Open access peer-reviewed chapter

On the Titration Curves and Titration Errors in Donor Acceptor Titrations of Displacement and Electronic Transference Reactions

Written By

Julia Martín, Laura Ortega Estévez and Agustín G. Asuero

Submitted: December 9th, 2016 Reviewed: March 23rd, 2017 Published: September 6th, 2017

DOI: 10.5772/intechopen.68750

Chapter metrics overview

3,021 Chapter Downloads

View Full Metrics


An overview of the state of the art concerning with earlier approaches to titration in redox systems is given in this chapter, in which an overview on redox bibliography has also been undertaken. Titration error has been the subject of a variety of excellent papers, but the number of papers dealing with titration error in redox titrations is scarce. However, a single hyperbolic sine expression for the titration error in donor/acceptor titration of displacement and electronic transference reactions is derived in this chapter. The titration error expression is applicable to symmetrical redox reactions, that is to say, those in which no polynuclear species are involved in the equilibria. The donor versus acceptor particle notation is chosen to accentuate the analogy with that used in the description of acids and bases following the steps given by the French School and other recognized authors (Budevsky, Butler, Charlot, Gauguin, Inczedy, Monnier, Rosset). A diagram for the titration error in function of the difference between the end and equivalence point (pX) is drawn in order to facilitate the graphical calculation of titration error. A detailed error analysis concerning with the propagation of systematic and random error propagation in the titration error is given.


  • titration curves
  • titration error
  • electronic transference reactions

1. Introduction

Titrimetry constitutes one of the former analytical methods, been applied since the late eighteenth century [1]. Based on reaction chemistry, it is still developing [25] and continues to be extensively employed in several analytical fields as well as in routine studies [6] due to its simplicity, with little sacrifice in accuracy and precision, and low cost [79].

Titrimetry is a fast technique easily automated and cheap in terms of equipment. It is together with gravimetry one of the two special existing methods to determine chemical composition on the basis of chemical reactions (primary method) being used for methodological and working levels [10, 11]. Independent values of chemical quantities expressed in SI units are obtained through gravimetry and titrimetry (classical analysis). In titrimetry the quantity of tested components of a sample is assessed through the use of a solution of known concentration added to the sample which reacts in a definite proportion. The reaction between the analyte and the reagent must be fast, complete (with an equilibrium constant very large), proceed according to a well‐defined (known stoichiometry) chemical equation (without side‐reactions), lead to a stable reaction product, and usually, and take place in solution, although in some cases precipitation reactions are involved. To identify the stoichiometric point, where equal amounts of titrant react with equal amounts of analyte, indicators are used in many cases to point out the end of the chemical reaction by a color change. The indicator reacts with either the analyte or the reagent to produce (in a clear and unambiguous way) a color change when the chemical reaction has been completed.

This is known as the “end point” of the titration, which should be as close as possible to the equivalence point. The difference between the end point and the equivalence point is the titration error. In order to reduce the uncertainty in the results, the end point can also be detected by instrumental rather than visual means, i.e., by potentiometry, where the potential difference is measured between a working electrode and a reference electrode; in photometry, where the light transmission is measured with a photometric sensor; or in coulometric, where the titrant is generated electrochemically and the amount of titrant is calculated from the current and time of reaction using Faraday’s law.

Titrimetric methods are classified into four groups depending on the type of chemical reaction involved: acid‐base, oxidation‐reduction, precipitation, and complexometric. Among the most common applications of titrimetry are the measurement of anionic and cationic species, and neutral molecules of both organic and inorganic substances. Although it has been extensively used for the measurement of pure substances, it also performs well when trace constituents in complex matrices want to be evaluated.

Before the experience, the titrant is standardized by titration of the primary standard, which must be stable against the influence of light, air, and temperature. The main primary standards are potassium hydrogen phthalate (KHP), benzoic acid, tris(hydroxymethyl)aminomethane (TRIS, THAM), or Na2CO3 for acid‐base titration, As2O3, Na2C2O4, K2Cr2O7, or KIO3 for redox titration, NaCl for precipitation titrations, and CaCO3, Pb(NO3)2, Zn, Ni or Cu for complexometry. During the experience, the temperature should be constant (for ordinary work ± 0.5°C). To reduce the overall uncertainty, titrimetry can also be performed on a mass/mass basis using a weight burette with a primary standard.

Titrimetric methods can also be classified based on the properties of the reaction: if the assay is performed with a known concentration of titrant or with an unknown concentration of titrant is called direct or indirect titration, respectively, whereas back titration involves an excess of titrant that reacts until the equivalence with a known concentration of sample.


2. Titration in redox systems: earlier approaches

Redox titrations were essential in volumetric analysis as well as in electrometric techniques. The first attempts to quantify titration curves, including redox systems, and data processing [12] were from 1920s to 1930s. As suggested by Goldman’s review [13], the first theoretical developments were due to Kolthoff [14], who provided the link between the application of physical chemistry and titrimetry [1517] (though it was criticized by his Dutch colleagues in his earlier career). The understanding of redox equilibria provided the platform to the development on electrode processes [13].

One can plot against pE (or pe) as variable either the logarithm of the concentration (or activity) of each species (maintaining the other concentration constant) or the logarithm of the ratio of the concentration (or activity) of the species [13] using as a reference one oxidation state. Then, the diagram, entirely composed of straight lines, is simple and easy to construct [18]. From log(concentration) versus E diagrams redox titrations may be drawn [1922], even the titration error by an appropriate modification [23, 24]. In fact, logarithmic diagrams, introduced by Bjerrum [25], have found a wide use in the study of titration systems, mainly in the acid‐base domain [23, 26, 27]. On the basis of logarithmic diagrams, Maccà and Bombi [28] discuss the symmetry properties of titration curves and the linearity of Gran plots. In the particular case of redox systems, distribution diagrams are not as commonly used as they are for acid‐base and complex ion systems [18, 2931].

