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A graphical approach to redox titrations Carlo Macc~ and G. Giorgio Bombi

Dipartimento di Chimica Inorganica, Metallorganica e Analitica, Universitfi di Padova, 1-35131 Padova, Italy

Graphische Behandlung yon Redox-Titrationen

Zusammenfassung. Ein Verfahren wird beschrieben zur Auf- stellung eines logarithmischen Diagramms, das eine kor- rekte Darstellung der Gleichgewichtskonzentrationen der verschiedenen Teilnehmer an einer Redoxtitration bietet. Der Gebrauch eines solchen Diagramms zur Auswertung des Titrationsfehlers und zur Berechnung der Titrations- kurve wird gezeigt. Aufgrund der logarithmischen Dia- gramme werden auch die Symmetrieeigenschaften yon Titra- tionskurven und der Linearit/itsbereich von Gran-Dia- grammen diskutiert.

Summary. A method for drawing a logarithmic diagram which gives a correct representation of the equilibrium con- centrations of the various species taking part in a redox titration is described. The use of this kind of diagrams for evaluating the titration error and for calculating the titration curve is illustrated. The symmetry properties of titration curves and the linearity range of Gran plots are also dis- cussed on the basis of the logarithmic diagrams.

Introduction

Logarithmic diagrams of equilibrium concentrations [1] offer for the study of ionic equilibria a very convenient tool [2-8] , the usefulness of which is only in part challenged by the increasing availability of programmable calculators and of personal computers.

Starting from their introduction by N. Bjerrum [9], loga- rithmic diagrams have been widely exploited for the study of titration systems, first to evaluate the theoretical titration error in acid-base titrations [9, 10] and subsequently also to compute the shape of various kinds of titration curves [6, 10-13]. In spite of this, their capabilities do not seem to have been fully developed, especially for the case of redox titrations. With a view of dealing with this subject, it is useful to restate explicitly some points underlying the use of graphical methods for the study of titrations.

In order to compute a titration curve it is necessary to solve, for each point of the titration, an algebraic system of as many independent equations as are the chemical species involved in the titration system. These equations comprise the mass law expressions for the relevant equilibria and the proper number of mass balance equations [4], and can be

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Fresenius Z Anal Chem (1986) 324:52-57 �9 Springer-VerIag 1986

solved exactly by readily available numerical methods. On the other hand, approximate solutions can be obtained from a number of "formulae", each of which is valid only in a given range of conditions, or through the use of graphical methods.

In order to be correct, a graphical treatment must be exactly equivalent to the algebraic one: the diagram must hence contain or exploit all the informations on the titration system contained in the corresponding equation system. In practice, all the equations but one are employed in building the diagram; the remaining equation, which must express a quantity related to the degree of advancement of the titration (the mass balance for the titrant or another equivalent mass balance equation, like e. g. the proton balance [1] for acid- base titrations) is employed to follow the course of the titration on the diagram.

Strong base - strong acid, weak acid - strong base, weak base - strong acid, precipitation and direct complexometric titrations can be accurately described by using a single logarithmic diagram of the Bjerrum type, provided the dilution effect due to the addition of titrant can be neglected [10-13]. On the contrary, in the case of complexometric back-titrations [14], of weak acid-weak base titrations, and ofredox titrations it is not possible to describe correctly (i. e. by including all the relevant equations) the whole titration by means of a plot based only on the usual Bjerrum diagrams, which presume a fixed analytical concen- tration. Referring in particular to the case ofredox titrations, the analytical concentration can be considered as a constant only for the titrated couple, but it is continuously varying during the titration for the titrant couple. A diagram drawn for the analytical concentration of titrant prevailing at the equivalence point [8, 10, 11], while allowing a correct identification of the latter point, cannot give the correct solution for all the other points of the titration.

In the present paper we will introduce a logarithmic diagram for redox titrations which exactly accounts for the concentration of all the species in the whole course of a redox titration (neglecting only, of course [1], the dilution effect). Furthermore, we will develop a formal treatment (valid for redox as well as for any other kind of titration) which, as it will be seen, allows to deduce from the relevant logarithmic diagram all the main features of a titration, including the complete titration curve. This treatment is based on the use of a simple algebraic expression which follows from the definition of titration error and is equiva- lent to the mass balance for the titrant system.

