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Principles of Titrimetric Analyses According to Generalized Approach to Electrolytic Systems (GATES)

Written By

Anna Maria Michałowska‐Kaczmarczyk, Aneta Spórna‐Kucab and Tadeusz Michałowski

Submitted: December 15th, 2016 Reviewed: April 14th, 2017 Published: September 27th, 2017

DOI: 10.5772/intechopen.69248

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The generalized equivalent mass (GEM) concept, based on firm algebraic foundations of the generalized approach to electrolytic systems (GATES) is considered, and put against the equivalent “weight” concept, based on a “fragile” stoichiometric reaction notation, still advocated by IUPAC. The GEM is formulated a priori, with no relevance to a stoichiometry. GEM is formulated in unified manner, and referred to systems of any degree of complexity, with special emphasis put on redox systems, where generalized electron balance (GEB) is involved. GEM is formulated on the basis of all attainable (and preselected) physicochemical knowledge on the system in question, and resolved with use of iterative computer programs. It is possible to calculate coordinates of the end points, taken from the vicinity of equivalence point. This way, one can choose (among others) a proper indicator and the most appropriate (from analytical viewpoint) color change of the indicator. Some interpolative and extrapolative methods of equivalence volume Veq determination are recalled and discussed. The GATES realized for GEM purposes provides the basis for optimization of analytical procedures a priori. The GATES procedure realized for GEM purposes enables to foresee and optimize new analytical methods, or modify, improve, and optimize old analytical methods.


  • equilibrium analysis
  • mathematical modeling
  • redox titration curves
  • equivalence volume
  • Gran methods

1. Introductory remarks

Titrimetry reckons to the oldest analytical methods, still widely used because of high precision, accuracy, convenience, and affordability [1]. Nowadays, according to Comité Consultatif pour la Quantité de la Matière (CCQM) opinion [2], it is considered as one of the primary methods of analysis, i.e., it fulfills the demands of the highest metrological qualities. Titration is then perceived as a very simple and reliable technique, applied in different areas of chemical analysis. A physical chemist may perform a titration in order to determine equilibrium constants, whereas an analytical chemist performs a titration in order to determine the concentration of one or several components in a sample.

In a typical titration, V0 mL of titrand (D), containing the analyte A of an unknown (in principle) concentration C0 is titrated with V mL of titrant (T) containing the reagent B (C); V is the total volume of T added into D from the very start up to a given point of the titration, where total volume of D + T mixture is V0 + V, if the volume additivity condition is fulfilled. Symbolically, the titration T → D in such systems will be denoted as B(C,V) → A(C0,V0). Potentiometric acid‐base pH titrations are usually carried out by using combined (glass + reference) electrode, responding to hydrogen‐ion activity rather than hydrogen‐ion concentration. Potentiometric titrations in redox systems are made with use of redox indicator electrodes (RIE), e.g. combined (Pt + reference) electrode [35]. For detection of specific ions in a mixture, ion-selective electrodes (ISE) are used [5]. The degree of advancement of the reaction between B and A is the fraction titrated [6], named also as the degree of titration, and expressed as the quotient Φ = nB/nA of the numbers of mmoles: nB = C·V of B and nA = C0·V0 of A, i.e.,


We refer here to visual, pH, and potentiometric (E) titrations. The functional relationships between potential E or pH of a solution versus V or Φ, i.e., E = E(V) or E = E(Φ) and pH = pH(V) or pH = pH(Φ) functions, are expressed by continuous plots named as the related titration curves. The Φ provides a kind of normalization in visual presentation of the appropriate system. In the simplest case of acid‐base systems, it is much easier to formulate the functional relationship Φ = Φ(pH), not pH = pH(Φ). In particular, the expression for Φ depends on the composition of D and T, see Appendix.

The detailed considerations in this chapter are based on principles of the generalized approach to electrolytic systems (GATES), formulated by Michałowski [9] and presented recently in a series of papers, related to redox [726] and nonredox systems [2732], in aqueous and in mixed‐solvent media [3337]. The closed system separated from its environment by diathermal walls secure a heat exchange between the system and its environment, and realize dynamic processes in quasistatic manner, under isothermal conditions.

The mathematical description of electrolytic nonredox systems within GATES is based on general rules of charge and elements conservation. Nonredox systems are formulated with use of charge (ChB) and concentration balances f(Yg), for elements/cores Yg ≠ H, O. The description of redox systems is complemented by generalized electron balance (GEB) concept, discovered by Michałowski as the Approach I to GEB (1992) and the Approach II to GEB (2006); GEB is considered as a law of a matter conservation, as the law of nature [7, 9, 11, 13, 25].

Formulation of redox systems according to GATES principles is denoted as GATES/GEB. Within the Approach II to GEB, based on linear combination 2·f(O) – f(H) of the balances: f(H) for H and f(O) for O, the prior knowledge of oxidation degrees of all elements constituting the system is not needed; oxidants and reductants are not indicated. Moreover, the linear independency or dependency of 2·f(O) – f(H) from other balances: ChB and f(Yg), is the general criterion distinguishing between redox and nonredox systems. Concentrations of the species within the balances are interrelated in complete set of equations for equilibrium constants, formulated according to mass action law principles. The GATES and GATES/GEB in particular, provide the best possible tool applicable for thermodynamic resolution of electrolytic systems of any degree of complexity, with the possibility of application of all physicochemical knowledge involved.

Several methods of equivalence volume (Veq) determination are also presented, in terms of the generalized equivalence mass (GEM) [8] concept, suggested by Michałowski (1979), with an emphasis put on the Gran methods and their modifications. The GEM concept has no relevance to a chemical reaction notation. Within GATES, the chemical reaction notation is only the basis to formulate the expression for the related equilibrium constant.