In 1960s titration was revisited to improve formulation of titration curves, in the papers by Bard and Simonsen, Bishop and Goldman’s papers as indicated in the review of Goldman [13], “formal” potentials, E'0i, being used in those approaches instead of standard potentials. de Levie [3234] pointed out the similarity between the profile of redox and acid‐base titration curves, related to a polyprotic acid titrated with NaOH. Equivalence points were also discussed. Some further papers were related to redox indicators and titration errors.

Anfält and Jagner [35] reviewed and evaluated the accuracy and precision of some frequently potentiometric end‐point indicator methods. Yongnian and Ling [36] used mathematical methods to assess the application and advancement of titration methods. Although it is common to find examples of applications to redox titrations [37, 38] their actual use in redox titrations has been scarce [3942]. The Gran methods were modified later [4345], without unnecessary simplifications.

The most common topics dealing redox titrations and published in the last two decades were: to calculate stoichiometric point potentials by different types of redox titration [46]; the use of linear regression analysis of potentiometric data through linearization of the titration function [47, 48]; approximated titration curves [49] by logarithmic function studying the prediction of random error in titration parameters [50]; as well as the factors affecting the accuracy [51] of redox titrations. Nonlinear regression analysis allows the simultaneous evaluation of several parameters from the data obtained [52] in a single redox titration. A general algorithm and a program for the calculation and construction of titration curves [53] have been proposed. Numerical and computer simulations [54, 55] and equations by nonlinear least squares with novel weighting functions [56] for redox potentiometric data were also evaluated.

The analogy between acid‐base and redox‐behavior, as particle exchange reactions (of protons or electrons, respectively) [57, 58], has been interpreted through the years [5967].

Titration error has been the subject of excellent publications, but the number of papers dealing with redox titration error is relatively scarce. The focus has been mainly put in acid‐base, precipitation, and complex formation reaction titrations. In this chapter, an attempt is given to devise a titration error theory applicable to donor‐acceptor titration of displacement and electronic transference (redox) reactions. The error will be formulated as a function of the titration parameters, in a hyperbolic sine expression way. As a matter of fact a detailed treatment of the analysis error issue is also carried out. The error analysis requires differentiating with respect to given variables leading at first sight to complex expressions, which at the end finally appears to be compact.


3. On the titration error in donor/acceptor titrations of displacement and electronic transference reactions

Acid‐base, complexation, and precipitation titrations have been extensively dealt with both in scientific and educational literature. Redox titrations, although being of primary importance, have received less attention [68]. Titration error has been the subject of several excellent papers, the emphasis being placed, however, mainly on acid‐base [6973] as well as on precipitation [7479], and complex formation titration reactions [80, 81]. A number of papers dealing with the topic of titration error in redox titration [13, 24, 82, 83] have also been appeared. Redox titrations hold an important place in simple, fast, low‐cost analysis of redox‐active‐species [56]. In this chapter, a single equation is presented for symmetrical reactions for which the reduced and oxidized species of each half‐reaction should be the same. Therefore the treatment is not strictly valid for such couples as Cr2O72−/Cr3+ or I2/I [81].

Let SD be a particle donor (weak or strong), to be titrated with a particle acceptor TA. The analyst can choose the titrant and will always use a strong one to obtain better results. The systems implied in the titration reaction and their corresponding equilibrium constants will be given by


X being the particle transferred in the semi‐reactions involved in the global titration reaction


to which corresponds the following equilibrium constant


Proton free equilibria are assumed first in the presentation for the sake of clarity though redox equilibria that are independent of pH are relatively few. Effects of such factors as hydrogen ion concentration and complexing ligands may be easily incorporated [84] in the corresponding conditional constants. The particle X may be a proton, electron, cation, or an uncharged molecule. This notation is chosen to accentuate the analogy with that used in the description of acids and bases by the French School and other recognized authors [19, 2931, 8592]. Anyway, as TA is a strong particle acceptor, it must occur that KT >> KS.

From Eq. (3) it is readily seen that in any moment of the titration it holds that (through the entire concentration range)


On the other hand, we may define the (relative) titration error as


where CT and CS are the analytical concentrations of titrant and analyte, respectively,


and T, the fraction titrated, is defined as the ratio between the amount of titrant added and the initial amount of analyte at any moment of the titration. Thus, depending on the nature of the titration, ΔT might be either positive or negative. When the titration is carried out in the reverse order the same result is obtained, but the equation now bears a minus sign.

Therefore, from Eqs. (6)(8), it follows that


By combining Eqs. (5) and (1), we have


and from Eq. (2) we obtain


By substituting the values of [TA] and [SD] given by Eqs (10) and (11)


As the molarity fraction of the species [SA] is given by


Eq. (12) may be transformed into


By multiplying and dividing the right hand side of Eq. (14) by (KS/KT) we get


In the equivalence point, when the exact stoichiometric amount of titrant has been added, in addition to Eq. (15), the following condition is satisfied


and then, from Eqs. (1) and (2), it follows that when the exact stoichiometric amount of titrant has been added


and then, the potential at the equivalence point is independent of the concentration of the reactants and thus unaffected by dilution. Note, however, that Eq. (18) is not perfectly general, because the simple relation of Eq. (16) for reactants and products is not always valid. The species involved in the equilibria may be polynuclear. The pX in this instance varies with dilution.

By substituting Eq. (17) into Eq. (15)


A chemical error will arise because of lack of agreement between the end point and equivalence point. The difference between the end point and the equivalence point of a titration is the source of systematic error of determination. Taking into account that




after some manipulation, the following expression may be easily obtained




where the shape coefficient W is depending on the particle concentration in the end point titration when asymmetrical titrations are being considered


Low values of the stoichiometric coefficients a, b, as well as low difference a−b values, and large KT values lead to lower errors.

