Any electrolytic system in aqueous media is described with use of charge balance, f₀ = ChB, and K elemental or core balances, f_k = f(Y_k) (k = 1,…,K). The balances: f₁ = f(H) and f₂ = f(O) are the basis to formulate the linear combination f₁₂ = 2f₂ – f₁ = 2f(O) - f(H). For a redox system with K–K* electron-active elements (players), f₁₂ is linearly independent on f₀, f₃,…, f_K, i.e., the redox system is described by K independent balances f₀, f₁₂, f₃,…, f_K. For a non-redox system (K*=K), f₁₂ is linearly dependent on f₀, f₃,…, f_K, i.e., a redox system is described by K–1 independent balances f₀,f₃,…,f_K. Consequently, the linear combination

$$f_{12}+f_0-{\displaystyle \sum {}_{k=3}^{k*}}{d}_k.f_k\iff {\displaystyle \sum {}_{k=1}^{k*}}{d}_k.f_k-f_0$$

with dk equal to the oxidation numbers of the related elements, is reducible to identity, 0=0, for a non-redox system, i.e., f12 is not the independent balance in this system. In a redox system, f12 is the primary form of Generalized Electron Balance (GEB). The paper is completed by description of typical redox and non-redox systems according to GATES/GEB principles.

Electrolytic redox systems, GATES/GEB, Approach I to GEB, Approach II to GEB, Potentiometric titration.

Any electrolytic system in aqueous media is described with use of charge balance, f₀ = ChB, and K elemental or core balances, f_k = f(Y_k) (k = 1,…,K). The balances: f₁ = f(H) and f₂ = f(O) are the basis to formulate the linear combination f₁₂ = 2f₂ – f₁ = 2f(O) - f(H). For a redox system with K–K* electron-active elements (players), f₁₂ is linearly independent on f₀, f₃,…, f_K, i.e., the redox system is described by K independent balances f₀, f₁₂, f₃,…, f_K. For a non-redox system (K*=K), f₁₂ is linearly dependent on f₀, f₃,…, f_K, i.e., a redox system is described by K–1 independent balances f₀,f₃,…,f_K. Consequently, the linear combination

$$f_{12}+f_0-{\displaystyle \sum {}_{k=3}^{k*}}{d}_k.f_k\iff {\displaystyle \sum {}_{k=1}^{k*}}{d}_k.f_k-f_0$$

with dk equal to the oxidation numbers of the related elements, is reducible to identity, 0=0, for a non-redox system, i.e., f12 is not the independent balance in this system. In a redox system, f12 is the primary form of Generalized Electron Balance (GEB). The paper is completed by description of typical redox and non-redox systems according to GATES/GEB principles.

Electrolytic redox systems, GATES/GEB, Approach I to GEB, Approach II to GEB, Potentiometric titration.

The article concerns fundamental regularities, obligatory for electrolytic systems of different complexity, considered from the viewpoint of Generalized Approach to Electrolytic Systems (GATES), formulated by Michałowski (1992) [1,2]. According to GATES, a balancing of any electrolytic system is based on rules of conservation of particular elements, and on a charge balance, expressing the rule of electro-neutrality of this system. We refer to aqueous media, W=H₂O, where the species$$\left[X_i^{Zi}\right]$$ exist as hydrates $$\left[X_i^{Zi}\right].{{\displaystyle n}}_{iW};{{\displaystyle Z}}_i=0$$ ± 1, ± 2,…is a charge, expressed in elementary charge units, e = F/N_A (F = Faraday’s constant, N_A = Avogadro’s number, n_i = n_iW = n_iH₂O ≥ 0 is a mean number of water (W=H₂O) molecules attached to $$X_i^{Zi}$$ For ordering purposes, we assume Y1 = H (hydrogen) and Y2 = O (oxygen), and formulate first the linear combination

$$f_{12}=2•f_2-f_1=2•f\left(O\right)-f\left(H\right)$$ (1)

of elemental balances: f₁ = f(H) for hydrogen (H) and f₂ = f(O) for oxygen (O), related to the system considered. Then the linear combination of f₁₂ with charge balance (f_0,) and elemental/core balances f_k = f(Y_k) (k=3,…,K) for elements/cores Y_k ≠ H, O will be formulated [3]. A core is considered as a cluster of different atoms with defined composition (expressed by chemical formula), structure and external charge, e.g. SO₄^-2 is a core with external charge –2, composed of two elements: S and O. The balances fk (k=1,…,K) interrelate the numbers of different elements/ cores in components forming a system, and in the species thus formed. The notation of the balances (f₀, f₁, …, f_K) applied here is as follows

$$f_k=\left\{S_{ik}\right\}-\left\{C_{jk}\right\}=0\iff -f_k=\left\{C_{jk}\right\}-\left\{S_{ik}\right\}=0\iff f_k:\left\{S_{ik}\right\}=\left\{C_{jk}\right\}\iff -f_k:\left\{C_{ik}\right\}=\left\{S_{jk}\right\}$$ (2)

where {c_jk}is referred to a set of some components, {s_jk}- to some species in k-th balance; {c_i0}=0, i.e., $$\left\{c_{i0}\right\}=\varphi$$ (empty set, for f₀). Numbers of components of j-th kind are denoted by N_0j (j=1,…,J), numbers of the species $$s_{ik}=X_i^{zi}$$ of i-th kind are denoted by N_i (i=1,…,I). For balancing purposes, the hydrated species $$\left[X_i^{Zi}\right].{{\displaystyle n}}_{iW}$$ in the system are specified as $$X_i^{zi}$$ (N_i, n_i); e.g., the notation HSO₄^-1 (N₅, n₅) applied in Example 1 (below) refers to N₅ ions of HSO₄^-1∙n₅H₂O involving: N₅(1+2n₅) atoms of H, N₅(4+n₅) atoms of O, and N₅ atoms of S. Molar concentration of the species $$X_i^{zi}.niW$$ be denoted by $$\left[X_i^{Zi}\right]$$

The f₁₂ (Equation 1) is the primary form of Generalized Electron Balance (GEB), f₁₂ = pr-GEB, formulated (2005) for redox systems according to Approach II to GEB, where prior knowledge of ONs of all elements in components and species is not needed. The Approach II to GEB is fully compatible with the Approach I to GEB [4-15] formulated (1992) and based on the principle of common pool of electrons involved with electron-active elements, named as ‘players’ and distinguished from electron-non-active elements considered as ‘fans’, when a redox system is considered according to card game principle [15]; ONs for elements in components and species are needed in the Approach I. The principles of GEB formulation, discovered (1992, 2005) by Michałowski and resolved according to Generalized Approach to Electrolytic Systems (GATES) as GATES/GEB [1,16], were unknown in earlier literature.

