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### Transcript of Understanding Voltammetry (Ch 1)

• July 5, 2007 b496 ch01 Understanding Voltammetry Fa

1

Equilibrium Electrochemistry andthe Nernst Equation

This chapter presents fundamental thermodynamic insights into electrochemicalprocesses.

1.1 Chemical Equilibrium

Thermodynamics predicts the direction (but not the rate) of chemical change.Consider the chemical reaction,

aA + bB + xX + yY + , (1.1)where the reactants A, B, . . . and products X, Y, . . . may be solid, liquid, or gaseous.Thermodynamics tells us that the Gibbs energy of the system, Gsys , is minimisedwhen it has attained equilibrium, as shown in Fig. 1.1.

Mathematically, at equilibrium, under conditions of constant temperature andpressure, this minimisation is given by

dGsys = 0. (1.2)Consider the Gibbs energy change associated with dn moles of reaction (1.1)proceeding from left to right

dG = {Gain in Gibbs energy of products}+ {Loss of Gibbs energy of reactants}

1

• July 5, 2007 b496 ch01 Understanding Voltammetry Fa

2 Understanding Voltammetry

Fig. 1.1

= {xxdn + yxdn . . .} {aAdn + bBdn + }= {xX + yY . . . aA bB}dn, (1.3)

where

j =(

G

nj

)T ,ni =nj

(1.4)

is the chemical potential of species j( j = A,B, . . . X ,Y , . . . ),T is the absolutetemperature (K) and ni is the number of moles of i (i = A,B, . . .X ,Y , . . . ). Thechemical potential of j is therefore the Gibbs energy per mole of j. It follows atequilibrium that

aA + bB + = xX + yY + , (1.5)

so that under conditions of constant temperature and pressure, the sum of thechemical potential of the reactants (weighted by their stoichiometric coefcientsa, b, . . . x , y , . . .) equals that of the products. If this were not the case, then theGibbs energy of the system would not be a minimum, since the Gibbs energy couldbe further lowered by either more reactants turning into products, or vice versa.

For an ideal gas,

j = oj + RT In(

PjPo

), (1.6)

where oj is the standard chemical potential of j , R is the universal gas constant

(8.313 J K1 mol1), Pj is the pressure of gas j and pois a standard pressure con-ventionally taken to be 105 Nm2 approximating to 1 atmosphere (atm), althoughstrictly speaking 1 atm = 1.01325 105 Nm2. It follows that oj is the Gibbsenergy of one mole of j when it has a pressure of 1.01325 105 Nm2. It follows

• July 5, 2007 b496 ch01 Understanding Voltammetry Fa

Equilibrium Electrochemistry and the Nernst Equation 3

from Eqs. (1.5) and (1.6) that at equilibrium

xoX + yoY + aoA boB = xRTInPXPo

yRTIn PYPo

+

+ aRTIn PAPo

+ bRTIn PBPo

, (1.7)

so thatGo = RTInKp , (1.8)

where Go = xoX + yoY + aoA boB is the standard Gibbs energychange accompanying the reaction and

Kp =(

PXPo

)x (PYPo

)y (PAPo

)a (PBPo

)b (1.9)is a constant at a particular temperature, because the standard chemical potentialo depends only on this parameter (unless the gases are not ideal, in which caseKp may become pressure dependant). Thus, for the gas phase reaction

2O2(g ) + N2(g ) 2NO2(g ) (1.10)equilibrium is denoted by the equilibrium constant

Kp =(

PNO2Po

)2(

PO2Po

)2 (PN2Po

) . (1.11)Note that if some of the reactants and/or products in reaction (1.10) are in solution,then the pertinent ideal expression for their chemical potentials are

j = oj + RTIn[j][ ]o , (1.12)

where [ ]o is a standard concentration taken to be one molar (one mole per cubicdecimetre). Applied to Eq. (1.1) this leads to a general equilibrium constant

KC =( [X ]

[ ]o)x ( [Y ]

[ ]o)y

( [A][ ]o

)a ( [B][ ]o

)b . (1.13)It follows that for the equilibrium

HA(aq) H+(aq) + A(aq), (1.14)

• July 5, 2007 b496 ch01 Understanding Voltammetry Fa

4 Understanding Voltammetry

where HA is, say, a carboxylic acid and A a carboxylate anion, the equilibriumconstant, Kc is given in terms of concentrations by

KC =( [H+]

[ ]o) ( [A]

[ ]o)

( [HA][ ]o

) . (1.15)In common usage, Eqs. (1.11) and (1.1) take the more familiar forms of

Kp = PxXP

yY

PaAPbB

and Kc = [X ]x [Y ]y

[A]a[B]b ,

where it is implicitly understood that pressure is measured in unitsof 105 Nm2 (or strictly 1.01325 105 Nm2) and concentrations inM (mol dm3) units.