In the vicinity of the equivalence point, +1>>[X]endaKS and so


In those cases in which the titration reaction is symmetrical, a=b, and then


and the following formula is obtained for the titration error


Note that the titration error may be formulated as a hyperbolic sine expression. Hyperbolic functions are of great worth in parameter estimation as shown by Asuero [93].

The methodology developed in this section of the chapter is going to be applied forward to some experimental situations characteristics of redox titration reactions. The calculations made by the hyperbolic sine method are checked with the procedure devised by de Levie [21, 26, 62]. A detailed treatment of systematic and random errors associated with the titration error is carried out in the following.

Note that the equations developed in this contribution can easily take into account the lateral reactions by using the corresponding lateral reaction coefficients and the conditional constants involved. A condition is however required, namely that the pH remains constant in the course of titration, which cannot always be achieved. Michalowski [4, 94100] has given a general and definitive solution to the problem of redox equilibria, which does not require any restriction.

The beginnings of the rigorous GATES/Generalized Electron Balance (GEB) approach of Michalowski, which can be interpreted as a new natural law, dates back from 1992 to 1995. This approach has recently been shown repeatedly in the bibliography solving complex chemical problems and requires the use of nonlinear regression and a high level language such as MATLAB. The equations developed here, although very modest, have an obvious didactic interest and can be seen in the case of the redox equilibria as an alternative route to that given by de Levie [62].


4. Electronic transfer reactions

In the following, some titration curves of typical oxide‐reduction reactions, involving Ce4+ and MnO4 as titrant, are the subject of study. The transferred particle, the electron, takes the place of [X] in Eq. (22). The numerical values obtained by applying the hyperbolic sine method proposed in this contribution are checked against the method devised by de Levie [62], thus verifying the identity of the results in all cases.

4.1. Fe2+ titration curve with Ce4+

The equilibrium constant and pe of the semi‐reaction Ce4+ + e = Ce3+ (E0T = 1.44 v) are given (Ce4+ is the acceptor) by


Also, for the half‐reaction Fe3+ + e = Fe2+ (E0S = 0.68 v) (Fe2+ is the donor)


The equilibrium constant of the overall reaction is expressed as


From Eq. (22) taking into account that [X] = e, and that a = b = 1


From the Nernst equations applied to the two half‐directions involved


and taking into account Eqs. (28) and (29) are reached


The value of peeq is calculated, Eq. (32), from the expression


whereby the potential at the point of equivalence is given by the expressions


Note that


which allows us to calculate the titration curve (Figure 1)

Figure 1.

Titration curve of Fe2+ with Ce4+ in acid medium (H2SO4 1 M).


or the graph of the titration error (Figure 2)

Figure 2.

Titration error diagram ΔT = fpX).


The required calculations are detailed (from 0.62 to 0.82 v) in Table 1. Note that when T = 0.5, the potential value, E = 0.68 v, coincides with the normal potential of the Fe3+/Fe2+. Continuing the calculations would prove that when T = 2, E = 1.44 v, normal potential value of the Ce4+/Ce3+ pair.

Ce(IV) + Fe(II) = Ce(III) + Fe(III)
E0T=1.44neT=1nHT=0f(pHT)= 0pH=−0.31M H2SO4
E0S=0.68neS=1nHS=0f(pHS)= 0E0T′=1.44
Ws[X]^(ab)/2Wsine hΔTTEXpXΔpXLOG (ABS(ΔT))

Table 1.

Titration curve of Fe(II) with Ce(IV): hyperbolic sine method.

4.2. Tl+ titration curve with Ce4+

For the system Tl3+ + 2e = Tl+ (E0S = 1.25 v) the equilibrium constant Ks and pe


The overall reaction and its equilibrium constant are expressed as


Applying the Nernst equation to the half‐reaction Tl3+/Tl+


The values of KS (Eq. 45), KT (Eq. 35), pe (Eq. 34) and peeq (Eq. 46) can be replaced in Eq. (22), taking into account that a = 2 and b = 1


The titration curve E = f (T) is shown in Figure 3, together with those corresponding to other half‐reactions exchanging a single electron, VO2−/VO2+ (E0 = 1.001 v), NO3/NO2 (E0 = 0.80 v), and Fe 3+/Fe2+ (E0 = 0.68 v).

Figure 3.

Titration curves of several redox systems with Ce4+ as a titrant.

4.3. Fe2+ + Tl+ titration curve with Ce4+

In this particular case, the total ΔT function (or total T) is additive, i.e., ΔT is the sum of the values of ΔT (T) corresponding to the individual titration of Fe+2 with Ce+4 and Tl+ with Ce+4. The reactions and equations involved have been previously described in Sections 4.1. and 4.2.. Thus, in this case we should only sum the values given by Eqs. 31 (Fe+2 with Ce+4) and 48 (Tl+ with Ce+4). The corresponding titration curve of a mixture of Fe2+ and Tl+ with Ce4+ calculated in this way is shown in Figure 4.

Figure 4.

Titration curve of a mixture of Fe2+ and Tl+ with Ce4+.

4.4. Fe2+ titration curve with MnO4 as titrant

For the half‐reaction MnO4− + 5 e + 8 H+ = Mn2+ + 4 H2O (E0T = 1.51 v)


and thus following the previous procedure


The terms of Eq. (54) are conditional (pH‐dependent). In the equivalence point


The expression for the valuation error will be given by (a = 1, b = 5)


The curve and the titration error are shown in Figures 5 and 6, respectively, at different pH values. Part of the necessary calculations, at pH = 0, is shown in Table 2. When T = 2 and pH = 0, E = 1.51 v, normal MnO4/Mn2+ system potential.

Figure 5.

Titration curve of Fe2+ with MnO4, at different pH values.

Figure 6.

Titration error diagram ΔT = f(E) at different pH values.

pH=0MnO4 + 5 Fe(II) + 8 H+ = Mn(II) + 5 Fe(III) + 4 H2O
E0T=1.51neT=5nHT=8f(pHT)= 0E0T′=1.51
E0S=0.77neS=1nHS=0f(pHS)= 0E0S′=0.77
Ws[X]^(ab)/2Wsine hΔTTEXpXΔpXLOG(ABS(ΔpX))

Table 2.

