The (Φ, E) values related to _{01} = 0 and other data presented in legend for Figure 1A.

## Abstract

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.

### Keywords

- 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, V_{0} mL of titrand (D), containing the analyte A of an unknown (in principle) concentration C_{0} 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 V_{0} + V, if the volume additivity condition is fulfilled. Symbolically, the titration T → D in such systems will be denoted as B(C,V) → A(C_{0},V_{0}). 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 [3–5]. 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 = n_{B}/n_{A} of the numbers of mmoles: n_{B} = C·V of B and n_{A} = C_{0}·V_{0} 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 [7–26] and nonredox systems [27–32], in aqueous and in mixed‐solvent media [33–37]. 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*(

Y

_{g}), for elements/cores Y

_{g}≠ 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) –

*(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

*(*f

Y

_{g}), 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 (V_{eq}) 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, V_{eq}, corresponding to the volume V = V_{eq} of T, where the fraction titrated (1) assumes the value

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

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

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 V_{e} [mL] value is the volume of T consumed up to the end (e) point, where the titration is terminated (ended). The V_{e} 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, pH_{e} value corresponds to the volume V_{e} (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 m_{A}. Namely, Eq. (8a) cannot be applied for the evaluation of m_{A}: V_{e} is known, but Φ_{e} unknown; calculation of Φ_{e} needs prior knowledge of C_{0} value; e.g., for the titration NaOH(C,V) → HCl(C_{0},V_{0}), see Appendix, we have

However, C_{0} is unknown before the titration; otherwise, the titration would be purposeless. The approximate pH_{e} value is known in visual titration. Also Eq. (8b) is useless: the “round” Φ_{eq} value is known exactly, but V_{eq} is unknown; V_{e} (not V_{eq}) is determined in visual titrations.

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

where Φ_{eq} is put for Φ_{e} in Eq. (8a), and

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

For Φ_{e} = Φ_{eq}, we get δ = 0 and m_{A}’ = m_{A}; thus Φ_{e} ≅ Φ_{eq} (i.e., V_{e} ≅ V_{eq}) corresponds to m_{A}’ ≅ m_{A}. 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

## 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 x_{0} of x in a sample tested. In N repeated trials made on this sample, we obtain x_{j} (j = 1, …, N) and then the mean value ^{2} are obtained

The accuracy can be defined by the absolute value |^{2})^{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 x_{j} — experimental (j = 1, …, N) and true (x_{0}) values for x, ^{2} — variance. The problem referred to accuracy and precision of different methods of V_{eq} 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 V_{eq} 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, C_{0}V_{0}/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].

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].

From inspection of Figure 2B, it results that the neighboring, * quasi*linear segments of the line (at C

_{Hg}= 0) intersect at the equivalent points Φ

_{eq1}= 2.5 and Φ

_{eq2}= 3.0. So, it might seem that the pH titration is an alternative to the potentiometric titration method for the V

_{eq}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].

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

Other examples of the nonmonotonicity were presented in Refs. [7, 9, 46–49]. 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 = V_{IP}, with the volume referred to inflection point (IP) [52, 53], i.e., the point (V_{IP}, y_{IP}) 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 = y_{1}(V) and d^{2}y/dV^{2} = y_{2}(V) on the ordinate. We have, by turns [54],

At η ≠ 0, from Eq. (18b), we get d^{2}V/dy^{2} = 0. Analogously to Eq. (16), we have

At the inflection point on the curve y = y(V), we have maxima for dy/dΦ and d^{2}y/dV^{2} = 0, see Figure 5 for y = E [55].

