Journal of

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Journal of Electroanalytical Chemistry 594 (2006) 96–104

ElectroanalyticalChemistry

Introduction of a model for describing the redox potentialin faradic electrodes

Juan Soto a,*, Roberto H. Labrador a, Marıa D. Marcos a, Ramon Martınez-Manez a,Carmen Coll a, Eduardo Garcıa-Breijo b, Luis Gil b

a Departamento de Quımica, Instituto de Quımica Molecular Aplicada, Universidad Politecnica de Valencia, Camino de Vera s/n, 46021 Valencia, Spainb Departamento de Ingenierıa Electronica, Instituto de Quımica Molecular Aplicada, Universidad Politecnica de Valencia, Camino de Vera s/n,

46021 Valencia, Spain

Received 8 November 2005; received in revised form 22 May 2006; accepted 30 May 2006Available online 20 July 2006

Abstract

We report here a general model for the description of the redox potential behaviour of certain electrodes in complex mixtures basedon the study of redox process. The model has been developed theoretically and has also been experimentally verified. The model allows tostate broad-spectrum equations which describe the behaviour of the potential of second and third order Faradic electrodes. Experimentalverification of the model was carried out using common redox electrodes. The model was used to predict the redox behaviours in thefollowing cases; (i) study of the potential variation of a Silver electrode as function of the pH, (ii) the redox potential of an Ag/AgClelectrode as function of the chloride concentration at different pH values, (iii) the potential of a Ag/AgBr electrode in the presence ofcertain amounts of chloride and (iv) the quantitative solid–solid transformation between AgBr and AgCl phases. There was found aremarkably good agreement between the experimental and theoretical values.� 2006 Elsevier B.V. All rights reserved.

Keywords: Faradic electrodes; Redox potential; Metal-based electrodes

1. Introduction

Metallic and metal-based electrodes have been usedwidely in a number of different applications. For example;gold or platinum electrodes have been employed for thedetermination of the redox potential [1,2] and platinum,iridium, titanium, lead, tin, ruthenium, etc. (as metal/metaloxide electrodes M/MO), used in the determination of thepH in aqueous solutions [3–6]. Moreover, some othersmetal/metal insoluble salt electrodes (M/MX), have beenused for the determination of anions and as reference elec-trodes [7,8]. Additionally, metal/metal oxide and metaloxide–glass composites [7,9,10] have been employed

0022-0728/$ - see front matter � 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.jelechem.2006.05.033

* Corresponding author. Tel.: +34 963877343; fax: +34 963877349.E-mail addresses: jsotoca@qim.upv.es (J. Soto), rmaez@qim.upv.es

(R. Martınez-Manez).

recently as electrode arrays in electronic tongues designfor the qualitative analysis of natural waters.

Systematic investigations on families of metal/metaloxide electrodes (M/MO) have been performed character-izing, among other parameters, processes of interferenceof some anions. But, as far as we know, the subject of inter-fering processes in M/MO electrodes has not been dealt indepth specially from a theoretical viewpoint in order todevelop models capable of justify the electrode behaviourin complex aqueous solutions. In fact, there are severaltypes of electrodes, [11–14] namely;

(i) Faradic or Redox electrodes: zero (Inert Metal/Redox Couple), first (M/Mn+), second (M/MX/X�)and third (M/MX/NX/N+) order electrodes (whereM = metal, M+, N+ = cations, X� = anion), whichgenerate an electric potential caused by a redoxprocess (Faradic Potentiometry).

J. Soto et al. / Journal of Electroanalytical Chemistry 594 (2006) 96–104 97

(ii) Non-Faradic Electrodes, with a potential generatedby processes of charge absorption or adsorption,ionic exchange in the electrode interface, or chargetransport.

For the charge transport electrodes, the electrode poten-tial is associated with the electrical work of the ionic trans-port through an interface or membrane. Those electrodeshave an appropriate model (the Nicolsky–Eisenman equa-tion) which is suitable for describing both the electrodebehaviour and interfering processes, and is also the basisof the most of selectivity coefficient (Kpot) calculating meth-odologies. In a previous IUPAC technical report [15],Umezawa and co-workers presented a thorough reviewabout the methodology for calculating Kpot, as well assome incongruence in some Kpot calculations.

For the specific case of solid-state ISEs, (with adsorp-tion and ionic exchange or metathetic reactions), Hulanickiand Lewenstam [16] proposed two different models for thetreatment of selectivity coefficients, for a system withoutdiffusion in the membrane (AgCl/Br� and AgBr/Cl�) andsystem with diffusion in the membrane (LaF3/OH�). Thefinal equation of the proposed model, expressed the selec-tivity coefficient as a function of apparent coverage of themembrane in the first system, and as a function of theapparent site-filling factor of the membrane for the secondsystem. Later, Michalska and Lewenstam [17] applied thetheory of the Hulanicki’s proposed model in the evaluationof the selectivity of p-doped polymer films.

