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electr ode, Heyrovsky assum ed (Ref. as a bove) tha t t he su rface of the electrode is occupiedby the evolved molecules of hydrogen, which reduce at the free surface on which thefollowing reaction takes place :

H + H+ 2H+ ...(12)Kut a [4], derived an equat ion for th e polarogra phic wave of hydrogen ions ba sed on

this assumption.Kutas experiments with a streaming mercury electrode [Kuta] [4] collection

Czechoslov. Chem. Commun s 23, 383 (1958) showed tha t t he concept of surface coverageby hydrogen molecules was not justified.

The best agreement with polar ographic results is obtained by assuming th e reaction:H+ + e H ...(13)

This is supposed to be the slowest step in the proposed mechanism. In fact the

influence of the electrode double layer on the discharge of hydrogen ions must also betaken into account.A brief survey of the experiment s on th e discha rge of hydrogen ions a t t he dr opping

mercury and the streaming mercury electrode is described below :(i) Current -voltage curves for hydrogen ions without concentr ation polarization.

Workers at about the same period found that in more concentrated solutions of str ong acids (e.g., 0.010.1 N H Cl) an d in t he a bsence of neutr al sa lts t he a bovedefined reduction potential of hydrogen shifts in accord with the relationship :

E = const + ln H RT F

+ ...(14)

whereas , with an excess of an indifferent electr olyte t he dependence is given by

E = const +2 RT

F ln [H+

] ...(15)In buffer solutions, with one unit increase in the buffer pH shifts the reductionpotential of hydrogen by 116 mV to more negative potential.On adding small amounts of neutral salts, the over-voltage is markedly increased.In other words the reduction potential is displaced to more negative potentials,whereas, with increasing concentr ation of the neut ral s alt, th e reduction potentialapproaches a constant value that depends on the hydrogen ion concentration.The valency of the neutral salt is also to be taken into account.

(ii) Polarographic Waves for H ydrogen Ions Given by Str ong Acids : Examining th eshape the current-voltage curve and its dependence on the hydrogen ionconcentr ation, it ha d been inferred th at these ar e the two funda menta l quantit iesin the theory of hydrogen over voltage. It has been stated that the hydrogenwave is asymm etr ic (Tomes [6],) an d th at its h alf-wave potent ial depends on th ehydrogen ion concentration (Tamamushi [5]).

(iii ) The behaviour of weak non-reducible acids which give redu ction waves forhydrogen ions is quite complicated. The limiting current is diffusion controlledfor acids with pK

a 25. The half-wave potential is displaced to more negative

Polarization 79

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80 In troducti on to P ola rograp hy a nd Al lied T echniques

values with increasing concentration of the acid and, particularly of its anions.At the same time, the slope of the wave decreases. Only with very weak boricacid a kinetic limiting current controlled by dissociation rate was observed bypast workers.

6.3 DOUBLE LAYER

The layer of negative charge at the surface of the electrode and the layer of positivelycharged solution adjacent to it constitute the so-called double layer.

To charge the double layer upto any potential E , a certain quantity of electricity isrequired. For an electrode of constant area A (in cm2) this is given by

q = xA ( E max E ) ...(16)whereq is the quantity of electricity in microcoulombs, x is the differential capacity of

the double layer in microfarads/square centimeter, and E max is the potential of theelectrocapillary maximum. The value of x is not independent of potential. In dilutesolutions of hydrochloric acid or alkali metal halides it is approximately 40f/cm2 atpotent ials more positive tha n E max but only about 18 f/cm2 at potentials more negativethan E max and in each of these regions there are definite further variations withpotential [1].

It is evident that the quantity of electricity described by the above equation ispositive when E is more negative than E max, zero when E and E max are equal, andnegative when E is more positive than E max.

The reversal of sign occurs because of the polarity of the double layer is reversed ingoing from one side of the electrocapillary maximum to the other, and the signs are sochosen as to accord with the customary polarographic convention, in which the flow of electrons into the dropping electrode is taken to constitute a positive current. It can aswell be said that the curent is taken to be positive when the dropping electrode is thecathode, and negative when it is the anode.

When a reaction is sa id to take place at th e sur face of an electr ode, it is not impliedthat the electroactive species must come into actual physical contact with the electrode.There is a potential gradient around an electrode immersed in an electrolyte solution.The potential difference between the electrode and the bulk of the solution may berepresented by ( E E max) where E is the potential of the electrode and E max is theelectr ocapillary m aximum potent ial or potential of zero char ge. If th is is n egative, thenany surface near the electrode will be at a more negative potential than the bulk of thesolut ion a nd t he potential will become more and more negative as the electr ode sur facemore closely approached. An ion a molecule diffusing toward the electrode will reach apoint (whose distance from the electrode surface is always negligible compared with the

th ickness of th e diffusion layer un der polarogra phic condition) at which t he potentia l issufficiently negative to bring about its reduction. All these points will lie on a sphericalsur face with a dr opping electrode at a ny instan t. There is a simple case in which th ereis no specific adsorption at the electrode surface and in which the potential variesmontonically with distance from the electrode. In more complicated cases specificadsorption does occur and in which the dependence of potential on distance from the

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electrode is no monotonic.These are depicted schemati-cally in the Figure 6.1.

Fig. 6.1 schemat ic diagramsof the electrical double layer atan electrode whose potential ism o r e n e g a t i v e t h a n t h epotential of the electrocapillarym a x i m u m . (a ) No specificadsorption, and (b ) when thespecifically adsorbed cationsa r e p r e s e n t i n t h e i n n e rHelmholtz plane.

On the bas i s o f t hepolarographic observations itcan be qualitatively describedthat the surface at which thereaction occurs coincides with t he boundar y between th e Helmholtz layer an d t he diffusedouble layer. Influence of structure of the electrode double layer on the rate of onirreversible process.

The double layer affects the rates of electrode processes by its influences on thesurface concentration of the depolarizer and on the kinetics of the electrode reaction.This effect was first considered by Frumkin [1] in the case of hydrogen ions.

According to Frumkin [1], the electrode process involves only ions in direct contactwith t he electr ode and t hat electr on t ran sfer t o more distan t ions is not likely. Thus, t heelectrochemical reaction occurs only in that part of the double layer, which does notexceed the effective ionic radius.

Several examples of irreversible processes in which the double layer plays animportant role are being described on time to time.

6.4 REDUCTION OF CATIONS

(a) R e d u c t i on o f H y d r o g e n I o n s a n d H y d r o g e n O v e r Vo lt a g e : Among the mosttypical and most frequently studied examples of an irreversible process is thereduction of hydrogen ions [4]. The over voltage of this process varies greatlywith t he na tu re of th e electr ode. The difference between th a h alf-wave poten tialof the irreversible wave and the standard potenital E 0 (which is virt ua lly equalto the half-wave potent ial of a r eversible wave) is the polar ogra phic over potent ial(over voltage)1/2,i.e. , 1/2 = ( E 1/2)irrev E 0 ...(17)Hydrogen overvoltage is usually investigated at constant current density j andis defined as the difference between the electrode potential E , at which hydrogenis reduced at this current density, and the potential E r of a reversible hydrogenelectrode in the same solution,i.e. , = E E

r .

Fig. 6.1

Polarization 81

+

++++++++++

+ ++

+ ++ +

+ +

+

+

+ ++ +

+

+

+

+

+ +

+

Electrode Electrode

Helmholtzlayer

Diffuselayer

Helmholtzdoublelayer

Diffusedoublelayer

(a) (b)

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The sequence of reaction could be2H+ + 2e 2H ...(i)

2H H2 ...(ii )Heyrovsky [3] suggested the following reaction mechanism. For the sake of simplicity, H+ is written instead of H3O+.

H+ e H ...(18)

H + H+ 2H+ ...(19)

2H+ + e H2 ...(20)

It is supposed tha t t he upda te of a sin gle electr on by a hydrogen ion occur s veryrapidly and that the rate determining step is reaction (19).This mechanism is in accordance with the discharge of hydrogen on someelectrodes.

(b) P o l a r o g r a p h i c Wa v e s f or H y d r o g e n I o n s a s g i v e n b y S t r o n g Ac i d s : Asregards th e shape of the curren t-voltage curve and its dependence on the hydrogenion concentration. These two are very important and fundamental quantities inthe theory of hydrogen over voltage. It has been stated, for example, that thehydrogen wave is asymmetric [1] and its half-wave potential depends on thehydr ogen ion concent ra tion [6].

(c) Polar ograp hic Waves for Hydr ogen Ions Given by Weak Non-Reducible Acids :The beha viour of weak n on-reducible acids tha t give redu ction wa ves for hydrogenions is quite complicated. For acids which havek a between 25, the resultinglimiting current is diffusion controlled. The half-wave potential is displaced to

more negative values with in creasing concentr at ion of the acid and par ticularlyof its anions and at the same time, the slope of the wave decreases.A further example characterized by the strong influence of the double layer on

polarographic behaviour could be represented by the system3Eu e++ Eu2 + ...(21)

The standard oxidation reduction potential for this system is 0.601 V (in 1 MNaClO4) so that the influence of the double layer may be studied on either side of theelectro-capillary zero.