GEB completes the set of K balances (f₀, f₁₂, f₃,…, f_K) as algebraic equations necessary for solving a redox system; K-1 balances (f₀, f₃,…, f_K) are required for solving a non-redox system; f₁₂ is omitted as derivative (not primary) balance. GEB is the law of Nature related to equilibrium, metastable or kinetic electrolytic redox systems, of any degree of complexity [1,15-24].

In this paper, the general properties of the linear combination of the balances (f₀, f₁₂, f₃,…, f_K) related to nonredox and redox systems of different degree of complexity, are discussed. Any redox system is involved with change of oxidation numbers of K-K* electron-active elements (players), where K* is the number of electron-non-active elements (fans). For a non-redox system, we have K*=K, i.e., K – K* = 0. Assuming that H and O participate the system as fans, we formulate the linear combination [15,16,25-33].

$$f_{12}+f_0-{\displaystyle {\sum }_{K=3}^{K*}{d}_k}.f_k=0\iff {\displaystyle {\sum }_{K=1}^{K*}{d}_k}.f_k-f_0=0$$ (3)

applicable for non-redox and redox systems, of different degree of complexity. For a non-redox system

$${\displaystyle {\sum }_{K=1}^{K*}{d}_k}.f_k=f_0$$ (3a)

is transformed into identity, 0 = 0. We prove that the equation (3)

(1^o) implies a general criterion distinguishing between non-redox and redox systems;

(2^o) defines d_k as oxidation numbers (ONs) of particular elements in components and species of a non-redox or redox system.

Synthesis of chemical and physical laws of conservation is expressed, respectively, by the equalities of left and right sides of Equations (3) and (3a).

Example 1: The (static) system is formed from the following components: Na2S·9H₂O (N₀₁), Na₂SO₄·10H₂O (N₀₂) (Sal Mirabilis), H₂O (N₀₃). The species in the system thus formed, namely:

H₂O (N₁), H⁺¹ (N₂, n₂), OH^-1 (N₃, n₃), Na⁺¹ (N₄, n₄), HSO₄^-1 (N₅, n₅), SO₄^-2 (N₆, n₆), H₂S (N₇, n₇), HS^-1 (N₈, n₈), S^-2 (N₉, n₉) are involved in the following balances:

f₀ = ChB:

N₂ – N₃ + N₄ – N₅ – 2N₆ – N₈ – 2N₉ = 0

f₁ = f(H):

2N₁ + N₂(1+2n₂) + N₃(1+2n₃) + 2N₄n₄ + N₅(1+2n₅) + 2N₆n₆ + N₇(2+2n₇) + N₈(1+2n₈) + 2N₉n₉

= 18N₀₁ + 20N₀₂ + 2N₀₃

f₂ = f(O):

N₁ + N₂n₂ + N₃(1+n₃) + N₄n₄ + N₅(4+n₅) + N₆(4+n₆) + N₇n₇ + N₈n₈ + N₉n₉ = 9N₀₁ + 14N₀₂ + N₀₃

f₁₂ = 2·f₂ – f₁:

= N₂ + N₃ + 7N₅ + 8N₆ – 2N₇ – N₈ = 8N₀₂

– f₃ = – f(Na):

2N₀₁ + 2N₀₂ = N₄

– 6·f₄ = 6·f(SO₄):

6N₀₂ = 6N₅ + 6N₆

2·f₅ = 2·f(S):

2N₇ + 2N₈ + 2N₉ = 2N₀₁

f₁₂ + f₀ – f₃ – 6·f₄ + 2·f₅

0 = 0

i.e., the linear combination (4) is transformed into identity; it can be rewritten into the form

(+1)·f₁ + (–2)·f₂ + (+1)·f₃ + (+6)·f₄ + (–2)·f₅ – f₀ = 0

(+1)·f(H) + (–2)·f(O) + (+1)·f(Na) + (+6)·f(SO₄) + (–2)·f(S) – ChB = 0

where the numbers in round brackets as multipliers for fi (i=1,…,5) are equal to ONs of the related elements. Within f12, and then within (5), N₁ and all n_iW values, within components and species, are cancelled. The species X_i^zi not involving H and/or O within X_i^Zi , are also cancelled within f12. More specifically, the f3 involving only one kind of species, is considered here as equality, not as equation.

In this system, symproportionation [21] of sulfur does not occur; from this viewpoint, the system is at a metastable state [1,13]. Consequently, all elements (H, O, Na, S) involved in this system are perceived as fans, i.e., K* = K = 4. Sulfide oxidation can occur in presence of sulfate-reducing bacteria [34]. Sulfide gives sulfate after oxidation with H₂O₂ [35], whereas sulfate ions are not reducible in usual procedure [36].

The elemental balance for S is f*(S) = f45 = f4 + f5.

The linear combination f₁₂ + f₀ – f₃ + p·f₄₅

is not transformable into identity 0 = 0, at any p-value. From (6) and (4) we get the contradiction p =6 and p =–2, i.e., f₄₅ is not applicable for checking f₁₂ (Eq. 1) as the criterion of independency of the related balances.

Example 2 (dynamic system). V₀ mL of titrand (D), containing ZnSO₄ (C₀) + NH₃ (C₁) + NH₄Cl (C₂) + NaH₂In (erio T, C_0In) is titrated with V mL of titrant (T) containing EDTA (C) [3].

