## 1. Introduction

According to the principles assumed in the generalized approach to electrolytic systems (GATES) introduced/formulated by Michałowski in 1992 [1], a balancing of any electrolytic system is based on the rules of conservation of particular elements/cores *Y*_{g} (*g* = 1,…, *G*), and on a charge balance (ChB) expressing the rule of electroneutrality of this system. The closed systems, separated from its environment by diathermal walls, are considered for modelling purposes. The elemental *f*(*E*_{g}) or/and core *f*(core_{g}) balances are denoted briefly as *f*(*Y*_{g}) (*f*(*E*_{g}) or *f*(core_{g}). The balance for the *g*th element (*E*_{g}) or core (core_{g}) is expressed by the equation interrelating the numbers of *g*th atom or core in components as units composing the system with the numbers of atoms or cores of *g*th kind in the species of the system thus formed. For ordering purposes, we assume *E*_{1} = H (hydrogen) and *E*_{2} = O (oxygen); then, we have *f*(H) for *Y*_{1} = *E*_{1} = H, *f*(O) for *Y*_{2} = *E*_{2} = O, etc. Free water particles and water bound in hydrates are included in balances for *f*(H) and *f*(O).

The ChB interrelates the charged species (ions) in this system. A core is a cluster of elements with defined composition, expressed by its chemical formula, structure and external charge, which remains unchanged in a system considered, e.g. SO_{4}^{−2} in Eq. (54).

The rules of conservation, formulated according to GATES principles [1–36], have the form of algebraic equations related to closed systems, composed of condensed (e.g. liquid, liquid + solid, etc.) phases separated from its environment by diathermal (freely permeable by heat) walls; it enables the heat exchange between the system and its environment. Any chemical process, such as titration, is carried out under isothermal conditions, in a *quasistatic* manner; constant temperature (*T* = constant) is one of the conditions securing constancy of equilibrium constant values. Any exchange of the matter (H_{2}O, CO_{2}, O_{2},…) between the system and its environment is thus forbidden for modelling purposes.

We refer first to aqueous media, where the species *z*_{i} = 0, ±1, ±2,… is a charge, expressed in terms of elementary charge unit, *e* = *F*/*N*_{A} (*F* = 96485.333 C/mol, Faraday’s constant; *N*_{A} = 6.022141·10^{23} mol^{−1}, Avogadro’s number; *n*_{i} (≡ *n*_{iW} ≡ *n*_{i}H_{2}O) ≥ 0 is a mean number of water (W = H_{2}O) molecules attached to *n*_{iW} = 0 is then also admitted. For ordering purposes, we assume *z*_{2} = 1, *z*_{3} = −1,…. Molar concentration of the species *n*_{i} = *n*_{iW} values are virtually unknown, even for *I*) of the solution. The _{2}O molecules involved, e.g. H^{+1}, H_{3}O^{+1}, H_{9}O_{4}^{+1}; H_{4}IO_{6}^{−1}, IO_{4}^{−1}; H_{2}BO_{3}^{−1}, B(OH)_{4}^{−1}; AlO_{2}^{−1} and Al(OH)_{4}^{−1} are considered equivalently [27], i.e. as the same species in this medium. The ChB interrelates charged species (ions, *z*_{j} ≠ 0) in the system.

From *f*(H) and *f*(O), the linear combination 2∙*f*(O) – *f*(H) is formulated and termed as the *pr*imary form of generalized electron balance (GEB), *pr*-GEB = 2∙*f*(O) – *f*(H), obtained according to approach II to GEB; this leitmotiv will be extended in further parts of this chapter. The GEB is the immanent part of GATES; the computer software applied to redox systems is denoted as GATES/GEB [1]. When related to redox systems, GATES is based on the generalized electron balance (GEB) [1–36] concept, perceived as a law of nature [1, 2, 14, 15, 22], as the hidden connection of physicochemical laws, as a breakthrough in the theory of electrolytic redox systems. The GATES refers to mono- and poly-phase, redox and non-redox, equilibrium and metastable [38–42] static and dynamic systems, in aqueous, non-aqueous and mixed-solvent media [26, 29] and in liquid-liquid extraction systems [16, 43].

The generalized electron balance (GEB) concept, discovered by Michałowski as the approach I in 1992 and approach II in 2006 to GEB, plays the key role in redox systems; both approaches are equivalent:

The GEB is fully compatible with charge balance (ChB) and concentration balances *f*(*Y*_{g}) (*g* = 3,…, *G*), formulated for different elements and cores *Y*_{g}. The elemental *f*(*E*_{g}) or/and core *f*(core_{g}) balances are transformed into concentration balances: CB(*Y*_{g}) for *g* = 3,…, *G*.

To avoid redundant terms/naming, the acronyms ChB and GEB are applied both to equations, expressed in terms of particular units (*N*_{i}, *N*_{0j}), or in terms of molar concentrations. On the basis of Eqs. (1a) and (1b) exemplified in Eqs. (2a) and (2b), this should not cause any misinterpretations. Then, *i* ∈ < 1, I > enumerate species, *j* ∈ < 1, *J* > enumerate components, *g* ∈ < 0, *G* > enumerate equations for ChB (*g* = 0) and elements/cores: *g* = 1 for H, *g* = 2 for O, *g* ∈ < 3, *G* > for other elements/cores, i.e. *Y*_{1} = H, *Y*_{2} = O,…, for ordering purposes.

The terms components and species are distinguished. In the notation applied here, *N*_{0j} (*j* = 1, 2,…, *J*) is the number of molecules of components of *j*th kind composing the static or dynamic D + T system, whereby the D and T are composed separately, from defined components, including water. The mono- or two-phase electrolytic system thus obtained involve *N*_{1} molecules of H_{2}O and *N*_{i} species of *i*th kind, *i* = 2, 3,…, *I*), specified briefly as *N*_{i}, *n*_{i}), where *n*_{i} ≡ *n*_{iW} ≡ *n*_{i}H_{2}O; then, we have H^{+1} (*N*_{2}, *n*_{2}), OH^{−1} (*N*_{3}, *n*_{3}),…. Thus, the components form a (sub)system, and the species *X*_{i}∙*n*_{iW} (precipitate, *z*_{i} = 0), as a species in a two-phase system, is marked by bold letters, e.g. **I**_{2(s)} and **AgCl**.

In *Example* 1 (Section 4.1), CuSO_{4}⋅5H_{2}O is one of components, and Cu(OH)_{3}^{−1}∙*n*_{9}H_{2}O is one of the species in the system. *N*_{01} molecules of CuSO_{4}⋅5H_{2}O involve 10*N*_{01} atoms of H, 9*N*_{01} atoms of O and *N*_{01} atoms of Cu, and *N*_{01} atoms of S. The notation Cu(OH)_{3}^{−1} (*N*_{9}, *n*_{9}) refers to *N*_{9} ions of Cu(OH)_{3}^{−1}∙*n*_{9}H_{2}O involving *N*_{9}(3 + 2*n*_{9}) atoms of H, *N*_{9}(3 + *n*_{9}) atoms of O and *N*_{9} atoms of Cu.

Molar concentration

(1) |

In a static or dynamic system, the balances are ultimately expressed in terms of molar concentrations of compounds and species, like the expressions for equilibrium constants. In particular, the charge balance (ChB) formulated as follows:

interrelates charged (*z*_{i} ≠ 0) species of this system. In terms of molar concentrations [mol/L] (Eq. (1a) or (1b)), the charge balance has the form

where *z*_{1} = 0 for *z*_{2} = +1 for *z*_{3} = –1 for

In non-aqueous and mixed-solvent media, with amphiprotic (co)solvent(s) involved, we assume/allow the formation of mixed solvates *A*_{s} (*s* = 1,…, *S*) molecules attached to *N*_{i} is a number of entities of these species in the system [25, 27, 28, 44–46].

## 2. Preliminary information

The balances *f*(*Y*_{g}) and ChB will be related to some dynamic redox and non-redox systems for comparative purposes. The balances for a given system are combined according to linear combination principles, and some general properties of the resulting balances are indicated.

The components and species in redox systems are involved in GEB, charge (ChB) and elemental balances. All the balances are founded on the well-established physical, physicochemical and chemical rules involved with charge and all elements conservation, and on the so-named mass action law (MAL), with its ‘old-fashioned’ principles. However, to be fully compatible with the GATES idea, introduced to avoid the stoichiometric reasoning, the equilibrium law (EL), suggested by Michałowski in 2016, was put instead of MAL; the principle of EL formulation is based on the idea of Lagrange multipliers for searching the local extrema of a function that subjects some constraints, expressed by GEB, ChB and concentration balances. The Gibbs free energy *G* is applied here as a function of the measurable, intensive properties: *p* and *T* and the numbers of constituents are convenient in the study of chemical reaction equilibrium. The problem in question is then consistent with the GATES formulation.

The generalized equivalent mass (GEM) concept, suggested by Michałowski in 1979 [11, 19, 47] is also based on the GATES principles, contrary to IUPAC recommendations [48] based on the stoichiometry of chemical reactions. Within GATES, the stoichiometric reaction notations are used only to formulate the expressions for the set of equilibrium constants related to the system in question, not to one ‘representative’ reaction, selected arbitrarily.

