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

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, © 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

distribution, and eproduction in any medium, provided the original work is properly cited.

system is based on the rules of conservation of particular elements/cores Y<sup>g</sup> (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(Eg) or/and core f(coreg) balances are denoted briefly as f(Yg) (f(Eg) or f(coreg). The balance for the gth element (Eg) or core (coreg) is expressed by the equation interrelating the numbers of gth atom or core in components as units composing the system with the numbers of atoms or cores of gth kind in the species of the system thus formed. For ordering purposes, we assume E<sup>1</sup> = H (hydrogen) and E<sup>2</sup> = O (oxygen); then, we have f(H) for Y<sup>1</sup> = E<sup>1</sup> = H, f(O) for Y<sup>2</sup> = E<sup>2</sup> = 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. SO4 �<sup>2</sup> 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 (H2O, CO2, O2,…) between the system and its environment is thus forbidden for modelling purposes.

We refer first to aqueous media, where the species Xzi <sup>i</sup> exist as hydrates Xzi <sup>i</sup> � niW; zi = 0, �1, �2, … is a charge, expressed in terms of elementary charge unit, e = F/N<sup>A</sup> (F = 96485.333 C/mol, Faraday's constant; <sup>N</sup><sup>A</sup> = 6.022141�1023 mol�<sup>1</sup> , Avogadro's number; ni (� ni<sup>W</sup> � niH2O) ≥ 0 is a mean number of water (W = H2O) molecules attached to Xzi <sup>i</sup> ; the case ni<sup>W</sup> = 0 is then also admitted. For ordering purposes, we assume Xz<sup>2</sup> <sup>2</sup> � <sup>n</sup>2W <sup>¼</sup> <sup>H</sup>þ<sup>1</sup> � <sup>n</sup>2<sup>W</sup> , Xz<sup>3</sup> <sup>3</sup> � <sup>n</sup>3W <sup>¼</sup> OH�<sup>1</sup> � <sup>n</sup>3W, …, i.e. <sup>z</sup><sup>2</sup> = 1, <sup>z</sup><sup>3</sup> <sup>=</sup> �1,… . Molar concentration of the species <sup>X</sup>zi <sup>i</sup> � ni<sup>W</sup> is denoted as <sup>½</sup>Xzi <sup>i</sup> �. The n<sup>i</sup> = ni<sup>W</sup> values are virtually unknown, even for Xz<sup>2</sup> <sup>2</sup> <sup>¼</sup> <sup>H</sup>þ<sup>1</sup> [37] in aqueous media, and depend on ionic strength (I) of the solution. The Xi zi s with different numbers of H2O molecules involved, e.g. H+1, H3O+1, H9O4 +1; H4IO6 �1 , IO4 �1 ; H2BO3 �1 , B(OH)4 �1 ; AlO2 �<sup>1</sup> and Al(OH)4 �<sup>1</sup> are considered equivalently [27], i.e. as the same species in this medium. The ChB interrelates charged species (ions, zj 6¼ 0) in the system.

From f(H) and f(O), the linear combination 2∙f(O) – f(H) is formulated and termed as the primary 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:

#### ∴ approach II to GEB ⇔ aapproach I to GEB

The GEB is fully compatible with charge balance (ChB) and concentration balances f(Yg) (g = 3, …, G), formulated for different elements and cores Yg. The elemental f(Eg) or/and core f(coreg) balances are transformed into concentration balances: CB(Yg) 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 (Ni, N0j), 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<sup>1</sup> = H, Y<sup>2</sup> = O,…, for ordering purposes.

The terms components and species are distinguished. In the notation applied here, N0j (j = 1, 2,…, J) is the number of molecules of components of jth 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<sup>1</sup> molecules of H2O and N<sup>i</sup> species of ith kind, Xzi <sup>i</sup> � ni<sup>W</sup> (<sup>i</sup> = 2, 3,…, <sup>I</sup>), specified briefly as Xzi <sup>i</sup> (Ni, <sup>n</sup>i), where <sup>n</sup><sup>i</sup> � <sup>n</sup>iW � <sup>n</sup>iH2O; then, we have H+1 (N2, <sup>n</sup>2), OH�<sup>1</sup> (N3, <sup>n</sup>3),…. Thus, the components form a (sub)system, and the species Xzi <sup>i</sup> � niW enter the system thus formed. A solid Xi∙niW (precipitate, zi = 0), as a species in a two-phase system, is marked by bold letters, e.g. I2(s) and AgCl.

In Example 1 (Section 4.1), CuSO4�5H2O is one of components, and Cu(OH)3 �1 ∙n9H2O is one of the species in the system. N<sup>01</sup> molecules of CuSO4�5H2O involve 10N<sup>01</sup> atoms of H, 9N<sup>01</sup> atoms of O and N<sup>01</sup> atoms of Cu, and N<sup>01</sup> atoms of S. The notation Cu(OH)3 �<sup>1</sup> (N9, n9) refers to N<sup>9</sup> ions of Cu (OH)3 �1 ∙n9H2O involving N9(3 + 2n9) atoms of H, N9(3 + n9) atoms of O and N<sup>9</sup> atoms of Cu.