References

1. Fru mkin A. N. : 2, Physik. Chem. 164 A, 121 (1933).

2. Grah am, D.C., J . Electrochem. Soc., 98, 343 (1951) an d J . Am. Chem. Soc., 71, 2975(1949).3. Heyrovsky, J., Chem. Listy 31, 440 (1937).4. Kuta J., Chem. Listy 50, 991 (1956).5. Tamam ushi R. : Bull Chem. Soc., Japa n 25, 287, 293 (1952) : 26, 56 (1953).6. Tomes J ., Collection Czechoslov. Chem. Commu ns. 9, 1950 (1937).

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CHAPTER 7

AMPEROMETRIC TITRATIONS

A conventional amperometric titration is one whose course is followed by measuring acurrentalmost always a limiting current at a voltammetric indicator electrode.Depending on t he potential of the electr ode an d th e voltam metr ic char acteristics of thechemical substances involved, the current may be proportional to the concentration of

the substances being titrated, to the concentration of the excess of reagent, or to theconcentration of one of the products of the reaction, or it may depend on two of theseconcentrations. The titration curve is a plot of the limiting current against the volumeof the reagent added. If necessary it may be corrected for the residual current and fordilution by the reagent. Ideally it consists of two straight lines intersecting at theequivalence point. Amperometric titrations can be used to determine many substances,such as phosphate a nd sulpha te th at are n ot electr oactive besides amperometric titrat ionsare usually less tedious and more precise.

7.1 TYPES OF AMPEROMETRIC TITRATIONS

7.1.1 Theory of Amperometric Titration Curves

The shape of an amperometric titration curve depends on the substance being titrated,the reagent used, and the potential applied to the indicator electrode. Many differentcombinations are possible but the features common to most titrations curves can beexplained by taking the titration of lead ion in weakly acidic supporting electrolyte withstandard potassium chromate, using dropping electrode at a potential where both leadion and chromium (vi ) yield their diffusion currents. For the sake of simplicity it isassu med t hat the solubility of lead chromat e is negligibly small in the medium employedand t hat th e concentra tion of support ing electr olyte is so high t hat the m igration cur rentcan be ignored. The chemical equilibrium is attained between the addition of eachaliquot of reagent and t he measur ement of the current , and t hat the diffusion currentof each ion is proportional to its concentration.

An a mperometric titra tion curve is a plot (i ir ) [V 0 + v] againstv, whereV 0 is thevolume of solution titrated and the volume of reagent added. The factor (V 0 + v)/ V 0

serves as a corr ection for dilution by t he r eagent. In case the r eaction proceeds virtua llyto completion at every point during titration, the plot consists of two straight lineswhich intersect the equivalence point. The slope of the line preceding the equivalencepoint depends on the value ofk i.e. , the ratio of diffusion current to concentration forth e subst ance being titra ted; tha t of the line following the equivalence point depends onthe value ofk for the reagent.

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The example of lead being titrated with chromate (that has been taken) (i) bothksare positive at potentials where both ions can be reduced and in this case the plot isV shaped. (i i ) In the same titration when it is performed at a potential where chromateion is reduced but lead ion is not. In this casek cr would be positive whilek pb would bezero. Then the current would remain equal to residual current until the equivalencepoint is rea ched, th e tit ra tion curve would be _/-shaped. (i i i ) If the chromate ion wouldbe titrated with lead ion at the latter potential, the current would decrease untilequivalence point is r eached and r emain zero after it ha d been passed. The curve wouldbe\ _-shaped. On titr at ing lead ion with su lphide at a potentia l where lead ion gives acathodic current while sulphide gives anodic one so thatk

Pb would be positive whilek

s

negative. The cat hodic curren t would decreas e to zero at th e equivalence point, an d th eanodic current would increase th ereafter if the t wok

s were numerically equal. A single

straight line intersecting the zero current axis at the equivalence point would be obtained.

There is a probability that their values would differ slightly. In such a case the curvewould consist of two straight lines intersecting each other (at a very obtuse angle) andthe zero current line at the equivalence point.

For the titration of lead ion with chromate, the point of intersection of the two linesegments is described by the rearranged equation:

0 0ib Pb

cr cr

V C v

C = ...(1)

which is exactly the volum e of the r eagent r equired to rea ch the equivalence point. Thisis true of all amperometric titrations. No matter what kind of chemical reaction may beinvolved, or what shape of the titration curve is attained and no matter what values of k

s may be there. The line segment should always intersect at the equivalence point.

Sources of error do remain in amperometric titration as well.Amperometric titrations are useful in cases where potentiometric titrations fail.

There are two kinds of interferences in amperometric titration. (i) substances thatconsume the reagent, that co-precipitates, (i i ) that take part in induced reactions withthe substance being determined. Interference in amperometric titrations takes place asis there in any other technique.

Amperometric titrations have often been employed to evaluate solubility productsand other equilibrium constants. For example, if a reducible ion M n + is titrated with aninert one X n to give the precipitate Mx , and if the concentration of both solutions areknown the value ofk t under conditions of the tit ra tion is easily obtained by appr opriat euse of the equations. The equilibrium-constant expression would be :

[ M n +] [ X n ] = 1/ k t

...(2)The apparatus and technique are described ahead.

7.2 KINDS OF AMPEROMETRIC TITRATIONS

(i) (a ) Redox Titrations, (b ) Complexometric and Chelometric Titrations, (c)Compensation and Diffusion-Layer Titrations.

(ii) Amperometric Titrat ions with Two Polarized Electrodes.

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7.2.1 Redox Titrations

The titra tion cur ve obta ined in an am perometr ic redox titr at ion may have any of afairly large number of shapes. The possibilities are more numerous than in a precipitationtitration because there are four substancesthe oxidized and reduced forms of each of th e two couples involved which ma y cont ribut e to th e curren t. E ach of th e four ma y givea cathodic current (k > 0) or an anodic one (k < 0), or may be electrolytically inert(k = 0), at t he potent ial selected. (k denotes th e ra tio of diffusion curr ent to concentr at ion).

However, two cases ar e more comm on th an others. In one of them, only one of thesefour substances yields a current, which may be either cathodic or anodic. In the other,the oxidized form of one couple yields a cathodic current while the reduced form of theyields an anodic one.

For the first case example which can be furnished is the titration of vanadium (iv )with vanadium (i i ) in 1 F sulphuric acid, using the dropping mercury electrode as theindicator electrode. The following equation describes the reaction :

VO++ + V++ + 2H+ = 2V+++ + H2O ...(3)The second kind of redox titration curve could be exemplified in a titration of iron

(111) with vanadium (11) in a neutral citrate medium. At 0.8V vs SCE iron (111) isreduced to iron (11) giving a cathodic cur rent while vanadium (11) is oxidized to vanadium(111) giving an anodic current.

Mercur y electrodes ar e specially advant ages in t itra tions with s tr ong reducing agentsbecaus e very negative potent ials can be at ta ined an d becau se th e electr ode surface doesnot catalyze the oxidation of the reagent by water or hydrogen ion.

The rotating platinum wire electrode in particular has been very widely used inamperometric titrations.

7.2.2 Complexometric and Chelometric Titrations

Amperometric techniques cna be used to find the end point of a complexometric orchelometric titration in several ways. If the metal ion is electroactive and the complexor chelonat e is fairly stable, it m ay be possible to find a potent ial at which t he un reactedmetal ion yields its limiting current while the reaction product is inert [6]. The titrationof cupric ion with ethylene diamine tetra acetate may be performed with a droppingelectr ode at a potentia l where cupric ion is r educed but th e chelonat e is not can be ta kenas an example. The current due to the reduction of unreacted cupric ion decreases asthe equivalence point is approached and becomes very small after it has been passed.

In a redox titra tion, e.g., 1 xO + Red2 = Red1 + 2 xO ; the mea sur ement of th e diffusion

current of 1 xO appr eciable kinetic cur rent s will be obtained near th e equivalence pointunless the reverse reaction is quite slow, although such a situation is quite rare.

One can titrate an ion giving a chelonate more stable than that of the indicator inthe presence of other giving less stable ones by appropriate selection of the indicator.

Am perom et ri c T it ra t ion s 85

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7.2.3 Compensation and Diffusion-Layer Titrations

Compensation titrations are those titrations in which the reagent and the substancebeing titrated give currents of opposite signs at some potential, but ideally do not reactwith each other at all. The end point of such a titration is the point where the cathodiccurrent due to the reduction of one of them is just equal to the anodic current due tothe oxidation of the other. In such a case the measured current is just equal to theresidual current.

This occurs when the electrode itself is oxidized in the anodic half-reaction. This isquite likely to be possible if the electrode is made of mer cury, alum inium , silver, or someoth er r elatively base metal [4] the titr at ion of oxygen with sulphide at a mer cur y electr ode[3] is a typical example.