The D is composed of N₀₁ molecules of ZnSO₄·7H₂O (goslarite), N₀₂ molecules of NH₃, N₀₃ molecules of NH₄Cl, N₀₄ molecules of NaH₂In = C₂₀H₁₂N₃O₇SNa, N₀₅ molecules of H₂O and the T is composed of N₀₆ molecules of EDTA = Na₂H₂L·2H₂O = C₁₀H₁₄N₂O₈Na₂·2H₂O and N₀₇ molecules of H₂O, at defined point of titration (V mL of T added). In the D+T system, the following species are formed:

H₂O (N₁), H⁺¹ (N₂, n₂), OH^-1 (N₃, n₃), HSO₄^-1 (N₄, n₄), SO₄^-2 (N₅, n₅), Cl^-1 (N₆, n₆), Na⁺¹ (N₇, n₇), NH₄ ⁺¹ (n₈, N₈), NH₃ (n₉,N₉), Zn⁺² (N₁₀, n₁₀), ZnOH⁺¹ (N₁₁, n₁₁), soluble complex Zn(OH)₂ (N₁₂, n₁₂), Zn(OH)₃^-1 (N₁₃, n₁₃), Zn(OH)₄^-2 (N₁₄, n₁₄),ZnNH₃ ⁺² (N₁₅, n₁₅), Zn(NH₃)₂⁺² (N₁₆, n₁₆), Zn(NH₃)₃⁺² (N₁₇,n₁₇), Zn(NH₃)₄ ⁺² (N₁₈, n₁₈), ZnCl⁺¹ (N₁₉, n₁₉); ZnSO₄ (N₂₀, n₂₀), C₂₀H₁₃N₃O₇S (N₂₁, n₂₁), C₂₀H₁₂N₃O₇S^-1 (N₂₂, n₂₂), C₂₀H₁₁N₃O₇S^-2 (N₂₃, n₂₃), C₂₀H₁₀N₃O₇S^-3 (N₂₄, n₂₄), C₂₀H₁₀N₃O₇SZn^-1 (N₂₅, n₂₅), (C₂₀H₁₀N₃O₇S)₂Zn^-4 (N₂₆, n₂₆), C₁₀H₁₀N₂O₈⁺² (H₆L⁺²)

(N₂₇, n₂₇), C₁₀H₁₇N₂O₈⁺¹ (H₅L⁺¹) (N₂₈, n₂₈), C₁₀H₁₆N₂O₈ (H₄L) (N₂₉, n₂₉), C₁₀H₁₅N₂O₈^-1 (H₃L^-1) (N₃₀, n₃₀), C₁₀H₁₄N₂O₈^-2 (H₂L^-2) (N₃₁, n₃₁), C₁₀H₁₃N₂O₈^-3 (HL^-3) (N₃₂, n₃₂), C₁₀H₁₂N₂O₈^-4 (L^-4) (N₃₃, n₃₃), C₁₀H₁₃N₂O₈Zn^-1 (ZnHL^-1) (N₃₄, n₃₄), C₁₀H₁₂N₂O₈Zn^-2 (ZnL^-2) (N₃₅, n₃₅), C₁₀H₁₃N₂O₉Zn^-3 (ZnOHL^-3) (N₃₆, n₃₆).

In particular, the complex ZnOHL^-3 is formed from ZnOH⁺¹ and L^-4.

The charge (ChB), elemental (f(H), f(O), f(Zn), f(Cl), f(Na)) and core (f(SO₄), f(NH₃), f(C₂₀H₁₀N₃O₇S), f(C₁₀H₁₂N₂O₈)) balances are written here as follows:

f0 = ChB:

N₂ – N₃ – N₄ – 2N₅ – N₆ + N₇ + N₈ + 2N₁₀ + N₁₁ – N₁₃ – 2N₁₄ + 2N₁₅ + 2N₁₆ + 2N₁₇ + 2N₁₈ + N₁₉ – N₂₂ – 2N₂₃

f₁ = f(H) :

2N₁ + N₂(1 + 2n₂) + N₃(1 + 2n₃) + N₄(1 + 2n₄) + 2N₅n₅ + 2N₆n₆ + 2N₇n₇ + N₈(4 + 2n₈) + N₉(3 + 2n₉) + 2N₁₀n₁₀

+ N₁₁(1 + 2n₁₁) + N₁₂(2 + 2n₁₂) + N₁₃(3 + 2n₁₃) + N₁₄(4 + 2n₁₄) + N₁₅(3 + 2n₁₅) + N₁₆(6 + 2n₁₆) + N₁₇(9 + 2n₁₇)

+ N₁₈(12 + 2n₁₈) + 2N₁₉n₁₉ + 2N₂₀n₂₀ + N₂₁(13 + 2n₂₁) + N₂₂(12 + 2n₂₂) + N₂₃(11 + 2n₂₃) + N₂₄(10 + 2n₂₄)

+ N₂₅(10 + 2n₂₅) + N₂₆(20 + 2n₂₆) + N₂₇(18 + 2n₂₇) + N₂₈(17 + 2n₂₈) + N₂₉(16 + 2n₂₉) + N₃₀(15 + 2n₃₀)

+ N₃₁(14 + 2n₃₁) + N₃₂(13 + 2n₃₂) + N₃₃(12 + 2n₃₃) + N₃₄(13 + 2n₃₄) + N₃₅(12 + 2n₃₅) + N₃₆(13 + 2n₃₆)

= 14N₀₁ + 3N₀₂ + 4N₀₃ + 12N₀₄ + 2N₀₅ + 18N₀₆ + 2N₀₇

f₂ = f(O) :

N₁ + N₂n₂ + N₃(1 + n₃) + N₄(4 + n₄) + N₅(4 + n₅) + N₆n₆ + N₇n₇ + N₈n₈ + N₉n₉ + N₁₀n₁₀

+ N₁₁(1 + n₁₁) + N₁₂(2 + n₁₂) + N₁₃(3 + n₁₃) + N₁₄(4 + n₁₄) + N₁₅n₁₅ + N₁₆n₁₆ + N₁₇n₁₇