2. Formulation of generalized equivalent mass (GEM)

The main task of titration is the estimation of the equivalence volume, Veq, corresponding to the volume V = Veq of T, where the fraction titrated (1) assumes the value


In contradistinction to visual titrations, where the end volume Ve Veq is registered, all instrumental titrations aim, in principle, to obtain the Veq value on the basis of experimental data {(Vj, yj) | j = 1,…,N}, where y = pH, E for potentiometric methods of analysis. Referring to Eq. (1), we have

C0V0= 103mA/MA E3

where mA [g] and MA [g/mol] denote mass and molar mass of analyte (A), respectively. From Eqs. (1) and (3), we get

mA= 103CMAV/ΦE4

The value of the fraction V/Φ in Eq. (4), obtained from Eq. (1),


is constant during the titration. Particularly, at the end (e) and equivalence (eq) points, we have


The Ve [mL] value is the volume of T consumed up to the end (e) point, where the titration is terminated (ended). The Ve value is usually determined in visual titration, when a preassumed color (or color change) of D + T mixture is obtained. In a visual acid‐base titration, pHe value corresponds to the volume Ve (mL) of T added from the start for the titration and


is the Φ‐value related to the end point. From Eqs. (4) and (6), one obtains:


This does not mean that we may choose between the two formulas: (8a) and (8b), to calculate mA. Namely, Eq. (8a) cannot be applied for the evaluation of mA: Ve is known, but Φe unknown; calculation of Φe needs prior knowledge of C0 value; e.g., for the titration NaOH(C,V) → HCl(C0,V0), see Appendix, we have

Φe=CC0·C0αeC+αe where  α(Appendix), and  αe=α(pHe)E9

However, C0 is unknown before the titration; otherwise, the titration would be purposeless. The approximate pHe value is known in visual titration. Also Eq. (8b) is useless: the “round” Φeq value is known exactly, but Veq is unknown; Ve (not Veq) is determined in visual titrations.

Because Eqs. (8a) and (8b), appear to be useless, the third, approximate formula for mA, has to be applied, namely:

mA103CVeMA/Φeq= 103CVeRAeqE10

where Φeq is put for Φe in Eq. (8a), and


is named as the equivalent mass. The relative error in accuracy, resulting from this substitution, equals to

δ = (mA mA)/mA=mA/mA 1 =Ve/Veq 1 =Φe/Φeq 1E12

For Φe = Φeq, we get δ = 0 and mA’ = mA; thus Φe ≅ Φeq (i.e., Ve ≅ Veq) corresponds to mA’ ≅ mA. A conscious choice of an indicator and a pH‐range of its color change during the titration is possible on the basis of analysis of the related titration curve. From Eqs. (10) and (8b), we get

mA=mA/(1 + δ) =mA(1 δ + δ2)E13

3. Accuracy and precision

In everyday conversation, the terms “accuracy” and “precision” are often used interchangeably, but in science—and analytical chemistry, in particular—they have very specific, and different definitions [38].

Accuracy refers to how close a result of measurement, expressed, e.g., by concentration x (as an intensive variable), agrees with a known/true value x0 of x in a sample tested. In N repeated trials made on this sample, we obtain xj (j = 1, …, N) and then the mean value x¯ and variance s2 are obtained


The accuracy can be defined by the absolute value |x¯x0|, whereas precision is defined by standard deviation, s = (s2)1/2; the accuracy and precision are brought here into the same units.

Accuracy and precision are the terms of (nearly) equal importance (weights: 1 and (1 – 1/N) for the weighted sum of squares [39]) when involved in the relation [40, 41]


where xj — experimental (j = 1, …, N) and true (x0) values for x, x¯ — mean value, s2 — variance. The problem referred to accuracy and precision of different methods of Veq determination has been raised, e.g., in Refs. [42, 43].

Accuracy and precision of the results obtained from titrimetric analyses depend both on a nature of D + T system considered, and the method of Veq evaluation. Here, kinetics of chemical reactions and transportation phenomena are of paramount importance.


4. The E = E(Φ) and/or pH = pH(Φ) functions

Relatively simple, functional relationships for Φ = Φ(pH), ascribed to acid‐base D + T systems, are specified in elegant/compact form in Refs. [6, 27, 28, 30], see Appendix.

In acid‐base systems occurred in aqueous media, pH is a monotonic function of V or Φ. From the relation,


it results that the Φ = Φ(pH) and pH = pH(Φ) relationships are mutually interchangeable, C0V0/C > 0. The relation (16) can be extended on other plots.

Explicit formulation of functional relationships: Φ = Φ(pH) and E = E(Φ), is impossible in complex systems, where two or more different kinds (acid‐base, redox, complexation, precipitation, liquid‐liquid phase equilibria [44, 45]) of chemical reactions occur, sequentially or/and simultaneously [8]. The E values are referred to SHE scale.

Monotonicity of pH = pH(Φ) and/or E = E(Φ) is not a general property in electrolytic redox systems. In Figure 1, the monotonic growth of E = E(Φ), i.e., dE/dΦ > 0, is accompanied by monotonic growth of pH = pH(Φ), i.e., dpH/dΦ > 0 [20].

Figure 1.

The collected (A) E = E(Φ) and (B) pH = pH(Φ) curves plotted for D + T system KMnO4 (C) → FeSO4 (C0) + H2SO4 (C01) at V0 = 100, C0 = 0.01, C = 0.02, and different C01 values, indicated in Figures (B), (C), and (D) (in enlarged scales), before and after Φ = Φeq = 0.2.

In Figure 2, the monotonic drops of E = E(Φ), i.e., dE/dΦ < 0, are accompanied by nonmonotonic changes of pH = pH(Φ) [9, 46, 47].

Figure 2.

The theoretical plots of (A) E = E(Φ) and (B) pH = pH(Φ) functions for the D + T system, with KIO3 (C0 = 0.01) + HCl (C01 = 0.02) + H2SeO3 (CSe = 0.02) + HgCl2 (CHg) as D, and ascorbic acid C6H8O6 (C = 0.1) as T; V0 = 100, and (a) CHg = 0, (b) CHg = 0.07.