In principle, a compensation titration could be performed even if no net chemicalreaction could occur at all. Example could be iron (i i i )iron (i i ). The couple behavesreversibly at a dropping mercury electrode in a weakly acidic solution containing citrate,ethylene diamine tetra acetate, oxalate or tartrate. In any of these media ferric ironcould be titr at ed with ferrous iron at a ny potent ial on th e rising part of their waves,where ferric iron would give a cathodic current and ferrous iron an anodic one.

Diffusion-layer tit ra tions can be exemplified by the t itra tion of oxygen with a s tr ongacid [2]. The titr at ion can be perform ed with a dropping mercur y electr ode at a potent ial( 1.8 V vs SCE) where both oxygen and hydrogen ar e reducible. A neut ra l but u nbufferedsupporting electr olyte should be used to suppr ess th e migrat ion curr ent of hydrogen ion.When oxygen is reduced, hydroxyl ions are formed at the electrode surface by the half reaction

O2 + 2H2O + 4e = 4OH

As these diffuse away from the electrode surface they react with hydrogen ionsdiffusing towards it. Hydrogen ions are thus prevented from reacting the electrodesurface until their flux equals that of hydroxyl ion, when D1/2 H + 2+ 1 2OH 2C 4D CO= .Upto this point the value (i i

r ) remains constant after correction is made for dilution.

After this it increases linearly with the volume of the acid added in excess.

7.3 AMPEROMETRIC TITRATIONS WITH TWO POLARIZED ELECTRODES

In s uch a case qu alitat ively different cur ves are obtain ed with t wo polarized electr odes,also known as dual-electrode amperometric titrations. These are titrations in which thecurrent-potential curve for each of the two electrodes changes as the composition of thetitration mixture changes. Case is easy to discuss in which two electrodes are identical,because then the behaviour of both can be explained with the aid of a single current-

potential curve. It is not necessary for these to be identical, they may have differentareas, the efficiencies of stirring at their surfaces may differ, and they may even bemade from different materials. In such an arrangement the difference between thepotent ials of the two indicator electr odes must be equal t o the a pplied potent ial (neglectingiR drop through the cell), and the current flowing through the indicator cathode mustbe equal to (but have the opposite sign from) that flowing through the indicator anode.

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The current varies as the titration proceeds and the shape of the titration curve dependson the reversibilities of the couples involved and on the magnitude of the potentialdifference applied.

In principle these titrations can be made with electrodes of any kind, but stationaryplatinum wire electrode in stirred solutions are much the most commonly used.

Amperometric titration with two polarized electrodes have sometimes been claimedto be more sensitive tha n t hose employing one polarized electr ode. For th is reas on t heyhave been widely used in coulometric and other titrations in which high sensitivity isneeded.

The pr inciples and applicat ions of these t itra tions ha ve been r eviewed by Stock [7],Delahay [10] and Lingane [5].

7.4 APPARATUS AND TECHNIQUES

Experimentally, amperometric titrations are much simpler than other polarographic orvoltamm etric techniques. Since th e curr ent h as t o be measured at only a single potent ial,simple apparatus is needed. Fewer variables need to be controlled because the exactvalue of the current at any point is not important : all that matters is, how it varies asthe reagent is added. The temperature, the composition of the supporting electrolyte,the height of the mercury column above a dropping mercury electrode or the rate of rotat ion or s tirr ing of another electr ode is used. Other factors which ar e very import antin other voltammetric technique are, th erefore, of little concern in amperometr ic titr at ions,what is necessary to avoid significant variations during the course of any one titration.

All amperometic titr ations ar e performed by measur ing the current after th e additionof each of a fairly small number of aliquots of the reagent. In order to avoid reductionwave of dissolved oxygen, inert gas is bubbled for sufficient time after each addition.More u seful would be t o use a microburet te equipped with a t hr ee-ways st opcock. If thesolution is sensitive to air precautions must be taken.

Mixing after each addition is necessary even when deaeration is not. A stream of inert gas can be used for stirring even if it is not required for deaeration. Stirring mustbe stopped before the curr ent is m easur ed except in t he case of rota ting wire an d stir redpool electrodes.

Amperometric titr at ion can be performed with any voltamm etric indicator electr ode.Widely used are rotating platinum wire electrodes.

In amperometric titrations concentrated reagents are to be employed. There are tworeasons for it. One, as is already indicated is to minimize the importance of thetheoretically necessary correction for dilution. The other is to minimize variations in theconcentrations of the supporting electrolyte, which would affect the diffusion currentconstants of the substances that contribute to the measured current. Regardless of theshape of the titration curve, failure to correct for dilution always causes the end pointfound by the u sual extr apolat ive procedure t o occur a litt le earlier th an it sh ould changesin the concentration of the supporting electrolyte may displace the end point in eitherdirection. The simplest example could be of a solution containing high concentration of sodium perchlorate as supporting electrolyte which is being titrated with a very dilutereagent.


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100 200 %c.p.0

ic

ia

c u r r e n t

Fig. 7.1 : Amperometric titration

curve for applied potential at E 1 and E 2

M

+

Most amperometric titrations are performedwith reagents having normalities at least 10 or20 times as large as those of the solutions titrated.Comprehensive reviews of the applications of amperometric titrat ions in pra ctical an alysis h avebeen given in the literature cited.

As already stated that an amperometrictitr ation is one in which the end point is determi-ned by the current resulting from a potentialapplied across the two electrodes.

7.4.1 The Working Electrode : ReferenceElectrode

In the case in which the potential of a workingelectrode is controlled relative to a referenceelectrode, the potential is applied so that alimiting current which is proportional to the concentration of one or more of the reactantsor products of the titration is measured. As a result a titration curve is obtained byplotting the limiting current as a function of volume of titr ant added. The shape of the t itrat ioncurve can be predicted from hydrodynamicvoltam mograms of the solut ion obtained a t var iousstages of the titration.

The Fig. 7.1, shows resulting amperometrictitr at ion curves for two values of applied potential.Their shapes are determined by the behaviour of the limiting current of the voltammogram at theparticular potential during the titration.

Fig. 7.2, described th e experimenta l set u p of titration assembly.

7.5 TWO WORKING ELECTRODES

In brief, a useful variation of the amperometrictitration involves meaning the current resultingfrom a small fixed potential applied across twoworking electrodes. One of these electrodesfunctions a s an anode and th e other as a cath ode.The expected current behaviour during a titr at ioncan be explained by mean s of hydrodynam ic voltam mograms . The advant age of th e twoworking electrode variation is the elimination of a reference electrode, which can betroublesome in non-aqueous solvents.

One advantage of amperometric titration is it ease of automation. A titrator can besignaled to shut off when a specified current level is reached.

Fig. 7.2 : Wiring diagram for atitration assembly without anexternal applied voltage (with arotat ing electrode)

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7.6 CHRONOAMPEROMETRY

The excitation signal in chronoamperometry is a square-wave voltage signal. This isshown in Fig. 7.3 (a ) which stepsthe po ten t i a l o f t he work ingelectr ode from a value at which nofaradaic current occurs, E

i, to a

potential, E s, at which the surface

concentration of the electroactivespecies is effectively zero. Thepotential can either be maintainedat E

s until the end of the experi-

ment or be stepped to a finalpotential E

f after some interest of

time T has passed. The lat terexper iment i s te rmed double-potential-step chronoamperometry.

Current as a function of timeis the system response as well asthe mon i to red r e sponse inchronoampe-rometry. A typicaldouble potential-step chronoam-perogram is shown by the solid linein Fig. 7.3 (b) (the dashed line shows the background response to the excitation signalfor a solut ion cont aining su pport ing electr olyte only. This curr ent decays r apidly whenthe electrode has been charged to the applied potential). The potential step initiates an

instantaneous current as a result of the reduction ofO to R . The curr ent then dropsas the electrolysis proceeds.It is important to note that the chargeQ passed across the interface is related to

the amount of material that has been converted, and the currenti is related to theinsta nt aneous ra te at which t his conversion occur s. Curr ent is physically defined as t herate of charge flow. Therefore,

Q = nFN ...(1)where N is the number of moles converted, and the instantaneous current at timet is

i t =t t

dQ dN n F

dt dt = ...(2)

The rate of conversion,d N/ d t , is directly proportional to the electrode area and tothe flux of material to the electrode as described by the following equation (which isderived from Ficks first law) :

it = nF AD 0

0

0 X , t

C X =

...(3)

Es

E iE f

Time

P o t e n t i a

l 0 C u r r e n t

Time

(a) Potential excitations igna l fo r doub lepotential step

(b) Current-time responsesignal (chronoampero-gram)

Fig. 7.3 : Chronoamperometry.


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where it = current at timet , A

n = number of electrons, eq/molF = Faradays constant, 96,485 C/eq

A = electrode area, cm2C = concentration ofO , mol/cm3 (mol/L!)

