Nernst equation is obtained from O by reduction, causing cathodic current. 

EXAMPLE 12.8 Using the Nernst equation to predict a cell potential... 

The electrochemical determination of pH using a pH meter is a particularly important application of the Nernst equation. Consider, for example, a cell with a hydrogen electrode as the anode and a second reference electrode as the cathode ... 

In the previous section the mercury electrode has been described. If no redox pairs (e.g. Fe2+ and Fe3+) are in solution and if we exclude gas reactions, the mercury electrode is completely polarizable. Polarizable means If a potential is applied, a current flows only until the electric double layer has formed. No electrons are transferred from mercury to molecules in the solution and vice versa. The other extreme is a completely reversible electrode, for which the Agl electrode is an example. Each attempt to change the potential of an Agl electrode leads to a current because the equilibrium potential is fixed by the concentrations of Ag+ or I according to the Nernst equation. 

An example is described here for the measurement of fluoride ions in solution. The fluoride electrode uses a LaF3 single crystal membrane and an internal reference, bonded into an epoxy body. The crystal is an ionic conductor in which only fluoride ions are mobile. When the membrane is in contact with a fluoride solution, an electrode potential develops across the membrane. This potential, which depends on the level of free fluoride ions in solution, is measured against an external constant reference potential with a digital pH/mv meter or specific ion meter. The measured potential corresponding to the level of fluoride ions in solution is described by the Nernst equation ... 

It is important to note that the electrode potential is related to activity and not to concentration. This is because the partitioning equilibria are governed by the chemical (or electrochemical) potentials, which must be expressed in activities. The multiplier in front of the logarithmic term is known as the Nernst slope . At 25°C it has a value of 59.16mV/z/. Why did we switch from n to z when deriving the Nernst equation in thermodynamic terms Symbol n is typically used for the number of electrons, that is, for redox reactions, whereas symbol z describes the number of charges per ion. Symbol z is more appropriate when we talk about transfer of any charged species, especially ions across the interface, such as in ion-selective potentiometric sensors. For example, consider the redox reaction Fe3+ + e = Fe2+ at the Pt electrode. Here, the n = 1. However, if the ferric ion is transferred to the ion-selective membrane, z = 3 for the ferrous ion, z = 2. 

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