From inspection of Figure 2B, it results that the neighboring, quasi linear segments of the line (at CHg = 0) intersect at the equivalent points Φeq1 = 2.5 and Φeq2 = 3.0. So, it might seem that the pH titration is an alternative to the potentiometric titration method for the Veq detection. It should be noted, however, that there are small pH changes within the pH range, where the characteristics of glass electrode is nonlinear, and an extended calibration procedure of this electrode is required. The opportunities arising from potential E measurement are here incomparably higher, so the choice of potentiometric titration is obvious.

In Figure 3, the nonmonotonic changes of E = E(V) are accompanied by nonmonotonic changes of pH = pH(V) [16].

Figure 3.

The theoretical plots of (A) E = E(V) and (B) pH = pH(V) functions for the system with V0 = 100 mL of NaBr (C0 = 0.01) + Cl2 (C02) as D titrated with V mL of KBrO3 (C = 0.1) as T, at indicated (a, b, c) C02 values.

The unusual shape of the respective plots for E = E(Φ) and pH = pH(Φ) is shown in Figure 4 [13].

Figure 4.

The plots of (A) E = E(Φ), and (B) pH = pH(Φ) functions for the system HI (C = 0.1) → KIO3 (C0 = 0.01).

Other examples of the nonmonotonicity were presented in Refs. [7, 9, 4649]. The nonmonotonic pH versus V relationships were also stated in experimental pH titrations made in some binary‐solvent media [33]. Then, the Gran’s statement “all titration curves are monotonic” [50] is not true, in general.


5. Location of inflection and equivalence points

Some of the E = E(Φ) and/or pH = pH(Φ) (or E = E(V) and/or pH = pH(V)) functions have inflection point(s), and then assume characteristic S‐shape (or reverse S‐shape), within defined Φ (or V) range [51].

Generalizing, let us introduce the functions y = y(V), where y = E or pH and denote V = VIP, with the volume referred to inflection point (IP) [52, 53], i.e., the point (VIP, yIP) of maximal slope |η|


on the related curve y = y(V) (y = E, pH), plotted in normal coordinates, (V, y) or their derivatives: dy/dV = y1(V) and d2y/dV2 = y2(V) on the ordinate. We have, by turns [54],


At η ≠ 0, from Eq. (18b), we get d2V/dy2 = 0. Analogously to Eq. (16), we have


At the inflection point on the curve y = y(V), we have maxima for dy/dΦ and d2y/dV2 = 0, see Figure 5 for y = E [55].

Figure 5.

The function (A) E = E(Φ) and the difference quotient DE/DΦ = (Ej+1Ej)/(Φj+1 − Φj) versus (Φj+1 + Φj)/2 relationships in the vicinity of Φ = 0.2 (B), and Φ = 0.5 (C) plotted for the system KIO3 (C = 0.1) --> KI (C0 = 0.01) + HCl (C01 = 0.2).

Referring to examples presented in Figures 1A and 2A, we see that the inflection points (ΦIP, EIP) have the abscissas close to the related equivalence points (Φeq, Eeq), namely:

(0.2, 1.034)—see Table 1 and Figure 1A;


Table 1.

The (Φ, E) values related to C01 = 0 and other data presented in legend for Figure 1A.

(2.5, 0.903), (3.0, 0.414)—see Table 2 and the curve a in Figure 2A;

CHg = 0CHg = 0.07

Table 2.

The (Φ, E) values related to the data presented in legend for Figure 2A.

(3.0, 0.652)—see Table 2 and the curve b in Figure 2A;

Then we can consider Φeq (Eq. (2)) as a ratio of small natural numbers: p and q, i.e.,

Φeq=pq   (p, qN)E19

e.g., Φeq = 1 (=1/1) for titration in D + T system with A = HCl and B = NaOH (see Eq. (9)); Φeq = 1/5 = 0.2 in Figure 1A (see Table 1); Φeq = 5/2 = 2.5 or Φeq = 3/1 = 3 in Figure 2A (see Table 2).

As we see (Eq. 12), the Φe values are compared, each time, with the “round” Φeq = p/q value for Φe. due to the fact that just Φeq is placed in the denominator of the expression for the equivalent mass, RAeq (Eq. (11)).

The Φe values, presented in Tables 1 and 2 refer—in any case—to the close vicinity of the Φeq value(s), see, e.g., Φeq1 = 2.5 and Φeq2 = 3.0.

Then from Figures 1A and 2A, it results that location of IP is an interpolative method and VIPVeq [56], but in practice, this assumption may appear to be a mere fiction, especially in context with accuracy of measurements.


6. The case of diluted solutions

The Veq and VIP do not overlap in the systems of diluted solutions. For titration of V0 mL of HB (C0) with V mL of MOH (C), we have [6, 57]




and KW = [H+1][OH−1]. Similar relationship occurs for AgNO3 (C,V) → NaCl (C0,V0) system; in this case, the relations [57]: Eq. (20) and


where Ksp = [Ag+1][Cl−1] are valid.


7. Some interpolative methods of Veq determination

7.1. The Michałowski method

Two interpolative methods, not based on the IP location, were presented by Fortuin [58] and Michałowski [6, 57]. The Fortuin method is based on an nomogram; an extended form of Fortuin’s nomogram was prepared by the author of Ref. [6]. The Michałowski and Fortuin methods are particularly applicable to NaOH (C,V) → HCl (C0,V0) and NaOH (C,V) → HCl (C0,V0) systems. However, the applicability of the Michałowski method is restricted to diluted D and T, where the Fortuin method is invalid. In the Michałowski method, Veq is the real and positive root of the equation




and A0, A1, A2, A3 are obtained from results {(Vj, Ej) | j = 1,…, N} of potentiometric titration, after applying the least squares method (LSM) to the function


A useful criterion of validity of the Veq value are: pK = – log K (K = KW or Ksp) and standard redox potential (E0), calculated from the formulas [57]:




7.2. The Fenwick–Yan method

The Yan method [59] is based on Newton’s interpolation formula



f1(xj)=f(xj)f(x0)xjx0 for  j = 1,2,, nE106
fi(xj)=fi1(xj)fi1(xi1)xjxi1for   j = i,, nE107

and on the assumption that VeqVIP. Putting n = 3 in Eq. (28) and setting d2f(x)/dx2=0 for IP, after rearranging the terms one obtains


Let xj = Vk+j, f(xj) = yk+j, j = 0, 1, 2, 3; y = pH or E. According to Yan’s suggestion, xIP Veq. Then, on the basis of 4 experimental points (Vk+j, yk+j) (j = 0, 1, 2, 3) taken from the immediate vicinity of Veq, we get


Volumes Vk+j of T added were chosen from the immediate vicinity of Veq. The best results are obtained if Vk+1 < Veq < Vk+2. The error in accuracy may be significant if Vk < Veq < Vk+1 or Vk+2 < Veq < Vk+3. Moreover, the following conditions are also necessary for obtaining the accurate results: (i) small and rather equal volume increments Vk+i+1 – Vk+I (ca. 0.1 mL) and (ii) concentrations of reagent in T and analyte in D are similar.

When the titrant is added in equal volume increments ΔV in the vicinity of the equivalence point, then Vk+j – Vk+i = (j – i)·ΔV, and Eq. (30) assumes the form


identical with one obtained earlier by Fenwick [60] on the basis of the polynomial function


(compare it with Eq. (25)). In Ref. [6], it was stated that a simple equation for x Veq can be obtained after setting n = 4 in Eq. (28). Then one obtains the following equation

6f4(Vk+4)·Veq2+ 3(f3(Vk+3) β·f4(Vk+4))·Veq+ f2(Vk+2) σ·f3(Vk+3) + γ·f4(Vk+4) = 0E33

where the parameters:

σ = Vk+ Vk+1+ Vk=2,β= σ + Vk+3, γ =i>j=03Vk+iVk+jE108

are obtained on the basis of 5 points {(Vk+j, yk+j) | j=0,…,4} from the close vicinity of Veq.


8. Standardization and titrimetric analyses

The amount of an analyte in titrimetric analysis is determined from the volume of a titrant T (standard or standardized solution) required to react completely with the analyte in D. Titrations are based on standardization and determination steps. During the standardization, the titrant T with unknown concentration C of the species B is added into titrand D, containing the standard S (e.g., potassium hydrogen phthalate, borax), with mass the mS (g) known accurately. In this context, different effects involved with accuracy of visual titrations will be discussed.

Discussion on the formula 12 in context with Eq. (15) will be preceded by detailed considerations, associated with (1°) selection of an indicator (pHe), (2°) volume V0 of titrand D; (3°) concentration C0In of indicator in D; (4°) buffer effect, (5°) drop error, being considered as a whole. These effects will be considered first in context with nonredox systems. One should also draw attention whether the indicator is present in D as the salt or in the acidic form [61]; e.g., methyl orange is in the form of sodium salt, NaIn = C14H14N3NaO3S, more soluble than HIn = C14H15N3O3S.

To explain the effects 1° and 2°, we consider first a simple example, where the primary standard sample S is taken as an analyte A, A = S.

Example 1. We consider first the titration of nS = 1 mmole of potassium hydrogen phthalate KHL solution with C = 0.1 mol/L NaOH, The equation for the related titration curve


is valid here [62], where α (Appendix), and


and C0 = 1/V0 (V0 in mL). The values for the corresponding equilibrium constants are: pKW = 14 for H2O (in α), and pK1 = 2.76, pK2 = 4.92 for phthalic acid (H2L).

The Φ = Φe values in Table 3 are calculated from Eq. (34), at some particular pHe values, which denote limiting pH‐values of color change for: phenol red (color change 6.4 ÷ 8.0), phenolphthalein (8.0 ÷ 10.0), and thymolphthalein (9.3 ÷ 10.5). A (unfavorable) dilution effect, expressed by different V0 values, is involved here, in context with particular indicators; at pHe = 6.4, the dilution effect is insignificant, but grows significantly at higher pHe values, e.g., 10.5. As we see, at pHe = 8.0, the Φ = Φe value is closest to 1, assumed as Φeq in this case. At pHe = 6.4 and 10.5, the Φe values differ significantly from 1. At V0 = 100, and phenolphthalein used as indicator, at first appearance of pink color (pH ≈ 8.0), from Eq. (34) we have Φe = 0.9993 ⇒ δ = – 0.07%. The dilution practically does not affect the results of NaOH standardization against potassium hydrogen phthalate, if pH titration is applied and titration is terminated at pHe ≈ 8.0 (Table 3).

V0 = 50V0 = 100V0 = 200

Table 3.

The Φe values for different pH = pHe, calculated from Eq. (34), at C0 and C values assumed in Example 1.

A properly chosen indicator is one of the components of the D + T system in visual titrations. As a component of D, having acid‐base properties, the indicator should be included in the related balances [6, 62, 63]. The indicator effect, involved with its concentration, is considered in examples 2 and 3. Moreover, the buffer effect is considered in Example 3.

Example 2. The equation of the titration curve for titration of V0 mL of D, containing nS = 1 mmole of borax in the presence of C0In mol/l methyl red (pKIn = 5.3) as an indicator with C = 0.1 mol/L HCl as T is as follows [49, 62]


where α (Appendix), C0 = C0S =1/V0, and


It should be noted that the solution obtained after introducing 1 mmole of borax into water is equivalent to the solution containing a mixture of 2 mmoles of H3BO3 and 2 mmoles of NaH2BO3; Na2B4O7 + 5H2O = 2H3BO3 + 2NaH2BO3, resulting from complete hydrolysis of borax [62]. The results of calculations are presented in Table 4.

C0InV0 = 50V0 = 100V0 = 200

Table 4.