D = Diffusion coefficient ofO , cm2 /st = time s

X = distance from the electrode, cmChr onoamper ometr y ha s pr oven u seful for th e mea sur ement of diffusion coefficient

of electroactive species. An average value of1 2t i over a range of time is determ ined atan electrode the area of which is accurately known and with a solution of knownconcentration. The diffusion coefficient can then be calculated from1 2t i via the Cattrellequation [8]. The electrode area can be physically measured, but the common practice

is to measure it electrochemically by performing the chronoamperometric experiment asa r edox species whose diffus ion coefficient is kn own [1]. The va lue of A is then calculatedfrom 1 2t i .

If the heterogeneous electron transfer of the redox species with the electrode itself is slow, the current after t he potential step in n ecessarily less t han in a syst em in whichthe electron transfer is rapid. This aspect of chronoamperometry has been used for themeasurement of heterogeneous rate constants [9] [7].

The behaviour of1 2t i as a function of time can be influenced substantially by thepresence of chemical reactions that are coupled to the electrode process. Consequently,characteristic variations of1 2t i vs t have been effectively utilized for the quantitativestudy of such homogeneous chemical reactions. The ECE reaction in which a chemicalstep exists between two electron transfer steps is one mechanism that has beeninvestigated by means of chronoamperometry :

E : O + e R

C : R k X E : X + e P .

As shown in the reaction sequence above, a rate determining chemical step ininterposed between the two electrode reactions. The two dashed lines in Fig. 7.3 (a )show hypothetical chronoamperograms for the 1e reduction ofO to R and for direct 2ereduction ofO t o P with no kinetic complications. The solid line shows a typicalchronoamperogram for an ECE mechanism. The current is intermediate between the 1eand 2e reductions, since the reduction of X to P is controlled by the rate of the chemicalreaction of R to generate X . The exact position of the solid line is determined by the

value of the rate constantk .Chronoamperometry ha s been applied to th e stu dy of several electr ode mechanisms[12,13]. In such studies different mechanisms may exhibit similar responses [1, 6].

Double-potent ial step chr onoamperometry is part icular ly suited for stu dying systemsthat follow EC [12] or dimerization [16] mechanism.

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References

1. Delahay, P., New Instrumenta l Methods in Electrochemistry, Interscience, N7. 1954.pp. 258264.

2. Kemula, W., and Siekierski, S., Collection Czechoslov. Chem. Commu ns.15 , 1069 (1950).3. Kolth off, I.M., and Miller, C.S., J. Am. Chem. Soc.62 , 2171 (1940).4. Kolthoff, I.M., and Sambyctti, C.J., Ana l Chem. Acta,21 71 (1959);22 , 253, 351 (1960).5. Lingane, J.J ., Electroana lytical Chemistr y, Inter science., N.Y., 2nd ed., 1958, pp. 280

295.6. Pribil, R., and Ma tyska, B., Collection Czechoslov. Chem. Commu ns.,16 , 139 (1951).7. Stock, J .T. , Anal. Chem.,36 , 355 R (1964).8. Cottr ell eq. p. 58 Kissinger P eter T., Heinman W.R., Laborat ory Technique Marcel Dekker

Inc. N.Y.9. William R. Heinman , Dept. of Chemistry Un iversity of Cincinna ti, Ohio Kissinger Peter

T., Dept. of Chem. Purdue University, Inc., West Lafayette, Indiana.10. Adams, R.N., Electrochemistry at Solid Electrodes, Marcel Dekker, New York, 1969.11. Bard, A.J, and F aulkner, L.R., Electrochemical Methods : Fundament als and Applications,

Wiley, New York, 1980.12. Albert, G.S., and Shain I., Anal. Chem.35 : 1859 (1963).13. Hawley, M.D., and Feldberg, S.W., J . Phys. Chem.70 : 3459 (1966).14. Feldberg, S.J. , Phys. Chem.,73 : 1238 (1969).15. Schwaz, W.M., and Shain I., J . Phys. Chem.69 : 30 (1965).16. Olmstead, M.L., and N icholson, R.S., Anal. Chem.41 : 851 (1969).


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CHAPTER 8

POLAROGRAPHY OFMETAL COMPLEXES

8.1 REVERSIBLE, DIFFUSION-CONTROLLED SYSTEMS DETERMINATIONOF FORMULAE AND STABILITY CONSTANTS OF COMPLEXED METAL

IONSE ffe c t s o f L i g a n d s o n P o l a r o g r a p h i c Wa v e s : When meta l ions in solut ion undergo

complexation with ligands other than water molecules, their polarographic reductionwaves appear in two quite distinct ways. Firstly, the half-wave potential is shifted tomore negative potential. This happensalmost invariably. Secondly, the diffusioncurrent charges and usual ly becomessmaller (Fig. 8.1).

S tudies of the s tab i l i t ies of meta lcomplexes polarographically, involve thede te rmina t ion o f sh i f t s i n ha l f -wavepotentials or limiting currents of metal ions

in the presence of increasing amounts of complexing ligands. The shift of half wavepotential of the reduction of a metal ionta king place reversibly, both in t he a bsenceand presence of complexing agents, shiftstowards cathodic direction. The shift increases with increasing ligand concentration.The rate of the electron exchange process remains relatively fast with respect to thatof diffusion so that the latter is still rate determining. It is also to be noted that in thepresence of the ligands, the reduction wave has a shape indicative of a greater degreeof reversibility tha n t hat observed for the r eduction of th e aquo ion, which is att enua tedand of lower slope than expected for a 2-electron reversible reduction.

The decrease of diffusion current with increasing ligand concentra tion is t o be expectedowing to the increased bulk of the complexed ions relative to that of aquo ions. Suchvariat ions of diffusion coefficient ha ve been used as th e basis of meth ods of determ iningstability constants of complexes in solution.

Methods for determining stability constants of metal complexes by polarographicmethod are being divided into three categories :

(i) Methods which a re applicable to reversible reductions only.

654321

0.3 0.4 0.5 0.6 0.7 0.8

i d ( A )

Fig 8.1 : Effect of complexation onredu ction wave of a m eta llic ion

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(ii) Methods applicable to irreversible reductions.(iii ) Methods which may be applicable to both r eversible and irr eversible systems.The polar ogra phic technique, involving the dir ect m easur ement of half-wave potentials

of aquo and complexed metal ions may be used to determine the stability constants of metal-ligand systems of a variety of types.

Cases where reversible reductions take place, three main types of metal-ligandsystems are applicable.

(i) Those in which a single complex species is formed to the virtu al exclusion of allothers over the entire ligand concentration working range.

(ii) Those in which several complexes are formed in st epwise manner, but whosestabilities differ sharply from one another between the limits of ligandconcentration.

(iii ) Cases in which a set of mobile equilibria exists between th e various complexes

and the aquo ion. This is being represented by the following equation :M(H2O)n MX(H2O)n1 MX2(H2O)n2 ::: MX4 ...(1)

In such systems, several types of complexes are present at every ligand concentration.The pr oport ion of higher species increases with increasing concentr at ion of ligand. Fina lly,the predominent species is that with the highest possible coordination number. Largenumber of systems have been studied polarographically in this category by previousauthors.

8.2 DETERMINATION OF STABILITY CONSTANTS AND COORDINATIONNUMBERS OF METAL COMPLEXES

(i) The method of Lingane [3].(ii) The method of DeFord an d Hum e [1] for determ ining consecutive overall stability

constants.(iii ) The method of Schaap a nd McMasters [5] Mixed ligand systems.

(i ) T h e M e t h o d o f L in g a n e : The reduction of a complexed ion to the metallic state(as an amalgam) is being considered at the dropping mercury electrode. The electrodereaction may be expressed as :

( )n jm f MX ne H g

+ + = M ( H g ) + j ( X m ) ...(2)where MX f is the complex of met al ion M with ligand X carrying a charge (n j m ).The charge on the metal beingn + and on the ligand species asm .

This overall process is represented as being made up of two reactions. The first oneinvolving pr ior dissociat ion of the complex ion in to aqu o ions an d ligand . The second oneis the electrochemical reaction of the aquo ion itself, thus :

MX f (n j m )+ n m M j X + ...(3)| n e, E0M

If these processes take place reversibly and much more rapidly than the rate of natural diffusion of ions to the electrode surface, then the potential of the DME at allpoints on the polarographic wave may be given by :

Polarography of Metal Com plexes 93

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00 0ln A A A

M M C RT E E

n F C = ...(4)

whereCA 0 is the concentration of amalgam formed on the surface of the DME and Aits activity coefficient 0 M C is t he concentra tion of the m etal ion M in th e solution at the

drop surface and M it s a ctivity coefficient .0 A E is the standard potential of the amalgam.

The overall thermodynamic stability constant, X M + , of the complex MX f (omittingthe charges) is given by the expression.