Titration curve of Fe(II) with MnO4(−).

4.5. Titration curve of V2+ with MnO4

Oxido‐reductor systems involving the vanadium oxidation states 2+, 3+, 4+, and 5+ are shown in Table 3. The V2+ ion undergoes various successive ionizations at 3+, 4+, and 5+ when MnO4 is added, being appreciated in the curve (Figure 7), the three corresponding jumps. The total ΔT function (or total T) is additive. The titration curve is the sum of the contributions of each individual reaction

Figure 7.

Titration curve of V2+ with MnO4, at pH 0, 1 and 2.

E0TMnO4 + 8 H+ + 5e = Mn(II) + 4 H2O
E032V(III) + e = V(II)
E043VO(II) + 2 H+ + e = V(III) + H2O
E054VO2(I)+ 2 H+ + e = VO(II) + H2OpH=0
E0T=1.51neT=5nHT=8f(pHT)= 0E0T'=1.51
E0S32=−0.255neS32=1nHS32=0f(pHS32)= 0E0S32'=−0.255
E0S43=0.337neS43=1nHS43=2f(pHS43)= 0E0S43'=0.337
E0S54=1.001neS54=1nH54=2f(pH54)= 0E0S54'=1.001

Table 3.

Head of the spreadsheet in EXCEL of the titration curve of V(II) with MnO4.


The head of the spreadsheet in EXCEL with all the necessary elements to carry out the numerical calculations applying the equations of the type of Eq. (22) is shown in Table 3. From the corresponding EXCEL sheet we have extracted a few columns, those corresponding to the values of E, pX, T1, T2, T3 and TSUM (1, 2, 3 refer to 32, 43, and 54, respectively, in Eqs. 57 and 58, that is, the oxidations of V2+ to V3+, from this to VO2+ and from this to VO2+), which are shown in Table 4.


Table 4.

Titration curves of V(II) with MnO4.

Potential values varying from 0.05 in 0.05 units have been varied in this table in order to cover the entire valuation curve. Figure 7 has been drawn instead with potential variations of only 0.01 v.


5. Criterion for the quantitative titration and the influence of side‐reactions

The principle of redox titrations is that the solution of a reducing agent is titrated with a solution of an oxidizing agent (or vice versa)


At the equivalence point


The criterion of the quantitative titration can be deduced if we consider that the substance to be determined must be oxidized (or reduced) during the titration to an extent of 99.9% [23]. This means that the amount of determinant remaining untitrated at the equivalence point should not exceed 0.1% of that originally present, i.e.,




From Eqs. (60) and (63) also we obtain


If both oxidation‐reduction systems in the titration involve two electrons, the difference between the log K values must be greater than 6.

In addition, if the oxidized or reduced product present in the solution containing the redox system takes part in a side‐reaction, and the equilibrium position of this reaction can be kept constant, by maintaining suitable experimental conditions, the conditional oxidation‐reduction constant, K', can be deduced and used similarly to those used in complex chemistry


[Red'] and [Ox'] are analytical concentrations without any respect to side‐reactions. The connection between the conditional and real constants is the following one


where αRed(A) and αOx(B) are the side‐reaction functions, and A and B denote the substances reacting with the reduced and oxidized substance, respectively.


[A] and [B] are concentrations of the species reacting with the reduced and oxidized form, respectively, β’s and β*’s are complex products or protonization constants products.

In all calculations concentration constants can be used if they are corrected to the corresponding ionic strength.

In practice, the most important side‐reactions are complex formation and protonation. The oxidized and reduced form of a metal ion may form complexes of different stabilities with the complexing ligand L.

So, if the criterion of the quantitative determination is not fulfilled by a suitable pH change or by the use of a complexing agent that shifts the values of the conditional constants, the titration may be realized. For example, according to Vydra and Pribil [101], cobalt (II) can be titrated with iron (III) ions if 1,10‐phenanthroline is added to the solution, and the pH is adjusted to 3 even though KCo >>> KFe.

On the other hand, if another component present in the solution has similar oxidizing or reducing properties, then interfering species can be masked, so that the conditional redox constant of the interfering system is changed to such an extent that it no longer interferes with the main reaction.

5.1. Practical examples

(1) Calculate the pH necessary for the accurate direct titration of potassium hexacyanoferrate(III) with ascorbic acid, given that log KFe(CN)6 = 6.1; the protonation constants of hexacyanoferrate(II) are log K1 = 4.17, log K2 = 2.22, log K3 < 1, log K4 < 1; the logarithms of all the protonation constants of hexacyanoferrate(III) are >1. The equilibrium constant of the dehydroascorbic acid‐ascorbinate redox system is log KA = −2.5; the protonation constants of the ascorbinate ion are log K1 = 11.56 and 4.17.

The criterion for the feasibility of the titration, according to Eq. (64) is:


From the protonation constants of hexacyanoferrate(II), if the pH > 5.5, then αFe(CN)6(H) = 1 and


Therefore 2 x 6.1 − logKA > 9; or logKA < 3.2. If the pH = 6, then αA(H) = 1 + 10−6 x 1011.57 + 10−12 x 1015.73 = 105.58


Thus, if the pH > 6, the titration can be performed with adequate accuracy.

(2) Calculate the [L] maximum for the accurate direct titration of Ce+4 with Fe2+, being L the organic complexant (acetylacetone) present in the solution and given that log KT = 24 and log KS = 11.3 (see Eqs. (28)(30)); the global constants of Fe‐(L)(II) are log β1 = 5.07; log β2 = 8.67.

The reactions involved would be


The criterion for the feasibility of the titration, according to Eq. (64) is


Thus, if the [L] < 0.1 M, the titration can be performed with adequate accuracy.