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Construction of the diagrams

In building logarithmic diagrams for redox systems, the equilibrium concentrations of the various species are conveniently plotted using as the master variable the normalized value of the potential pE defined [1 a] as:

pE = EF/(RTlnlO) (= E/O.O59at25~ (1)

Let us indicate with Aox and A~d the two members of the titrated (analyte) couple

Aox + Zne ~ Ard (2)

(standard potential1 pE~), and with Tox and Trd the two members of the titrant couple

Tox + zTe ~ Tra (3)

(standard potential pE~). When pE~, < pE~, the "oxida- tion" titration of A,d with Tox will be feasible.

The log [i] vs. pE curves for the species of the titrated couple, whose total concentration CA is taken as constant, are plotted in the usual way [l a]. Their equations are obtained by combining the Nernst equation and the mass balance for the couple. In practice, the log [i] vs. pE curves are represented for most values of pE by straight lines having slope 0 or _+ Zn and intersecting at log [i] = log Ca, pE = pE~. For example, we have

log [Aox] = log CA -- zapE~ + zapE (4)

for pE < pE~ -- 1.3/ZA (if an approximation of _+ 5% is required [3]), and

log [Ao=] = log C m (5)

for pE > pE~, + 1.3/ZA. In order to obtain the equations of the curves for the

titrant couple, T, the titrant mass balance cannot be used as the stoichiometric condition, since the value of CT is the variable which measures the progress of the titration, and cannot be included from the beginning into the diagram (see the Introduction). The Nernst equation:

Z T p E = zTpE~r + log[Tox] - 1og[Trd] (6)

must be combined with the electron balance condition [3]:

ZT [Trd] = ZA [Aox ]. (7)

For the "reagent" form of the titrant couple we obtain

log[To~] = - zTpE~- + Iog(zA/ZT) + 1og[Ao~] + ZTpE. (8)

Recalling alternatively Eqs. (4) and (5) and indicating with CT,eq = (ZA/ZT)CA the stoichiometric concentration of the couple T at the equivalence point, we have

log[To~] = logCT,eq -- ( z a p E 2 -F zxpE~-) + (ZA + ZT)pE (9)

for pE < pE~ -- 1.3/ZA and

log[Tox] = logCT.~q -- ZTpE~ + zxpE (10)

for pE > pE~ + 1.3/ZA. The straight line segments represented respectively by

Eqs. (9) (slope ZA + ZT) and (10) (slope ZT) are connected around pE = pE~ by a curved section, the shape of which

I The denomination "s tandard potential" and the symbol pE ~ is employed here for simplicity. In general, the standard potential has to be replaced with the proper formal or conditional potential, pE ~

o - \ :--. Fe(ll)

I Fec'U I -8 ~~ -12

/ \ 0 4 8 12 DE 16

Fig. l. Logarithmic diagram for the t i tration of a l.O0 • 10 -2 M solution ofiron(III) (E ~ = 0.77 V) with a very concentrated solution oftin(II) (E ~ = 0.15 V)

can be computed by substituting [Aox] in Eq. (8) with its exact expression. The prolongements of these two segments intersect in the point having coordinates pE = pE~ and log[i] = zT(pE~ -- pE~) + 1OgCT,eq; the true value of log[Tox] at the same abscissa is 0.30 units lower than the point of intersection. It is worth noting that in most cases of practical importance the value of pE~, - pE~ is a rather large negative number, so that the segment of Eq. (9) lies very low in the diagram and needs not to be traced.

As regards the "product" form of the titrant couple, from Eq. (7) we obtain

1og[Trd] = 1og(2A/ZT) -F log [Aox]. (11)

Therefore, the log[T,d] line coincides with the log[Aox] line for an isovalent [15] reaction (ZA = ZT, CT,eq = CA) and is parallel to it in the general case.