The Φe values calculated from Eqs. (36) to (38) for different pH = pHe, C0In and V0 (mL) values assumed in Example 2. The pHe values are related to the pH‐interval <4.4 ÷ 6.2> corresponding to the color change of methyl red (HIn).

In context with Table 4, we refer to the one‐drop error. For this purpose, let us assume that the end point was not attained after addition of V’ mL of titrant T, and the analyst decided to add the next drop, of volume ΔV mL, of the T. If the end point is attained this time, i.e., Ve = V’ + ΔV, the uncertainty in the T volume equals ΔV. Assuming ΔV = 0.03 mL and applying Eq. (1), we have:

Φ =CV/(C0V0),Φe= CVe/(C0V0) and thenΔΦ=ΦeΦ = CVe/(C0V0) CV/(C0V0) = CΔV/(C0V0).At V0 = 100 mL, C0 = 0.01 mol/L, C = 0.1 mol/L, and ΔV = 0.03 mL, we have

ΔΦ= CΔV/(C0V0) = 0.003E39

Taking the value Φe = 2.0048 in Table 4, which refers to V0 = 100 mL, C0 = 0.01 mol/L, C = 0.1 mol/L, C0In = 10−5 mol/L, pHe = 4.4, we see that |2.0048 – 2| = 0.0048 > 0.003, i.e., the discrepancy between Φeq and Φe is greater than the one assumed for ΔΦ = 0.003; it corresponds to ca. 1.5 drop of the titrant. At pHe = 6.2 and other data chosen as previously, we get |1.9973 – 2| = 0.0027 < 0.003, i.e., this uncertainty falls within one–drop error.

The indicator effect stated in Table 4, for V0 = 100, C0 = 0.01, C = 0.1 and pHe = 4.4 equals, in Φ‐units: |2.0048 – 2.0047| = 0.0001 at C0In = 10−5 or |2.0058 – 2.0047| = 0.0011, i.e., it appears to be insignificant in comparison to ΔΦ = 0.003, and can therefore be neglected.

Example 3. The solution of ZnCl2 (C0 = 0.01) buffered with NH4Cl (C1) and NH3 (C2), C1 + C2 = CN, r = C2/C1, is titrated with EDTA (C = 0.02) in presence of Eriochrome Black T (CIn = p·10−5, p = 2, 4, 6, 8, 10) as the indicator changes from wine red to blue color. The curves of logy versus Φ relationships, where

y=x2x1 and:x1=i=03[HiIn],x2=[ZnIn]+2[ZnIn2]E109

are plotted in Figure 6, where (A) refers to r = 1, (B) refers to r = 4. It is stated that at CN = 0.1, the solution becomes violet (red + blue) in the nearest vicinity of Φeq = 1, and the color change occurs at this point. At CN = 1.0, the solution has the mixed color from the very beginning of the titration (Figure 7). At CN > 1.0, the solution is blue from the start of the titration. This system was discussed in more details in Refs. [9, 37, 49, 62].

Figure 6.

The logy versus Φ relationships in the close vicinity of Φeq = 1, for CIn = p · 10-5 mol/L (p = 2, 4, 6, 8, 10); curves ap correspond to CNH3 = 0.1 mol/L, curves bp correspond to CNH3 = 1.0 mol/L; (A) refers to r = 1, (B) refers to r = 4.

Figure 7.

The logy versus Φ relationships plotted at CN = 1 mol/L, and r = 1 (curve 1b), and r = 4 (curve 4b).


9. Intermediary comments

If a concentration C of the properly chosen reagent B in T is known accurately from the standardization, the B (C mol/L) solution can be used later as titrant T, applied for determination of the unknown mass mA of the analyte A in D. The B (C) reacts selectively with an analyte A (C0 mol/L) contained in the titrand (D). This way, NaOH is standardized as in Example 1, and HCl is standardized as in Example 2. In Example 3, the standard solution of EDTA can be prepared from accurately weighed portion of this preparation, without a need for standardization, if EDTA itself can be obtained in enough pure form.

The reaction between A and S, B and A, or S and A should be fast, i.e., equilibrium is reached after each consecutive portion of T added in the titration made with use of calibrated measuring instrument and volumetric ware.

In pH or potentiometric (E) titration, the correct readout with use of the proper measuring instrument needs identical equilibrium conditions, at the measuring electrode and in the bulk solution, after each consecutive portion of T added in quasistatic a priori manner, under isothermal conditions assumed in the D + T system.

The quasistaticity assumption is fulfilled only approximately; however, the resulting error in accuracy is affected by a drift involved with retardation of processes occurred at the indicator electrode against ones in the bulk solution, where titrant T is supplied. Then, the methods based on the inflection point (IP) registration give biased results, as a rule. This discrepancy can be limited, to a certain degree, after slowing down the titrant dosage. Otherwise, the end point lags behind the equivalence point because of a slow response of the electrode.

In modern chemical analysis, titrations are performed automatically, and the titrant is introduced continuously. In this context, the transportation factors concerning the response of the indicating system are of paramount importance. At low concentration of analyte, the degree of incompleteness of the reaction is the highest around the equivalence point, and then the methods based on the inflection point registration give the biased results, as a rule. The results like ones obtained with precision 0.02% within 5 min of the potentiometric titration performed with use of an ion–selective electrode or alike (according to some literature reports), can be considered only as a mere fiction.

In this context, for the reasons specified above, it is safer to apply extrapolative methods of titrimetric analyses. Such a requirement is fulfilled by some methods applied in potentiometric analysis; the best known ones are the Gran methods, considered, e.g., in Refs. [3, 6, 65, 77]. The Gran methods of Veq determination can replace the currently used first‐derivative method in the potentiometric titration procedure.

In the mathematical model applied for Veq evaluation, it is tacitly assumed that activity coefficients and electrode junction potentials are invariable during the titration. The slope of indicator electrode should be known accurately; the statement that the slope should necessarily be Nernstian [66] is not correct. In reference to acid‐base titrations, T and D should not be contaminated by carbonate; it particularly refers to a strong base solution used as T [67, 68].