(3) Calculate the [SO4−2] maximum present in the solution for the accurate direct titration of Ce4+ with Tl+, given that log KT = 24 and log KS = 41.6 (see Eqs. (28), (42), (43)); the global constants of Ce‐SO4(IV) are log β1 = 3.5; log β2 = 8.0; log β3 = 10.4.

The criterion for the feasibility of the titration, according to (64) is:


Thus, [SO4−2] should be < 0.39 M, to perform the titrimetry with accuracy.


6. Final comments

As a matter of fact redox titrations play a prominent role in volumetric analysis of redox actives species. A systematic study of the bibliography is undertaken in order to ascertain the state of the art concerning to redox titration curves. A method for the determination of titration error in donor/acceptor titrations of displacement and electronic transference reactions has been devised; a hyperbolic sine expression being derived for the titration error, applicable to symmetrical reactions (no polynuclear species being involved in one side of a half‐reaction). The hyperbolic sine expression developed is compact and allows calculating the entire titration curve without piecemeal approximations, as usually occurs by dividing the titration curve in three parts: before, in, and beyond the equivalence point regions.

The method has been applied to some experimental systems characteristics of redox titration reactions. The method proposed is also applicable to mixtures of analytes, e.g., Fe(II) + Tl(I), as well as to multistep redox titrations, e.g., V(II)/V(III)/V(IV)/V(V) system. The forms of the redox titration curves are independent of the concentrations. However, when the concentrations involved are very low the responses of the electrodes are not appropriate. All calculations involved have been checked with the method proposed by “de Levie” [62] for the sake of comparison, and no differences were found in the numerical values obtained by both methods. A diagram for the titration error in function of the difference between the end and equivalence point (pX) is drawn in order to facilitate the graphical calculation of titration error.

Automatic titrators enable recording automatically the change with potential (E) or pH in titre during a given titration. The accuracy of the measurements can increase with the help of on‐line microcomputer for the control and data acquisition, allowing among the possibility for curve‐smoothing and differentiation.

The extension of the method to nonhomogeneous systems of the type Cr2O72−/Cr3+, I2/I, or S4O62−/S2O32− remains a challenge, this being a complex problem involving a complete reformulation of the presented equations, which implies a higher level of difficulty.

At the end of the chapter an appendix including a detailed study of the propagation of systematic and random errors on redox titration error has been carried out and spite of the complex expression obtained first on differentiation, the final expressions formulated were very compact. This topic is still under study and it will be dealt in further calculus.



Error analysis

The exact calculation of standard deviations of nonlinear function of variables that are subject to error is generally a problem of great mathematical complexity. A linearization based on a Taylor expansion of the nonlinear portion of the expansion allows to obtain approximate estimates of standard deviations [102]; this approximation is quite adequate for most practical applications.

ΔT is a function of several variables, i.e. pX, log KS, log KT all independent of each other. The systematic error present in pX, log KS, and log KT, are propagated to give an overall systematic error in a calculate quantity


provided the errors Esys(pX), Esys(log KS), Esys(log KT) are small enough for higher order derivatives to be discarded.

For random error [103], the variance of ΔT can be calculated according to the propagation of variance


where spX2, slogKs2, and slogKT2 are the variances of the components pX, log KS, and log KT, respectively. The partial derivatives are taken, as before, as values equal or closest to the measured values.

For any measurement the total absolute error EabsT) is related to the different types of error present by


where ΔT is the value of the measurement, Δτ the true value, Erand the random error, Esys is the systematic error, and Ebl is the error due to blunders.

In any case, in order to know the proper error, it is necessary to know the standard deviation of the experimentally measured quantities, i.e., Esys(i) and si.

For the sake of convenience Eq. (22) may be put in the form




Make now




in order to may differentiate easily ΔT against d[X]. Thus


and we get for the derivative of a product


The derivative of u will be given by

u=A(p[ X ]p1(1+[ X ]aKS)KSa[ X ]a1[ X ]p(1+[ X ]aKS)2)=A(p[ X ]p1(1+[ X ]aKS)KSa[ X ]a[ X ]p1(1+[ X ]aKS)2)=A[ X ]p11+[ X ]aKS(paKS[ X ]a1+[ X ]aKS)E90

On the other hand


Taking into account Eqs. (87)(91), we get


and then


By multiplying through [X] and taking into account Eq. (31) we get


and so


Eq. (22) may be presented in the form


Differentiation of Eq. (96) with respect to KS gives


On the other hand


By combining Eqs. (97) and (99) we obtain


which on rearranging gives


Differentiating now ΔT against log KS leads to


In order to differentiate ΔT against KT we put (from Eq. (22))


From Eqs. (18), (20), and (34), we get


By differentiating 1/√KT against KT in Eq. (52) and combining the resulting expression with Eq. (53) we get


As before with the case of [X] and KS, we may express the partial derivative of ΔT against log KT as a function of the derivative against KT and then


By combining Eqs. (28), (44), (51), and (55), we get for the standard deviation of systematic error


In the same way, from Eqs. (29), (44), (51), and (55), we may obtain the variance of the random error


Note that in spite that relative complex expressions are involved in the required differentiations carried out with the purpose to propagate the systematic and random errors implied in the donor/acceptor titration, the algebra involved is simple, and the final expressions obtained are compact.