I f pE~ >> pE~, the "reduction" titration of Aox with T,d will be feasible. The diagram for the analyte couple is of course the same as above; as regards the titrant system, using Eq. (6) and the electron balance equation

ZT [Tox] = ZA [Ara] (12)

the following equations are obtained:

log[Tra] = logCT,eq + zApE+A + ZTpE~" -- (ZA + ZT)pE (13)

for pE > pE~ + 1.3/ZA and

log [Trd] = log CT.eq + ZT pE~ -- ZT pE (14)

for pE < pE~, -- 1.3/ZA; the log[Tox] line is parallel to (or identical with) the log [Ara] line.

In Fig. 1 the logarithmic diagram for the titration of Fe(III) with Sn(II) is reported as an example. The system has been represented as having a total concentration of Fe of 1.00 x 10 -2 M, but the same diagram can be used for other concentrations by simply shifting the ordinate scale. The diagram for the titrations of Sn(II) with Fe(III) (reverse of the above) is reported in Fig. 2 (dashed curves).

53

Or cjina pa#evs 0

Fe(ll) o . . . . . . . . . . . w . . . . . . o

,, _ \ / " -4 7' - 8 ~ ,'ep

-8

-12 ,," ~ ,' ~ Sn(ll)

,' ~ - 12 t t t

-16 [ I t ~ I O 4 8 12 pE 16

Fig. 2. Logarithmic diagram of the equilibrium concentrations (dashed lines) and of the titration error (solid lines) for the titrations of a 5.00 • 10 3 solution oftin(II) with a very concentrated solution of iron(III)

Ti tra t ion error and t i trat ion curves

The correct shape of a titration curve can be obtained from the corresponding logarithmic diagram by a simple, yet rig- orous method, which unifies the treatment of titration curves with that of titration errors.

Let us consider the general titration

a A + tT -~ (products) (15)

where A represents, as before, the titrated species. Indicating with nA the initial number of moles o f A, with nT the number of moles of T added at any given point o f the titration, and with CA and CT the respective analytical concentrations, the relative error ~ which would be made if the titration was stopped at that point is given by

(a / t ) nT -- nA (a/t) CT -- CA - - (16)

nA C A

Making use of the mass balance equations, the difference (a/t) CT - CA can easily be expressed in terms of the equilibrium concentrations of some of the species present in the titration system. These concentrations can be read from the logarithmic diagram and the value of e can be easily evaluated as a function of the master variable of the diagram.

Taking as an example the oxidation titration of Sn(II) with Fe(III), using the mass balances for iron and tin and the electron balance condition Eq. (16) becomes

0.5 [Fe(III)] - [Sn(II)] = 0 7 )

Cs,

This equation is equivalent to the mass balance for the titrant couple, which has not been employed in building the logarithmic diagram. Therefore it represents the nth equation which, together with the (n - 1) equations already incorporated into the diagram, allows the solution of the equilibrium problem for each particular point of the titration (see Introduction).

Logarithmic plots of the relative (e) or absolute (r CA) titration error were developed by Incz6dy [16] on the basis of an algebraic treatment. Such a plot can he immediately

(

Iogiel - 2

l l

/ l

-4 l /

/ /

/ /

-6 /

C107 / -1 ,#

0 . . . . . . . . . . . . :

1 I i i i i

14 la p E

-8

\\

\\ p" \

7 .... -k ....

B r - ~ '

b 22

C !, o 8

0

l og [ J ]

-2

Fig. 3 a - e . Use of the logarithmic diagram for deducing the theoret- ical features of the titration of 0.1 M hypochlorite with bromide at p H = 10.0 . pE~] = 19.0 , pE~'r = 16 .9 (pER' = p E ~ - p H ) . a Logarithmic diagram of the equilibrium concentrations (dashed lines) and of titration error (solid lines), b Titration curve measured with an inert metal redox electrode, e Titration curves calculated for an ideally specific bromide electrode (solid line) and taking into account the chloride interference (dashed line)

obtained from the logarithmic diagram for the titration system by applying Eq. (17) (Fig. 2). For most values o fpE , the log(l~lCA) vs. pE line is seen to coincide either with the log[Fe(III)] + log(zFe/ZSn ) line or with the log [Sn(II)] line, so that it has to be actually computed by points only in a region around the pE of the equivalence point, namely for

pEeq - 1.3/(zve + Zs,) < pE < pEeq 4- 1.3/(ZFe + ZSn ). (18)