10. The Gran methods

10.1. Introductory remarks

The Gran methods is an eponym of the well–known methods of linearization of the S–shaped curves of potentiometric E or pH titration [6971]. In principle, there are two original Gran methods, known as Gran I method (abbr. G(I)) [72] and Gran II (abbr. G(II)) method [73, 74].

In current laboratory practice, only G(II) is applied, mainly in alkalinity [75] (referred to seawaters, as a rule) and acid–base titrations, in general. The presumable reasons of G(I) factual rejection (this statement was nowhere pronounced in literature) were clearly presented in the chapter [65], where G(I) and G(II) were thoroughly discussed. It was stated that the main reason of rejection was too high error, inherent in the simplified model that can be brought to the approximation

ln(1 + x)xE40

to the first term of the related Maclaurin’s series [76]


The relation Eq. (40) is valid only at |x| << 1. To extend the x range, Michałowski suggested the approximation [6]


that appeared to be better than expansion of ln(1+x) into the Maclaurin series, up to the 18th term, at |x| ≤ 1 [65], see Figure 8.

Figure 8.

Comparison of the plots for: (1) f1(x) = ln(1 + x), (2) f2(x) = x/(1 + x/2), and (3) f3(x) = x at different x‐values, 0 < x ≤ 1.

It is noteworthy that some trials were done by Gran himself [50] to improve G(I), but his proposal based on some empirical formulas was a kind of “prosthesis” applied to the defective model. In further years, the name “Gran method” (in singular) has been factually limited to G(II), i.e., in literature the term “Gran method” is practically perceived as one tantamount with G(II).

10.2. The original Gran methods: G(I) and G(II)

The principle of the original Gran methods can be illustrated, in a modified form [6], starting from titration of V0 mL of C0 mol/L HCl with V mL of C mol/L NaOH, taken as a simplest case. From charge and concentration balances, and C0V0 = CVeq (i.e., Φeq = 1 in Eq. (2), we get


Applying the notations: h = γ· [H+1], ph = −log h, at [H+1] >> [OH−1] (acid branch), i.e., V < Veq, from Eq. (42), we have the relations:

phln10 = ln(V0+V) lnG1+ ln(Veq V)E44

10.2.1. G(I) method

Applying Eq. (44) to the pair of points: (Vj, pHj) and (Vj+1, pHj+1), we have, by turns,




Applying the approximation Eq. (40), we have:


Then we have, by turns,


where P1 = G1Veq, and


From Eq. (50) and LSM, we get the formula


where =j=1N, and yj is expressed by Eq. (52); it is the essence of G(I).

10.2.2. G(II) method

Eq. (43) can be rewritten into the regression equation



G2= γ·CE55a
P2= γ·C·Veq= G2·VeqE55b

Applying the least squares method (LSM) to ph titration data {(Vj, phj) | j=1,…,N}, from (55b) we get


similar to Eq. (53), where yj is expressed by Eq. (56), this time; it is the essence of G(II).

10.3. The modified Gran methods

10.3.1. MG(I) method

Applying Eq. (41) to Eqs. (45a) and (46), we have


From Eqs. (58) and (45a) we have, by turns,


where G1 and Vj* are as in Eq. (51), and:


Application of Vj* in Eqs. (59) and (62), suggested in Ref. [6], improves the results of analyses, when compared with Eqs. (50) and (52).

10.3.2. New algorithms referred to Fe+2 + MnO4–1 system

The algorithms applied below are referred to the system where V0 ml of the solution containing FeSO4 (C0) and H2SO4 (C01) as D is titrated with V ml of KMnO4 (C). The simplest form of GEB related to this system has the form [3, 46]

[ Fe+2 ] + [ FeOH+1 ] + [ FeSO4 ]  (5[ MnO41 ] + 4[MnO42] + [Mn+3] + [MnOH+2])= (C0V0 5CV)/(V0+V) = (1  5Φ)·C0V0/(V0+V)E64

Concentration balance for Fe has the form

[ Fe+2 ] + [ FeOH+1 ] + [ FeSO4 ] + [ Fe+3 ] + [ FeOH+2 ] + [ Fe(OH)2+1 ] + 2[ Fe2(OH)2+4 ]+ [ FeSO4+1 ] + [ Fe(SO4)21 ] = C0V0/(V0+V)E65

On the basis of Figure 9, at Φ < Φeq = 0.2, and low pH‐values, Eqs. (64) and (65) assume simpler forms:

Figure 9.

Dynamic speciation curves plotted for (A) Fe‐species; (B) Mn‐species in D + T system where V0 = 100 mL of T (FeSO4 (C0 = 0.01) + H2SO4 (C01 = 1.0) is titrated with V ml of KMnO4 (C = 0.02).

[Fe+2] + [FeSO4] = (1 5Φ)·C0V0/(V0+V)E66
[Fe+2] + [FeSO4] + [Fe+3] + [FeSO4+1] + [Fe(SO4)21] = C0V0/(V0+V)E67

These simplifications are valid at low pH‐values (Figure 6). Eqs. (66) and (67) can be rewritten as follows:

[Fe+2]b2= (1 5Φ)·C0V0/(V0+V)E68
[Fe+2](b2+f23b3) = C0V0/(V0+V)E69

valid for Φ < Φeq = 0.2, where:

b2= 1 + K21·[SO42]E70a
b3= 1 + K31·[SO42] + K32·[SO42]2E70b
f23=[Fe+3][Fe+2]=10A(E – E0)E71a

and [FeSO4] = K21[Fe+2][SO4–2], [FeSO4+1] = K31[Fe+3][SO4–2], [Fe(SO4)2−1] = K32[Fe+3][SO4–2]2. From Eqs. (68) and (69), we have, by turns,

10A(E – E0)b3b2=15ΦE72b

As results from Figure 10, the term ln(b3/b2) drops monotonically with Φ (and then V) value

Figure 10.