Following our recent interest in the development of elec-trodes [18–20], in this paper, we propose a model for explain-ing the electrical potential of Faradic or Redox electrodes,based on a study of the activity changes of different specieson the electrode surface where an heterogeneous redox reac-tion occurs. The model is additionally tested in some selectedproblems. The final model leads to a better understanding ofthe redox behaviour on these electrodes and particularlyallows to describe and to simulate the electrode potentialobserved in the presence of certain interferents.

2. Chemical potential and activity

The starting point of our model for explaining thepotential of Faradic or Redox electrodes is to study thechanges observed on the electrode when heterogeneousreactions (such as metathetic reactions) occurs via massand charge balance combined with the equilibrium con-stants at the surface. For this purpose we will start withan apparent oddity in order to consider that the activityof the solids would change as consequence of the transfor-mation of one solid into another.

It is recognized for a liquid or solid in the standard state,that their activity is equals to 1. It is also well-known (fromthe Colligative properties in terms of both the chemicalpotential and the activity of the solvent) that for a idealmixture of liquids (a binary system: for instance liquidA + liquid B in equilibrium with its vapour), the liquid

activity of A is equal to the molar fraction of liquid A(and also to the vapour A activity)

aðvapourÞA ¼P vA

P 0vA

¼ vA � P 0vA

P 0vA

¼ vA ¼ aðliquidÞA

In a similar manner, the activity (molar fraction) of a cer-tain solid C that is partially transformed to solid D willchange accordingly with the change of the molar fractionof the solids. Thus, for instance, we will assume that whena metathetic reaction occurs (such as the transformation ofAgCl into AgBr in presence of bromide), the activity ofAgCl would vary from 1 (when Br� is not present) to 0(when all Cl� is replaced by Br�)

aðAgClÞ ¼ vAgCl ¼nAgCl

nAgCl þ nAgBr

¼ nAgCl

n0i! aðsolidÞi ¼

ni

n0i

ð1ÞThis is in agreement with the definition of activity as a

dimensionless number ai equals to the quotient (cCi/Ci0),where c is the activity coefficient. When we assume an idealbehaviour (c = 1) the activity is equals to moles or concen-trations quotient

ai ¼Ci

C0i¼ ni=V

n0i=V¼ ni

n0ið2Þ

where Ci is the concentration of i and C0i is the reference con-centration (normally 1 M). Obviously, the relation is onlytrue if the occupied volume by i remains constant or nearconstant during the reaction. As we will see below, this pro-posed definition of the solid activity is very useful for thesolution of certain problems. The starting point of the modelobeys the general rules of redox and solubility equilibrium,joined with a combination of mass balance, charge balance,thermodynamic constants and the concept of solid activity(ai = ni/n0i, only 1 if no reactions were present). Conse-quently the results offer a broad number of applications(for example: the study of M/MO or M/MX systems in thepresence of certain anions, redox-pH (Pourbaix diagrams),solubility, co-precipitation, metathetic reactions, etc.).

3. Metal oxide in the presence of an anion which forms aninsoluble salt

To start with, we will study a general situation in whicha metal electrode (M), partially coated with its metal oxide(MO) is in contact with an aqueous solution containingand anion (X�) that forms an insoluble precipitate (MX2)at a certain pH. Let us suppose the total volume of thesolid remains near constant. A resume of this situation isshown in Schemes 1 and 2.

If we assume, for convenience, that the electrode M/MOresponds to the pH, then we can define the interferencereaction as

MðOHÞ2 ðsÞ þ 2X� ðaqÞ ¡ MX2 ðsÞ þ 2OH� ðaqÞ

KpotI ¼

aMX2ðOH�Þ2

aMðOHÞ2ðX�Þ2

ð3Þ

M+2M(OH)2(s)

MO(s)

M0(s)

Kps

K0

+2e- -2e-

KIpot

MX2

Scheme 1. Simplified equilibrium in water for a divalent metal M inpresence of X�.

M+2

Ks0

MOH+

M0(s)

M(OH)3-

Kf3Kf2Kf1

+2e- -2e-

MX3-

KX3

MX+

KX1

KX2

M(OH) 2

MO(s)

M(OH)2(s)

K0

Kps

MX2 (s)

MX2

Ks0x

Kpsx

Scheme 2. Involved species in a typical equilibrium for a divalent metallicion in water.

M(s)

MO(s)

H2O

MX2(s)

M(s)

MO(s)

H2OX-

M(s)

MO(s)

H2OM(s)

MO(s)

H2O

MX2(s)

M(s)

MO(s)

H2O

MX2(s)

M(s)

MO(s)

H2OX-

Fig. 1. Equilibrium displacement of the system M/MO, when aninterferent X� is added.