[ ][ ][ ] f

f MX f

MX

M X = ...(5)

here the bracketed terms represent activities. The concentration of the complex in thebulk of the solution, for a given concentration of metal ion and ligand may be writtenas :

[ ] f f

f

f MX M M

MX MX

C X C

=

...(6)

At the electrode surface the metal ion will have a concentration given by theexpression :

[ ]00 f f

f

f MX m M

MX MX

C X C

=

....(7)

where, as before, the superscript refers to the values at the surface. This equation isvalid if it be assumed th at the concentr at ion of th e ligand is large and const ant th roughoutthe solution with the same value of activity coefficient both in the bulk and at theelectrode surface.

The sh ift in t he h alf-wave potential, pr oduced by th e presen ce of an excess of ligan d X , is given by :

( E 1/2)S ( E 1/2)C = E 1/2 =[ ]M2303 log f f

f

f MX MX

M MX

I X . RT .nF I

...(8)

If it is assumed that the diffusion current constants I M , I M X are approximately equal.Thus, the equation in its simplified form as was originally used by Lingane [3] is

given a s :

E 1/2 = ( )2 3 0 3 00591 00591log log log at 25 C f f MX X ' MX x. RT . .C j c

nF n n = + ...(9)

The equation :

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E = ( E 1/2)c 00591

log at 25 Cd . i

n i i ...(10)is seen to be identical with the Heyrovsky- Ilkovic equat ion for th e reduction of an a quoion. The equation thus holds both for simple and complexed ions provided that thereduction of both proceeds reversibly.

A more simplified equation based on the assumption that the half-wave potential isindependent of the concentration of the complex species is given below :

( E 1/2)c =0 00591 00591log log

f A MX X . .

E j C n n

...(11)

It is evident t hat th e rat e of cha rge of ha lf-wave potential with ligan d concentr at ionmay be expressed as,

( )1 2 00591logc

X

d E . jd C n

= a t 25 C ...(12)

Thu s, a plot of (E1/2)c versu s logC X should be linear of slope j (0.0591/ n ) from whichthe coordination number, j, of the complex, f X M , may be found. Once j is found, it issimple to calculate

Mx f from the equation given below :

E 1/2 =00591 00591log logC 25 C

j MX X . .

j at n n

...(13)

This equation was originally used by Lingane.

(ii ) The method of DeFord and Hume [1] for determining consecutive overallstability constants

DeFord and Hume made the first attempt in polarography, regarding the step-equilibria between successfully formed complexes in solution.

The concentration of each complex species, for a given free ligand concentration, isgiven by th e expression of th e form of equat ions (6 an d 7) which a re a lready given. Theequation 7 on summing up over all possible (mononuclear) species, now takes the form:

[ ]0 00 0

f

f N N MX

MX M f

X C C M

M X

= ...(14)

where j = 1, 2, 3, ..., N .The shift in half-wave potential is now finally expressed by :

E 1/2 =[ ]

0

2 3 0 3log f

f

f N MX

C M M MX

X I . RT nF I

...(15)

In order to calculate the individual overall constants, it is convenient to rearrangethe above equation and finally express in the form :


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F 0 [ X ] = 0 + 1 [ X ] [ ] ( ) [ ] ( )2

222

X X N

N N M X M X M X

N M MX M

X ... X + + + ...(16)

In t he equation (16) the left h and side has been writt en asF 0 [ X ] which denotes tha tit is a function of the free ligand concentration. The summation, on the right hand sidehas been given in the expanded form. The initial term0, is the st ability const ant of thezero complex which by definition, has the value unity.

It is, therefore, possible in principle to calculate the N stability constants form N values of the fun ctionF 0[ X ] corresponding t o the r an ge of [ X ] values. In actua l pra ctice,it is necessary to use many more values of [ X ] than the normal value of N .

8.2.1 Calculation of Individual Complex Stability Constants

In t he equat ion (16) the functionF 0[X] expresses in terms of the concentration of freeuncomplexed ligand and a lso a set of activity coefficients. Work ing in solutions of constantionic strength, the values of activity coefficients can be maintained constant. Not onlyso, in condition of constant ionic strength, it is assumed that the activity coefficient inthe equation (16) may be dropped and finally the latter takes the form :

F 0[ X ] = 1 +1[ X ] + 2[ X ]2 + ... + N [ X ] N ...(17)In normal practice it is desirable to determine the shift in half-wave potential of a

given metal ion for about twelve values of ligand concentration in the range of 0.1 to2.0 M depending upon the ligands solubility. The metal ion concentration which isnormally to be used is 5 104M to 103M. The choice has a latitude as the half-wavepotent ial shifts ar e independent of it. It should, however, normally be maint ained st rictlyconstan t over a ny par ticular r un of ligand concentr at ion in order t o corr ectly and easilyallow for the variation of the limiting current. After making suitable corrections to theobserved E 1/2 values, the variousF 0[ X ] functions are calculated for each value of [ X ].

To determine1, ..., N , the graphical extrapolation method devised by Leden [2] isapplied.

On plotting the derivedF 1[ X ] values against corresponding values of X , a limitingslope is obtained, a s [ X ] tends of zero, of2 and a n intercept, on t heF 1[X] axis of1. Thusa confirmative estimation of1 is possible and in addition, a preliminary value of2 isobtained. A functionF 2[X] can thus be defined similarly. In order to account for N complexes, this procedure is continued. As such theF N 1[ X ] versus [ X ] plot gives is astraight line and indicating directly, that the penultimate function has been reached.

It is a usual practice to attempt measurement of half-wave potentials to the nearest0.1 mV.

8.3 MIXED LIGAND SYSTEMSTHE METHOD OF SCHAAP ANDM C MASTERS

The met hod signifies th e logical extension of th e DeFord an d H ume meth od. The m ethodwas applied in cases where the metal ions complex with two ligand species aresimultaneously present in the solution.

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Considering a complexing reaction of the type :

M + i X + jY i X f M Y ...(18)

in which,i, j are stoichiometry numbers and X , Y are two different ligands species. Inthis method [Y] is regarded as maintained constant while [ X ] is varied. A relation of identical form holds under such conditions that [ X ] is constant and [Y] varied.

This principle was applied by Schaap and McMasters to the copper and cadmium-ethylene diamin e-oxalate s ystems. The oxalate concentra tion being held constan t whilethat of ethylene diamine was varied. For this system, three mixed complexes are possible,

e.g.,Cd (en ) (OX ), Cd (en )2 (OX ) andCd (en ) ( )22OX . By using t he conventional DeFord-

Hume method values of the two constants could be determined from studies on thesimple cadmium-ethylene diamine and cadmium oxalate systems. The other constant

could be found fromF 10 function by obtaining the intercept from a plot ofF 10 (en, OX)versus [en ].For more details the readers are advised to consult valid literature on the subject.The polar ogra phic technique, involving the dir ect m easur ement of half-wave potentials

of aquo and complexed metal ions, may be used to determine the stability constants of metal-ligand systems of a variety of types. All these methods are applicable when theelectrode process of both simple and complexed species occur reversible. Irreversiblereductions have been t aken up la ter on. The meth od of Schwar zenbach [6] and Ringhbomand Eriksson [4] have been quite popular and useful.

References

1. Deferd, D.D., Hume, D.N., J. Am. Soc.73 , 5321 (1951).

2. Leden, I. , Z. Phys. Chem.188, 160 (1941).3. Lingane, J .J . , Chem. Rev.29, 1 (1941).4. Ringbom, A., and Eriksson, L., Acta. Chem. Scand.7 , 1105 (1953).5. Schaap, W.B., and McMasters, D.L., J. Am. Soc.83 , 4699 (1961).6. Schwarzenbech, G., and Akerma nn, H., Helv. Chim. Acta.35 , 485 (1952).


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CHAPTER 9

POLAROGRAPHY OFORGANIC COMPOUNDS

The polarographic activity of organic compounds is due to the presence of one or morefunctional groups in th e molecule. Out of the more importa nt compounds a re a ldehydes,

unsaturated ketones, ni t ro and ni troso-compounds, disulphides, polymerhydrocarbons, halides, and peroxides.As a ru le, the redu ction waves of organic substan ces ar e governed by the r at e of th e

electrode process and by diffusion.In t he redu ction of organic subst ances, the hydr ogen ion concentr at ion ha s a sim ilar

significance as the concentration of the complex forming agent in the reduction of complexes. For this reason, the half-wave potential is usually a function of pH, e.g.,

OX + pH+ HP P X O + ne H1 Red. ...(1)

As already described applications of polarography are based on the measurementand inter pret at ion of curren t volta ge cur ves. In 1922, however. Dr. J ar oslov Heyrovskyof Charles University, Prague, introduced the dropping mercury electrode. This electrode

consists of mercury drop, hanging for a few seconds at the orifice of a glass capillaryfrom wh ich t he mer cury regular ly drops out . The electr ode an d a reference electr ode ar eimmersed into the solution to be electrolysed. When an external voltage is appliedacross these two electrodes controlled potential electrolysis can be carried out (Fig. 9.1).The variation of current with a continuously increasing voltage can be measured by aninstrument in the circuit. This measurement gives courrent-voltage curvesi.e. , i vs E .