Within GATES, the mass conservation law is not limited only to components/species of one, specified/ ‘responsible’ chemical reaction equation but relates to all components/species of an electrolytic system. The mass change involved with an (exo- or endothermal) effect is negligible when compared with the total mass of the system. For example, the mass change, Δ*m*, involved with enthalpy Δ*H*° of the reaction H_{2(g)} + 0.5O_{2(g)} = H_{2}O_{(l)} (Δ*H*° = – 286 kJ/mol H_{2}O), equal Δ*m* = Δ*H*°/*c*^{2} = –3.18∙10^{−9} g, is negligible (not measurable) when compared with 18 g of H_{2}O; *c* = 299,792,458 m/s is the speed of light in vacuum. The neutralization or dilution gives much smaller heat effects. The resulting law of mass preservation is then fulfilled, irrespectively on whether stoichiometric or non-stoichiometric chemical reactions occur (or do not occur) in the system.

All considerations made in this chapter refer, in principle, to the systems where the elements are formed by stable/non-radioactive isotopes [2], i.e. where no nuclear (α, β^{+}, β, or electron capture) transformations occur [7], with emission of γ and/or X-ray radiation. However, it is possible to extend the description of redox systems on the systems with radioactive elements, in which the concentrations of the respective components are expressed by dependencies, identified as Bateman’s system of linear differential equations [49–52], binding quantitatively the radioactive elements with products of their decay. Radioactive elements and their decay products are also included in the balances for other elements and dependencies for the equilibrium constants.

## 3. Linear dependence of algebraic equations and transformation into identity

Linear combination of algebraic equations plays a fundamental/decisive role in thermodynamics of electrolytic systems, considered according to GATES [1]. An elementary information related to linear combination, perceived (mainly) from mathematical/algebraic viewpoint, can be found in Ref. [23]. It should be noted that the results of simple addition, i.e. *f*_{1} + *f*_{2} and simple subtraction, i.e. *f*_{1} − *f*_{2} or *f*_{2} − *f*_{1}, are also linear combinations of any equations *f*_{1} and *f*_{2}.

We refer here to the problem of linear dependency of balances, analogous to the problem of dependency of linear equations, considered in elementary algebra, see the picture below. In this context, the general property of linear independency, inherently involved with redox systems, will be emphasized.

(1) For the beginning, let us take the set of linear equations:

completed by the linear combination of these equations, i.e.

(5) |

Applying the matrix algebra, we see that the determinant

has zero value

irrespectively on the *c*_{1} and *c*_{2} values; at *D* = 0, calculation of *x*_{1}, *x*_{2} and *x*_{3} is then impossible.

(2) Let us consider the set of *G* + 1 equations: *f*_{g}(**x**) = φ_{g}(**x**) − *b*_{g} = 0, where *g* = 0,1,…, *G*, **x**^{T} = (*x*_{1},…, *x*_{I}) – transposed (^{T}) vector **x**, composed of independent (scalar) variables *x*_{i} (*i* ϵ < 1, I>); *a*_{gi}, *b*_{g} ɛ **x**. After multiplying the equations by the numbers *ω*_{g} ϵ

Formation of linear combinations is applicable to check the linear dependency or independency of the balances. A very useful/effective manner for checking/stating the linear dependence of the balances is the transformation of an appropriate system of equations to the identity, 0 = 0 [2, 23]. For this purpose, we will try, in all instances, to obtain the simplest form of the linear combination. To facilitate these operations, carried out by cancellation of the terms on the left and right sides of equations after changing sides of these equations, we apply the equivalent forms of the starting equations *f*_{g}(**x**) = 0:

In this notation, *f*_{g}(**x**) will be essentially treated not as the algebraic expression on the left side of the equation *f*_{g}(**x**) = 0, but as an equation that can be expressed in alternative forms presented above.

We refer now to the set of linear, algebraic equations

where *a*_{gi} are the coefficients and *b*_{g} is the free terms. Multiplying Eq. (9) by *ω*_{g}, after subsequent summation and rearrangement, we have

Assuming

from Eqs. (10) and (11) we have

Referring to the problem in question, and placing *x*_{i} = *N*_{i}, *x*_{0j} = *N*_{0j} in Eq. (11), we write

Then for ChB (Eq. (2a)), where the right side equals zero, we have

The charge balance (ChB) is expressed by Eqs. (2a) and (2b), where *f*(H), *f*(O) and *f*(*Y*_{g}) (*Y*_{g} ≠ H, O, *g* = 3,…, *G*) are written as follows:

(15) |

where *a*_{gi} and *b*_{gj} are the numbers of atoms/cores of *g*th kind in the *i*th species as a constituent of the system and in the component of *j*th kind, respecively. Denoting, for a moment, *ω*_{1} = –1, *ω*_{2} = 2, we transform the balance

In Eq. (17), *f*_{1}^{*} and *f*_{2}^{*} have the shape similar to the general expression for *f*_{g} (*g* = 3,…, *G*) in Eq. (15).

In the balances related to aqueous media, the terms involved with water, i.e. *N*_{0j} (for *j* related to H_{2}O, as the component), *N*_{1}, and all *n*_{i} = *n*_{iW} are not involved (in *f*_{0}, *f*_{3},…, *f*_{G}) or are cancelled within *f*_{12} (Eq. (17)). Other species, such as CH_{3}COOH transformable (mentally, purposefully) into C_{2}H_{4}O_{2} ≡ C_{2}(H_{2}O)_{2}, and the species in which H and O are not involved in *f*_{12}.

On the basis of relations (13) and (17), the linear combination of G + 1 balances *f*_{0}, *f*_{12}, *f*_{3},…, *f*_{G} expressed by Eq. (14) can be presented in equivalent forms:

All multipliers at *N*_{i} and *N*_{0j} in Eq. (18a) are cancelled simultaneously, if we have

for all *i* = 1,…, *I* and *j* = 1,…, *J*; then Eq. (18) is transformed into identity

Transformation of a set of the equations into the identity, 0 = 0, proves the linear dependence between the equations considered. Then from Eq. (18a), we have

i.e. *f*_{12} is the dependent balance.

Briefly, from G + 1 starting balances: *f*_{0}, *f*_{1}, *f*_{2}, *f*_{3}, …, *f*_{G} we obtain G balances: *f*_{0}, *f*_{12}, *f*_{3},…, *f*_{G}. If *f*_{12} is the dependent balance, we have G − 1 independent balances: *f*_{0}, *f*_{3},…, *f*_{G}; it is the case related to non-redox systems. If *f*_{12} is the independent balance, we have G independent balances: *f*_{0}, *f*_{12}, *f*_{3},…, *f*_{G}, that will be rearranged, optionally, as the set (*f*_{12}, *f*_{0}, *f*_{3},…, *f*_{G}) related to GEB, ChB and *f*(*Y*_{g}) (*g* = 3,…, *G*), respectively. The number of elemental/core balances *f*(*Y*_{g}) (*Y*_{g} ≠ H, O, *g* = 3,…, *G*) and then the number of concentration balances CB(*Y*_{g}) in redox systems equals *K* = *G* − 2.

As stated above, the linear combination of 2∙*f*(O) − *f*(H) with ChB and elemental/core balances *f*(*Y*_{g}) (*g* = 3,…, *G*) provides the criterion distinguishing between non-redox and redox systems of different complexity [3, 23]. To obtain the simplest form of the linear combination, some useful/general rules allowing to select multipliers *ω*_{g} of the corresponding balances are suggested. Namely, the *ω*_{g} values in Eq. (10) are equal to the oxidation numbers of the electron-non-active elements in the related system. The proper linear combination of the balances is reducible (a) to the identity, 0 = 0, for non-redox systems or (b) to the simplest equation different from (not reducible to) the identity, but involving some terms related to the species and components of the system. This way we state that 2∙*f*(O) − *f*(H) is linearly dependent on the other balances in non-redox systems, i.e. it is not a new/independent balance in such systems. In redox systems, the balance 2∙*f*(O) − *f*(H) and its linear combinations with ChB and *f*(*Y*_{g}) (*g* = 3,…, *G*) are the new equation, completing, as the generalized electron balance (GEB), the set of equations needed for resolution of electrolytic redox systems.

Static and dynamic non-redox and redox systems, with water as the main component, are considered. A static system of volume *V*_{0} mL is obtained by disposable mixing different components. The dynamic system is realized according to the titrimetric mode. At defined point of the titration B (*C, V*) ⇒ *A* (*C*_{0}, *V*_{0}), denoted briefly as T ⇒ D, *V* mL of titrant T containing the component B (C mol/L) is added into *V*_{0} mL of titrand D containing the component A (*C*_{0} mol/L), and *V*_{0} + *V* mL of D + T system/mixture is thus obtained, if the assumption of volume additivity is valid/tolerable; then D and T are sub-systems of the D + T system. The progress of the titration is represented by the fraction titrated Φ [11, 19, 47, 53, 54] value

For comparative purposes, we first refer to non-redox systems, where the derivation of relevant formulas will be carried out in detail also for training purposes. The regularities resulting from these examples, namely linear dependency of (linear) algebraic equations stated for non-redox electrolytic systems, will provide the reference point for further parts of this chapter, where redox systems are considered. Further examples in this chapter, related only to redox systems, will be presented in a synthetic manner, indicating the similarities and differences in the respective equations. The conclusions drawn here will provide the basis for further, important generalizations.