98 J. Soto et al. / Journal of Electroanalytical Chemistry 594 (2006) 96–104

where KpotI is the selectivity coefficient (or constant of equi-

librium of interference). For the simple M/M2+ system thecorresponding Nernst equation is

E ¼ E0M2þ=M

� 59:16

2log

aM

ðM2þÞð4Þ

Under ideal potentiometric conditions, where the intensitythrough the electrode is null, no changes in the surface ofmetal can be expected and then

From Eq: ð1Þ: aM ¼nM

ðnMÞ0¼ 1

Consequently E ¼ E0 � 59:16

2log

1

ðM2þÞðE0 ¼ E0

M2þ=MÞ

where it has been assumed that the activity of the metal Mis equal to unity. In general, this approximation is suitablefor a solid in contact with a simple solution. However, asstated above, when a mixture of solids is present simulta-neously at the same electrode and they can be convertedto each other this approximation is not acceptable. Sucha situation is for instance displayed in Fig. 1 that shows

a metal oxide (MO) transformed to the correspondinginsoluble salt (MX2) when is in contact with a solution con-taining the anion X�.

In this case, at the equilibrium, we can write

ðnMOÞ0 ¼ nM2þ þ nMðOHÞ2 þ nMO þ nMX2ð5Þ

where (nMO)0 is the initial amount of moles of MO at themetal surface. Then we have

1 ¼ V ðM2þÞðnMOÞ0

þnMðOHÞ2ðnMOÞ0

þ nMO

ðnMOÞ0þ nMX2

ðnMOÞ0

¼ ðM2þÞ

C0

þ aMðOHÞ2 þ aMO þ aMX2ð6Þ

where according with (1), aMðOHÞ2 , aMO and aMX2are the

activity of solids M(OH)2, MO and MX2, respectively.Additionally, we have defined C0 as the ratio (nMO)0/V thisterm, C0, will be related further with the called ‘‘manufac-ture conditions of the electrode’’. In principle, C0 can bedefined for each state of oxidation of the redox process,referred to any species involved in the equilibrium. Work-ing out the value of the activity of MX2 from Eq. (3)

aMX2¼ Kpot

I �aMðOHÞ2 � ðX

�Þ2

ðOH�Þ2ð7Þ

Additionally for equilibrium:

MO ðsÞ þH2O ¡ MðOHÞ2 ðsÞWe have

K0 ¼aMðOHÞ2

aMOaH2O

¼aMðOHÞ2

aMO

and aMO ¼aMðOHÞ2

K0

ð8Þ

Kps ¼ðM2þÞðOH�Þ2

aMðOHÞ2working out

aMðOHÞ2 : aMðOHÞ2 ¼ðM2þÞðOH�Þ2

Kps

ð9Þ

Afterwards, replacing Eqs. (7)–(9) in Eq. (6) and workingout the value of the (M2+) result in Eq. (10).

ðM2þÞ ¼ 1

1C0þ ðOH�Þ2

Kpsþ ðOH�Þ2

K0Kpsþ Kpot

IðX�Þ2

Kps

ð10Þ

This expression can be rearranged with Eq. (4), to give

E ¼ E0M2þ=M

� 59:16

2log

1þK0

K0Kps

ðOH�Þ2 þ 1

C0

þKpotI

Kps

ðX�Þ2� �

ð11Þ

J. Soto et al. / Journal of Electroanalytical Chemistry 594 (2006) 96–104 99

This equation shows the interference of an anion on thepotentiometric response of a M/MO system acting as pHelectrode. A simple thermodynamic cycle allows to obtaina relation between Kpot

I and the equilibrium solubility con-stants of both, the hydroxide M(OH)2 (Kps) and the insol-uble salt MX2 ðKMX

ps ÞMðOHÞ2 ðsÞ¡ M2þ ðaqÞ þ 2OH� ðaqÞ Kps

M2þ ðaqÞ þ 2X� ðaqÞ¡ MX2 ðsÞ 1=KMX2ps

MðOHÞ2 ðsÞ þ 2X� ðaqÞ¡ MX2 ðsÞ þ 2OH� ðaqÞ KpotI

KpotI ¼

Kps

KMX2ps

ð12ÞIs important to remark that Eqs. (11) and (12) are a

particular solution for a specific problem and therefore, adifferent electrochemical system will have a differentsolution. Eq. (11) can be rearranged as

E ¼ E0M2þ=M

� 59:16

2log

1þ K0

K0Kps

� 59:16

2log ðOH�Þ2 þ K0Kps

C0ð1þ K0Þþ K0Kpot

I

1þ K0

ðX�Þ2� �

ð13ÞIf K0 is larger than 1, the quotient K0/(1 + K0) � 1. ThenEq. (13) can be written as

E ¼ E0M2þ=M

� 59:16

2log

1

Kps

� 59:16

2log ðOH�Þ2 þ Kps

C0

þ KpotI ðX�Þ

2

� �ð14Þ

It is easy to verify that

E0MðOHÞ2=M ¼ E0

M2þ=M� 59:16

2log

1

Kps

ð15Þ

Replacing (15) in (14)

E ¼ E0MðOHÞ2=M �

59:16

2log ðOH�Þ2 þ Kps

C0

þ KpotI ðX�Þ

2

� �ð16Þ

Both, Eqs. (13) and (16) are able to predict and interpretthe response of a M/MO/MX2/H+, X� electrode as a func-tion of the pH and the concentration of anion X�. Addi-tionally, considering the OH� as the primary ion, Eq.(16) indicates that the response of the M/MO electrodecan be interfered by anions that form an insoluble salt withthe metallic ion and allows to determine the corresponding‘‘interference constant’’. Besides, Eq. (16) establishes amathematic relation similar than the Nicolsky–Eisenman(NE) equation, with some remarkable differences:

(i) Eq. (16) has been obtained from a model based onthe existence of an heterogeneous redox equilibriumM/MO/MX2/H+, X�, where the different physicalphases have been taken into account.