  1. 1. Felber H, Rezzonico S, Máriássy M. Titrimetry at a metrological level. Metrologia. 2003;40(5):249–254
  2. 2. Asuero AG, Michalowski T. Comprehensive formulation of titration curves for complex acid‐base systems and its analytical implications. Critical Reviews in Analytical Chemistry. 2011;41(2):151–187
  3. 3. Michałowski T, Asuero AG. New approaches in modeling carbonate alkalinity. Critical Reviews in Analytical Chemistry. 2012;42(3):220–244
  4. 4. Michałowski T, Toporek M, Michałowska‐Kaczmarczyk AM, Asuero AG. New trends in studies on electrolytic redox systems. Electrochimica Acta. 2013;109:519–531
  5. 5. Winkler‐Oswatitsch R, Mangen M. The art of titration: from classical end points to modern differential and dynamics analysis. Angewandte Chemie International Edition. 1979;18(1):20–49
  6. 6. Zhang CL, Narusawa Y. A new titration method based on concentration‐variable patterns‐principles and applications to acid‐base and redox titrations. Bulletin of the Chemical Society of Japan. 1997;70(3):593–600
  7. 7. Terra J, Rossi AV. Sobre o desenvolvimento da análise volumétrica e algumas aplicações antais. Química Nova. 2005;28(1):166–171
  8. 8. Zhan X, Li C, Li Z, Yang X, Zhong S, Yi T. Highly accurate nephelometric titrimetry. Journal of Pharmaceutical Sciences. 2004;93(2):441–448
  9. 9. Van Hulle SWH, De Meyer S, Vermeiren TJL, Vergote A, Hogie J, Dejans P. Practical application and statistical analysis of titrimetric monitoring of water and sludge samples. Water SA. 2009;35(3):329–333
  10. 10. King B. Review of the potential of titrimetry as a primary method. Metrologia. 1997;34(1):77–82
  11. 11. Quinn TJ. Primary methods of measurement and primary standards. Metrologia. 1997;34(1):61–65
  12. 12. Kolthoff IM, Furman NH. Potentiometric Titrations. 2nd ed. New York: Wiley; 1947. pp. 45–60. Chapter 3
  13. 13. Goldman JA. Oxidation reduction equilibria and titration curves. In: Kolthoff IM, Elving PJ, editors. Treatise on Analytical Chemistry, Part 1, Theory and Practice. Volume 3, Section D, Solution Equilibria and Chemistry (Continued). New York: Wiley; 1983. pp. 1–79. Chapter 24
  14. 14. Kolthoff IM. Die oxydopoteniiometrischc Bestimmung von Ferro. Chemisch weekblad. 1919;16:408
  15. 15. Beck CM. Toward a revival of classical analysis. Metrologia. 1997;34(1):19–30
  16. 16. Beck CM. Classical analysis. A look of the past, present and future. Analytical Chemistry. 1994;66(4):224A‐230A, 233A, 239A
  17. 17. Sillen LG. Graphic Presentation of Equilibrium Data. New York: Interscience; 1959
  18. 18. Sillen LG. Redox diagrams. Journal of Chemical Education. 1952;29(12):600–608
  19. 19. Budevski O. Foundations of Chemical Analysis. Chichester, UK: Ellis Horwood; 1979
  20. 20. Budevski O. Graphical method for construction of titration curves. In: Ringbom A, Wänninen E, editors. Essays on Analytical Chemistry: In Memory of Professor Ringbom A, and Wänninen E. Oxford: Pergamon Press; 1977. pp. 169–174
  21. 21. de Levie R. Redox buffer strength. Journal of Chemical Education. 1999;76(4):574–577
  22. 22. Tabbut FD. Titration curves from logarithmic concentration diagrams. Journal of Chemical Education. 1966;43(5):245–249
  23. 23. Inczédy J. Analytical Applications of Complex Equilibria. New York: Wiley; 1976
  24. 24. Inczédy J. Representation of titrations errors in logarithmic diagrams. Journal of Chemical Education. 1970;47(11):769–772
  25. 25. Bjerrum N. Die Theorie der alkalimetrischen und azidimetrischen Titrierungen; Samlung chemischer und chemisch‐technischer Vorträge Band XXI, 1–128. Stuttgart: Verlag von Ferdinand Enke; 1914. pp. 91, 94, 97, 103
  26. 26. de Levie R. Principles of Quantitative Chemical Analysis. New York: McGraw‐Hill; 1997
  27. 27. Kahlert H, Scholz F. Acid‐Base Diagrams. Heidelberg: Springer; 2013
  28. 28. Maccà C, Bombi GG. A graphical approach to redox titrations. Fresenius’ Journal of Analytical Chemistry. 1986;324(1):52–57
  29. 29. Vicente‐Pérez S. Química de las Disoluciones. Diagramas y Cálculos Gráficos. UNED: Madrid; 1997. p. 424
  30. 30. Vicente‐Perez S, Durand JS, Montes F. Titration of weak reductants (or oxidants) with weak oxidants (or reductants)—The study with logarithmic diagrams. Anales de Quimica (Madrid). 1992;88(7–8):688–693
  31. 31. Vicente‐Perez S, Losada J, Espinosa A. E‐pC diagrams for monoelectronic‐redox equilibrium system—Applications. Anales de Quimica (Madrid). 1990;86(7):751–761
  32. 32. de Levie R. Advanced Excel for Scientific Data Analysis. 3rd ed. New York: Oxford University Press; 2012
  33. 33. de Levie R. How to Use Excel in Analytical Chemistry and in General Scientific Data Analysis. Cambridge: Cambridge University Press; 2001
  34. 34. de Levie R. Explicit expressions of the general form of the titration curve in terms of concentration. Writing a single‐closed form expression for the titration curve for a variety of titrations without using approximations or segmentation. The Journal of Chemical Education. 1993;70(3):209–217
  35. 35. Anfalt T, Jagner D. The precision and accuracy of some current methods for potentiometric end‐point determination with reference to a computer‐calculated titration curve. Analytica Chimica Acta. 1971;57:165–176
  36. 36. Yongnian N, Ling J. Application and advancement of titration assisted with mathematical methods. Chinese Journal of Analytical Chemistry. 1996;24(10):1219–1226