The ln(b3/b2) versus Φ relationships for the D + T system where V0 = 100 mL of T (FeSO4 (C0 = 0.01) + H2SO4 (C01) is titrated with V ml of KMnO4 (C = 0.02). The lines are plotted at different concentrations (C01) of H2SO4, indicated at the corresponding curves.


The value for β in (73b) is small for higher C01 values, ca. 1 mol/L; in Ref. [77], it was stated that β = 1.7 · 10−3 at C01 = 1.0 mol/L; this change is small and can be neglected over the V‐range covered in the titration. The assumption ln(b3/b2) = const is applied below in the simplified Gran models. For lower C01 values, this assumption provides a kind of drift introduced by the model applied, and then in accurate models, the formula Eq. (72c) is used.

From Eqs. (1) and (2), we have Φ/Φeq = V/Veq; at Φeq = 0.2, we get 5Φ = V/Veq. Then applying Eq. (71b), we have


valid for V < Veq, with the parameters: ω, α, β and a assumed constant within the V‐range considered.

10.3.3. Simplified Gran I method

For jth and j+1th experimental point, from Eq. (72), we get:

Ej=E0alnb3b2+aln(5Φj)aln(1j); Ej+1=E0alnb3b2+aln(5Φj+1)aln(1j+1)Ej+1Ej=alnΦj+1Φjaln1j+11jE75

Applying in Eq. (69) the identities: Φj+1=Φj+Φj+1Φj and 1j=1j+1+5(Φj+1Φj) we have




Applying the approximation Eq. (41) [6] forx=x1jandx=x2j in Eq. (69) and putting Φj=CVj/(C0V0), Φj+1=CVj+1/(C0V0), we get, by turns,


where Vj* (Eq. (61)), and


P1 and G1 in Eq. (80) is obtained according to LSM, as previously.

10.3.4. Accurate Gran I method

Applying analogous procedure based on Eqs. (67) and (68), we get, by turns,




The parameters: B, a and Veq are then found according to iterative procedure; Vj* is defined by Eq. (61).

10.3.5. Simplified Gran II method

From Eqs. (1), (2) and (72a), we have, by turns


In this case, the fraction b3/b2 is assumed constant. From Eqs. (88) and (71a), we get, by turns,


If b3/b2 is assumed constant, then G2 = b2/b3·10AE0= const, and




where P2 and G2 are calculated according to LSM from the regression equation (90).

10.3.6. MG(II)A method

At β⋅V << 1, and then we write


From Eqs. (89) and (92), we get


where G2=eα10AE0= const, and real slope ϑ of an electrode is involved, after putting 1/ϑ for A. From Eq. (93), we have



P = G2βE95a
Q = G2(βVeq+1)E95b
R = G2·Veq E95c

The P, Q, and R values in Eqs. (95a,b,c) are determined according to the least squares method, applied to the regression equation




Then we get, by turns,


Eq. (96) is the basis for the modified G(II) method in its accurate version, denoted as MG(II)A method [77]. This method is especially advantageous in context of the error of analysis resulting from greater discrepancies |ϑc – ϑp| between true (correct, ϑc) and preassumed (ϑp) slope values for RIE has been proved; the error in Veq is significantly decreased, even at greater |ϑc – ϑp| values [77].

Numerous modifications of the Gran methods, designed also for calibration of redox indicator electrodes (RIE) purposes, were presented in the Refs. [46, 77]. Other calibration methods, related to ISE electrodes, are presented in Ref. [5].

10.4. Modified G(II) methods for carbonate alkalinity (CA) measurements

The G(II) methods were also suggested [28] and applied [78] for determination of carbonate alkalinity (CA) according to the modified CAM method. The CAM is related to the mixtures NaHCO3 + Na2CO3 (system I) and Na2CO3 + NaOH (system II), see Table 5. In addition to the determination of equivalence volumes, the proposed method gives the possibility of determining the activity coefficient of hydrogen ions (γ). Moreover, CAM can be used to calculate the dissociation constants (K1, K2) for carbonic acid and the ionic product of water (KW) from a single pH titration curve. The parameters of the related functions are calculated according to the least squares method.

No.pH intervalGran type functions
System ISystem II
apH > pK2(V0+V)10ph=C/KW*(VaV)
bpK2 –Δ < pH ≈ pK2(Vb+V)10ph=(K2*)1(VcV)(VVa)10ph=(K2*)1(VbV)
cpK1 –Δ ≤ pH ≤ pK1(VdV)10ph=K1*(VVc)(VdV)10ph=K1*(VVc)
dpH < pK1–Δ(V0+V)10ph=γC(VVd)(V0+V)10ph=γC(VVd)
Sequence of operationsd → c and bd → c and b, a
RelationshipsVd = Veq1 + Veq2Vd = Veq2 + Veq3
Vc = Veq2/2Vc = Vb = Veq2/2 + Veq3
Vb = Veq1Va = Veq3

Table 5.

The modified Gran functions (CAM) related to the systems I and II (see text).

11. A brief review of other papers involved with titrimetric methods of analysis

11.1. Isohydric systems

Simple acid‐acid systems are involved in isohydricity concept, formulated by Michalowski [31, 32, 79]. For the simplest case of acid‐acid titration HB (C,V) → HL (C0, V0), where HB is a strong acid, HL is a weak monoprotic acid (K1), the isohydricity condition, pH = const, occurs at


where pK1 = −logK1.

In such a system, the ionic strength of the D + T mixture remains constant during the titration, i.e., the isohydricity and isomolarity conditions are fulfilled simultaneously and independently on the volume V of the titrant added. On this basis, a very sensitive method of pK1 determination was suggested [31, 32]. The isohydricity conditions were also formulated for more complex acid‐acid, base‐base systems, etc.