+ N₁₈n₁₈ + N₁₉n₁₉ + N₂₀(4 + n₂₀) + N₂₁(7 + n₂₁) + N₂₂(7 + n₂₂) + N₂₃(7 + n₂₃) + N₂₄(7 + n₂₄)

+ N₂₅(7 + n₂₅) + N₂₆(14 + n₂₆) + N₂₇(8 + n₂₇) + N₂₈(8 + n₂₈) + N₂₉(8 + n₂₉) + N₃₀(8 + n₃₀)

+ N₃₁(8 + n₃₁) + N₃₂(8 + n₃₂) + N₃₃(8 + n₃₃) + N₃₄(8 + n₃₄) + N₃₅(8 + n₃₅) + N₃₆(9 + n₃₆)

= 11N₀₁ + 7N₀₄ + N₀₅ + 10N₀₆ + N₀₇

– 2·f3 = – 2·f(Zn)

2N₀₁ = 2N₁₀ + 2N₁₁ + 2N₁₂ + 2N₁₃ + 2N₁₄ + 2N₁₅ + 2N₁₆ + 2N₁₇ + 2N₁₈ + 2N₁₉ + 2N₂₀ + 2N₂₅ + 2N₂₆ + 2N₃₄ + 2N₃₅ + 2N₃₆

f₄ = f(Cl):

N₆ + N₁₉ = N₀₃

– f₅ = – f(Na):

N₀₄ + 2N₀₆ = N₇

– 6·f₆ = – 6·f(SO₄):

6N₀₁ = 6N₄ + 6N₅ + 6N₂₀

3·f₇ = 3·f(NH₃):

3N₈ + 3N₉ + 3N₁₅ + 6N₁₆ + 9N₁₇ + 12N₁₈ = 3N₀₂ + 3N₀₃

– f₈ = – f(C₂₀H₁₀N₃O₇S):

N₀₄ = N₂₁ + N₂₂ + N₂₃ + N₂₄ + N₂₅ + 2N₂₆

0·f₉ = 0·f(C₁₀H₁₂N₂O₈):

0∙N₂₇ + 0∙N₂₈ + 0∙N₂₉ + 0∙N₃₀ + 0∙N₃₁ + 0∙N₃₂ + 0∙N₃₃ + 0∙N₃₄ + 0∙N₃₅ + 0∙N₃₆ = 0∙N₀₆

The balance

f₁₂ = 2·f(O) – f(H):

– N₂ + N₃ + 7N₄ + 8N₅ – 4N₈ – 3N₉ + N₁₁ + 2N₁₂ + 3N₁₃ + 4N₁₄ – 3N₁₅ – 6N₁₆ – 9N₁₇ – 12N₁₈ + 8N₂₀ + N₂₁ + 2N₂₂ + 3N₂₃ + 4N₂₄ + 4N₂₅ + 8N₂₆ – 2N₂₇ – N₂₈ + N₃₀ + 2N₃₁ + 3N₃₂ + 4N₃₃ + 3N₃₄ + 4N₃₅ + 5N₃₆

= 8N₀₁ – 3N₀₂ – 4N₀₃ + 2N₀₄ + 2N₀₆

f₀ + f₁₂ – 2·f₃ + f₄ – f₅ – 6·f₆ + 3·f₇ – f₈ + 0·f₉

0 = 0

i.e., the the linear combination (10) is transformed into identity, 0 = 0. It can be rewritten into the form

(+1)·f₁ + (–2)·f₂ + (+2)·f₃ + (–1)·f₄ + (+1)·f₅ + (+6)·f₆ + (–3)·f₇ + (+1)·f₈ + (0)·f₉ – f₀ = 0

i.e. the numbers in round brackets as multipliers for f_i (i=1,…,9) are equal to ONs of the elements or charges ascribed to S in SO₄^-2, N in NH₃, C₂₀N₃S in all the species of erio T, C₁₀N₂ in all the species of EDTA.

This system needs some further comments. All elements (H, O, Na, Cl, Zn, S, N, C) in involved in the components and species of the system are considered as fans, i.e., K* = K = 8. The number, 8, of the elements is lower than the total number of charge and elemental/core balances (K=1+9=10).

The formula C₂₀H₁₃N₃O₇S for the neutral species of erio T can be rewritten into the form (C₂₀N₃S)(H₁₃O₇) (C₂₀N₃S)₁(H₂O)₆OH. The group of elements within C₂₀N₃S has the net charge x calculated from the equation 1∙x +6∙(0) + 1∙(-2+1) = 0 ⇒ x=+1. Similarly, in C₁₀H₁₆N₂O₈ = (C₁₀N₂) (H₁₆O₈) = (C₁₀N₂)1(H₂O)₈, the net charge x of the C₁₀N₂ group is calculated from the equation 1∙x + 8∙0 = 0, i.e., x = 0. The C₂₀N₃S⁺¹ and C₁₀N₂ can be also (optionally) considered as cores.

The f₆, f₇, f₈, f₉ are specified separately, for different cores: SO₄^-2, NH₃, C₂₀H₁₀N₃O₇S^-3, C₁₀H₁₂N₂O₈ ^-4, resp. Note that S enters the compounds and species in f₆, f₈; N enters the compounds and species in f₇, f₈, f₉; C enters the compounds and species in f₈, f₉. Furthermore, none transformations occur between the cores of the species belonging to separate concentration balances.

Referring again to the species involved with erio T, one can write the elemental balances: N₂₁ + N₂₂ + N₂₃ + N₂₄ + N₂₅ + 2N₂₆ = N₀₄ (for S); 3N₂₁ + 3N₂₂ + 3N₂₃ + 3N₂₄ + 3N₂₅ + 6N₂₆ = 3N₀₄ (for N); 20N₂₁ + 20N₂₂ + 20N₂₃ + 20N₂₄ + 20N₂₅ + 40N₂₆ = 20N₀₄ (for C).