## 4. Linear combination of balances for non-redox systems

### 4.1. Examples

*Example* 1 (static): *V*_{0} mL of CuSO_{4} solution is prepared from *N*_{01} molecules of CuSO_{4}⋅5H_{2}O and *N*_{02} molecules of H_{2}O. The resulting solution consists of the following species: H_{2}O (*N*_{1}), H^{+1} (*N*_{2}, *n*_{2}), OH^{−1} (*N*_{3}, *n*_{3}), HSO_{4}^{−1} (*N*_{4}, *n*_{4}), SO_{4}^{−2} (*N*_{5}, *n*_{5}), Cu^{+2} (*N*_{6}, *n*_{6}), CuOH^{+1} (*N*_{7}, *n*_{7}), Cu(OH)_{2} (*N*_{8}, *n*_{8}), Cu(OH)_{3}^{−1} (*N*_{9}, *n*_{9}), Cu(OH)_{4}^{−2} (*N*_{10}, *n*_{10}) and CuSO_{4} (*N*_{11}, *n*_{11}). The components and species are involved in the balances for particular elements, *Y*_{g}: H, O, Cu, S, i.e.

(22) |

(23) |

Simple addition of the elemental/core balances (Eqs. (24)–(27)) gives the identity

As we see, the multipliers *ω*_{g} in the transformed identity are equal to oxidation numbers of the indicated elements.

*Example* 2 (dynamic): *V*_{0} mL of CuSO_{4} solution, as a titrand D composed of *N*_{01} molecules of CuSO_{4}⋅5H_{2}O and *N*_{02} molecules of H_{2}O, is titrated with *V* mL of titrant T, composed of *N*_{03} molecules of NaOH and *N*_{04} molecules of H_{2}O, added up to a defined point of the titration. In the resulting D + T mixture (two-phase system of volume ca. *V*_{0} + *V* mL), we have the species as in Example 1 and, additionally: **Cu(OH)**_{2} (*N*_{12}, *n*_{12}), Na^{+1} (*N*_{13}, *n*_{13}). From the balances:

That means we obtain the identity, 0 = 0, for

The terms within (…)’s in Eqs. (29) and (30) are the same as in Eqs. (22) and (23).

### 4.2. A comment

The equations for *f*(H) and *f*(O) in involve the numbers *n*_{i} = *n*_{iW} of hydrating water molecules, attached to particular species. In the linear combinations 2∙*f*(O) – *f*(H), all the *n*_{iW} are cancelled, together with *N*_{1}, i.e. the *n*_{iW} and *N*_{1} do not enter the equation for 2∙*f*(O) – *f*(H). The *n*_{iW} is not involved in ChB and in *f*(*Y*_{j}) for *Y*_{j} ≠ H, O and then they are not introduced in the linear combinations of 2∙*f*(O) – *f*(H) with these equations. The multipliers *ω*_{g} in Eq. (10) are equal to oxidation numbers of particular elements involved in the related elemental/core balances. Then, the linear combination of the balances related to a non-redox system, when put in context with Eqs. (19) and (20) gives the identity 0 = 0. This regularity is obligatory for all non-redox systems considered above (*Examples* 1 and 2), and elsewhere, e.g. [22, 37, 55]. Note that 2∙*f*(O) – *f*(H) involves –*N*_{2} + *N*_{3}, whereas ChB involves *N*_{2} – *N*_{3} = – (–*N*_{2} + *N*_{3}), and the related terms are cancelled within the combination 2∙*f*(O) – *f*(H) + ChB. Moreover, all components related to the species *f*(O) – *f*(H).

In further sections, we formulate the balances for redox systems containing one, two or three electron-active elements in redox systems (aqueous media), where the complete set of expressions for independent equilibrium constants interrelating concentrations of different species is involved. We start our considerations from the systems with one electron-active element involved with disproportionation and symproportionation [5, 13, 15, 16, 19]. These properties are appropriate for the elements that form compounds and species at three or more oxidation degrees. In particular, bromine (Br) forms the species on five oxidation degrees (–1, –1/3, 0, 1, 5), see **Figure 1**, e.g. 0 (for Br_{2}) and 1 (for HBrO or NaBrO) ϵ (–1, 5). There are possible transitions between different bromine species, associated with changes of the oxidation states of this element, see **Figure 1**.

## 5. Disproportionating systems

### 5.1. Formulation of balances for NaOH (*C, V*) ⇒ Br_{2} (*C*_{0}, *V*_{0}) system

#### 5.1.1. Approach II to GEB

Note that *V*_{0} mL of the titrand D composed of Br_{2} (*N*_{01}) + H_{2}O (*N*_{02}) is titrated with *V* mL of titrant T composed of NaOH (*N*_{03}) + H_{2}O (*N*_{04}). For the *V*_{0} + *V* mL of the D + T mixture, with the set of the species completed by Na^{+1} (*N*_{11}, *n*_{11}).

The balances can be formulated as follows:

(37) |

(38) |

Then we formulate the linear combinations:

For comparative purposes, we formulate also the linear combination *Z*_{Br}∙*f*_{3} – *f*_{0123}

where *Z*_{Br} = 35 is the atomic number for bromine (Br).

All the linear combinations (42)–(46) are not reducible to identity, 0 = 0; in the simplest case, we have six constituents (species, components) involved in *f*_{0123} (Eq. (44)). Eqs. (42)–(46) are equivalent forms of GEB, expressed in terms of particular constituents (components, species). To express them in terms of molar concentrations, we apply the relations: (1a) and *C*_{0}∙*V*_{0} = 10^{3}∙*N*_{01}/*N*_{A}, *CV* = 10^{3}∙N_{03}/*N*_{A}. In this way, we consider the titration of *V*_{0} mL of Br_{2} (*C*_{0} mol/L) solution with *V* mL of C mol/L NaOH solution added up to a defined point of the titration. In particular, from Eq. (46), we have the balance for GEB

(46a) |

completed by the balances for ChB, CB(Br) and CB(Na):

#### 5.1.2. Approach I to GEB

The (optional) formulas for GEB, expressed by Eqs. (42)–(46), were obtained according to approach II to GEB, based on *pr*-GEB = 2∙*f*(O) – *f*(H). In this section, we apply the approach I to GEB, known also as the ‘short’ version of GEB, where prior knowledge of oxidation numbers of all elements in the system is assumed/required.

In the NaOH (*C, V*) ⇒ Br_{2} (*C*_{0}, *V*_{0}) system considered in Section 4.2, bromine is the only electron-active element, considered as the carrier of its own, bromine electrons. One atom of Br has *Z*_{Br} bromine electrons, and then one molecule of Br_{2} has 2Z_{Br} bromine electrons, i.e. *N*_{01} molecules of Br_{2} involve 2Z_{Br}∙*N*_{01} bromine electrons. The oxidation degree *x* of an atom in simple species, such as ones formed here by bromine, is calculated on the basis of known oxidation degrees: +1 for H and –2 for O and external charge of this species. Then for HBrO_{3}, we have, by turns: 1∙1 + 1∙*x* + 3∙(–2) = 0 ⇒ *x* = 5; for BrO_{3}^{−1}: 1∙*x* + 3∙(–2) = –1 ⇒ *x* = 5; for HBrO: 1∙1 + 1∙*x* + 1∙(–2) = 0 ⇒ *x* = 1, etc.

The oxidation number is the net charge resulting from the presence of charge carriers, inherently involved in an atom: protons in nuclei and orbital electrons, expressed in elementary charge units—as +1 for protons and –1 for electrons. The number *y* of bromine electrons in one molecule of HBrO_{3} is calculated from the formula: Z_{Br}∙(+1) + *y*∙(–1) = 5, i.e., bromine in HBrO_{3} involves *y* = Z_{Br} – 5 bromine electrons, etc. On this basis, we state that [15]

Balancing the bromine electrons, we get Eq. (46), and then Eq. (46a), considered as the GEB obtained immediately from the approach I to GEB; this proves the equivalency of approaches I and II to GEB.

#### 5.1.3. Preparation of an algorithm and computer program

Let us refer again to balances (46a)–(46d) interrelating concentrations [*X*_{i}^{zi}] of the species *X*_{i}^{zi} with the total concentrations *C*_{0}, *C* of indicated components of the system. In this place, one should distinguish between the terms equation and equality. The term equation is related here to the balance where at least two species are involved; these species are interrelated by expressions for the corresponding equilibrium constants values (**Table 1**). Such a requirement is fulfilled by Eqs. (46a)–(46c), whereas in Eq. (46d) we have concentration of one species; at any *V* value, [Na^{+1}] is a number (not variable) and, as such, it enters immediately ChB (Eq. (46b)).

where log*K*_{e1} = 6AE_{01}, log*K*_{ei} = 2AE_{0i} (*i* = 2, 3, 4); *A* = 16.92; p*K*_{51} = −log*K*_{51}, p*K*_{11} = −log*K*_{11}, p*K*_{W} = −log*K*_{W}.