  37. 37. Gran G. Determination of the equivalence point in potentiometric titration. Acta Chemica Scandinavica. 1950;4:559–577
  38. 38. Gran G. Determination of the equivalence point in potentiometric titrations. Part II. Analyst. 1952;67(11):661–671
  39. 39. Turyan Y, Ivanova GV, Pokhodzei VF. The gran method in redox potentiometric titrations. Journal of Analytical Chemistry. 1992;47(4):527–533
  40. 40. Maccà C, Bombi GG. Linearity range of gran plots for the end‐point in potentiometric titrations. Analyst. 1989;114:463–470
  41. 41. Johansson A. Choice of chemical conditions in order to obtain linear titration curves in potentiometry. Talanta. 1975;22(12):945–954
  42. 42. Dyrssen D, Jagner D, Wengelin F. Computer Calculation of Ionic Equilibria and Titration Procedures. Stockholm: Almqvist and Wiksell; 1968
  43. 43. Michalowski T, Kupiec K, Rymanowski M. Numerical analysis of the gran methods. A comparative study. Analytica Chimica Acta. 2008;606(2):172–183
  44. 44. Ponikvar M, Michalowski T, Kupiec K, Wybraniec S, Rymanowski M. Experimental verification of the modified gran methods applicable to redox systems. Analytica Chimica Acta. 2008;628(2):181–189
  45. 45. Michalowski T, Baterowicz A, Madel A, Kochana J. Extended gran method and its applications for the simultaneous determination of Fe(II) and Fe(III). Analytica Chimica Acta. 2001;442(2):287–293
  46. 46. Zhang C, Fu Y, Long C-Y. The discussion of the calculation about stoichiometric point potential for redox-titration. Journal of Dezhou University. 2003;04.
  47. 47. Maryanov BM. Linear regression analysis of potentiometric titration data for asymmetric redox titrations. Journal of Analytical Chemistry. 1997;52(6):508–513
  48. 48. Maryanov BM. Processing of potentiometric redox titration data by regression‐analysis with reduction of the number of variables. Journal of Analytical Chemistry. 1992;47(3):396–398
  49. 49. Krotopov VA. Approximation of redox titration curves by logarithmic functions. Journal of Analytical Chemistry. 1998;53(8):701–703
  50. 50. Krotopov VA. Approximation of potentiometric titration curves by logarithmic functions: Prediction of random errors in titration parameters. Journal of Analytical Chemistry. 2000;51(5):449–453
  51. 51. Krotopov VA. Approximation of potentiometric titration curves by logarithmic functions. Factors affecting the accuracy of redox titration. Journal of Analytical Chemistry. 2000;55(2):60–164
  52. 52. Meites L, Fanelli N. Factors affecting the precision of a new method for determining the reduced and oxidized forms of a redox couple by a single potentiometric titration. Analytica Chimica Acta. 1987;194:151–162
  53. 53. Federov AA, Shmata TS. Computer‐assisted calculation and graphical presentation of titration curves. Journal of Analytical Chemistry. 2004;59(5):402–406
  54. 54. Krotopov VA. Numerical simulation of redox titrations. Journal of Analytical Chemistry. 1993;48(2):167
  55. 55. Huang X-Z, Chen H-L. The computer simulation of redox titration. Journal of Nanyang Institute of Technology. 2011;2.
  56. 56. Allnutt MI. The use of conjugate charts in transfer reactions: a unified approach. The Journal of Chemical Education. 2007;84(10):1659–1662
  57. 57. Desbarres J, Bauer D. Simulation des courbes de dosage potentiomètriques par emploi d’une equation universelle. Dosage par echange d’une seule particule. Talanta. 1975;22(10–11):877–879
  58. 58. Maccà C. The formulation of the electron and proton balance equations for solving complicated equilibrium problems in redox titrations. Fresenius’ Journal of Analytical Chemistry. 1997;357(2):229–232
  59. 59. Pereira CF, Alcalde M, Villegas R, Vale J. Predominance diagrams, a useful tool for the correlation of the precipitation‐solubility equilibria with other ionic equilibria. Journal of Chemical Education. 2007;84(3):520–525
  60. 60. Charlot M. Les Réactions Chimiques en Solution Aqueuse. Paris: Elsevier‐Masson; 1997
  61. 61. Charlot G. Les Réactions Chimiques en Solution Aqueuse et Caractérisation des Ions. 7th ed. Paris: Masson; 1983
  62. 62. de Levie R. A simple expression for the redox titration curve. Journal of Chemical Education. 1992;323(1–2):347–355
  63. 63. Elenkova NG. General treatment of conjugated acid‐base, redox and complexation equilibria. Talanta. 1980;27(9):699–704
  64. 64. Chaston SHH. Double scales for equilibria. Journal of Chemical Education. 1979;56(1): 24–26
  65. 65. Pacer RA. Conjugate acid‐base and redox theory. Journal of Chemical Education. 1973; 50(3):178–180
  66. 66. Hazlehurst TH. Acid‐base reactions. Their analogy to oxidation‐reduction reactions in solution. Journal of Chemical Education. 1940;17(10):466–468
  67. 67. Rosset R, Bauer D, Desbarres J. Chimie Analytique des Solutions et Informatique. 2nd ed. Paris: Masson; 1991.
  68. 68. Bishop E. Some theoretical considerations in analytical chemistry. V. A simple method of calculating acid‐base titration errors. Analytica Chimica Acta. 1960;22:205–213
  69. 69. Butler JN. Calculating titration errors. Journal of Chemical Education. 1963;40(2):66–69
  70. 70. Butcher J and Fernando QJ. Theoretical error in acid‐base titrations. Journal of Chemical Education. 1966;43(10):546–550
  71. 71. Gonzalez GG, Jimenez AM, Asuero AG. Titration errors in acid‐base titrations. Microchemical Journal. 1990;41(1):113–120