All the equations are identical and equivalent to Eq. (8), because the core C₂₀H₁₀N₃O₇S is unchanged in reactions occurred during the titration. Similarly, the species involved with EDTA fulfill the relations: 10N₂₇ + 10N₂₈ + 10N₂₉ + 10N₃₀ + 10N₃₁ + 10N₃₂ + 10N₃₃ + 10N₃₄ + 10N₃₅ + 10N₃₆ = 10N₀₆ (for C), and 3N₂₇ + 3N₂₈ + 3N₂₉ + 3N₃₀ + 3N₃₁ + 3N₃₂ + 3N₃₃ + 3N₃₄ + 3N₃₅ + 3N₃₆ = 3N₀₆ (for N). Both equations are equivalent to f₉, Eq. (9).

Denoting the elemental balances for S, N and C as f^*(S) = f₆₈, f(N) = f₇₈₉, f(C) = f₈₉, we have the relations, expressed in the matrix form as follows

$$\left[\begin{array}{c}{f}_{68}\\ f_{789}\\ f_{89}\end{array}\right]=\left[\begin{array}{ccc}1& 0& 1\\ 0& 1& 3\\ 0& 0& 20\end{array}\begin{array}{c}0\\ 2\\ 10\end{array}\right]\left[\begin{array}{c}{f}_6\\ f_7\\ f_8\\ f_9\end{array}\right]$$

One can prove that the linear combination

f₀ + f₁₂ – 2·f₃ + f₄ – f₅ + p∙f₆₈ + q∙f₇₈₉ + r∙f₈₉

is not transformable into identity 0 = 0, at any of the (p, q, r) values. Namely, from Equations (10) and (12) we get

p (f₆ + f₈) + q(f₇ + 3f₈ + 2f₉) + r(20f₈ + 10f₉) = p∙f₆ + q∙f₇ + (p+3q+20r)∙f₈ + (2q+10r)∙f₉

– 6∙f₆ + 3∙f₇ – f₈ + 0∙f₉ p = –6, q = 3, and p+3q+20r = –1, 2q+10r = 0

r = (–1 – (–6) – 3∙3)/20 = –0.2; r = –2∙3/10 = –0.6.

The r = -0.2 and r = -0.6 are contradictory values. Then the option given by Eq. 12, with f₆₈, f₇₈₉, f₈₉, is not applicable for checking f₁₂ (Eq. 1) as the criterion of independency of the related balances.

Example 3: We consider here

(1o) T subsystem (volume V mL), composed of KMnO₄ (N₀₁) + H₂O (N₀₂) + CO₂ (N₀₃);

(2o) D subsystem (V₀ mL), composed of FeSO₄∙7H₂O (N0₄) + H₂SO₄ (N₀₅) + H₂O (N₀₆) + CO₂ (N₀₇);

and

(3o) D+T redox system (V0+V mL), as the mixture of D and T, where the following species are formed:

H₂O (N₁); H⁺¹ (N₂, n₂), OH^-1 (N₃, n₃), HSO₄^-1 (N₄, n₄),SO₄^-2 (N₅, n₅), H₂CO₃ (N₆, n₆), HCO₃^-1 (N₇, n₇),CO₃^-2 (N₈, n₈), Fe⁺² (N₉, n₉), FeOH⁺¹ (N₁₀, n₁₀), FeSO₄ (N₁₁, n₁₁),Fe⁺³ (N₁₂, n₁₂), FeOH⁺² (N₁₃, n₁₃), Fe(OH)₂⁺¹ (N₁₄, n₁₄), Fe₂(OH)₂⁺⁴ (N₁₅, n₁₅); FeSO₄⁺¹ (N₁₆, n₁₆), Fe(SO₄)₂^-1 (N₁₇, n₁₇), K⁺¹ (N₁₈, n₁₈), MnO₄^-1 (N₁₉, n₁₉), MnO₄^-2 (N₂₀, n₂₀), Mn⁺³ (N₂₁, n₂₁), MnOH⁺² (N₂₂, n₂₂), Mn⁺² (N₂₃, n₂₃), MnOH⁺¹ (N₂₄, n₂₄), MnSO₄ (N₂₅, n₂₅)

The presence of CO₂ in T and D is considered here as an admixture from air, to imitate real conditions of the analysis, on the step of preparation of D and T; the titration T(V) ⟹ D(V₀) is realized in the closed system, under isothermal conditions. Precipitation of MnO₂ does not occur at sufficiently low pH value [1].

The D+T dynamic redox system is then composed of nonredox static subsystems: D and T. On this basis, some general properties involved with non-redox and redox systems will be indicated. Different forms of GEB, resulting from linear combinations of charge and elemental balances related to D+T system, will be obtained.

To avoid (possible) disturbances, the common notation (subscripts) assumed in the set (14) of species will be applied for components and species in T, D and D+T. In context with the dynamic D+T system, T and D are considered as static (sub)systems.

The T subsystem: We get here the balances:

f₀ = ChB:

N₂ – N₃ – N₇ – 2N₈ + N₁₈ – N₁₉ = 0

f₁ = f(H):

2N₁ + N₂(1+2n₂) + N₃(1+2n₃) + N₆(2+2n₆) + N₇(1+2n₇) + 2N₈n₈ + 2N₁₈n₁₈ + 2N₁₉n₁₉ = 2N₀₂

f₂ = f(O):

N₁ + N₂n₂ + N₃(1+n₃) + N₆(3+n₆) + N₇(3+n₇) + N₈(3+n₈) + N₁₈n₁₈ + N₁₉(4+n₁₉)

= 4N₀₁ + N₀₂ + 2N₀₃

-4·f₃ = -4·f(CO₃) :

4N₀₃ = 4N₆ + 4N₇ + 4N₈

-f₄ = -f(K) :

N₀₁ = N₁₈

-7·f₅ = -7·f(Mn) :

7N₀₁ = 7N₁₉

f<12 = 2f2 – f1 = 2f(O) – f(H)

–N₂ + N₃+ 4N₆ + 5N₇ + 6N₈ + 8N₁₉ = 8N₀₁ + 4N₀₃

f₁₂ + f₀ – 4f₃ – f₄ – 7f₅ = 0 ⟺ (+1)∙f₁ + (–2)∙f₂ + (+4)∙f₃ + (+1)∙f₄ + (+7)∙f₅ – f₀ = 0

(+1)∙f(H) + (–2)∙f(O) + (+4)∙f(CO₃) + (+1)∙f(K) + (+7)∙f(Mn) – ChB = 0

0 = 0

i.e., the the linear combination (15) is transformed into identity, 0 = 0. The coefficients/multipliers at the balances f(Y_k) are equal to ONs of elements in the corresponding species.