From the interrelations obtained on the basis of equilibrium data [56] collected in **Table 1** we have

(47) |

where the uniformly defined (scalar) variables: *E*, pH and pBr, forming a vector **x** = (*E*, pH, pBr)^{T}, are involved; *A*∙*E* = – log[e^{–1}], pH = –log[H^{+1}], pBr = –log[Br^{–1}]. All the variables are in the exponents of the power for 10 in [e^{–1}] = 10^{-AE}, [H^{+1}] = 10^{-pH}, [Br^{–1}] = 10^{-pBr}. The number of the (independent) variables equals to the number of equations, *K* = 2 + 1 = 3; this ensures a unique solution of the equations, at a pre-set *C*_{0}, *C* and *V*_{0} values, and the *V*-value at which the calculations are realized, at a defined step of the calculation procedure, according to iterative computer program presented below.

Computer program for the NaOH_Br2 system

function F = NaOH_Br2(x)

global V C0 V0 C yy

E = x(1);

pH = x(2);

pBr = x(3);

H = 10^(-pH);

Kw = 10^-14;

pKw = 14;

OH = Kw/H;

A = 16.92;

Br = 10^-pBr;

ZBr = 35;

Br2=Br^2*10^(2*A*(E-1.087));

Br3=Br^3*10^(2*A*(E-1.05));

BrO=Br*10^(2*A*(E-0.76)+2*pH-2*pKw);

BrO3=Br*10^(6*A*(E-1.45)+6*pH);

HBrO=10^8.6*H*BrO;

HBrO3=10^0.7*H*BrO3;

Na=C*V/(V0+V);

F = [%Charge balance

(H-OH+Na-Br-Br3-BrO-BrO3);

%Concentration balance for Br

(Br+3*Br3+2*Br2+HBrO+BrO+HBrO3+BrO3-2*C0*V0/(V0+V));

%Electron balance

((ZBr+1)*Br+(3*ZBr+1)*Br3+2*ZBr*Br2+(ZBr-1)*(HBrO+BrO)…

+(ZBr-5)*(HBrO3+BrO3)-2*ZBr*C0*V0/(V0+V))];

yy(1)=log10(Br);

yy(2)=log10(Br3);

yy(3)=log10(Br2);

yy(4)=log10(HBrO);

yy(5)=log10(BrO);

yy(6)=log10(HBrO3);

yy(7)=log10(BrO3);

yy(8)=log10(Na);

end

#### 5.1.4. Preparation of the computer program for NaOH (*C, V*) ⇒ HBrO (*C*_{0}, *V*_{0}) system

Modification of the computer program makes it possible to carry out calculations for other systems associated with bromine, e.g. for the system NaOH (*C, V*) ⇒ HBrO (*C*_{0}, *V*_{0}). In this case, the changes are made in the following lines:

Computer program for the NaOH_HBrO system

Function F = NaOH_HBrO(x)

---

(Br+3*Br3+2*Br2+HBrO+BrO+HBrO3+BrO3-C0*V0/(V0+V));

---

+(ZBr-5)*(HBrO3+BrO3)-(ZBr-1)*C0*V0/(V0+V))];

---

#### 5.1.5. Graphical presentation of the data and discussion

The systems considered here are characterized by distinctly marked jumps in *E* and pH values (**Figures 2A** and **B**) at the vicinity of the equivalence points, occurring at Φ_{eq1} = 1 for NaOH (*C, V*) → HBrO (*C*_{0}, *V*_{0}) (system I), and at Φ_{eq2} = 2 for NaOH (*C, V*) → Br_{2} (*C*_{0}, *V*_{0}) (system II), see **Table 2** [15, 16].

Reactions occurred in the systems I and II can be formulated from the analysis of the respective speciation diagrams. As results from **Figure 2C**, the disproportionation reaction in the system I occurs according to the scheme

i.e. Φ_{eq} = 2 ⇒ *CV*_{eq1} = 2*C*_{0}*V*_{0}. After crossing the equivalent point, the main products of this disproportionation reaction are Br^{−1} and BrO_{3}^{−1}, not Br^{−1} and BrO^{−1} corresponding to the reaction

with the same stoichiometry, 3 : 6 = 1 : 2. From detailed calculations at Φ = 2.5, i.e. at an excess of NaOH added, we find [BrO_{3}^{−1}]/[BrO^{−1}] = 10^{−2.574}/10^{−6.7584} = 1.53∙10^{4}, i.e. efficiency of the reaction (48) is more than 4 orders of magnitude greater than the efficiency of reaction (49).

Scheme of Br_{2} disproportionation affected by NaOH (C) can also be calculated [16]. At Φ = 2.5, we have [BrO_{3}^{−1}]/[Br^{−1}] = 10^{−2.574}/10^{−1.875} = 10^{−0.699} = 0.2 = 1:5, i.e. at an excess of NaOH added (see **Figure 2c**), the disproportionation occurs mainly according to the scheme indicated by reaction (48) from the products side.

In the system II, disproportionation of HBrO affected by NaOH (C) added according to the titrimetric mode is presented in **Figure 2**D [16]. The [Br^{−1}]/[BrO_{3}^{−1}] ratio equals: 10^{−2.2553}/10^{−2.5563} at Φ = 2.0; 10^{−2.2730}/10^{−2.5740} at Φ = 2.5, i.e. 10^{0.3010} = 2 = 2:1, corresponding to the stoichiometric ratio of products of this reaction. As results from **Figure 2C**, the disproportionation, at an excess of NaOH added, occurs mainly according to reaction 3HBrO + 3OH^{−1} = 2Br^{−1} + BrO_{3}^{−1} + 3H_{2}O (stoichiometry 3:3 = 1:1), resulting from half reactions: HBrO + 2e^{−1} + H^{+1} = Br^{−1} + H_{2}O, HBrO – 4e^{−1} + 2H_{2}O = BrO_{3}^{−1} + 5H^{+1}, and 3H^{+1} + 3OH^{−1} = 3H_{3}O. The (Φ, pH, *E*) values from the close vicinity of the corresponding equivalence points on the curves in **Figures 2A** and **B** are collected in **Table 2**. The Br_{2} solution is acidic, as results, e.g. from the ChB (Eq. (29a)): at [Na^{+1}] = 0 (Φ = 0) we have [H^{+1}] – [OH^{−1}] = [BrO_{3}^{−1}]+[BrO^{−1}]+[Br_{3}^{−1}]+[Br^{−1}] > 0, i.e. [H^{+1}] > [OH^{−1}]; this inequality is also obtained from Eq. (42), at *N*_{03} = 0: [H^{+1}] – [OH^{−1}] = 5[HBrO_{3}]+6[BrO_{3}^{−1}] + [HBrO] + 2[BrO^{−1}] > 0. Br_{2} is an acid with a strength comparable to that of acetic acid; at *C*_{0} = 0.01, pH equals 3.40 for Br_{2} and 3.325 for CH_{3}COOH (pK_{1} = 4.65). Disproportionation of Br_{2} occurs initially to a small extent (several %), according to the scheme Br_{2} + OH^{−1} = HBrO + Br^{−1}, compare with [57].

In *C*_{0} = 0.01 mol/L HBrO solution, more than 90% HBrO disproportionates occur according to the reaction 5HBrO = BrO_{3}^{−1} + 2Br_{2} + 2H_{2}O + H^{+1}; at *V* = 0, we have [Br_{2}] = 10^{−2.4406}, [BrO_{3}^{−1}] = 10^{−2.7442}, i.e. [Br_{2}]/[BrO_{3}^{−1}] = 10^{0.3036} ≈ 2, which confirms this stoichiometry of the reaction. The H^{+1} ions formed in this reaction acidify the solution significantly: at *C*_{0} = 0.01 and *V* = 0, we have pH = 2.74, although HBrO itself is a relatively weak acid.

The numerical values of the concentrations given here are taken from the corresponding files with results of iterative calculations.

### 5.2. Titration in NaOH (*C, V*) ⇒ HIO (*C*_{0}, *V*_{0}) system

The curves plotted in **Figure 3a**–**c** are related to titration of *V*_{0} = 10 mL of HIO (*C*_{0} = 0.1 mol/L) with *V* mL of *C* = 0.1 mol/L NaOH. At the initial part of the titration, we have the reactions:

In the following, at Φ ca. 0.20–0.22, a pronounced increase in [*I*^{−1}] occurs, as a result of reaction

The increase in [*I*^{−1}] is accompanied by an increase in [*I*_{3}^{−1}]

This leads to the gradual disappearance of *I*_{2(s)} (which is ultimately ended at Φ = 0.5347) and lowering of [*I*_{2}] and [*I*_{3}^{−1}]; all them disproportionate

At [**I**_{2(s)}] > 0, we have [*I*_{2}] = *s* = const; *s* = 1.33⋅10^{−3} mol/L is the solubility of *I*_{2(s)} in water, at 20°C. Finally, the disproportionation of HIO, affected by NaOH, can be expressed by the equation

Note that the stoichiometry of the reaction (54) is 3 : 3 = 1 : 1, which corresponds to the jump on the curves presented in **Figures 3a**,**b**, occurring at Φ = 1. For Φ > 1, we have [I^{−1}]/[IO_{3}^{−1}] = 2, i.e. the stoichiometry of the products of reaction (54) equals to 1 : 2. The jumps on the curves *E* = *E*(Φ) and pH = pH(Φ) (**Figures 3a**,**b**) occur at Φ ca. 0.2 (which corresponds to the stoichiometry 1:5 of the reaction (50)) and at Φ ca. 1 (which corresponds to the stoichiometry 3:3 = 1:1 of the reaction (54)); the maxima on the corresponding derivative curves in **Figures 4a**,**b** fit the stoichiometric ratios.