Any information required for drawing the titration curve is already explicitly contained in the above error diagram. In fact, the value of e can be assumed as a measure of the progress of the titration, being related to the more frequently employed "titration ratio" ~b by the equation ~b = aCT/ (tCA) = I + e. The use o f e i n the place ofq~ has the advantage of giving directly more precise values (both for graphical and for numeric computations) in the region around the equivalence point. In order to draw the titration curve it is then sufficient to read on the diagram at a conveniently large number of values of pE the values of [Fe(III)] and of [Sn(II)], or directly the values of e and to plot pE vs. e.

As a further example, we will now discuss a hypothetical titration based on a reaction having an equilibrium constant much smaller than most reactions usually exploited for re- dox titrations.

It is known [5] that hypochlorite reacts rapidly with an excess of bromide in neutral or basic solution to give the stoichiometric amount of hypobromite:

C10- + Br- ~ C1- + B r O - . (19)

The equilibrium constant for this reaction can be calculated from the standard potentials of the redox couples involved [17]: log K = 2 (PE~I -- pE~r) = 4.20. We want now to verify whether this reaction can in principle be used for the titration of hypochlorite. The logarithmic diagram for the titration system at pH = 10 is reported in Fig. 3a.

54

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The expression for the titration error is

C.~ - Cc,o [Br-] - [CIO-] e - - (20)

Cclo Cclo

and the curve representing loglel is given by the heavy line in the same figure. In Fig. 3 b, the titration curve built by taking from the above diagram several significant points is reported. For instance, at point P, with abscissa pE = 18.50, we have l o g ( - e ) = log[C10-] = - 1 . 0 0 . The cor- responding point on the titration curve is P ' , with abscissa

= - 0 A 0 0 and ordinate pE = 18.50. The equivalence point is immediately identified by the condition e = 0, i. e. [Br-] = [C10-] = 1 0 - 3 . 6 at pE = 17.95. (For e < - 0.5, the relationship

[Br-] - [CI-] e = 1 (21)

Cclo

obtained from Eq. (20) and from the mass balance for the analyte system has been used to improve the precision.)

As shown in Fig. 3a, a fairly definite potential drop arises at the equivalence point of the titration curve, which could however be exploited provided an electrode material can be found on which the two redox systems involved give a sufficiently reversible response. In the probable event that such a material cannot be found, one could speculate whether the titration can be followed by means of a bromide electrode. The relevant titration curve (Fig. 3c) can be obtained as follows: taking again, e. g., point P, the concen- tration o f bromide prevailing at this stage of the titration is given by the ordinate of point P" (log [Br-] = - 4.20). The corresponding point of the titration curve is P '" .

The titration curve log [Br-] vs. e shows a larger jump at the equivalence point than does the pE vs. e curve. It could be followed for instance by means of a single-crystal bromide selective electrode (which has been found to be not sensitive to redox interferences [18]), provided the thermodynamically favored reaction

AgBrs + C10- ~ AgC% + B r O - (22)

is not too fast at the surface of the electrode. The titration curve would however be altered with re-

spect to its theoretical shape in its first part owing to the partial selectivity o f the bromide electrode towards chloride ions [19, 20]. The correct shape of the curve (broken line in Fig. 3 c) can easily be deduced from the diagram, by plotting the calculated value of

log[Br-]* = log([Br-] + k~~ (23)

where [Br-]* is the "apparent" bromide concentration mea- sured by the electrode.

Symmetry properties of redox titration curves

As another application of logarithmic diagrams to the study of titration systems, we will now discuss the symmetry properties of redox titration curves. This topic, besides its formal interest, is also of practical importance, as several of the methods commonly employed for the location of the end point in potentiometric titrations presume a symmetric titration curve [21].