The D subsystem: We get here the balances:

f₀ = ChB

N₂ – N₃ – N₄ – 2N₅ – N₇ – 2N₈ + 2N₉ + N₁₀ = 0

f₁ = f(H)

2N₁ + N₂(1+2n₂) + N₃(1+2n₃) + N₄(1+2n₄) + 2N₅n₅ + N₆(2+2n₆) + N₇(1+2n₇) + 2N₈n₈ + 2N₉n₉ + N₁₀(1+2n₁₀) + 2N₁₁n₁₁ = 14N₀₄ + 2N₀₅ + 2N₀₆

f₂ = f(O)

N₁ + N₂n₂ + N₃(1+n3) + N₄(4+n₄) + N₅(4+n₅) + N₆(3+n₆) + N₇(3+n₇) + N₈(3+n₈) + N₉n₉ +

N₁₀(1+n₁₀) + N₁₁(4+n₁₁) = 11N₀₄ + 4N₀₅ + N₀₆ + 2N₀₇

–4·f₃ = –4·f(CO₃)

4N₀₇ = 4N₆ + 4N₇ + 4N₈

–6·f6 = –6·f(SO₄)

6N₀₄ + 6N₀₅ = 6N₄ + 6N₅ + 6N₁₁

–2·f₇ = –2·f(Fe)

2N₀₄ = 2N₉ + 2N₁₀ + 2N₁₁

f₁₂ + f₀ – 4·f₃ – 6·f₆ – 2·f₇

0 = 0

i.e., the the linear combination (16) is transformed into identity, 0 = 0. From transformation of (16)

(+1)·f₁ + (–2)·f₂ + (+4)·f₃ + (+6)·f₆ + (+2)·f₇ – f₀

(+1)·f(H) + (–2)·f(O) + (+4)·f(CO₃)+ (+6)·f(SO₄) + (+2)·f(Fe) – ChB

we see again that the coefficients/multipliers at the balances f(Y_k) are equal to ONs of elements in the corresponding species.

For the D+T system, from (14) we have the balances:

f₀ = ChB

N₂ – N₃ – N₄ – 2N₅ – N₇ – 2N₈ + 2N₉ + N₁₀ + 3N₁₂ + 2N₁₃ + N₁₄ + 4N₁₅ + N₁₆ – N₁₇ + N₁₈ – N₁₉ – 2N₂₀ +

3N₂₁ + 2N₂₂ + 2N₂₃ + N₂₄ = 0

f₁ = f(H)

2N₁ + N₂(1+2n₂) + N₃(1+2n₃) + N₄(1+2n₄) + 2N₅n₅ + N₆(2+2n₆) + N₇(1+2n₇) + 2N₈n₈ + 2N₉n₉ + N₁₀(1+2n₁₀) + 2N₁₁n₁₁ + 2N₁₂n₁₂ + N₁₃(1+2n₁₃) + N₁₄(2+2n₁₄) + N₁₅(2+2n₁₅) + 2N₁₆n₁₆ + 2N₁₇n₁₇ + 2N₁₈n₁₈ + 2N₁₉n₁₉ + 2N₂₀n₂₀ + 2N₂₁n₂₁ + N₂₂(1+2n₂₂) + 2N₂₃n₂₃ + N₂₄(1+2n₂₄) + 2N₂₅n₂₅

= 2N₀₂ + 14N₀₄ + 2N₀₅ + 2N₀₆

f₂ = f(O)

N₁ + N₂n₂ + N₃(1+n₃) + N₄(4+n₄) + N₅(4+n₅) + N₆(3+n₆) + N₇(3+n₇) + N₈(3+n₈) + N₉n₉ +

N₁₀(1+n₁₀) + N₁₁(4+n₁₁) + N₁₂n₁₂ + N₁₃(1+n₁₃) + N₁₄(2+n₁₄) + N₁₅(2+n₁₅) + N₁₆(4+n₁₆) + N₁₇(8+n₁₇) + N₁₈n₁₈ + N₁₉(4+n₁₉) + N₂₀(4+n₂₀) + N₂₁n₂₁ + N₂₂(1+n₂₂) + N₂₃n₂₃ + N₂₄(1+n₂₄) + N₂₅(4+n₂₅)

= 4N₀₁ + N₀₂ + 2N₀₃ + 11N₀₄ + 4N₀₅ + N₀₆ + 2N₀₇

– 4·f₃ = – 4·f(CO₃):

4N₀₃ + 4N₀₇ = 4N₆ + 4N₇ + 4N₈

– f₄ = – f(K) :

N₀₁ = N₁₈

– f₅ = – f(Mn)

N₀₁ = N₁₉ + N₂₀ + N₂₁ + N₂₂ + N₂₃ + N₂₄ + N₂₅

– 6·f₆ = – 6·f(SO₄)

6N₀₄ + 6N₀₅ = 6N₄ + 6N₅ + 6N₁₁ + 6N₁₆ + 12N₁₇ + 6N₂₅

– f₇ = – f(Fe)

N₀₄ = N₉ + N₁₀ + N₁₁ + N₁₂ + N₁₃ + N₁₄ + 2N₁₅+ N₁₆ + N₁₇

Then we have, by turns,

f₁₂ = 2·f(O) – f(H) :