### 5.3. Titration in HCl (*C, V*) ⇒ NaIO (*C*_{0}, *V*_{0}) system

*V*_{0} = 10 mL of *C*_{0} = 0.01 mol/L NaIO is titrated with *C* = 0.1 mol/L HCl. The related curves are presented in **Figures 5a**‐**d**. Initially, the reaction

and then, the reaction

occurs. Then *I*^{−1} from Eq. (55) and *I*_{2} from Eq. (56) form *I*_{3}^{−1} in the reaction *I*_{2} + *I*^{−1} = *I*_{3}^{−1} and [*I*_{3}^{−1}] increases. At Φ = 0.4654, *I*_{2(s)} appears as the solid phase

At [*I*_{2(s)}] > 0, we have [*I*_{2}] = const. The increase in [Cl^{−1}], resulted from addition of HCl, causes an increase in [I_{2}Cl^{−1}], and—to a lesser extent—the increase in [ICl_{2}^{−1}] and [ICl]. The addition of HCl lowers pH of the solution, and then [HIO] becomes larger than [IO^{−1}]; [HIO_{3}] also increases. In effect, the summary concentration [HIO] + [IO^{−1}] after addition of an excess of HCl is higher than in the starting NaIO solution.

In the algorithm, we have allowed the participation of Cl^{−1} ions from HCl solution in the redox reaction. However, the concentration of Cl_{2} and HClO as the main products of Cl^{−1} oxidation (**Figure 5b**) is quite negligible. This way, one can state that the Cl^{−1} ions practically do not participate in the redox reaction as a reducing agent. From linear combination of reactions (55) and

(multiplication by 5 and 2 respectively), cancellations and division by 3, we get the reaction

with stoichiometry 4:5 = 0.8, which corresponds to Φ = 4:5 = 0.8, where the inflection point on the curves in **Figures 5c** and **d** are observed. The I^{−1} and I_{3}^{−1} ions are consumed in reactions (58) and

See **Figure 5a**.

Computer program for HCl (*C, V*) ⇒ NaIO (*C*_{0}, *V*_{0}) system

function F = System_NaIO_HCl(x);

%NaIO<-HCl

% Titration of V0 mL of NaIO (C0) with V mL HCl (C).

global V Vmin Vstep Vmax V0 C C0 fi H OH pH E Kw pKw A aa

global I I3 I2 I2s HIO IO HI5O3 I5O3 H5I7O6 H4I7O6 H3I7O6 Na

global logI logI3 logI2 logI2s logHIO logIO logHI5O3 logI5O3 logH5I7O6

global logH4I7O6 logH3I7O6 logNa

global Cl Cl2 HClO ClO HCl3O2 Cl3O2 Cl4O2 Cl5O3 Cl7O4 I2Cl ICl ICl2

global logCl logCl2 logHClO logClO logHCl3O2 logCl3O2 logCl4O2 logCl5O3

global logCl7O4 logI2Cl logICl logICl2 pI pCl

E=x(1);

pH=x(2);

pI=x(3);

pCl=x(4);

H=10.^-pH;

pKw=14;

Kw=10.^-14;

OH=Kw./H;

I=10.^-pI;

Cl=10.^-pCl;

A=16.92;

ZCl=17;

ZI=53;

I2=I.^2.*10.^(2.*A.*(E-0.621));

I3=I.^3.*10.^(2.*A.*(E-0.545));

IO=I.*10.^(2.*A.*(E-0.49)+2.*pH-2.*pKw);

HIO=IO.*10.^(10.6-pH);

I5O3=I.*10.^(6.*A.*(E-1.08)+6.*pH);

HI5O3=I5O3.*10.^(0.79-pH);

H5I7O6=I.*10.^(8.*A.*(E-1.24)+7.*pH);

H4I7O6=H5I7O6.*10.^(-3.3+pH);

H3I7O6=I.*10.^(8.*A.*(E-0.37)+9.*pH-9.*pKw);

Cl2=Cl.^2.*10.^(2.*A.*(E-1.359));

ClO=Cl.*10.^(2.*A.*(E-0.88)+2.*pH-2.*pKw);

HClO=ClO.*10.^(7.3-pH);

Cl3O2=Cl.*10.^(4.*A.*(E-0.77)+4.*pH-4.*pKw);

HCl3O2=Cl.*10.^(4.*A.*(E-1.56)+3.*pH);

Cl4O2=Cl.*10.^(5.*A.*(E-1.5)+4.*pH);

Cl5O3=Cl.*10.^(6.*A.*(E-1.45)+6.*pH);

Cl7O4=Cl.*10.^(8.*A.*(E-1.38)+8.*pH);

I2Cl=I2.*10.^(0.2-pCl);

ICl=I2.^0.5.*10.^(A.*(E-1.105)-pCl);

ICl2=ICl.*10.^(2.2-pCl);

Na=C0.*V0./(V0+V);

if I2>1.33e-3

I2s=I2-1.33e-3;

I2=1.33e-3;

aa=1;

else

aa=0;

I2s=0;

end;

%Charge balance

F=[(H-OH+Na-I-I3-IO-I5O3-H4I7O6-2.*H3I7O6-Cl-ClO-Cl3O2-Cl5O3-Cl7O4…

-I2Cl-ICl2);

%Concentration balance for I

(I+3.*I3+2.*(I2+aa.*I2s)+HIO+IO+HI5O3+I5O3+H5I7O6+H4I7O6+H3I7O6+…

2.*I2Cl+ICl+ICl2-C0.*V0./(V0+V));

%Concentration balance for Cl

(Cl+2.*Cl2+HClO+ClO+HCl3O2+Cl3O2+Cl4O2+Cl5O3+Cl7O4+I2Cl+ICl…

+2.*ICl2-C.*V./(V0+V));

%Electron balance

((ZI+1).*I+(3.*ZI+1).*I3+2.*ZI.*(I2+aa.*I2s)+(ZI-1).*(HIO+IO)…

+(ZI-5).*(HI5O3+I5O3)+(ZI-7).*(H5I7O6+H4I7O6+H3I7O6)+(ZCl+1).*Cl+…

2.*ZCl.*Cl2+(ZCl-1).*(HClO+ClO)+(ZCl-3).*(HCl3O2+Cl3O2)…

+(ZCl-4).*Cl4O2+(ZCl-5).*Cl5O3+(ZCl-7).*Cl7O4+(2.*ZI+ZCl+1).*I2Cl+…

(ZI+ZCl).*ICl+(ZI+2.*ZCl+1).*ICl2…

-((ZI-1).*C0.*V0+(ZCl+1).*C.*V)./(V0+V))];

logI=log10(I);

logI3=log10(I3);

logI2=log10(I2);

logI2s=log10(I2s);

logHIO=log10(HIO);

logIO=log10(IO);

logHI5O3=log10(HI5O3);

logI5O3=log10(I5O3);

logH5I7O6=log10(H5I7O6);

logH4I7O6=log10(H4I7O6);

logH3I7O6=log10(H3I7O6);

logCl=log10(Cl);

logCl2=log10(Cl2);

logHClO=log10(HClO);

logClO=log10(ClO);

logHCl3O2=log10(HCl3O2);

logCl3O2=log10(Cl3O2);

logCl4O2=log10(Cl4O2);

logCl5O3=log10(Cl5O3);

logCl7O4=log10(Cl7O4);

logI2Cl=log10(I2Cl);

logICl=log10(ICl);

logICl2=log10(ICl2);

logNa=log10(Na);

% The end of program

** A remark**: Some notations applied in this program are as follows: [HIO

_{3}] → HI5O3, [H

_{3}IO

_{6}

^{−2}] → H3I7O6, [ClO

_{2}

^{−1}] → Cl3O2, [ClO

_{2}] → Cl4O2, etc.

### 5.4. Disproportionation in static redox systems

In disproportionating redox systems, it is advisable to check an effect of dilution of the corresponding solutes on values of the corresponding variables. It is advisable to plot the desired relationships in the figures with the values *pC* = –log*C* on the abscissa, where *C* is the concentration [mol/L] of the solute considered. Formally, the related plots correspond to presentation, in extended logarithmic scale, the results of titration of the initial solution of this solute with pure water as a titrant T.

The static systems with *C* mol/L solutions of (1) HIO and (2) NaIO are shown graphically in **Figures 6a**‐**c** and **7a**‐**c**, where pH, *E* and log *C* of the corresponding solutes with the values *pC* = −log*C* on the abscissa.