It is well known that heterovalent redox titrations (i. e., titrations for which ZA = ZT) give rise to asymmetrical titra- tion curves; on the other hand, titrations based on isovalent

P Eeq

h , , . . . .

A(~)o o A(-% A

Fig. 4. Enlarged portion of Fig. 3a, showing the "approximate symmetry region" of the titration curve

limit of

reactions (with za = z~) are frequentely termed "symmetric", giving rise to the misconception that these titrations give also symmetric titration curves. Logarithmic diagrams can help to show immediately that even isovalent redox titration curves are asymmetric, and possess only a limited apparent symmetry range around the equivalence point.

A titration curve can be said to be symmetric [22] with respect to the equivalence point, whose coordinates are e = 0 and pE = p E e q if, for any value of A = pE - p E e q , it is

e(A) = - e ( - A ) (24)

or also

log le(A)l = log le ( - A)I. (25)

Referring again to Fig. 2 and to Fig. 3 a, it is evident that the latter condition would be satisfied if each of the two branches of the titration error curve were the mirror image of the other with respect to the line pE = p E e q , i. e. A =

0. This is obviously not true for the heterovalent titration Sn(II) - Fe(III) (Fig. 2). In the case o f the titration hypochloride - bromide, condition (25) is approximately verified in the region in which the two lines representing log [Br-] and log [C10]- can be identified with two straight lines having opposite slopes.

The limits of this region of "approximate symmetry" can be set at the value of _+ e for which the difference between the two corresponding values o f A is equal to the uncertainty in the measurement of pE. If a numerical value is assumed for the latter uncertainty (e. g., _+ 0.02 units, or _+ 1 mV at 25~ the "critical" value of le] can be easily identified on a large scale diagram as that of Fig. 4. To avoid the necessity of drawing an ad hoc diagram, we can observe (Fig. 4) that, at the "critical" value of I el, the vertical distance h between lines a and b, which represent respectively the actual of log l e[ and the value which would give rise to a symmetrical curve, is equal to 0.02 x 2, the latter numerical factor arising from the value of the slope of the straight line. With the aid of a little mathematics it can be verified that the condition h = 0.04 is satisfied when log[C1-] - log[C10-] - 1.0. Thereby, using a normally sized logarithmic diagram, the interval of approximate symmetry is found to be - 10% < ~ < + 10%. The amplitude of this interval (which is clearly independent

55

@r 9 Ha paper8

of the difference between the pE ~ values of the two couples involved) can be seen, by the same procedure, to be approximately - 5% < e < 4- 5% for z = I and - 15% < e < -t-15% for z = 3 (always for an uncertainty of _+ 1 mV in the measured potential values).

G r a n p lo t s

In several instances, and in particular in the case oftitrations based on "incomplete" reactions, it is advantageous to use the well-known Gran method [23] to linearize the titration curve. In applying this method, it is of particular interest to ascertain beforehand which part of the titration is sufficiently "qantitative" to give a linear Gran plot, which can be safely extrapolated towards the equivalence point. Logarithmic diagrams can help also for this purpose.

Taking as an example a reduction titration, it can be shown [24] that the Gran plot gives a relative error of the order of 6 or less if only points for which

z~-[Tra] < 6 Z A [Ard ] (26)

are employed for the extrapolation before the equivalence point and

zA [Aox] < ~ ZT [To\] (27)

after the equivalence point. We have now only to look on the diagram for the titration system for the point where

log[Tra] = log[A,a] -t- Iog(6ZA/ZT) (28)

or, respectively,

log[Aox] = log[To\] 4- Iog(6zT/ZA). (29)

AS an example, we will refer once again to the hypotheti- cal titration of hypochlorite with bromide. From the log- arithmic diagram, Fig. 3, it is seen that the condition log [Br-] < logiC1-] - 2 [Eq. (29) with c5 = 0.01] is satisfied for pE > 17.8, while the condition log[C10-] < log[BrO] - 2 [Eq. (28)] is satisfied for pE < 18.2. As the first value follows the equivalence point and the second preceedes it, both Gran plots are linear, within the specified limits, in their complete ranges.