Referring to examples presented in Figures 1A and 2A, we see that the inflection points (Φ_{IP}, E_{IP}) have the abscissas close to the related equivalence points (Φ_{eq}, E_{eq}), namely:

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

Φ | E |
---|---|

0.19800 | 0.701 |

0.19900 | 0.719 |

0.19980 | 0.761 |

0.19990 | 0.778 |

0.19998 | 0.820 |

0.20000 | 1.034 |

0.20002 | 1.323 |

0.20010 | 1.365 |

0.20020 | 1.382 |

0.20200 | 1.442 |

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

_{Hg} = 0 | _{Hg} = 0.07 | ||||
---|---|---|---|---|---|

Φ | E | Φ | Φ | ||

2.45 | 1.004 | 2.95 | 0.632 | 2.95 | 0.97 |

2.475 | 1 | 2.975 | 0.62 | 2.975 | 0.96 |

2.49 | 0.995 | 2.99 | 0.607 | 2.99 | 0.947 |

2.492 | 0.994 | 2.992 | 0.604 | 2.992 | 0.944 |

2.494 | 0.992 | 2.994 | 0.6 | 2.994 | 0.94 |

2.496 | 0.989 | 2.996 | 0.595 | 2.996 | 0.935 |

2.498 | 0.983 | 2.998 | 0.586 | 2.998 | 0.926 |

2.5 | 0.903 | 3 | 0.414 | 3 | 0.652 |

2.502 | 0.809 | 3.002 | 0.38 | 3.002 | 0.379 |

2.504 | 0.791 | 3.004 | 0.371 | 3.004 | 0.371 |

2.506 | 0.781 | 3.006 | 0.365 | 3.006 | 0.365 |

2.508 | 0.774 | 3.008 | 0.362 | 3.008 | 0.362 |

2.51 | 0.768 | 3.01 | 0.359 | 3.01 | 0.359 |

2.525 | 0.744 | 3.03 | 0.345 | 3.03 | 0.345 |

2.55 | 0.727 | 3.06 | 0.336 | 3.06 | 0.336 |

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

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

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

As we see (Eq. 12), the Φ_{e} values are compared, each time, with the “round” Φ_{eq} = p/q value for Φ_{e}. due to the fact that just Φ_{eq} is placed in the denominator of the expression for the equivalent mass, R_{A}^{eq} (Eq. (11)).

The Φ_{e} values, presented in Tables 1 and 2 refer—in any case—to the close vicinity of the Φ_{eq} value(s), see, e.g., Φ_{eq1} = 2.5 and Φ_{eq2} = 3.0.

Then from Figures 1A and 2A, it results that location of IP is an interpolative method and V_{IP}_{eq} [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 V_{eq} and V_{IP} do not overlap in the systems of diluted solutions. For titration of V_{0} mL of HB (C_{0}) with V mL of MOH (C), we have [6, 57]

where

and K_{W} = [H^{+1}][OH^{−1}]. Similar relationship occurs for AgNO_{3} (C,V) → NaCl (C_{0},V_{0}) system; in this case, the relations [57]: Eq. (20) and

where K_{sp} = [Ag^{+1}][Cl^{−1}] are valid.

## 7. Some interpolative methods of V_{eq} 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 (C_{0},V_{0}) and NaOH (C,V) → HCl (C_{0},V_{0}) 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, V_{eq} is the real and positive root of the equation

where

and A_{0}, A_{1}, A_{2}, A_{3} are obtained from results {(V_{j}, E_{j}) | j = 1,…, N} of potentiometric titration, after applying the least squares method (LSM) to the function

A useful criterion of validity of the V_{eq} value are: pK = – log K (K = K_{W} or K_{sp}) and standard redox potential (E_{0}), calculated from the formulas [57]:

where

### 7.2. The Fenwick–Yan method

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

where

and on the assumption that V_{eq}_{IP}. Putting n = 3 in Eq. (28) and setting

Let x_{j} = V_{k+j}, f(x_{j}) = y_{k+j}, j = 0, 1, 2, 3; y = pH or E. According to Yan’s suggestion, x_{IP} _{eq}. Then, on the basis of 4 experimental points (V_{k+j}, y_{k+j}) (j = 0, 1, 2, 3) taken from the immediate vicinity of V_{eq}, we get

Volumes V_{k+j} of T added were chosen from the immediate vicinity of V_{eq}. The best results are obtained if V_{k+1} < V_{eq} < V_{k+2}. The error in accuracy may be significant if V_{k} < V_{eq} < V_{k+1} or V_{k+2} < V_{eq} < V_{k+3}. Moreover, the following conditions are also necessary for obtaining the accurate results: (i) small and rather equal volume increments V_{k+i+1} – V_{k+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 V_{k+j} – V_{k+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 _{eq} can be obtained after setting n = 4 in Eq. (28). Then one obtains the following equation

where the parameters:

are obtained on the basis of 5 points {(V_{k+j}, y_{k+j}) | j=0,…,4} from the close vicinity of V_{eq}.