#### 5.4.1. C mol/L HIO

As results from the speciation diagram in **Figure 6c**, in more concentrated HIO solutions, i.e. at lower *pC* values, the predominating reactions are as follows: 5HIO = 2(*I*_{2(s)}, *I*_{2}) + IO_{3}^{−1} + 2H_{2}O + H^{+1} and 5HIO = 2(*I*_{2(s)}, *I*_{2}) + HIO_{3} + 2H_{2}O; solubility of **I**_{2(s)} *s* = 1.33∙10^{−3} mol/L (25°C). At further dilution of HIO, the reaction 3HIO = 2I^{−1} + IO_{3}^{−1} + 3H^{+1} occurs in an increasing degree. This change in disproportionation scheme, more significant at *pC* 4–5, resulted in a change of the shapes of the plots: *E* = *E*(pC) (**Figure 6a**) and pH = pH(*pC*) (**Figure 6b**).

#### 5.4.2. C mol/L NaIO

The disproportionation of IO^{−1} introduced by NaIO proceeds mainly according to the scheme 3HIO = IO_{3}^{−1} + 2I^{−1} + 3H^{+1} (see **Figure 7c**), where [I^{−1}]/[IO_{3}^{−1}] ≅ 2. Concentration of I_{2} is lower than 10^{−6} mol/L and then solid *I*_{2(s)} is not formed.

## 6. Symproportionating systems

### 6.1. Titration in KBrO_{3} (*C, V*) ⇒ NaBr (*C*_{0}, *V*_{0}) system

In this case, symproportionation practically does not occur (**Figure 8a**); concentration of HBrO, as the major product formed in the symproportionation reaction

is ca. 10^{−6} mol/L. The potential *E* increases monotonically (**Figure 8b**), whereas pH first increases, passes through maximum and then decreases (**Figure 8c**). The relevant pH and *E* changes are small. Binding the H^{+1} ions in reaction (61) causes a weakly alkaline reaction (**Figure 8b**).

### 6.2. Titration in KBrO_{3} (C, V) ⇒ NaBr (*C*_{0}) + H_{2}SO_{4} (*C*_{03}) V_{0} system

The stoichiometry 1:5, i.e. Φ_{eq} = 0.2, stated for *C*_{03} values indicated at the curves plotted in **Figure 9** (column a), results from reaction

For Φ > 0.2, an increase of efficiency of the competitive reaction

is noted. A growth of *C*_{03} value causes a small extension of the potential range in the jump region, on the side of higher *E* values (**Figure 9**, column b). With an increase in the *C*_{03} value, the graphs of pH vs. Φ resemble two almost straight-line segments intersecting at Φ_{eq} = 0.2 (**Figure 9**, column c). However, the pH ranges covered by the titration curves are gradually narrowed (**Figure 9**, column c).

## 7. Redox systems with two electron-active elements

In the redox systems considered above, one electron-active element was involved in disproportionation or symproportionation reactions affected by NaOH (in dynamic systems) or water (in static systems). In the HCl (*C, V*) ⇒ NaIO (*C*_{0}, *V*_{0}) system, where HCl was used as disproportionating reagent, the possibility of oxidation of Cl^{−1} ions was allowed *a priori*, in which HCl was used as a disproportionating reagent, i.e. chlorine in HCl was treated as an electron-active element. It turned out, however, that the oxidation of Cl^{−1} occurred only in an extremely small extent, ca. 10^{−16}/10^{−2} ≈ 10^{−14} part of Cl^{−1} ions was oxidized. So, we can consider that it is virtually the disproportionation of IO^{−1} ions originating from the NaIO.

In this section, we consider the systems where two electron-active elements are factually present.

### 7.1. Titration in K_{2}Cr_{2}O_{7} (*C, V*) ⇒ KI (*C*_{0}) + H_{2}SO_{4} (*C*_{01}) V_{0} system

Note that *V*_{0} mL of titrand D composed of KI (*N*_{03}) + H_{2}SO_{4} (*N*_{04}) + H_{2}O (*N*_{05}) is titrated with *V* mL of K_{2}Cr_{2}O_{7} (*N*_{01}) + H_{2}O (*N*_{02}) as titrant T added up to a defined point of the titration. In *V*_{0} + *V* mL of D + T mixture, we have the following species:

On this basis, we formulate *f*_{1} = *f*(H), *f*_{2} = *f*(O) and then:

(64) |

(68a) |

(68b) |

(69) |

where *C*_{0}*V*_{0} = 10^{3}∙*N*_{03}/*N*_{A}, *CV* = 10^{3}∙*N*_{01}/*N*_{A}. Note that Eq. (69) was obtained only from linear combination of electron-non-active elements (fans) in this system. The balances for electron-active elements are as follows:

(71) |

(73) |

Subtraction of 3∙*f*(Cr) (Eq. (72)) from Eq. (68a) and further operations give

(74) |

The simplest/shortest form of GEB, obtained from Eqs. (68), (70) and (72) is the relation

(75b) |

Note that the numbers (*N*_{01}, *N*_{03}) of components forming the system are not involved in Eqs. (75a) and (75b).

Applying the atomic numbers: *Z*_{I} = 53 for I and *Z*_{Cr} = 24 for Cr, we obtain the linear combination

(76) |

(77) |

Eqs. (69), (74), (75b) and (77) (and other linear combinations, as well) are equivalent forms of GEB for this system. Note that Eq. (77) is identical with the one obtained immediately on the basis of the approach I to GEB [4]. The *E* = *E*(Φ) and pH = pH(Φ) and some speciation curves for iodine and chromium species are plotted in **Figures 10** and **11**, where Φ is the fraction titrated.

For *C*_{01} = 0.01, *I*_{2}, *I*_{2(s)} and *I*_{3}^{−1} are formed in reactions:

where predominating products are involved. Binding the H^{+1} ions corresponds to the pH increase, which is largest for low *C*_{01} value (relatively low buffer capacity of the solution, compare with [53, 54, 58–60]). In reactions (78)–(80), protons are consumed and then *d*pH/*d*Φ > 0; moreover, *d*E/*d*Φ > 0 for Φ > 0.

At *C*_{01} = 0.02, *E* = *E*(Φ) and [*I*_{2(s)}] pass through maximum at Φ ca. 0.2. The plot of pH = pH(Φ) shows a slight distortion of the course at Φ = 1/6, and *D*pH/*D*Φ > 0 for Φ > 0. The [IO_{3}^{−1}] is comparable with [*I*_{2(s)}] and [*I*_{2}]; [*I*_{3}^{−1}] is small here because [*I*^{−}] < 10^{−6}. The reactions

are clearly indicated.

At *C*_{01} = 0.05, the function *E* = *E*(Φ) has a complex course: it first increases, reaches a maximum at Φ = 0.295, then decreases, reaches a minimum at Φ = 1, increases again, passes through a flat maximum at Φ ca. 1.6 and decreases. The curve pH = pH(Φ) breaks at Φ = 1/6 and Φ = 1. The *I*_{2(s)} exists as the equilibrium solid phase at 0.0934 < Φ < 0.762. A decrease in [*I*_{2}] value is more expressed at Φ > 1. The ratio [HIO_{3}]/[IO_{3}^{−1}] grows with growth of *C*_{01} value. The stoichiometry at Φ = Φ_{eq2} = 1 is described by the reaction

and by reactions where HSO_{4}^{−1}, SO_{4}^{−2}, CrSO_{4}^{+1} and HIO_{3} are involved.

At *C*_{01} ≥ 0.1, a jump on the *E* = *E*(Φ) curve at Φ = 1/6 is clearly marked. The growth of jump at Φ = 1 results from a more significant decrease in the [*I*_{2}] value at Φ > 1. The *E*-range covered by the jump at Φ = 1 extends with an increase in the *C*_{01} value (**Figure 10a**). For more details, see Ref. [4].