With the bromide electrode, the condition for the linearity of the Gran plot after the equivalence point is

log[C10-] < log[C1-] - 2 (30)

which, from Fig. 3a, is seen to be satisfied for pE < 18.0; in practice, also in this case, the Gran plot is linear in its whole range. (It is worth noting that in this part of the titration, the bromide indicator electrode is not affected by the possible occurrence of reaction (22) or by chloride interference.)

C o n c l u s i o n s

It is instructive to compare the use of the diagrams we propose for redox titrations with that of "twin" Bjerrum diagrams, drawn assuming both reagents to be at the analyti- cal concentration prevailing at the equivalence point, pre- viously used by several authors [8, 10, 11]. In Fig. 5, which can be compared with Fig. 3 a, the Bjerrum diagrams for a 0.1 M solution of C10- 4, C1- and for a 0.1 M solution of BrO- 4- Br- are plotted. As it is seen, the curves for log [BrO-] and those for log [Br-] are practically coincident

56

- //// \

r / \\

CI 0"/ \ Br" -e ' [ ' ' ' ' ' \ '

18 pE 22

Fig. 5. Logarithmic diagrams of 0.1 M hypochlorite + chloride, and for 0.1 M hypobromite + bromide. Solid lines show the parts which coincide with the curves of the logarithmic diagram for titra- tion of 0.1 M hypochlorite with bromide, Fig. 3 a

in the two figures in the vicinity of the equivalence point, but are very different in a large part of the diagram, which corresponds to a very large part of the titration. Whether this difference is important or not, depends on the use for which the diagram is intended. The twin Bjerrum diagrams will give correct results in the calculation of the titration error near the equivalence point. However, they cannot cor- rectly account for the complete shape of the titration curve, for its symmetry properties, and so on.

Redox titrations also include types of reactions which give rise to forms of logarithmic diagrams different from those discussed in the present paper (at variance, e. g., with acid-base titrations, which are all described by the same kind of diagrams). These include the case in which one of the couples involves a dimeric species (e. g., Cr(III)/CrzOg-), the case in which one of the species produced by the reaction is present, possibly in a large amount, from the beginning of the titration (as I - in iodometric titrations), and the case of the titration of a mixture of several analytes. The logarithmic diagrams for these titration systems are however easily built and exploited on the basis of the general principles reported in the Introduction, following (with the necessary modifications) the lines illustrated above.

The advantages of using logarithmic diagrams of true equilibrium concentrations in combination with the error equation for the study of titrations should be evident from the preceeding discussion. The use, in place of the error equation, of other equations, e.g. the expression of the titrated fraction as a function of equilibrium concentrations, the mass balance of the titrant, the electroneutrality balance of the solution, and (in the case of acid-base titrations) the proton balance, is a matter of preference, since all these equations are mutually equivalent from the algebraic point of view. However, the approach described here is unique in giving a complete and integrated vision of a number of different features of the titration.

At variance with other approaches (with the obvious exception of rigorous algebraic treatments), the approach we are supporting does not require extra approximations in the procedure of calculating the titration curve, all the approximations introduced being those inherent in the construction of the diagram and those explicitly "suggested" by the diagram itself. In particular, it accounts correctly also for reactions having a small equilibrium constant (which are

@ 'igiHa!arieile

becoming more and more frequently used, at least for acid- base t i t rat ions with the aid o f micro-computer ized ti trators), for t i t rat ions of mixtures of several analytes (very impor tan t for acid-base and for complexometr ic reactions) and for back-t i t ra t ions [14].

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Ellis Horwood Chichester 17. Pourbaix M (1966) Atlas of electrochemical equilibria in

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Pergamon, Oxford, p 168 20. Maccfi C, Cakrt M (1983) Anal Chim Acta 154:51-60 21. Ebel S, Seuring A (1977) Angew Chem Int Ed 16:157-168 22. Macc& C, Bombi GG (1983) J Chem Educat 60:1026-1030 23. Gran G (1952) Analyst 77:661 --671 24. Maccfi C, Bombi GG, to be published

Received July 31, 1985; revised December 3, 1985

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