– N₂ + N₃ + 7N₄ + 8N₅ + 4N₆ + 5N₇ + 6N₈ + N₁₀ + 8N₁₁ + N₁₃ + 2N₁₄ + 2N₁₅ + 8N₁₆ + 16N₁₇ + 8N₁₉ + 8N₂₀ + N₂₂ + N₂₄ + 8N₂₅ = 8N₀₁ + 4N₀₃ + 8N₀₄ + 6N₀₅ + 4N₀₇

f₁₂ + f₀ – 4·f₃ – f₄ – 6·f₆:

2(N₉+N₁₀+N₁₁) + 3(N₁₂+N₁₃+N₁₄+2N₁₅+N₁₆+N₁₇) + 7N₁₉ + 6N₂₀ + 3(N₂₁+N₂₂) + 2(N₂₃+N₂₄+N₂₅) = 7N₀₁ + 2N₀₄

From transformation of (26) we have

(+1)ff₁ + (-2)ff₂ + (+4)ff₃ + (+1)ff₄ + (+6)ff₆ – f₀

(+1)·f(H) + (-2)·f(O) + (+4)·f(CO₃) + (+1)·f(K) + (+6)·f(SO₄)-ChB

We see that the coefficients/multipliers at the related balances f_k (k=1,2,3,4,6) are equal to ONs of elements in the corresponding species; the balances f_k in (28) are related to fans.

Applying the relations:

[ X_i^Zi ]·(V₀+V) = 10³·N_i/N_A , CV = 10³·N₀₁/N_A ,

C₀V₀ = 10³·N₀₄/N_A , C1V = 10³·N₀₃/N_A ,

C₀₁V₀ = 10³·N₀₅/N_A , C₀₂V₀ = 10³·N₀₇/N_A.

we rewrite Eq. (26) in the more explicit form as

2([Fe⁺²] + [FeOH⁺¹] + [FeSO₄]) + 3([Fe⁺³] + [FeOH⁺²] + [Fe(OH)₂⁺¹] + 2[Fe₂(OH)₂⁺⁴] + [FeSO₄⁺¹] + [Fe(SO₄)₂ ^-1])

+ 7[MnO₄^-1] + 6[MnO₄^-2] + 3([Mn⁺³] + [MnOH⁺²]) + 2([Mn⁺²]+ [MnOH⁺¹] + [MnSO₄])

= (7CV + 2C₀V₀)/(V₀+V)

Eq. (30), obtained from (25,28,29), consists only of the species, where players are involved. Other linear combinations were also used. Applying atomic numbers: Z_Mn = 25 and Z_Fe = 26, we have

ZFe·f7 + ZMn·f5 - (f12 + f0 - 4·f3 - f4 - 6·f6) :

(Z_Fe–2)·(N₉+N₁₀+N₁₁) + (Z_Fe– 3)·(N₁₂+N₁₃+N₁₄+2N₁₅+N₁₆+N₁₇) + (Z_Mn^–7)·N₁₉ + (Z_Mn^–6)·N₂₀ + (Z_Mn^–3)·(N₂₁+N₂₂) + (Z_Mn^–2)·(N₂₃+N₂₄+N₂₅) = (Z_Fe–2)·N₀₄ + (Z_Mn^–7)·N₀₁

(Z_Fe–2)·([Fe+2] + [FeOH⁺¹] + [FeSO₄]) + (Z_Fe–3)·([Fe⁺³] + [FeOH⁺²] + [Fe(OH)₂⁺¹]+2[Fe₂(OH)₂⁺⁴] + [FeSO₄⁺¹] + [Fe(SO₄)₂ ^-1])+(Z_Mn–7)·[MnO₄^–1] + (Z_Mn^–6)·[MnO₄^–2] + (Z_Mn–3)·([Mn⁺³] + [MnOH⁺²]) + (Z_Mn–2)·([Mn⁺²]+[MnOH⁺¹] + [MnSO₄])

= ((Z_Fe–2)·C₀V₀ + (Z_Mn–7)·CV)/(V₀+V)

Equation (32) results innediately from the Approach I to GEB, see [6].

The least extended (the most compact) form is as follows [8,25]

3·f₇ + 2·f₅ – (f₁₂ + f₀ – 4·f₃ – f₄ – 6·f₆):

(N₉ + N₁₀ + N₁₁) – (5N₁₉ + 4N₂₀ + N₂₁ + N₂₂) = N₀₄ – 5N₀₁ · [Fe⁺²] + [FeOH⁺¹] + [FeSO₄] – (5[MnO₄^–1] + 4[MnO₄^–2] +[Mn⁺³] + [MnOH⁺²])

-(C₀V₀ – 5·CV)/(V₀+V)=0

Eq. (25) for f₁₂ (24), considered as pr-GEB (Eq. 1), can also be rewritten in terms of concentrations; we have

– [H⁺¹] + [OH^-1] + 7[HSO₄^-1] + 8[SO₄^-2] + 4[H₂CO₃] +5[HCO₃ ^-1] + 6[CO₃^-2] + [FeOH⁺¹] + 8[FeSO₄] +[FeOH⁺²] +2[Fe(OH)₂ ⁺¹] + 8[Fe(SO₄⁺¹] + 16[Fe(SO₄)₂^-1] + 8[MnO₄^-1] + 8[MnO₄^-2] + [MnOH⁺²] + [MnOH⁺¹] + 8[MnSO₄]

= (8CV + 8C₀V₀ + 4C₁V + 6C₀₁V₀ + 4C₀₂V₀)/(V₀+V)

Equations 30, 32–34 are equivalent forms of GEB for this system. Other linear combinations of the balances are also admitted/possible for this purpose; none of them are reduced to the identity 0 = 0. However, the shortest Eq. (33), chosen arbitrarily, seems to be the most useful for calculation purposes–for obvious reasons.

The T and D are non redox subsystems of the redox D+T system; this is not the general regularity, of course. In some other systems, D or T or both (D and T) can form redox subsystems. For example, the Br₂ solution considered in [5,6,37,38] is the redox subsystem D; I₂ + KI solution is the redox subystem T in [39].