### 7.2. A comment

Redox systems with two electron-active elements were widely discussed from the GATES/GEB viewpoint. From the earlier literature, one can recall the simulated systems: KMnO_{4} (*C, V*) ⇒ VSO_{4} (*C*_{0}) + H_{2}SO_{4} (*C*_{01}) *V*_{0} [18]; K_{2}Cr_{2}O_{7} (*C, V*) ⇒ VSO_{4} (*C*_{0}) + H_{2}SO_{4} (*C*_{01}) *V*_{0} [18]; KMnO_{4} (*C, V*) ⇒ FeSO_{4} (*C*_{0}) + H_{2}SO_{4} (*C*_{01}) [18]; Ce(SO_{4})_{2} (*C*) + H_{2}SO_{4} (*C*_{1}) V ⇒ FeSO_{4} (*C*_{0}) + H_{2}SO_{4} (*C*_{01}) *V*_{0} [18]; Cl_{2} (*C, V*) ⇒ KI (*C*_{0}, *V*_{0}) [19]; KMnO_{4} (*C, V*) ⇒ KBr (*C*_{0}), H_{2}SO_{4} (*C*_{01}) *V*_{0} [19]. Except the NaOH ⇒ Br_{2} and NaOH ⇒ HBrO systems, in [16] were simulated also the systems: NaOH (*C*) + CO_{2} (*C*_{1}) *V* ⇒ I_{2} (*C*_{0}) + KI (*C*_{01}) + CO_{2} (*C*_{02}); NaOH (*C*) + CO_{2} (*C*_{1}) V ⇒ I_{2} (*C*_{0}) + KI (*C*_{01}) + CO_{2} (*C*_{02}) *V*_{0} + CCl_{4} (V*) (dynamic liquid-liquid extraction system). In [16], an interesting qualitative reaction where MgSO_{4} solution was added into *I*_{2} + KI solution previously alkalinized with an excess of NaOH. This addition causes precipitation of **Mg(OH)**_{2} and a decrease of pH value, a growth of solubility of this precipitate and a shift of equilibrium (symproportionation) involved with formation of iodide (*I*_{2}) that adsorbes on the **Mg(OH)**_{2} precipitate turning it red-brown (similarity with starch action in visual iodometric titrations). The dynamic solubility curve was plotted, i.e. the MgSO_{4} addition is formally treated as titration. The speciation diagrams in all the systems considered were plotted. Among others, it is stated that the disproportionation of iodine (*I*_{2(s)}, *I*_{2}) occurs according to the scheme (A) 3(*I*_{2(s)}, *I*_{2}) + 6OH^{−1} = IO_{3}^{−1} + 5I^{−1} + 3H_{2}O, not the scheme (B) (*I*_{2(s)}, *I*_{2}) + 2OH^{−1} = IO^{−1} + I^{−1} + H_{2}O, as stated in almost all contemporary textbooks. One can calculate [16] that the yield of the reaction (A) is 2.5∙10^{9} times higher than the yield of reaction (B) of the same stoichiometry, 3:6 = 1:2. It is interesting to add that 80 years ≈ 2.5∙10^{9} s. Thus, the yield of the reaction B relative to A is in such a ratio as 1 s in relation to the average length of human’s life.

It should also be added that GATES/GEB enables to verify experimental results. Such a case was mentioned in Ref. [18] to the system, where KIO_{3} (*C*_{0}) + HCl (*C*_{01}) + H_{2}SeO_{3} (*C*_{02}) + HgCl_{2} (*C*_{03}) was titrated with ascorbic acid (C_{6}H_{8}O_{6}). Authors of the work [61] cited in there committed, among other things, a simple mistake resulting from improper recalculation of numerical values of potentials; for more details, see Refs. [1, 17, 18].

In all instances, full attainable (quantitative+qualitative) physicochemical knowledge was involved in the related algorithms. The quantitative knowledge was related to the complete set of equilibrium constants values, whereas qualitative knowledge was helpful in aspect of metastable and kinetic systems.

### 7.3. An illustrative presentation of the approach I to GEB

A redox reaction can also participate two or more electron-active elements. In this convention, the approach I to GEB can be perceived as the card game, with electron-active elements as ‘players’, electron-non-active elements—as ‘fans’, and electrons—as ‘money’ transferred between ‘players’, see the picture from Ref. [62]. The ‘players’ provide a common pool of their own electrons into the system in question. The ‘money’ is transferred between ‘players’. As a result of the game, the players’ accounts are changed, while the fans’ accounts remain intact/unchanged, in this convention.

In *C*_{0} mol/L aqueous solution of Br_{2}, bromine as one electron-acitve element in Br_{2} can be perceived as a ‘distributor’ of its own electrons, and H, O as ‘fans’. For example, Eq. (42) involves H and O as ‘fans’, Eq. (43) involves Na as ‘fan’, whereas Eq. (44) does not involve ‘fans’. Generally, ‘fans’ are eliminated after the proper (i.e. indicated above) combination of 2∙*f*(O) – *f*(H) with the balances for elements considered as ‘fans’. Further simplification can also result from further combination with the balance for ‘player(s)’. But, in any case, a linear combination of balances related to a redox system does not lead to the identity 0 = 0.

As usually happens in the ‘card game’ practice, the players devote to the game only a part of their cash resources. Similarly, in redox reactions, electrons may participate from the valence shells of atoms of electron-active elements; the electrons from the valence shell of the reductant atoms are transferred onto the valence shell of the oxidant atoms. However, this restriction to the valence electrons is not required here. For example, replacing *Z*_{Br} in Eq. (46a) by ζ_{Br} (ζ_{Br} < *Z*_{Br}), we have

(85) |

Replacing *Z*_{I} by ζ_{I} and *Z*_{Cr} by ζ_{Cr} (ζ_{I} < *Z*_{I}, ζ_{Cr} < *Z*_{Cr}) in Eq. (77), we have

(86) |

In particular, we can put ζ_{Cr} = ζ_{Cr} = 0 in Eq. (86). Obviously, we get the relation

(87) |

identical to the one obtained from Eq. (68a). If we put ζ_{Br} = 0 in Eq. (85), we get the relation

(88) |

identical to the ones obtained from Eq. (44). This way, we recall the card game without ‘live cash’ but with ‘debt of honor’—in not accidental reference to the title of the thriller novel by T. Clancy; btw, *nota bene*, the “Debt of honor” was published in 1994, like the papers [15, 16, 18].

## 8. Conclusions

The GEB formulated according to approach I was named first as electron prebalance and presented, in totally mature form, in three papers issued in 1994 [15, 16, 18] and then followed by further articles and other communications. Currently, in context with the approach II to GEB, it is named as the approach I to GEB, fully equivalent to the approach II to GEB, explicitly related to the law of conservation of H and O.

The linear combination 2⋅*f*(O) – *f*(H) is a keystone for the overall thermodynamic knowledge on electrolytic systems. It can be formulated both for non-redox and redox systems, in aqueous, non-aqueous and mixed-solvent systems, with amphiprotic (co)solvent(s) involved. The 2⋅*f*(O) – *f*(H) is linearly independent on ChB and other balances, for elements/cores *f*(*Y*_{m}) ≠ H, O, in any redox system. For any non-redox system, 2⋅*f*(O) – *f*(H) is linearly dependent on those balances. Then, the linear independency/dependency of 2⋅*f*(O) – *f*(H) on the other balances is the general criterion distinguishing between redox and non-redox systems. The equation for 2⋅*f*(O) – *f*(H), considered as the *pr*imary form of GEB,

Ultimately, GEB, ChB and elemental/core balances are expressed in terms of molar concentrations—to be fully compatible/congruent with expressions for equilibrium constants, interrelating molar concentrations of defined species, on the basis of the mass action law applied to correctly written reaction equation. The mass action law is the one and only chemical law applied in GATES. A complete set of independent (non-contradictory [22]) relations for the equilibrium constants is needed for this purpose.

When compared with the approach I, the approach II to GEB offers several advantages. Although derivation of GEB according to the approach II is more laborious, it enables to formulate this balance without prior knowledge of oxidation numbers for the elements involved frequently in complex components and species of the system. Only the composition (expressed by chemical formula) of components forming the system and composition of the species formed in the system, together with their external charges, are required, i.e. it provides an information sufficient to formulate the GEB; it is the paramount advantage of the approach II to GEB. Anyway, the oxidation number, representing the degree of oxidation of an element in a compound or a species, is commonly perceived as a contractual concept. In this regard, formulation of GEB according to approach II is more useful than the approach I when applied to complex organic species in redox systems of biological origin [63–67]. The approach II to GEB is advantageous/desired, *inter alia*, for redox systems where radical and ion-radical species are formed. What is more, the ‘players’ and ‘fans’, as ones perceived from the approach I to GEB viewpoint, are not indicated *a priori* within the approach II to GEB. The approach I, considered as a ‘short’ version of GEB, is more convenient when oxidation numbers for all elements of the system are known beforehand. Within the approaches I and II to GEB, the roles of oxidants and reductants are not ascribed *a priori* to particular components forming the redox system and to the species formed in this system. In other words, full ‘democracy’ is established *a priori* within GATES/GEB, where the oxidation number, oxidant, reductant, equivalent mass and stoichiometric reaction notation are the redundant concepts only.

The 1 + K balances composed of ChB and K concentration balances related to equations (not equalities, see e.g. Eq. (40)) for elemental balances form a complete set of equations related to a non-redox system. The set of 1 + K variables in **x** = (pH, pX_{1},…, pX_{K})^{T} is involved in the algorithm applied for calculation purposes. For example, 1 + K = 1 + 2 = 3 balances are related to non-redox systems presented in Examples 1 and 2, with independent variables represented by components of the vector **x** = (pH, pCu, pSO_{4})^{T}. In this context, Eq. (35) is considered as equality, not equation.

The 2 + K balances composed of GEB, ChB and K concentration balances related to equations (not equalities) for elemental balances form a complete set of equations related to the redox system. The set of 2 + K variables in **x** = (*E*, pH, pX_{1},…, pX_{K})^{T} is involved in the algorithm applied for calculation purposes. In this chapter, some results of the simulations of electrolytic redox systems, made according to the GATES/ GEB principles, with use of iterative computer programs, are graphically presented and discussed. The computer simulation realized within GATES with use of iterative computer programs, e.g. MATLAB, provides quite a new quality in knowledge gaining. It enables to follow the details of the process, registered with use of measurable quantities, such as pH and/or potential E. All these calculations are made under assumption that the relevant reactions take place in *quasi*-static manner under isothermal conditions. The reactions proceeding in the respective systems were formulated under assumption that all equilibrium constants found in the relevant tables and then used in the calculations are correct.