11.2. pH titration in isomolar systems

The method of pH titration in isomolar D + T systems of concentrated solutions (ionic strength 2–2.5 mol/L) is involved with presence of equal volumes of the sample tested both in D and T. The presence of a strong acid HB in one of the solutions is compensated by a due excess of a salt MB in the second solution [8090]. In the systems tested, acid‐base and complexation equilibria were involved. The method enables to calculate concentrations of components in the sample tested together with equilibrium constants and activity coefficient of hydrogen ions. This method was applied for determination of a complete set of stability constants for mixed complexes [9194].

11.3. Carbonate alkalinity, total alkalinity, and alkalinity with fulvic acids

Ref. [29] was referred to complex acid‐base equilibria related to nonstoichiometric species involved with fulvic acids and their complexes with other metal ions and other simpler species present in natural waters. For mathematical description of such systems, the idea of Simms constants was recalled from earlier issues, e.g., Refs. [27, 28, 8488], and the concept of activity/basicity centers in such systems was introduced.

11.4. Binary‐solvent systems

Mutual pH titrations of a weak acid solutions of the same concentration C in D and T formed in different solvents were applied [3335] to formulate the pKi = pKi(x) relationships for the acidity parameters, where x is the mole fraction of a cosolvent with higher molar mass in D + T mixture. The pKi = pKi(x) relationship was based on the Ostwald’s formula [95, 96] for monoprotic acid, or on the Henderson‐Hasselbalch functions for diprotic and triprotic acids. The systems were modeled with the use of different nonlinear functions, namely Redlich‐Kister, and orthogonal (normal, shifted) Legendre polynomials. Asymmetric functions by Myers‐Scott and the function suggested by Michałowski were also used for this purpose.

11.5. pH‐static titration

Two kinds of reactions are necessary in Veq registration according to pH–static titration; one of them has to be an acid–base reaction. The proton consumption or generation occurs in redox, complexation, or precipitation reactions [47], for example in titration of: 1. arsenite(+3) solution with I2 + KI solution [18]; zinc salt solution with EDTA [97]; cyanide according to a (modified) Liebig‐Denigès method [65, 102, 103].

11.6. Titration to a preset pH value

A cumulative effect of different factors on precision of Veq determination was considered in [98] for pH titration of a weak monoprotic acids HL with a strong base, MOH. The results of calculations were presented graphically.

11.7. Dynamic buffer capacity

The dynamic buffer capacity, βV, concept, involving the dilution effect in acid-base D + T system, has been introduced [99] and extended in further papers [27, 28, 30, 100].

11.8. Other examples

The errors involved with more complex titrimetric analyses of chloride (mercurimetric method) [101], and cyanide (modified) Liebig‐Denigès method) [97, 102, 103]. A modified, spectro‐pH‐metric method of dissociation constants determination was presented in Ref. [104]. An overview of potentiometric methods of titrimetric analyses was presented in Ref. [64]. The titration of ammonia in the final step of the Kjeldahl method of nitrogen determination [105, 106] was discussed in Ref. [107].

The proton consumption or generation occurs in redox, complexation, or precipitation reactions [47], for example in titration of: 1. arsenite(+3) solution with I2 + KI solution [18]; zinc salt solution with EDTA [97]; cyanide according to a (modified) Liebig‐Denigès method [65, 102, 103].

Three (complexation, acid‐base, precipitation) kinds of reactions occur in the Liebig‐Denigès method mentioned above. Four elementary (redox, complexation, acid‐base, precipitation of I2) types of reactions occur in the D + T system described in the legend for Figure 2 and in less complex HCl → NaIO system presented in Ref. [21]. Other examples of high degree of complexity are shown in the works [9, 11, 12, 1416]. One of the examples in Ref. [12] concerns a four‐step analytical process, with the four kinds of reactions, involving three electroactive elements.

12. Final comments

The Generalized Approach To Electrolytic Systems (GATES) provides the possibility of thermodynamic description of equilibrium and metastable, redox and non-redox, mono- and two-phase systems of any degree of complexity. It gives the possibility of all attainable/pre-selected physicochemical knowledge to be involved, with none simplifying assumptions done for calculation purposes. It can be applied for different types of reactions occurring in batch or dynamic systems, of any degree of complexity. The generalized electron balance (GEB) concept, discovered (1992, 2006) by Michałowski [11, 13], and obligatory for description of redox systems, is fully compatible with charge and concentration balance(s), and relations for the corresponding equilibrium constants.

The chapter provides some examples of dynamic electrolytic systems, of different degree of complexity, realized in titrimetric procedure that may be considered from physicochemical and/or analytical viewpoints. In all instances, one can follow measurable quantities (potential E, pH) in dynamic and static processes and gain the information about details not measurable in real experiments; it particularly refers to dynamic speciation. In the calculations made according to iterative computer programs, all physicochemical knowledge can be involved.

This chapter aims to demonstrate the huge/versatile possibilities inherent in GATES, as a relatively new quality of physicochemical knowledge gaining from electrolytic systems of different degree of complexity, realizable with use of iterative computer programs.

Expressions for Φ related to some D + T acid‐base systems [6]; M+1 = Na+1, K+1; B−1 = Cl−1, NO3−1; k = 0,…,n (nos. 1–10), k = 0,…,q − n (no. 11); l = 0,…,m.

No.ABΦ =

The symbols:


enable to get a compact form of the functions, where:

[HiL+in]=KLiH[H+]i[Ln] (i = 0,…,q); [HiŁ+im]=KŁiH[H+]im](i = 0,…,p) ; [NH4+1] = K1NH[H+][NH3] (logK1NH=9.35); KL0H = KŁ0H = 1; M+1 = K+1, Na+1 ; [H+1] = 10−pH

and the ubiquitous symbol


termed as “proton excess” is used; pKW = 14.0 is assumed here.


D, titrand; T, titrant; V0, volume of D; V, volume of T; all volumes are expressed in mL; all concentrations are expressed in mol/L.


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Written By

Anna Maria Michałowska‐Kaczmarczyk, Aneta Spórna‐Kucab and Tadeusz Michałowski

Submitted: December 15th, 2016 Reviewed: April 14th, 2017 Published: September 27th, 2017