These curves, obtained with a dropping mercury electrode or even with any otherelectrode with a periodically renewed surface are completely reproducible and are calledpolarographic curves. These curves depend only on the composition of the electrolysedsolution, with experimental conditions kept constant. In the presence of substanceswhich undergo reduction or oxidation at the surface of the dropping electrode, orsubstances that catalytically effect the electrode process, or those that form stablecompounds with mercury, an increase in cathodic or anodic current on the current-voltage curve is observed. The curr ent rises in a given potential r ange an d th is increaseis followed by a region of potential in which the current has reached a limiting value.Th e S -shaped portion of the current-voltage curve is called a polarographic wave.

The sh ape a nd position of polar ogra phic waves provides us with inform at ion on boththe quantitative and qualitative composition of the electrolysed solution. The difference

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between the limiting currrent and the current before the waves rise, is called the waveheight. This usually depends in the concentration of the electroactive substance insolution. Most analytical applications of both organic and inorganic polarography arebased on the increase in wave height with concentration.

Another important quantity is known as half-wave potential. This is the point on apolarographic curve at which the current reaches half of its limiting value. The waveheight, although, it depends on concentration, the half-wave potential is practicallyindependent of the concentration of the electroactive species. But its value does dependson the kind of organic compound involved. The half-wave potential is a quantity thatcan be used for qualita tive char acterization of organ ic substan ces because, t he h alf-wavepotential depends on the nature of electrolysed solution and also on the composition of the solution.

Measurements of both wave heights and half-wave potentials are important in solving

problems in fundamental organic chemistry. The measurements of wave heights can beapplied to the study of some slowly established equilibria, of reaction kinetics and henceof optimum conditions for any desired syntheses. The half-wave potentials offerinformation about rapidly established equilibria, reactivity towards nucleophilic attackand similar reactions, and the presence of certain structural groups in the molecule.

Commonly used in chemical analysis, the polarographic technique can solve someproblems of structure, reactivity, mechanism and synthesis in organic chemistry.

9.1 STRUCTURAL EFFECTS

As already indicated in Ch apter 1 tha t for system s in which the equilibrium between t heoxidised and reduced forms is rapidly established at the surface of the electrode. Theseare called reversible systems and the half-wave potentials measured polarographicallya re p rac t i ca l ly equa l t o s t anda rd ox ida t ion - reduc t ion po ten t i a l s measu redpotentiometrically. In these cases the half-wave potentials are a function of theequilibrium constants of the oxidation-reduction equilibrium.

The possibility of characterising oxidation reduction properties of numerous systemsis offered by polarography whereas application of potentiometry is not possible. Systemswhich involve a st ep with a high activation energy, are called irr eversible systems. Thehalf-wave potential is a function of the rate constant of the electrode process involved.

It is not surprising to find correlations between the values of half-wave potentialsand th e str uctur e of organ ic compounds. This is becaus e of th e relat ionsh ip of half-wavepotentials to equilibrium or rate constants. Among structural factors that effect half-wave potentials are : (i) the nature of the electroactive group,i.e. , the group wherecleavage or formation of bond occurs during electrolysis, (i i ) stereochemistry, and (i i i )the nature of substituents. In general, the shift towards more positive potentials of acathodic wave, corresponding to a reduction process, indicates that the reduction isproceedings more readily. A shift toward more negative potential indicates that theredu ction is proceeding with great er d ifficulty. Similarly, for anodic waves, correspondingto an oxidat ion pr ocess, the sh ift t owar ds more negat ive values indicate th at th e oxidationproceeds more readily.

Polarography of Organic Compounds 99

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9.2 NATURE OF ELECTROACTIVE GROUP

The decisive factors for determining the polarographic behaviour of the organic moleculeare : th e nat ure of atoms in th e electroactive group, their ar ran gement in space, and t hetype of bond cleaved or formed. The polarographic behaviour also depends on themolecular frame to which the electroactive group is bound, the substituents present,and in particular on the groups present in the immediate vicinity of the electroactivegroup.

Some such groups are more susceptible to the effects of the molecular framework.Valid quantitative comparison of half-wave potential is possible when the electrodeprocesses follow the same mechanism for all the systems to be composed.

It is generally true that reduction proceeds more easily when conjugation of doubleor t riple bonds of the electr oactive group with a m ultiple bond or with an ar omat ic ringoccurs. Taking the example of the reduction observed in4 3 ketosteroids is shiftedto more positive potential for1.4 3 ketosteroids by 0.15 to 0.22 volts. But for4.6 3ketosteroids by 0.29 to 0.45 volt. Similarly as the number of condensed aromatic ringsincreases, the r eduction is m ade easier. Single C X bonds a re u sua lly reduced at morenegative potentials than C = C CX, but at more positive potentials than C C = C X (X is halogen).

A rigorous comparison of electroactive groups is restricted to a sequence of closelyrelated groups such as

C F < C Cl < C Br < C Iin which the ease of reduction of the C X bond increases with the increasingpolar izability of the ha logen X . There are some other examples also. Examples of simplifiedreduction processes are given in the Table 9.1 and 9.2.

TABLE 9.1

T h e s e b on d t y p e s c a n b e r e d u c e d a t a d r o p p i n g m e r c u r y e l e ct r o d e

C C C N C O C S C XC = C C = N C = O C = SC C C N

N N N O N SN = N N = O

O O O S O XS S S X

X = Halogen

Condens ed benzenoid ringsCarbonium ionsSome heterocyclic rings

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TABLE 9.2

A sys tem of con juga te m ul t ip le bonds rep resen t s a s ing le e lec t roac t ive g roup

C = C C C C = OElectroactive Group

< 12

Reaction coordinate

400 300 200 100

100 200 300 400

. mV

1.0

0.8

0.6

0.4

0.2

0.2

0.4

0.6

0.8

1.0

i a

i 0

E eq

i l

i li/i l

Total currentic

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a detecta ble cur rent , mass t ran sfer becomes rat e-limiting and th e elecrt ron tr ansferkinetics no longer contr ols the experiment. In few systems, mass tr ansfer is not an issueand kinetics can be measured over very wide range of potential. In cases involvingsurface-bound electroactive species, large variation of with potential have also beenobserved.

The behaviour predicted in the last equat ion is depicted in t he ensu ing Fig. 17.6 Thesolid curve sh ows th e actual t ota l curr ent which is t he sum of the componentsi c and i a ,as a re sh own in da shed t ra ces. For large negative overpotent ials, the a nodic componentis negligible. Hence the total current curve merges with that fori c. At large positveoverpotent ials, the cathodic component is negligible, and the total curr ent is essent iallythe same asi a . In going either direction from E eq, the magnitude of the current risesrapidly, because exponential factors dominate the behaviour, but at extreme, thecurrent levels off. In these level regions, the current is limited by mass transfer rather

than heterogeneous kinetics. The exponential factors in the forgoing equation are thenmoderated by the factorsC 0 (o,t )/

*oC and C R (O , t )C R *, which manifest the reactant supply.

17.7 APPROXIMATE FORMS OF THE i EQUATION

(a ) No Mass-Transfer Effects

If the solution is well stirred or currents are kept so slow that the surfaceconcentrations do not differ appreciably from the bulk valves, then the previous equationbecomes

( )1 f f eoi i e

= ...(33)

which is historically known as the Butler-Volmer equation.Since mass-transfer effects are not included here, the overpotential associated with

any given current serves solely to provide the activation energy required to drive theheterogeneous process at the rate reflected by the current. The lower the exchangecurrent, the more sluggish the kinetics. Hence the larger this activation overpotentialmust be for any particular net current.

(b ) Linear characteristic at small

For small values of x; the exponential e x can be approximated as 1 + x, hence forsufficiently small, the above equation can be expressed as

i = i0 + f ...(34)which sh ows tha t t he net curr ent is linearly related to overpotent ial in a n arr ow potentialrange near E eq.The ratio / i has units of resistance and is often called the chargetransfer resistance, R

ct

0ct

RT R

Fi= ...(35)

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F ig. 17.7 : Effect of exchange current density on the activation overpotential required to delivernet current densities. (a ) j o = 103 A/cm2 (Curve is indistinguishable from the current axis),(b) j0 = 106 A/cm2, (c) j0 = 109 A/cm2. For all cases the reaction is O +e R with = 0.5and T = 298 K

This parameter is the negative reciprocal slope of the in curve where that curvepasses thr ough t he origin ( = 0, i = 0). It can be evaluated directly in some experiment s,and it ser ves as a convenient index of kinetic facility. For very lar geK 0, it approacheszero Fig. 17.7.