(ii) The sign that proceeds the logarithm term in Eq. (11)is always negative, in accordance with the IUPAC,

meanwhile NE depends on the sign of the transportedion.

(iii) In Eq. (16), there are three terms, the first is relatedwith the primary ion, the second with the quotientKps/C0 and the last term related with the interferingspecies: an anion (X�). NE is only conceived fortwo terms for both primary and interfering ion. Theterm Kps/C0 depends on the history of the electrodeand can contribute remarkably in the electroderesponse.

(iv) The proposed model establishes that the selectivitycoefficient Kpot

I can be calculated from the commonequilibrium constants related with the formed spe-cies (i.e. Kps, Kw). The Kpot

I included in NE is relatedwith the ionic mobilities u+ or u� through themembrane.

3.1. Others equivalent expressions

An equivalent relation can be established from Eqs. (13)and (16).

Replacing (OH�) in Eq. (16)

Kw ¼ ðOH�ÞðHþÞ ! ðOH�Þ2 ¼ K2w

ðHþÞ2

It is easy to verify that

E ¼ E0MðOHÞ2=M �

59:16

2log K2

w

� 59:16

2log

1

ðHþÞ2þ Kps

C0K2w

þ KpotI

K2w

ðX�Þ2" #

ð17Þ

This equation can be rearranged in a mathematic structuresimilar than NE where the primary ion is H+ and the inter-ferent is X�

E ¼ E02 �

59:16

2log

1

ðHþÞ2þ Kps

C0K2w

þ Kpot0

I ðX�Þ2

" #ð18Þ

Comparing Eqs. (17) and (18), is obvious that

E02 ¼ E0

MðOHÞ2=M � 59:16 log Kw ð19Þ

and

Kpot0

I ¼ KpotI

K2w

ð20Þ

A similar relation can be obtained when we consider X�

as the primary ion and OH� as the interfering species. Toachieve this, we work out Kpot

I from expression (16) tofinally give Eq. (21)

E ¼ E03 �

59:16

2log ðX�Þ2 þ Kps

C0KpotI

þ Kpot00

I ðOH�Þ2� �

ð21Þ

where

E03 ¼ E0

MðOHÞ2=M �59:16

2log Kpot

I ð22Þ

Table 1Equations for describing the standard potential of a M/MO/MX2/H+, X� system at 25 �C

Primary ion Interf. E (mV) KpotI E0 (mV)

OH� X� E01 � 59:16

2 log ðOH�Þ2 þ Kps

C0þ Kpot

I ðX�Þ2

h iKpot

I E01

H+ X� E02 � 59:16

2 log 1ðHþÞ2 þ

Kps

C0K2wþ Kpot0

I ðX�Þ2

h iKpot

I

K2w

E01 � 59:16pKw

X� OH� E03 � 59:16

2 log ðX�Þ2 þ Kps

C0KpotI

þ Kpot00

I ðOH�Þ2� �

1Kpot

I

E01 � 59:16

n log KpotI

X� H+ DE04 � 59:16

2 log ðX�Þ2 þ Kps

C0KpotI

þ Kpot000I

ðHþÞ2

� �K2

w

KpotI

E01 � 59:16

n log KpotI

M2+ H+ E00 � 59:16

2 log 1C0þ Kpot

I1

ðHþÞ2

h ið1þK0ÞKw

K0KpsE0

Mnþ=M

M2+ X�i E00 � 59:16

2 log 1C0þP

KpotI ðX�Þ

2i

h i1

KMX2ps

E0Mnþ=M

KpotI ¼ K

MðOHÞnps

KMXnps

, E01 ¼ E0

MðOHÞn=M and E00 ¼ E0

Mnþ=M.

100 J. Soto et al. / Journal of Electroanalytical Chemistry 594 (2006) 96–104

and

Kpot00

I ¼ 1

KpotI

ð23Þ

A similar relation can be obtained for M2+ as primaryion and H+ or OH� as interferent (see Table 1).