The number of electron-active elements (considered as ‘players’, in terms of the approach I to GEB) in a redox system, is practically unlimited and adapted according to current needs; among others, the systems with one, two, three [13] or four [1, 18] ‘players’ were considered.

The GATES and GATES/GEB (in particular) can be applied for thermodynamic resolution of systems of any degree of complexity. An example is the four-step process involved with iodometric method of copper(+2) determination, considered in detail in Refs. [14, 68], where 47 species are involved in the system with three electron-active elements, eight equations and two equalities, interrelated in 35 independent expressions for equilibrium constants. The systems with similar complexity were resolved. The complexity is limited, however, by factors of the qualitative and quantitative nature, that is, the knowledge of species the knowledge of equilibrium constants interrelating concentration of complex species with concentrations of their constituting parts. The physicochemical data are incomplete or not reliable, in many cases. This problem was raised lately in Ref. [12].

Each of the components of **x** = (pH, pX_{1},…, pX_{K})^{T} is informally ascribed to the corresponding balance: ChB, and the corresponding concentration balances for elements/cores ≠ H, O. Each of the components of **x** = (E, pH, pX_{1},…,pX_{k})^{T} is informally ascribed to the corresponding balance: GEB, ChB, and the corresponding concentration balances for elements/cores ≠ H, O. A unequivocal solution of equations is obtained when the number of independent equations equals to the number of independent variables; it is the ‘iron rule’ obligatory in mathematics; *pe* = – log[*e*^{−1}] = *A∙E*, where 1/*A* = *RT*/*F*∙ln10, pH = –log[H^{+1}], *pX*_{i} = –log

GATES avoids the necessity of quantitative inferences based on fragile/rachitic chemical reaction notation, involving only some of the species existing in the system; it is only a faint imitation of a true, algebraic notation. From the GATES viewpoint, the ‘stoichiometry’ can be perceived as a mnemonic term only. In calculations, the metastable state is realized by omission of potential products in the related balances, whereas ‘opening’ a reaction pathway in metastable state is based on insertion of possible (from equilibrium viewpoint) products in the related balances [17, 18]. One can also test the interfering effects of different kinds.

All the inferences made within GATES/GEB are based on firm, mathematical (algebraic) foundations. The proposed approach allows us to understand far better all physicochemical phenomena occurring in the system in question and improve some methods of analysis. All the facts testify very well about the potency of simulated calculations made, according to GATES, on the basis of all attainable physicochemical knowledge. Testing the complex redox and non-redox systems with use of iterative computer programs deserves wider popularization among physicochemists and chemists-analysts.

The GATES/GEB is put in context with constructivistic and deterministic principles, and GEB is perceived as the general law of nature, referred to as electrolytic (aqueous media) redox systems. It is proved that stoichiometry of reactions is not a primary concept in chemistry, and its application provides false results, for obvious reasons. From the GATES viewpoint, the stoichiometric reactions are only the basis to formulate the related equilibrium constants. GATES/GEB referred to modeling of redox titration curves in context with earlier approaches to this problem. The GATES/GEB is also presented in three other chapters issued in 2017 within InTech [68–70].

The dependency/independency criteria ascribed to 2∙*f*(O) – *f*(H) distinguishing between the relevant (non-redox and redox) systems are the properties of the equations obtained from the linear combination of the balances for H and O. Namely, the resulting equation is not independent of non-redox systems, since it is a linear combination of the remaining (charge and concentration) balances, whereas in the case of redox systems, this equation is linearly independent of those balances. This is a general property of nature, independent of the complexity of the system under consideration, which is the electrolytic system. GATES and GATES/GEB, in particular, are clear confirmation of the fact that the nature is mathematically designed and the true laws of nature are mathematical. In other words, the quantitative, mathematical method became the essence of science. To paraphrase a Chinese proverb, one can state that ‘the lotus flower, lotus leaf and lotus seed come from the same root’ [2]. Similarly, the three kinds of balances: GEB, charge and concentration balances come from the same family of fundamental laws of preservation. This compatibility is directly visible from the viewpoint of the approach II to GEB. The equivalent equations for GEB, based on a reliable law of the matter conservation, are equally robust as equations for charge and concentration balances. The complementarity of the GEB (approaches I and II) to other balances is regarded as the expression of harmony of nature and GATES/GEB as an example of excellent epistemological paradigm.

All earlier (dated from the 1960s) efforts made towards formulation of electrolytic redox systems were only clumsy attempts of resolution of the problem in question, as stated in review papers [2, 10–14]. These approaches were slavishly related to the stoichiometric reaction notations, involving only two pairs of indicated species participating in redox reaction; there were usually minor species of the system considered. The species different from those involved in the reaction notation were thus omitted in considerations. Moreover, the charge balance and concentration balances for accompanying substances were also omitted. Theoretical considerations were related to virtual cases, not to real, electrolytic redox systems.

## 9. Afterword: reductionism

The principles of science gaining are based on a conviction that complex phenomena occurred in nature, e.g. in electrolytic systems, that can be explained in terms of some general laws of the matter conservation; it is the basic assumption of reductionism [2]. These laws are expressed in terms of mathematical equations, valid for the systems of any degree of complexity. Reductionism is very similar to and has its roots in Occam’s razor principle that gives precedence to simplicity, i.e. the explanation which requires the fewest assumptions. The power of reductionism lies in prediction and formulation; it is perceived as a good approximation of the macroscopic world. The knowledge thus obtained is verifiable and based on logical premises. This way the quantitative knowledge gained from the study of relatively simple systems can be synthesized in the knowledge obtainable from more complex systems. From this viewpoint, the knowledge obtained from physicochemical analysis involved, e.g. with determination of the stability constants of complex species formed in a particular system, can be perceived as a ‘stone’ used in construction of more complex systems. In this place, one can recall H. Poincaré who stated that ‘Science is facts; just as houses are made of stones, so science is made of facts’. Any complex species must be equipped with its equilibrium constant value; a qualitative knowledge only (e.g. chemical formula) is insufficient in this respect. However, to construct the knowledge on more complex systems, these stones should be arranged according to a defined scheme (design), based on a set of compatible balances. Closely associated with reductionism is determinism—the philosophy that everything has a cause, and that a particular cause leads to a unique effect.

In any complex system, many particular reactions occur; the resultant reaction is the combination (superposition) of these elementary reactions which occur with different efficiencies that are known only after thorough physicochemical examination of the system in question. Similarly, the vibration in a polyatomic molecule is a superposition of normal vibrations. The physicochemical/thermodynamic knowledge on electrolytic systems is based on equilibrium constant values, referred to as the equilibrium system and all information about possible paths of particular reactions occurred in the system in question. Not all paths of chemical reactions are accessible, under defined conditions of analysis, involved with temperature and/or the presence of catalytic agents. This problem has been raised in Ref. [1], in context with GATES/GEB.

The approach II to GEB, preceded by the approach I to GEB, indicated new insight into redox systems and electrolytic systems in general. This way, the thermodynamic knowledge about redox systems was built in 1992 practically from scratch. The combination 2∙*f*(O) – *f*(H) of elemental balances for H and O (approach II to GEB) is the quintessence of the generalized electron balance (GEB) that is the link needed for mathematical/algebraic description of electrolytic redox systems of any degree of complexity within the generalized approach to electrolytic systems as GATES/GEB.

Within GATES, a resolution of electrolytic systems is realized with use of iterative computer programs. Formally, the manner of resolution of this task is even easier than the one based on the formulation of some functional dependencies that require some simplifications, as a rule. The simplifications are not necessary in iterative methods.

Summarizing, the generalized electron balance (GEB), perceived within GATES as GATES/GEB, fulfils all the requirements imposed on reductionism. Its formulation is possible for any electrolytic redox systems—static and dynamic (titration), mono- and multiphase, equilibrium, metastable and non-equilibrium systems, of any degree of complexity, provided that all necessary physicochemical knowledge is attainable. Simply, it is the best possible thermodynamic approach to electrolytic redox systems.

GEB, within the context of GATES/GEB, confirms the validity of some statements expressed in the past:

*A hidden connection is stronger than an obvious one*(Heraclitus)*All truths are easy to understand once they are discovered; the point is to discover them*(Galileo Galilei)*Everything should be made as simple as possible, but not simpler*(A. Einstein)*Minds are like parachutes: they function only when they are open*(J. Dewar)*A good theory is the best practical tool*(J.C. Maxwell)*Nothing is too wonderful to be true if it be consistent with the laws of nature*(M. Faraday).*A new scientific truth does not triumph by convincing its opponents (…), but rather because its opponents eventually die, and a new generation grows up that is familiar with it*(M. Planck).*Art of reasoning is nothing more than a well-ordered language**Telling the truth in a time of universal deceit is a revolutionary act*(G. Orwell)*Man learns throughout his life, with the exception of school years**Do not believe in any messages just because they are in force for a long time, in many countries. Do not believe in something just because a lot of people say/write it for a long time. Do not accept anything just because someone else said/wrote it. Do not believe in something just because it sounds plausible. Have confidence in what you have accepted as true after a long experience/checking, what brings success to you and others*(own apprenticeship).