(c ) Tafel Behaviour at Large

For large values of (either negative or positive), one of the bracketed terms inequation becomes negligible. For example at large negative overpotentials,

exp (af ) >> exp[(1) f ] and the equation Fig. 17.8 becomes: f

oi i e = ...(36)

or

o RT RT

l n i l n iF F

= ...(37)

A successful model of electrode kinetics must explain the frequent validity of theabove equation, known asTa f e l e q u a t i o n . The empirical tafel constants : equation

= a + b logi can be identified from theory as

=

2 3 logoi

. RT F

=

2 3. RT b

F ...(38)

Kinetics of Electrode Process 253

400 300 200 100 100 200 300 400

. mV

0.8

0.6

0.4

0.2

0.2

0.4

0.6

0.8

j, A/cm 2

(c)

(a)

(b)

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The tafel form can be expected to hold whenever the back reaction (i.e. , the anodicprocess, when a net reduction is considered, an d vice versa) cont ributes less th an 1% of th e current . Tafel relat ionsh ips cannot be observed fro such cases, becaus e th ey requireth e absence of mass-tr ans fer effects on the cur rent . When electr ode kinetics are sluggishand significant activation over potentials are required, good Tafel relationships can beseen. This point explains the fact t ha t Ta fel behaviour is an indicator of totally irreversiblekinetics. Systems in that category allow no significant current flow except at highoverpotentials, where the faradaic process is effectively unidirectional and, therefore,chemically irreversible.

(d ) Tafel Plots

A plot oflog i . vs. , known as Tafel plot, is a useful device for evaluating kineticparameters. In general, there is an anodic branch with a slope (1)F/2.3RT and acathodic branch with slope F/ 2 .3RT. As shown in Fig. 17.8 both linear segmentsextrapolate to an intercept oflog i 0.

Fig. 17.8 : Tafel plots for anodic and cathodic branches of the current-overpotential curve forO + e R with = 0.5,T = 298 K, and j0 = 10-6 A/cm2.

9Note that for = 0.5,b = 0.118V , a value that is sometimes quoted as typicalTafel slope.

The plots deviate sh ar ply from linear behaviour as approches zero, this is becausethe back reactions can no longer be regarded as negliglible. The transfer co-efficientand t he exchan ge cur rent ,i 0 are obviously readily accessible from this kind of presentation

when it can be applied.Exchange Current Plots (Tafel Plots)

This kind of plot is useful for obtaining from experiments in whichi 0 is measuredessentially directly. From the equations it recognized that the exchange current can berelated as :

200 150 100 50 50 100 150 200

log i0

5.5

4.5

3.5

6.5

Slope =(1 )F2.3RT

Slope = F2.3RT

log | i|

, mV

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2 3 2 3o * o'

o o eqF F

log i log FAk log C E E . RT . RT

= + + ...(39)

Therefore, a plot of logi 0 vs. E eq at constant C* should be linear with a slope of 2 3F / . RT . The equilibrium potent ial E eq can be varied experimen ta lly by cha nging the

bulk concent ra tion of species R , while that of speciesO is held constant. This kind of plot is useful for obtaining from experiments in whichi 0 is measured essentiallydirectly.

17.8 EFFECTS OF MASS TRANSFER

A more complete 1- relation can be obtained from

( ) ( ) ( )1 f o R f o * *o R

C o,t C O,t

i i e eC C

= ...(40)

(where = E Eeq) by substituting for ( ) *o oC O,t C and ( ) * R RC O,t C

According to( )0 R xo

*

C

Co

=1

iil

...(41)

and( )0

1 x

R

R*

C iC il,a

== ...(42)

( )11 1e

f

o

i i ie

i i l,c i l ,a

= ...(43)

This equat ion can be rea rr an ged easily to givei as an explicit function of over t hewhole range of.

For small overpotentials, a linearized relation can be used.In the Tafel regions, other useful forms of the above equation can be obtained. For

the cathodic branch at high values, th e anodic contribution is insignificant and as suchthis equation becomes

( )1 1 1 f o l c l,a

i i ie e fn

i i , i= ...(44)

This equation can be useful for obtaining kinetic parameters for systems in whichthe normal Tafel plots are complicated by mass-transfer effects.

Kinetics of Electrode Process 255

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References

1. B.E. Conway, Theory and P rinciples of Electrode Processes, Rona ld, New York,1965, Chap. 6.

2. H.R. Thirsk, A Guide to the Study of Kinetics, Academic, New York, 1972, Chap. 1.3. J . Om. Bockris a nd A.K.N. Reddy, Modern Electrochemist ry, Vol. 2, Plenu m, New

York, 1970, Cha p. 8.4. K. J. Vetter, Electrochemical Kinetics, Academic, New york, 1967, Chap. 2.5. C.N. Reilley in Treat ise on Analytical Chemistry, Pa rt I, Vol. 4, I.M. Kolth off an d P.J .

Elving Eds ., Wiley-Int erscience, 1963, Cha p. 42.6 . T. Erdey-Gruz, Kinetics of Electrode P rocesses, Wiley-Int erscience, New York , 1972,

Chap.1 & 4.7. W.J . Albery, Electrode Kinetics, Clarendon, Oxford, 1975.

J .E.B. Ran dles,Trans. Faraday Soc., 48 ,828 (1952).H. Kojima and A.J. Bard, J . Am. Chem. Soc. , 97 , 6317 (1975).

8 . M. E . Peover, E lect roa n al . Ch em ., 2 , 1 (1967).N. Koizum i an d S. Aoyagui, J . E lect roa nal . Ch em ., 55 ,452 (1974).

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CHAPTER 18

BIOELECTRODICS

INTRODUCTION

Electrochemistry as we have come to know it consists in the study of ionic solutions, andelectr odes wher e ions and electr ons combine an d separ at e. Galvani from Bologna, It aly,

in 1791, put forth that bio electrochemistry has been a part of electrochemistry.It would be not too much to say that there are electrochemical events going on inliving systems where ever you pry into them. The nervous system is certainly based onthe flow of electric currents and it is not of all fanciful to see nerves as the wires thatru n bet ween th e enzymes, the electrodes of the body. Bodies ar e full of membra nes, 100,and so are electrochemical cells. Some reactions in the body baffle chemists by going uptree energy gradients, but again that is just what happens in electrolysis, inelectrochemical reactors.

We can stu dy electr ochemical phenomena in t he imm ense complexity of living systemswhen all we know is how to explain simple systems like fuel cells and corrosion seemsto be the crassest arrogance.

The science of biology is a truly gigantic edifice, so big, in fact, that it includes all

of organic chemistry and uses it to explore very very complicated interactions.USEFUL PRELIMINARIES

The aminoacid glycine i.e., NH3+CH2COO would be simplest example excluding of course H3O+, H2O, and O2 of an entity that takes part in bioreactions,

A Structural element within the amino acid is the peptide group N C

H O| |

which is important because when many of these groups occur in a chain, and suchchains form a polymer, one has a protein. Proteins form skin, nails, and skeletalstr uctures. E nzymes biocat alysts a re pr oteins, Hem oglobin, which carr ies O2 around thebody is a protein. Proteins can have molecular weights as law as 10,000 but some arereally very large with molecular weights of 50 106. The corresponding radii of thelarger of these ent ities (when th ey get form ed spherically), would be in th e hun dreds of angstorms. Examples of some amino acids are shown in the Fig. 18.1. Structures that

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Fig . 18.1 Exam ples of some a mino acids. (Reprint ed with permission from Accoun ts of ChemicalResearch21 copyright 1989, Fig. 1, Amer ican Ch emical Society)

a protein, for exam ple in glycine. The proteins found in n at ur e are ma de up of onlyabout 20 different individual a mino acids. On t he other ha nd, a typical protein consist sof several h undr eds of th ese 20 distinguisha ble amino acids. These pr oteins (cont ainingseveral hu ndr eds of th e 20 special am ino acids) might be t hought a t first to be long, longchain s cont aining r epeated pept ide groups (see last fig.) 18.2 which can be writt en m ore

F ig. 18.2 Peptide

explicitly as where the Rs may be hydrocarbon chains on other peptide elements andth e sha ded area denotes th e bond between t he t wo peptides. However th ese long flexiblechains are neither linear nor random in shape. They coil and stretch in a way thatgreatly affects the properties of the proteini.e. , how it does its work. Among the mostimportant of these structures is an arrangement called the-helix (according to paulingand Corey 1951) but of the element s in cells, the m itochondrion. These mitochondria ar ethe entities in cells where energy is made from the oxidation of organics derived fromintake of food and oxygen.

H 2NCCOOH

H|

|H

H 2 NCCOOH

H|

| CH

CH 3 CH 2CH 3

H 2NCCOOH

H

||

CH 2|

OH

H 2NCCOOH

H|

| (CH 2 4)

| NH 2

H 2NCCOOH

H|

|

CH 2

NH

H 2NCCOOH

H|

| CH 2

| COOH

Glycine (gly) Isoleucine(ile) Serine (ser)

Tryptophan (trp) Aspartic acid (asp) Lysine (lys)

Peptide

HNCCNCCOH

R|

|H

|H

||O

|H

|H

||O

R|

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Membrane Potentials

The measurement and interpretation of the protentials across biological membraneshas been going on for about a century. A remarkable suggestion was put forward byBois-Raymond in 1868, whose concept was that a cell surface could be well looked at asthough it were an electrode.