  72. 72. Hulanicki A. Reactions of Acids and Bases in Analytical Chemistry. Chichester: Ellis Horwood; 1987.
  73. 73. Carr PW. Intrinsic end‐points errors in precipitation titrations with ion selective electrodes. Analytical Chemistry. 1971;43(3):525–430
  74. 74. Christopherson HL. Theoretical titration error in potentiometric asymmetrical precipitation titrations. Journal of Chemical Education. 1963;40(2):63–65
  75. 75. Fernando Q, Butcher J. Calculation of titration error in precipitation titrations. A graphical method. Journal of Chemical Education. 1967;44(3):166–168
  76. 76. Goldman JA. Potentiometric titration curves for single ion combination reactions. Journal of Chemical Education. 1963;40(10):519–522
  77. 77. Schultz FA. Titration errors and curve shapes in potentiometric titrations employing ion‐selective indicator electrodes. Analytical Chemistry. 1971;43(4):502–508
  78. 78. Kosonem PO and Hakoila EJ. The titration error in potentiometric precipittion titration. Talanta. 1975;22(12):1045–1047
  79. 79. Ringbom A. Complexation in Analytical Chemistry. New York: Interscience; 1963
  80. 80. Johansson E, Olin A. A complexometric method for the determination of small amounts of an alkaline earth metal in mixture with other metals. Talanta. 1980;27(2):165–168
  81. 81. Hulanicki A and Głab S. Total systematic error in redox titrations with visual indicators. I: Basic principles. Talanta. 1975;22(4–5):363–370
  82. 82. Perez‐Ruiz T, Hernández Córdoba M, Martínez‐Lozano C, Sánchez‐Pedreño C. Nuevas consideraciones teóricas sobre las valoraciones de oxido‐reducción. Química Analítica. 1984;3:138–146
  83. 83. Edens GJ. Redox titration of antioxidant mixtures with n‐bromosuccinimide as titrant: Analysis by non‐linear least‐squares with novel weighting function. Analytical Chemistry. 2005;21(11):1349–1354
  84. 84. Umetsu K, Itabashi H, Saloh K, Kawashima T. Effect of ligands on the redox reactions of metal ions and the use of a ligand buffer for improving end‐point detection in the potentiometric titration of vanadium(V) with iron(II). Analytical Chemistry. 1991;7(1):115–118
  85. 85. Butler JN. Ionic Equilibrium. Solubility and pH Calculations. New York: Wiley; 1998
  86. 86. Charlot G, Gauguin R. Les Méthodes d’Analyse des Réactions en Solution. Paris: Masson; 1951
  87. 87. Garric M. Chimie Générale. Vol. 1, Paris: Dunod; 1970. pp. 439–442
  88. 88. King DW. A general approach for calculating speciation and posing capacity of redox systems with multiple oxidation states: Application to redox titrations and the generation of pε‐pH diagrams. The Journal of Chemical Education. 2002;79(9):1135–1140
  89. 89. Monnier D, Haerdi W, Buffle Y, Ruscony Y. Chimie Analytique. Application aux Méthodes Instrumentales, Radiochimiques et a la Chimie de l’Environnent.Genève: Georg; 1979
  90. 90. Monnier D, Haerdi W, Ruscony Y. Chimie Analytique. Analyse Qualitative Minérale. Eléments de Radiochimie.Genève: Georg Genève; 1968. pp. 118–132. Chapter IX
  91. 91. Stumm W, Morgan JJ. Oxidation and Reduction. Equilibria and Microbial Mediation. In Aquatic Chemistry. An introduction emphasizing chemical equilibria in natural waters. Edited by Stumm W and Morgan JJ. In 3rd ed. New York: Wiley; 1996. pp. 425?515. Chapter 7
  92. 92. Tremillon B. Reactions in Solution: An Applied Analytical Approach. New York: Wiley; 1997
  93. 93. Asuero AG. Buffer capacity of a polyprotic acid: first derivative of the buffer capacity and pKa values of single and overlapping equilibria. Critical Reviews in Analytical Chemistry. 2007;37(4):269–301
  94. 94. Michalowski T, Lesiak A. Formulation of generalized equations for redox titration curves. Chemia Analityczna (Warsaw). 1994;39:623–637
  95. 95. Michalowsi T, Wajda N, Janecki D. A unified approach to electrolytic systems. Chemia Analityczna (Warsaw). 1996;41:667–685
  96. 96. Michałowski T. The generalized approach to electrolytic systems: I. Physicochemical and analytical implications. Critical Reviews in Analytical Chemistry. 2010;40(1):2–16
  97. 97. Michałowski T, Michałowska‐Kaczmarczyk AM, Toporek M. Formulation of general criterion distinguishing between non‐redox and redox systems. Electrochimica Acta. 2013;112:199–211
  98. 98. Michalowka‐Kaczmarcyk AM, Asuero AG, Toporek M, Michalowski T. “Why not stoichiometry” versus “stoichiometry—why not?” Part II: Gates in context with redox systems. Critical Reviews in Analytical Chemistry. 2015;45(3):241–269
  99. 99. Toporek M, Michalowka‐Kaczmarcyk AM, Michalowski T. Symproportionation versus disproportionation in bromine redox systems. Electrochimica Acta. 2015;171:176–186
  100. 100. Michalowka‐Kaczmarcyk AM, Toporek M, Michalowski T. Speciation diagrmas in dynamic iodide + dichromate system. Electrochimica Acta. 2015;155:217–227
  101. 101. Vydra F, Přibil R. New redox systems-IV1 1 Part III-see reference 4. Oxidation of cobaltIII with ironII chloride in 2,20-bipyridyl. Talanta. 1961;8:124
  102. 102. Still ER. Statistical adjustment of parameters for potentiometric titration data. Talanta. 1980;27(7):573–582
  103. 103. Asuero AG, Gonzalez G, de Pablos F, Gomez-Ariza JL. Determination of the optimum working range in spectrophotometric procedures. Talanta. 1988;35(7):531–537

Written By

Julia Martín, Laura Ortega Estévez and Agustín G. Asuero

Submitted: December 9th, 2016 Reviewed: March 23rd, 2017 Published: September 6th, 2017