In the case that the reactants in Eq. (1.1) are pure solids or pure liquids,

j oj . (1.16)That is to say, the chemical potential approximates (well) to a standard chemicalpotential. Note that unlike gases or solutions, Gibbs energy per mole depends onlyon the temperature and pressure; changing the amount of material changes thetotal Gibbs energy, but not the Gibbs energy per mole.

It follows from Eq. (1.16) that, since the chemical potentials of pure liquidsand solids are independent of the amount of material present, there are no corre-sponding terms in the expression for equilibrium constants in which these speciesparticipate. So for the general case

aA(g ) + bB(aq) + cC(s) + dD(l) wW (g ) + xX(aq) + yY (s) + zZ (l), (1.17)the equilibrium constant will be

K = PwW [X ]xPaA [B]b

, (1.18)

where it is understood that the pressures are measured in units of 105 Nm2 andthe concentrations in units of moles dm3. The pure solids C and Y , and pureliquids D and Z do not appear. Illustrative real examples follow:

First, for

AgCl(s) Ag+(aq) + Cl(aq),K = [Ag+][Cl]. (1.19)

• July 5, 2007 b496 ch01 Understanding Voltammetry Fa

Equilibrium Electrochemistry and the Nernst Equation 5

Second, forCaCO3(s) CaO(s) + CO2(g ), (1.20)

K = PCO2 .Last,

Fe2+(aq) + 1/2Cl2(g ) Fe3+(aq) + Cl(aq) (1.21)leads to

K = [Fe3+][Cl ]

[Fe2+]P1/2Cl2.

1.2 Electrochemical Equilibrium: Introduction

In the previous section, we considered various forms of chemical equilibriuminvolving gaseous, liquid, solution phase and solid species. We now turn to elec-trochemical equilibrium and, as a paradigm case, focus on the following process

Fe(CN )36 (aq) + e Fe(CN )46 (aq). (1.22)Such an equilibriumcanbe establishedbyrst preparing a solution containing bothpotassiumhexacyanoferrate(II),K4Fe(CN)6, andpotassiumhexacyanoferrate(III),K3Fe(CN)6 dissolved in water and then inserting a wire (an electrode) made ofplatinum or another inert metal into the solution (Fig. 1.2).

The equilibrium in Eq. (1.22) is established at the surface of the electrodeand involves the two dissolved anions and the electrons in the metal electrode.The establishment of equilibrium implies that the rate at which Fe(CN )46 gives

Fig. 1.2 A platinum wire immersed into an aqueous solution containing both ferrocyanideand ferricyanide.

• July 5, 2007 b496 ch01 Understanding Voltammetry Fa

6 Understanding Voltammetry

up electrons to the metal wire or electrode is exactly balanced by the rate atwhich electrons are released by the wire to the Fe(CN )36 anions, which are saidto be reduced. Correspondingly, the Fe(CN )46 ions losing electrons are said tobe oxidised. That a dynamic equilibrium of this type is established implies that,once established, no further change occurs. Moreover, the net number of electronsthat are transferred in one direction or another is innitesimally small, such thatthe concentrations of Fe(CN )46 and Fe(CN )

36 are not measurably changed from

their values before the electrode is introduced into the solution.Equation (1.22) has a signicant difference from the chemical equilibrium con-

sidered in section 1.1. In particular, the reaction involves the transfer of chargedparticles, electrons, between the metal and solution phases. As a result, when equi-librium is attained, there is likely to be a net electrical charge on each of these twophases. If reaction (1.22) lies to the left when equilibrium is reached, in favour ofFe(CN )36 and electrons, then the electrode will bear a net negative charge and thesolution a net positive charge of equal magnitude. Conversely, if the equilibriumfavours Fe(CN )46 and lies to the right, then the electrode will be positive and thesolution negatively charged.

Irrespective of the position of the equilibrium of reaction (1.22), it canbe recognised that there will likely exist a charge separation between theelectrode and the solution phases. Accordingly there will be a potential dif-ference (difference in electrical potential) between the metal and the solu-tion. In other words, an electrode potential has been established at the metalwire relatively to the solution phase. The (electro)chemical reaction given in(1.22) is the basis of this electrode potential and it is helpful to refer to thechemical processes which establish electrode potentials as potential determiningequilibria.

Other examples of electrochemical processes capable of establishing a potentialon an electrode in aqueous solutions include the following:

(a) The hydrogen electrode, shown in Fig. 1.3, comprises a platinumblack electrodedipping into a solution of hydrochloric acid.

The electrode may be formed by taking a bright platinum ag electrode andelectro-depositing a ne deposit of platinum black from a solution containing asoluble platinum compound, typically K2PtCl6. Hydrogen gas is bubbled over thesurface of the electrode and the following potential determining equilibrium isestablished:

H+(aq) + e(m) 1/2H2(g ), (1.23)where (m) reminds us that the source of electrons resides in the metal electrode.

(b) The silver/silver chloride electrode comprises a silver wire coated with a porouslayer of silver chloride. The latter is almost insoluble in water and can be formed

• July 5, 2007 b496 ch01 Unde