The principal elements in biological cells is shown in Fig. 18.3. A Section of a yeastcell with its membrane is shown in Fig. 18.4.

Fi g. 18.3 : The scheme of an a nima l cell. F ig. 18.4 : Electron micrograph of a sectionreprinted from J . Koryta, Ions, E lect rod es from a yeast cell. The outer envelope is the

And Mem bran es F ig. 69. Copyr igh t cell wa ll. Th e in ner dou ble lin e is th e cyt o-J . Wiley & Sons , Ltd . 1991. Reproduced p lasmatic membrane. (Repr in ted fromwit h per mis sion of J . Wiley & Son s, Lt d.) J . Kor yt a, Ion s, E lect rod es an d Mem bran es,

Fig. 70. Copyright J . Wiley & Sons, Ltd.1991. Reproduced with perm ission of J. Wiley & Sons, Ltd.)

From th ese two figures it is inferred th at t he anima l cells ar e rat her complex, eachone containing the heriditary material and in particular the entities known asmitochondria. It is infact the energy producing properties of these are discussed inforthcoming sections.

Membranes, the subject of this section, can be relatively thick (0.1mm) of made

chemically). Biological membranes are very much thinner (50100) of the same(35 nm) range as that of passive oxide. The figure shows the essential constituents of the biological membran es. These are lipids and proteins. In m yetin membra ne t he lipidcontent is 80% while at the other end the range, in mitochondria there is an innermembrane containing only about 20% lipid. There are many kinds of lipids as well asvery many kinds of proteins. In membranes are usually phospholipids Fig 18.5. The

1 m

B ioelectrod ics 259

nucleolus

nucleus

mitochondrium

endoplasmaticreticulum Golgi

complex

Cytoplasmicmembrane

Lysosome

approx. 10 m

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F ig. 18.6: Prepar ation of abilayer lipid membrane.(Reprinted from J. Koryta,

I on s , E l ect r od es a n d M em b r a n es , Fig. 84.Copyright J. Wiley &Sons Ltd. 1991. Reproducedwith per mission of J . Wiley& Sons, Ltd.)

IB 1IN

IM

IC

P

IB 2

polar headHydrophobic end

Teflon septum

window

aqueoussolution

(a)

drop of lipid solution

BLM

(b)

(c)

Plateau-Gibbs boundary

H 2COCR

O

||

H 2COPOX

RCOCH|

|

|O

O||

O||

Fig. 18.5: Typical biologicalmembran e str uctures. A liquid-mosaic model in the formproposed by A. Kortya withdifferent types of disposition of membrane p ro te in ) (phos -pholipids are shown as darkcircles with two wavy tails); IB1,integral, membrane-bridgingprotein, with a single poly-peptide span; IB2, the samewith several spans; IN and IC,integral noncytoplasmic andcy top lasmic p ro te ins ; IM,integral buried proteins; P,peripheral protein. (Reprintedfrom J . Koryta , Ion s, El ectrodes

a n d M e m b r a n e s , F i g . 8 1 .Copyright Ltd. 1991. Repro-duced with permission of J.Wiley & Sons, Ltd.)

F ig. 18.5(a ): A schem e of th ebilayer lipid membra ne. Theblack circles indicate thepolar heads (the hydrophilicpart) consisting of phosphoricacid, ethanol amine, andanlogue derivatives. Thewavy lines are the long alkylchains of fatty acids (thehydrophobic par t) (Reprint edfrom J. Koryta,ions, Electro-des and Membranes, above.Copyright J . Wiley & SonsLtd. 1991.

structure often contains an H atom and this allows the phosphoric acid element toionize. In t he membra ne str ucture, the a lkyl groups R a nd R ar e directed inwar d whilethe popular groups are on the surface.

A scheme of bilayer and lipid membrane. The black circles indicate the polar heads(the hydrophilic part) consisting of phosphoric acid, ethanol amine; and analoguederivatives. The wavy lines are the long alkyl chains of fatty acids (the hydrophoblicpart).

[Taken from J. Koryta, Ions Electrodes and Membranes.]The most common model system to act as a smiplified biological membrane is thebilayer Lipid membrane (BLM) which was first prepared Mueller in 1962. It consistsof two lipid molecules ta il to ta il Fig. 18.6 with t he polar groups orient ed to face thesolution. In the basic BLM individual, compounds of a biological system can be built andexamined.

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The BLMs, although very thin exhibit very high resistance, up to 1010 ohms.Nevertheless, some pores do develop in these membranes and water, followed by ions,enters there and reduces the resistance. Application of a potential increases the flow of ions through the pores and the number of pores. This further reduces the resistance.

Fi g. 18.7 : Prepa rat ion of a bilayer lipid membra ne[Taken from J . Koryta Ions, Electrodes an d Membran es.

The method of measuring a membrane potential is simple. A calomel electrode isplaced in a solution (if the solution contains Cl) on either side of membrane, whichusually occupies a hole about 1 mm in diameter in a teflon sheet. Since the potentialof the calomel electrode is accurately known and varies accoding to Nernst potentialwith logdCl-, the difference in potential ar ising from t he t wo different Cl concentra tionsin each side of the membrane is easily known and can be subtracted from the totalpotential difference registered between the two electrodes to give the value due to themembra ne. Most of th e membra ne potentials recorded in th e litera tu re lie within valuesof tens to hundreds of milli volts.

Simplistic Theory

The th eory of an equilibrium on one species between each side of the membr ane wa sformulat ed by Donnan in 1925 until 1955, it reigned as t he th eory of membrane potent ials.Its dem ise came when ra diotr acer measur ements showed that all relevant ions (e.g., K+,Na+, and Cl) perm eated m ore tha n a dozen actu al biological mem bran es, although each

ion had a characteristic permeability coefficient in each membrane according to Hodgkin a nd Keynes 1953.Until 1950s Some bio-electrochemists confidently explained membrane potentials by

assuming that only one ion like K+ in KCl permeated the membrane. If so, then = + = + + In iu i nF * ic RT a nF (1)

Teflon septumwindow

aqueoussolution

(a)

drop of lipid solution

BLM

(b)

(c)

Plateau-Gibbs boundary

B ioelectrod ics 261

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would be t he expression for electrochem ical potent ial of th e perm eable ion i, on one side, , on the other side, the corresponding expression for electrochemical potential of thesame ion, there would be

oji ib F b R T ln nF

= + = + +

As long as equilibrium can be assum ed between permeat ing ions,i on each side e.g,on the and the side respectively

ibi b =

Hence R ii

RT ln

F

= ...(2)

Being developed in parallel with the rise and fall of the Donan equilibrium theoryof membrane potentials was the application of liquid function. Potential theory tomembr anes. This was pr oposed as far back as 1888 by great N ernst himself. The theorygrew by application of the Nernst-Planck equation to take into account the drivingforces due to concentration and potential gradients.

Modern Approaches to the Theory of Membrane Potentials

According to (Jahn, 1962) there is found to be a poor match between theory andexperiment. Ionic concentration differences alone, then, do not completely determinemembrane potentials in living systems.

Some membrane po ten t i a l s a r ea ff ec t ed by l i gh t , j u s t a s i f t he

membranes were semi-conductors. Thisis entirely outside the capabilities of theories that depend on the interplay of potent ial and concent rat ion gra dients in1949 Albert Szent-Gyogyi made asuggestion that some biomaterials mightbe regarded a s semiconductors. This wascriticized and another view was thatbiomaterials could be thought of aspossible electronic conductors a nd henceelectrodes. This was consistent with theidea t hat the first step in photosynth esisis the photo-electrochemical decomposi-tion of water, and it would account inprinciple, for the photosensitivity of membranes.

In case Szent Gyorgyis concept isaccepted to interpret the electronic

F ig. 18.8: Schemat ic diagram showing the oxidat ionand reduction reactions occuring at a membranesolut ion interface on sides 1 an d 2 of the m embra ne.(Reprinted from M.A. Habibi and OM. Bockris,J . Bioelectrici ty2 :66 (1984). Reprinted byperm ission of Mar cel Dekker .)

E = Ea' 2 R 3e

B

B A

A

Membrane

E = E AA R1

E 1 E 2

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condu ction of some biomat erials, t heidea that arose was that electronict r ans fe r a t t he so l id / so lu t ioninterface could occur with the solidbeing a biomaterial. Jahn in 1962was the f i r s t to come up wi thd i ff e ren t t heo ry o f membranepotentials. Jahns concepts picturedthe membrane as a bioelectrode,with each side the site 1 differing(but coupled) redox reactions, themembrane itself acting somewhatlike the m embran e in a fuel cell F

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