E ¼ E0M2þ=M

� 59:16

2log

1

C0

þ 1

Kps

ðOH�Þ2 þ KpotI

Kps

ðX�Þ2� �

ð24ÞAs it can be seen from these equations, when the primaryion is a cation, the electrode potential is proportional tothe logarithm of the inverse of cation activity, whereas ifthe primary ion is an anion, the electrode potential is pro-portional to the logarithm of the anion activity. This ruleshall also be applied according to the nature of the interfer-ing ions. Table 1 summarizes these expressions for the rela-tion between E and different combinations of primary and‘‘interfereing’’ ions for a general M/MO/MX2/H+, X�

system.By this moment, it should be convenient to state some

considerations. Each time that an M/MO/MX2/H+, X�

electrode is used for potential measurements in a solution,its surface will be transformed by the formation of a mix-ture of different kind of ‘‘sparingly’’ soluble solids (i.e oxi-des, hydroxides, MX, MY, where X and Y are precipitantanions), which influences on the electrode response in pres-ence of a new environment. The above statement is relatedwith the term C0 that determines the so called ‘‘history ofthe electrode’’.

4. A more general model for describing metal electrodes

In the model developed above a M/MO electrode isstudied in function of pH and also in presence of anions(X�) able to form an insoluble salt with the cation M2+.However, following a more general approach, a metal elec-trode in contact with a solution can suffer a number ofadditional equilibria such as the re-dissolution at basic

pH due to the formation of hydroxocomplexes or the for-mation of soluble MX�m

nþm species in the presence of suitableanions. The difficulty level can be considerably increased, ifwe admit that the metal electrode surface could be attackedby transfer electron reaction between the ions of the envi-ronment and the metal (i.e oxidative reaction of the metalwith H+). In order to avoid the latter problem, we will sup-pose that the metal is noble enough to be non-corroded orthat the study is limited to a certain pH range.

Following a general approximation, on the metal sur-face the chemical and electrochemical equilibriums shownin Scheme 2 can take place:

As above, we can state for this system that

E ¼ E0M2þ=M

� 59:16

2log

aM

ðM2þÞand

ðnMOÞ0 ¼ nM2þ þ nMðOHÞþ þ nMðOHÞ2 þ nMðOHÞ2 ðsÞ þ nMOðsÞ

þ nMðOHÞ�3 þ nMXþ þ nMX2þ nMX�

3þ nMX2ðsÞ ð25Þ

Applying the same methodology in Section 3 or 3.1, wecan transform Eq. (25) in expression (26) that only depen-dent of the term (M2+).

Working out (M2+) and replacing in the Nernst equa-tion, we obtain Eq. (26), which is a cubic potential functionof (OH�) and (X�)

E ¼ E0 � 59:16

2log

1

C0

ð1þ Kf 1ðOH�Þ þ Kf 2ðOH�Þ2�

þKf 3ðOH�Þ3 þ Kx1ðX�Þ þ Kx2ðX�Þ2 þ Kx3ðX�Þ3Þ

þ ðOH�Þ2 1

Kps

þ 1

K0Kps

� �þ Kpot

I ðX�Þ2

�ð26Þ

We can generalize Eq. (26) as

E ¼ E0� 59:16

2log

1

C0

1þXn

i¼1

KfiðOH�ÞiþXn

i¼1

KxiðX�Þi !"

þ ðOH�Þ2

Kps

1þ 1

K0

� �þKpot

I ðX�Þ2

#ð27Þ

J. Soto et al. / Journal of Electroanalytical Chemistry 594 (2006) 96–104 101

From this equation, is easy to obtain the equivalents trans-formations shown en Table 1, and thus, raising the elec-trode behaviour as a function of H+ or X� (as primaryions).

Is feasible to identify the terms involved in the logarithmterm: the first parenthesis is associated with the species insolution: M2+, M(OH)+, M(OH)2, MðOHÞ�3 , MX+ andMX2, MX�3 , the second term associated with the solidphases MO and M(OH)2 in equilibrium, and the last termis related with the insoluble salt MX2(s). Eqs. (26) or (27)also show that the ‘‘selectivity coefficient or constant ofinterference’’ does not depend only on the interfering activ-ity (as demonstrated in the previous simplified model, orthe NE equation) but also depends really on a virial seriesof the involved species. Finally, the equations show that thepotential of a metal electrode depends on two factors: thepH of the dissolution and the initial quantity of the oxi-dized metal (C0) formed in the system. This equation alsoallows to plot the E vs. pH equilibrium lines (Pourbaix dia-gram [21]) of a metal as a continuous mathematic function,instead stretch by stretch, as is usually done.

5. Experimental verification of the model

In order to validate/verify the model described above,some common M/M+n, M/MO, M/MX2 and M/MO/MX2/H+, X� systems were studied using the equationsdeveloped above.

The first system studied is an Ag/Ag+ electrode. Forsuch system the possible chemical and electrochemicalequilibriums are shown in Scheme 3. As a consequence ofthe stechiometry of the formed oxide (Ag2O), the modelis mathematically different to those previous, that’s whyis studied.