## 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 m_{S} (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 (pH_{e}), (2°) volume V_{0} of titrand D; (3°) concentration C_{0In} 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 = C_{14}H_{14}N_{3}NaO_{3}S, more soluble than HIn = C_{14}H_{15}N_{3}O_{3}S.

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 n

_{S}= 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 C_{0} = 1/V_{0} (V_{0} in mL). The values for the corresponding equilibrium constants are: pK_{W} = 14 for H_{2}O (in α), and pK_{1} = 2.76, pK_{2} = 4.92 for phthalic acid (H_{2}L).

The Φ = Φ_{e} values in Table 3 are calculated from Eq. (34), at some particular pH_{e} 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 V_{0} values, is involved here, in context with particular indicators; at pH_{e} = 6.4, the dilution effect is insignificant, but grows significantly at higher pH_{e} values, e.g., 10.5. As we see, at pH_{e} = 8.0, the Φ = Φ_{e} value is closest to 1, assumed as Φ_{eq} in this case. At pH_{e} = 6.4 and 10.5, the Φ_{e} values differ significantly from 1. At V_{0} = 100, and phenolphthalein used as indicator, at first appearance of pink color (pH ≈ 8.0), from Eq. (34) we have Φ_{e} = 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 pH_{e} ≈ 8.0 (Table 3).

pH_{e} | Φ_{e} | ||
---|---|---|---|

_{0} = 50 | _{0} = 100 | _{0} = 200 | |

6.4 | 0.9679 | 0.9679 | 0.9678 |

8.0 | 0.9992 | 0.9993 | 0.9994 |

9.3 | 1.0012 | 1.0022 | 1.0051 |

10.0 | 1.0060 | 1.0010 | 1.0260 |

10.5 | 1.0190 | 1.0349 | 1.0825 |

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 V

_{0}mL of D, containing n

_{S}= 1 mmole of borax in the presence of C

_{0In}mol/l methyl red (pK

_{In}= 5.3) as an indicator with C = 0.1 mol/L HCl as T is as follows [49, 62]

where α (Appendix), C_{0} = C_{0S} =1/V_{0}, 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 H_{3}BO_{3} and 2 mmoles of NaH_{2}BO_{3}; Na_{2}B_{4}O_{7} + 5H_{2}O = 2H_{3}BO_{3} + 2NaH_{2}BO_{3}, resulting from complete hydrolysis of borax [62]. The results of calculations are presented in Table 4.

pH_{e} | Φ_{e} | |||
---|---|---|---|---|

_{0In} | _{0} = 50 | _{0} = 100 | _{0} = 200 | |

4.4 | 0 | 2.0027 | 2.0047 | 2.0087 |

10^{−5} | 2.0028 | 2.0048 | 2.0089 | |

10^{−4} | 2.0033 | 2.0058 | 2.0109 | |

5.3 | 0 | 1.9999 | 2.0001 | 2.0006 |

10^{−5} | 2.0001 | 2.0006 | 2.0016 | |

10^{−4} | 2.0024 | 2.0051 | 2.0106 | |

6.2 | 0 | 1.9964 | 1.9964 | 1.9965 |

10^{−5} | 1.9968 | 1.9973 | 1.9983 | |

10^{−4} | 2.0008 | 2.0053 | 2.0142 |

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., V_{e} = V’ + ΔV, the uncertainty in the T volume equals ΔV. Assuming ΔV = 0.03 mL and applying Eq. (1), we have:

_{0} = 100 mL, C_{0} = 0.01 mol/L, C = 0.1 mol/L, and ΔV = 0.03 mL, we have

Taking the value Φ_{e} = 2.0048 in Table 4, which refers to V_{0} = 100 mL, C_{0} = 0.01 mol/L, C = 0.1 mol/L, C_{0In} = 10^{−5} mol/L, pH_{e} = 4.4, we see that |2.0048 – 2| = 0.0048 > 0.003, i.e., the discrepancy between Φ_{eq} and Φ_{e} is greater than the one assumed for ΔΦ = 0.003; it corresponds to ca. 1.5 drop of the titrant. At pH_{e} = 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 V_{0} = 100, C_{0} = 0.01, C = 0.1 and pH_{e} = 4.4 equals, in Φ‐units: |2.0048 – 2.0047| = 0.0001 at C_{0In} = 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 ZnCl

_{2}(C

_{0}= 0.01) buffered with NH

_{4}Cl (C

_{1}) and NH

_{3}(C

_{2}), C

_{1}+ C

_{2}= C

_{N}, r = C

_{2}/C

_{1}, is titrated with EDTA (C = 0.02) in presence of Eriochrome Black T (C

_{In}= 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

are plotted in Figure 6, where (A) refers to r = 1, (B) refers to r = 4. It is stated that at C_{N} = 0.1, the solution becomes violet (red + blue) in the nearest vicinity of Φ_{eq} = 1, and the color change occurs at this point. At C_{N} = 1.0, the solution has the mixed color from the very beginning of the titration (Figure 7). At C_{N} > 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].

## 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 m_{A} of the analyte A in D. The B (C) reacts selectively with an analyte A (C_{0} 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 V_{eq} determination can replace the currently used first‐derivative method in the potentiometric titration procedure.

In the mathematical model applied for V_{eq} 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 [69–71]. 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

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.

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 V_{0} mL of C_{0} mol/L HCl with V mL of C mol/L NaOH, taken as a simplest case. From charge and concentration balances, and C_{0}V_{0} = CV_{eq} (i.e., Φ_{eq} = 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 < V_{eq}, from Eq. (42), we have the relations:

#### 10.2.1. G(I) method

Applying Eq. (44) to the pair of points: (V_{j}, pH_{j}) and (V_{j+1}, pH_{j+1}), we have, by turns,

where:

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

Then we have, by turns,

where P_{1} = G_{1}V_{eq}, and

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

where _{j} 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

where:

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

similar to Eq. (53), where y_{j} 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 G_{1} and V_{j}^{*} are as in Eq. (51), and:

Application of V_{j}^{*} 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} + MnO_{4}^{–1} system

The algorithms applied below are referred to the system where V_{0} ml of the solution containing FeSO_{4} (C_{0}) and H_{2}SO_{4} (C_{01}) as D is titrated with V ml of KMnO_{4} (C). The simplest form of GEB related to this system has the form [3, 46]

Concentration balance for Fe has the form

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

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

valid for Φ < Φ_{eq} = 0.2, where:

and [FeSO_{4}] = K_{21}[Fe^{+2}][SO_{4}^{–2}], [FeSO_{4}^{+1}] = K_{31}[Fe^{+3}][SO_{4}^{–2}], [Fe(SO_{4})_{2}^{−1}] = K_{32}[Fe^{+3}][SO_{4}^{–2}]^{2}. From Eqs. (68) and (69), we have, by turns,

As results from Figure 10, the term ln(b_{3}/b_{2}) drops monotonically with Φ (and then V) value

The value for β in (73b) is small for higher C_{01} values, ca. 1 mol/L; in Ref. [77], it was stated that β = 1.7 · 10^{−3} at C_{01} = 1.0 mol/L; this change is small and can be neglected over the V‐range covered in the titration. The assumption ln(b_{3}/b_{2}) = const is applied below in the simplified Gran models. For lower C_{01} 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/Φ_{eq} = V/V_{eq}; at Φ_{eq} = 0.2, we get 5Φ = V/V_{eq}. Then applying Eq. (71b), we have

valid for V < V_{eq}, 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:

Applying in Eq. (69) the identities:

where:

Applying the approximation Eq. (41) [6]

where V_{j}^{*} (Eq. (61)), and

P_{1} and G_{1} 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,

where

The parameters: B, a and V_{eq} are then found according to iterative procedure;

#### 10.3.5. Simplified Gran II method

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

In this case, the fraction b_{3}/b_{2} is assumed constant. From Eqs. (88) and (71a), we get, by turns,

If b_{3}/b_{2} is assumed constant, then G_{2} = b_{2}/b_{3}·

Then

where P_{2} and G_{2} 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

where:

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

where

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 |ϑ_{c} – ϑ_{p}| between true (correct, ϑ_{c}) and preassumed (ϑ_{p}) slope values for RIE has been proved; the error in V_{eq} is significantly decreased, even at greater |ϑ_{c} – ϑ_{p}| values [77].

Numerous modifications of the Gran methods, designed also for calibration of redox indicator electrodes (RIE) purposes, were presented in the Refs. [4–6, 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 NaHCO_{3} + Na_{2}CO_{3} (system I) and Na_{2}CO_{3} + 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 (K_{1}, K_{2}) for carbonic acid and the ionic product of water (K_{W}) from a single pH titration curve. The parameters of the related functions are calculated according to the least squares method.

No. | pH interval | Gran type functions | |
---|---|---|---|

System I | System II | ||

a | pH > pK_{2}+Δ | – | |

b | pK_{2} –Δ < pH ≈ pK_{2} | ||

c | pK_{1} –Δ ≤ pH ≤ pK_{1}+Δ | ||

d | pH < pK_{1}–Δ | ||

Sequence of operations | d → c and b | d → c and b, a | |

Relationships | V_{d} = V_{eq1} + V_{eq2} | V_{d} = V_{eq2} + V_{eq3} | |

V_{c} = V_{eq2}/2 | V_{c} = V_{b} = V_{eq2}/2 + V_{eq3} | ||

V_{b} = V_{eq1} | V_{a} = V_{eq3} |

## 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 (C_{0}, V_{0}), where HB is a strong acid, HL is a weak monoprotic acid (K_{1}), the isohydricity condition, pH = const, occurs at

where pK_{1} = −logK_{1}.

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 pK_{1} 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 [80–90]. 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 [91–94].

### 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, 84–88], 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 [33–35] to formulate the pK_{i} = pK_{i}(x) relationships for the acidity parameters, where x is the mole fraction of a cosolvent with higher molar mass in D + T mixture. The pK_{i} = pK_{i}(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 V_{eq} 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 I_{2} + 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 V_{eq} 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, β_{V}, 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 I_{2} + 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 I_{2}) 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, 14–16]. 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}, NO_{3}^{−1}; k = 0,…,n (nos. 1–10), k = 0,…,q − n (no. 11); l = 0,…,m.

No. | A | B | Φ = |
---|---|---|---|

1 | HCl | MOH | |

2 | MOH | HB | |

3 | M_{k}H_{n‐k}L | MOH | |

4 | M_{k}H_{n‐k}L | HB | |

5 | (NH_{4})_{k}H_{n‐k}L | MOH | |

6 | (NH_{4})_{k}H_{n‐k}L | HB | |

7 | M_{k}H_{n‐k}L | M_{l}H_{m‐l}Ł | |

8 | M_{k}H_{n‐k}L | (NH_{4})_{l}H_{m‐l}Ł | |

9 | (NH_{4})_{k}H_{n‐k}L | M_{l}H_{m‐l}Ł | |

10 | (NH_{4})_{k}H_{n‐k}L | (NH_{4})_{l}H_{m‐l}Ł |

The symbols:

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

_{4}^{+1}] = ^{+}][NH_{3}] (^{+1} = K^{+1}, Na^{+1} ; [H^{+1}] = 10^{−pH}

and the ubiquitous symbol

termed as “proton excess” is used; pK_{W} = 14.0 is assumed here.

D, titrand; T, titrant; V_{0}, volume of D; V, volume of T; all volumes are expressed in mL; all concentrations are expressed in mol/L.