Molar concentration <sup>½</sup>Xzi <sup>i</sup> � of Xzi <sup>i</sup> � ni<sup>W</sup> is as follow:

system is based on the rules of conservation of particular elements/cores Y<sup>g</sup> (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(Eg) or/and core f(coreg) balances are denoted briefly as f(Yg) (f(Eg) or f(coreg). The balance for the gth element (Eg) or core (coreg) is expressed by the equation interrelating the numbers of gth atom or core in components as units composing the system with the numbers of atoms or cores of gth kind in the species of the system thus formed. For ordering purposes, we assume E<sup>1</sup> = H (hydrogen) and E<sup>2</sup> = O (oxygen); then, we have f(H) for Y<sup>1</sup> = E<sup>1</sup> = H, f(O) for Y<sup>2</sup> = E<sup>2</sup> = O, etc. Free water particles and water bound in hydrates are

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,

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 (H2O, CO2, O2,…) between the system and its

… is a charge, expressed in terms of elementary charge unit, e = F/N<sup>A</sup> (F = 96485.333 C/mol,

; H2BO3

considered equivalently [27], i.e. as the same species in this medium. The ChB interrelates

From f(H) and f(O), the linear combination 2∙f(O) – f(H) is formulated and termed as the primary 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

�1

, B(OH)4

�<sup>2</sup> in Eq. (54).

<sup>i</sup> exist as hydrates Xzi

<sup>2</sup> � <sup>n</sup>2W <sup>¼</sup> <sup>H</sup>þ<sup>1</sup> � <sup>n</sup>2<sup>W</sup> , Xz<sup>3</sup>

�1 ; AlO2

, Avogadro's number; ni (� ni<sup>W</sup> � niH2O) ≥ 0 is a

<sup>2</sup> <sup>¼</sup> <sup>H</sup>þ<sup>1</sup> [37] in aqueous media, and depend on

s with different numbers of H2O molecules involved,

<sup>i</sup> � ni<sup>W</sup> is denoted as <sup>½</sup>Xzi

<sup>i</sup> � niW; zi = 0, �1, �2,

<sup>3</sup> � <sup>n</sup>3W <sup>¼</sup> OH�<sup>1</sup> � <sup>n</sup>3W,

<sup>i</sup> �. The n<sup>i</sup> =

�<sup>1</sup> are

<sup>i</sup> ; the case ni<sup>W</sup> = 0 is then also

�<sup>1</sup> and Al(OH)4

included in balances for f(H) and f(O).

10 Redox - Principles and Advanced Applications

which remains unchanged in a system considered, e.g. SO4

environment is thus forbidden for modelling purposes. We refer first to aqueous media, where the species Xzi

mean number of water (W = H2O) molecules attached to Xzi

…, i.e. <sup>z</sup><sup>2</sup> = 1, <sup>z</sup><sup>3</sup> <sup>=</sup> �1,… . Molar concentration of the species <sup>X</sup>zi

�1 , IO4 �1

+1; H4IO6

zi

Faraday's constant; <sup>N</sup><sup>A</sup> = 6.022141�1023 mol�<sup>1</sup>

admitted. For ordering purposes, we assume Xz<sup>2</sup>

ni<sup>W</sup> values are virtually unknown, even for Xz<sup>2</sup>

ionic strength (I) of the solution. The Xi

charged species (ions, zj 6¼ 0) in the system.

e.g. H+1, H3O+1, H9O4

$$\begin{aligned} \text{(a)} \ [\mathbf{X}\_i^{\natural}] &= 10^3 \cdot (\mathbf{N}\_i / \mathbf{N}\_A) / V\_0 \text{ for a static system, or} \\ \text{(b)} \ [\mathbf{X}\_i^{\natural}] &= 10^3 \cdot (\mathbf{N}\_i / \mathbf{N}\_A) / (V\_0 + V) \text{ for a dynamic D} + \text{T system} \end{aligned} \tag{1}$$

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:

$$\sum\_{i=2}^{l} z\_i \cdot N\_i = 0 \tag{2}$$

interrelates charged (zi 6¼ 0) species of this system. In terms of molar concentrations [mol/L] (Eq. (1a) or (1b)), the charge balance has the form

$$\sum\_{i=2}^{I} \mathbf{z}\_{i} \cdot [\mathbf{X}\_{i}^{z\_{i}}] = \mathbf{0} \tag{2a}$$

where z<sup>1</sup> = 0 for Xz<sup>1</sup> <sup>1</sup> <sup>¼</sup> H2O, <sup>z</sup><sup>2</sup> = +1 for <sup>X</sup><sup>z</sup><sup>2</sup> <sup>2</sup> <sup>¼</sup> <sup>H</sup>þ<sup>1</sup> , z<sup>3</sup> = –1 for X<sup>z</sup><sup>3</sup> <sup>3</sup> <sup>¼</sup> OH�<sup>1</sup> ,…

In non-aqueous and mixed-solvent media, with amphiprotic (co)solvent(s) involved, we assume/ allow the formation of mixed solvates Xi zi � ni1ni2…niS, where nis <sup>¼</sup> niAs (≥0) are the mean numbers of A<sup>s</sup> (s = 1,…, S) molecules attached to Xi zi . We apply the notation Xi zi ðNi;niA<sup>1</sup> ;niA<sup>2</sup> ;…;niAS Þ, where N<sup>i</sup> is a number of entities of these species in the system [25, 27, 28, 44–46].