The corresponding Nernst expression is

DE ¼ DE0Agþ=Ag �

59:16

1log

aAg

aAgþ

And taking into account similar considerations as thosediscussed above

ðnAgþÞ0 ¼ nAgþ þ nAgðOHÞ�2 þ 2nAg2O þ nAgðOHÞ ð28Þ

and

Ag+ Ag(OH)(s)

Ag2O(s)

Ag0(s)

Ag(OH)2-

KfKps

K0

+1e- -1e-

Scheme 3. Involved species in a simplified equilibrium for a Ag/Ag+

electrode in water.

1 ¼ ðAgþÞðAgþÞ0

þ ðAgðOHÞ�2 ÞðAgþÞ0

þ aAgðOHÞ þ 2aAg2O

¼ ðAgþÞC0

þ ðAgðOHÞ�2 ÞC0

þ aAgðOHÞ þ 2aAg2O ð29Þ

From the Kf, K0 and Kps constants we can obtain

ðAgþÞ2 þ 1

2

K20K2

ps

C0ðOH�Þ2þ

Kf K20K2

ps

C0

þ K20Kps

ðOH�Þ

!ðAgþÞ

�K2

0K2ps

2ðOH�Þ2¼ 0 ð30Þ

Denominating B to the following expression:

B ¼ 1

2

K20K2

ps

C0ðOH�Þ2þ

Kf K20K2

ps

C0

þ K20Kps

ðOH�Þ

!

and the Nernst equation is

E ¼ E0Agþ=Ag �

59:16

1log

aAg

aAgþ

¼ E0Agþ=Ag �

59:16

1log

2

�BþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB2 þ 2

K20K2

ps

ðOH�Þ2

r ð31Þ

Eq. (31) allows to calculate the potential of the electrodeAg/Ag+ as function of both: the pH of the dissolutionand the initial activity of Ag+ (C0). This equation can besimplified taking into account that the second order termin Eq. (30) would be negligible in front of the linear term.Then, it can be admitted that

1

2

K20K2

ps

C0ðOH�Þ2þ

Kf K20K2

ps

C0

þ K20Kps

ðOH�Þ

!ðAgþÞ� 1

2

K20K2

ps

ðOH�Þ2� 0

ð32Þand therefore

E ¼ E0Agþ=Ag �

59:16

1log

1

C0

þ Kf ðOH�Þ2

C0

þ ðOH�ÞKps

!

ð33ÞIn order to validate Eq. (33) we have studied the potentialvariation of the Ag/Ag+ electrode as function of pH. Forthis purpose, a silver foil (ALDRICH, 0.1 mm thick,99%) in a PVC cylinder as electrode body, was immersedin an aqueous solution containing a concentration of10�5 or 10�3 mol dm�3 of Ag+ (KNO3 0.01 M as support-ing electrolyte), and the variation of the electrode potentialvs. pH (from (2)–(13)) was studied. Fig. 2 shows the exper-imental and theoretical values using Eq. (33).

In the presence of an interferent such as X� able toform the insoluble precipitate AgX(s) Eq. (33) is trans-formed to

E¼E0Agþ=Ag�

59:16

1log

1

C0

þKf ðOH�Þ2

C0

þðOH�ÞKps

þðX�Þ

KAgXps

!

ð34Þ

0

100

200

300

400

500

0 2 4 6 8 10 12 14

pH

E(m

V)

Fig. 2. Electrode potential vs. pH at 298 K using Eq. (33) (solid line) andexperimental values (symbols), for a Ag electrode in aqueous solutionscontaining Ag+ concentrations (C0) of 10�5 (h) and 10�3 (n) mol dm�3.Potential is referred to the Ag/AgCl electrode. (pKps(AgCl) = 9.75,logKf = 3.18).

0

50

100

150

200

250

300

6 8 10

p Cl-

E (

mV

)

pH 6

pH 12.5

0 2 4

Fig. 3. Experimental (symbol) and theoretical values using Eq. (35) (solidline) of the redox potential of a Ag/AgCl electrode at two different valuesof pH.

102 J. Soto et al. / Journal of Electroanalytical Chemistry 594 (2006) 96–104

and following a similar procedure to that shown above thefinal Nernst equation is

E � E0Agþ=Ag � 59:16 log

1

C0

þ ðOH�ÞKps

þ ðX�Þ

KAgXps

!

¼ E0Agþ=Ag � 59:16 log

1

C0

þXn

i¼1

ðX�ÞiKMXi

ps

!ð35Þ

This final Eq. (35) is similar to that described above inTable 1. As Ag/AgX system we have choose a Ag/AgClelectrode. That electrode was prepared by oxidation of aAg wire in a 0.1 mol dm�3 solution of NaCl. A total cur-rent of 6 C gave a total of 6.22 · 10�5 moles of AgCl (s)on the metal surface. This electrode was further immersedin a total volume of 50 mL and a study of its potential attwo different pHs (6 and 12.5) as a function of the chlorideconcentration was studied. In the theoretical simulation,the factors such as ‘‘contact potential’’, ionic coefficientof Debye–Huckel (c±) and Cl� concentration caused bythe own electrode re-dissolution were taken into account.Under these experimental conditions C0 for Eq. (35) is1.24 · 10�3 (calculated as the ratio n0/V). Fig. 3 showsthe experimental (symbol) and theoretical values (line)using Eq. (35) of the potential of the Ag/AgCl electrodeat two different pH values. As it can be seen, there is aremarkable coincidence between experimental and theoret-ical data that demonstrates the suitability of this model toassess the potential behaviour of these electrodes.

6. A Ag/AgBr electrode in the presence of Cl�

The following example consists of an Ag/AgBr systemused as Br� selective electrode that is immersed in a solu-tion containing the anion chloride that also can form theinsoluble precipitate AgCl (s). In this case we will havethe following equilibrium:

AgBr ðsÞ þX�¡AgX ðsÞ þ Br� with KI ¼aAgXðBr�ÞaAgBrðX�Þ

And the equation that describes the electrode behaviourwill be

E ¼ E0Agþ=Ag � 59:16 log

1

C0

þ ðOH�ÞKps

þ ðCl�ÞKAgCl

ps

þ ðBr�ÞKAgBr

ps

!

¼ E0Agþ=Ag � 59:16 log

1

C0

þX ðX�Þ

KAgXps

!ð36Þ

The Ag/AgBr electrode was prepared by oxidation of aAg wire in a 0.1 mol dm�3 solution of NaBr. A total cur-rent of 6 C gave finally a total of 6.22 · 10�5 moles of AgBr(s) on the silver surface. For the studies described belowthis electrode was immersed in a total volume of 50 mLof water at pH 6 containing different chloride concentra-tions ([Cl�] = 1.0, 0.1 and 0.01 mol dm�3) and the redoxpotential of the electrode was studied as a function of theconcentration of bromide. For this system, C0 is 1.24 ·10�3 (calculated as the ratio n0/V, with n0 = 6.22 · 10�5

moles and V = 50 · 10�3 L). Fig. 4 shows the experimental(symbol) and theoretical values (line) using Eq. (36) of thepotential of the Ag/AgBr as a function of the bromide con-centration in the presence of different fixed concentrationof chloride. Here, the factor ‘‘contact potential’’, ioniccoefficient of Debye–Huckel (c±) and Cl� concentrationcaused by the own electrode re-dissolution were also takeninto account. As it can be seen, there is a remarkable coin-cidence between experimental and the behaviour predictedby Eq. (36).

The presence of chloride induces a drift of potential ineach case, with respect to the standard potential of theAg/AgBr electrode (235 mV under the experimental condi-tions). The detection limit of Br� (defined as the intersec-tion of the extrapolated linear regions of the calibrationgraph) of the Ag/AgBr electrode in absence of Cl� canbe determined as the square root of the Kps of the AgBrsalt, whereas in presence of interferent Cl�, the detection

-100

0

100

200

300

0 2 4 6 8 10

pBr-

E (

mV

)

Fig. 4. Experimental (symbol) and theoretical values (solid line) calcu-lated with Eq. (36) of the redox potential of an Ag/AgBr electrode inpresence of chloride concentrations of 1.00 (s), 0.10 (r) and 0.01 M (M)at pH = 6. The first curve (�) shows the redox potential in absence ofchloride (pKps(AgBr) = 12.27).

Table 2Final molar fraction of AgCl on the surface of Ag/AgBr electrodes incontact for 24 h with certain solutions containing different chlorideconcentrations

Experiment Theoretical (%) Experimental (%)

1 61.58 65.02 49.78 42.43 35.64 40.94 26.86 30.05 9.45 8.056 3.09 1.9

Experiments 1, 2, 3, 4, 5 and 6 refer to solutions of chloride of 1.0, 0.5, 0.2,0.1, 0.01 and 0.001 mol dm�3.

J. Soto et al. / Journal of Electroanalytical Chemistry 594 (2006) 96–104 103

limit of the electrode diminishes proportionally to the con-centration of the interferent.

6.1. Solubility equilibrium in the Ag/AgCl/AgBr system

In the previous case, the modification of the electrodepotential is due to a partial transformation of the AgBrsolid into the corresponding AgCl. In this section we willstudy in more detail the equilibrium between a sparinglysoluble salt MX that is in the presence of concentrationsof an anion Y� that also forms a sparingly soluble salt withthe metal cation M+. This kind of calculation might be ofinterest for the evaluation of the potential electrode ininterfering processes. We suppose the following equilibri-ums can take place at the surface of a certain electrode M

MX ðsÞ ¡ Mþ ðaqÞ þX� ðaqÞ KMXps ¼

ðMþÞðX�ÞaMX

MY ðsÞ ¡ Mþ ðaqÞ þY� ðaqÞ KMYps ¼

ðMþÞðY�ÞaMY

and the following balance matter for M and X is

n0 ¼ nMþ þ nMX þ nMY ð37Þn0 ¼ nX� þ nMX ð38Þ

working out n0 in (37) and (38) gives

1 ¼ ðMþÞ

C0

þ aMX þ aMY ð39Þ

1 ¼ ðX�Þ

C0

þ aMX ð40Þ

working out aMX from equilibrium equation and combin-ing with (40) we have

ðX�Þ ¼ 11

C0þ ðM

þÞKMX

ps

ð41Þ

and from balance matter for M we have

1 ¼ ðMþÞ 1

C0

þ ðX�Þ

KMXps

þ ðY�Þ

KMYps

( )ð42Þ

Replacing (41) in (42) and working out M+, we have thefollowing second order equation:

ðMþÞ2 1

KMXps

þ C0ðY�ÞKMX

ps KMYps

( )þðMþÞ 1

C0

þðY�Þ

KMYps

( )� 1¼ 0

ð43ÞThe solution of this equation allows to represent the activ-ity diagrams for the solid–solid equilibrium. In these dia-grams, the variation of �logaMX, �logaMY and�log(M+) can be represented as function of �log(Y�).The values of logaMX and logaMY are calculated replacing(M+) (obtained from Eq. (43)) in equilibrium constantsequations.

In order to check the proposed model, six silver foils of1.6 cm2 (ALDRICH, 0.1 mm thick, 99%) were electrodizedfor 20 min in solutions of KBr 0.1 M at pH 6, with a cur-rent density of 6.25 mA/cm2 (total transferred charge:12 C). After electrolysis, each foil was dipped in 50 ml ofa KCl solution of a certain concentration (see Table 2).After no less than 24 h, to make sure that the thermo-dynamic equilibrium was reached, they were rinsed withdistilled water and dried. The composition data of theresulting solid surfaces was determined using ElectronicMicroscopy. Table 2 summarizes both the experimentalfinal molar fraction of the solid AgCl in the surface andtheoretic results predicted by using previous equations.

As it can be observed there is a fairly good agreementbetween the calculations with the proposed equations andthe experimental data. Fig. 5 shows a plot of the progres-sive transformation of the AgBr solid in AgCl caused bythe presence of Cl� in the aqueous solution. The figureshows both, the experimental date (symbols) and the theo-retical curve from the equations. This figure indicates thatAgBr is the majority phase until concentrations of Cl�

equals 10�2 M. For [Cl�] = 0.1 M, the mixed precipitateis around 50% of each phase, and finally, for [Cl�] = 1 M,the main phase is AgCl (s). Obviously, we can also deducethat the presence of metallic ions (M+n) in dissolution,which form insoluble salts with the precipitant anion

0

0.2

0.4

0.6

0.8

1

-log (Cl-)

i

0 2 4 6

Fig. 5. Diagram of transformation of the system Ag/AgBr as a function ofp(Cl�), at 25 �C for 10�5. Experimental values of the composition of AgCl(�) and AgBr (N) are compared with the theoretic values (solid line).

104 J. Soto et al. / Journal of Electroanalytical Chemistry 594 (2006) 96–104

(Br�), would affect the composition of the solid, and alsowould produce a certain grade of exchange of the principalcation Ag+ by M+n. This information should be taken intoaccount in the design of regeneration methodologies (ormaintenance) of redox or faradic electrodes.

7. Experimental

7.1. Reagents and apparatus

All used reagents were of analytical degree. All the aque-ous solutions and HEPES buffer were prepared with deion-ized–bidistilled water (Milli-Q water purification system).Experiments were carried out under argon atmosphere at25 �C. The metallic electrodes were composed by a foil of ametal (From ALDRICH) introduced in a cylinder of PVCas electrode body and connected to a wire, and previouslyelectrolysed in solutions of the corresponding soluble salt0.1 M using a Potentiostat–Galvanostat Tacussel IMT-1.

7.2. Emf measurements

The reference electrode was a Ag/AgCl/KCl 3,5 M. ThepH measurements were carried out using a GLP22 CrisonpH/mV meter. The measurements of the electrode poten-tial were taken using the OPA129P operational amplifierwith high input impedance (1015 X) configured as tensionbuffer and a data acquisition card ADLINK-9112 usinga program developed with LabView. The resolution ofthe acquisition of measurements is ±0.6 mV. Activitycoefficients were calculated according to Debye–Huckelapproximation.

8. Conclusions

In this paper, a general model for describing the redoxpotential behaviour of certain electrodes based on the

study of superficial redox process has been developed andexperimentally verified. The model allows to state broad-spectrum equations which describe the behaviour of thepotential of second and third order Faradic electrodes.The model is applied to the study of several systems includ-ing the variation of the potential of the Ag/Ag+ electrodewith the pH, the redox potential of a Ag/AgCl electrodeas a function of the chloride concentration at differentpH values, the potential of a Ag/AgBr electrode in thepresence of certain amounts of chloride and the quantita-tive solid–solid transformation between AgBr and AgClphases. There was found a good agreement between theexperimental and theoretical values.

Acknowledgements

We thank the Spanish Ministerio de Ciencia y Tecno-logıa (MAT2003-08568-C03-02) for support. We alsothank the Polytechnic University of Valencia for support.

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