Equation 33 is completed by charge and concentration balances, obtained from Equations 18, 19, 21, 22, 23 and relations (29). We have, by turns,

[H⁺¹] – [OH^-1] – [HSO₄^-1] – 2[SO₄^-2] – [HCO₃^-1] – 2[CO₃^-2] + 2[Fe⁺²] + [FeOH⁺¹] + 3[Fe⁺³] + 2[FeOH⁺²] + [Fe(OH)₂⁺¹]+ 4[Fe₂(OH)₂⁺⁴] + [FeSO₄⁺¹] – [Fe(SO₄)₂^-1] + [K⁺¹] – [MnO₄ ^-1] –2[MnO₄^-2] + 3[Mn⁺³] + 2[MnOH⁺²] + 2[Mn⁺²] + [MnOH⁺¹] = 0

[H₂CO₃] + [HCO₃^-1] + [CO₃^-2] – (C₀₂V₀+C₁V)/(V₀+V) = 0

[MnO₄^-1] + [MnO₄^-2] + [Mn⁺³] + [MnOH⁺²] + [Mn⁺²] + [MnOH⁺¹] + [MnSO₄] – CV/(V₀+V) = 0

[HSO₄^-1] + [SO₄^-2] + [FeSO₄] + [FeSO₄⁺¹] + 2[Fe(SO₄)₂^-1] + [MnSO₄] – (C₀+C₀₁)V₀/(V₀+V) = 0

[Fe⁺²] + [FeOH⁺¹] + [FeSO₄] + [Fe⁺³] + [FeOH⁺²] +[Fe(OH)₂⁺¹] + 2[Fe₂(OH)₂⁺⁴]+[FeSO₄⁺¹] + [Fe(SO₄)₂-1] – C₀V₀/(V₀+V) = 0

The equality (not equation!)

[K⁺¹] = CV/(V₀+V)

can enter immediately Eq. 18a like a number, at defined V-value.

Concentrations of the species involved in the set of 6 equations: 33, 18a, 19a, 21a, 22a, 23a, are compatible with the complete set of equilibrium constants, specified as follows:

[H⁺¹] [OH^-1] = 10^-14.0; [HSO₄^-1] = 10^1.8[H⁺¹][SO₄^-2];

[H₂CO₃] = 10^16.4[H⁺¹]²[CO₃²];[HCO₃^-1] = 10^10.1[H⁺¹] [CO₃^-2];

[Fe⁺³] = [Fe⁺²]∙10^{A(E – 0.771)}; [FeOH⁺¹] =10^4.5[Fe⁺²][OH^-1];

[FeOH⁺²] = 10^11.0[Fe⁺³][OH^-1]; [Fe(OH)₂⁺¹] =

10^21.7[Fe⁺³][OH^-1]²; [Fe₂(OH)₂⁺⁴] = 10^21.7[Fe⁺³]²[OH^-1]²;

[FeSO₄] = 10^2.3[Fe⁺²][SO₄^-2]; [FeSO₄⁺¹] = 10^4.18[Fe⁺³][SO₄^-2];

[Fe(SO₄)₂^-1] = 10^7.4[Fe⁺³][SO₄^-2]²;

[MnO₄^-1] = [Mn⁺²]∙10^{5A(E – 1.507) + 8pH};

[MnO₄^-2] = [Mn⁺²]∙10^{4A(E – 1.743) + 8pH};

[Mn⁺³] = [Mn⁺²]∙10^{A(E – 1.509)};

[MnOH⁺²] = 10^14.2[Mn⁺³][OH^-1]

The electrode potentials E [V] are put in context with standard electrode potentials E_0i, expressed in SHE scale [40].

6 independent variables, chosen as components of the vector:

x = [x₁,…,x₆]^T = [E,pH,pMn2,pFe2,pSO4,pH2CO3]^T

where E – potential [V], pH = – log[H⁺¹], pMn2 = – log[Mn⁺²], pFe2 = – log[Fe⁺²], pSO4 = – log[SO₄ ^-2], pH2CO3 = – log[H₂CO₃].

All the variables, put in logarithmic scale, are considered as functions of volume V of the titrant T added during the titration T ⇨ D, xi = x i(V). The V [mL] is considered as parameter of the D+T system. On this basis, the mole fraction values

$$\varphi =\frac{C.V}{C_0.V_0}$$

are calculated at pre-assumed V0, C, C0 values. The Φ provides a kind of normalization (independence on V0 value) in the system, and is taken as the independent variable on the abscissa of the related plots. The knowledge of the xi = xi (V) values, allows also to calculate the concentrations $$\left[X_i^{Zi}\right]$$ of the different species of the D+T system on the basis of relations (35) and present these changes as plots on the relevant speciation diagrams log $$\left[X_i^{Zi}\right]={\theta }_i\left(\varphi \right)$$ , together with E = E(Φ) and pH = pH (Φ) relationships. Graphical presentation of the data provides an excellent tool for qualitative and quantitative evaluation of processes, occurring at any point of the titration.

All these data for the D+T system are obtained on the basis of calculations realized with use of the MATLAB [1] computer program.

function F = Function_MnO4_Fe(x)

global V Vmin Vstep Vmax V0 C C1 C0 C01 C02 H OH fi pH E

global Kw pKw A K logK

global HSO4 SO4 logHSO4 logSO4

global H2CO3 HCO3 CO3

global logH2CO3 logHCO3 logCO3

global Mn7O4 Mn6O4 Mn3 Mn3OH

global logMn7O4 logMn6O4 logMn3 logMn3OH

global Mn2 Mn2OH Mn2SO4

global logMn2 logMn2OH logMn2SO4

global Fe2 Fe2OH Fe2SO4

global logFe2 logFe2OH logFe2SO4

global Fe3 Fe3OH Fe3OH2 Fe32OH2 Fe3SO4 Fe3SO42

global logFe3 logFe3OH logFe3OH2 logFe32OH2 logFe3SO4 logFe3SO42

E=x(1);

pH=x(2);

Mn2=10.^-x(3);

Fe2=10.^-x(4);

SO4=10.^-x(5);

H2CO3=10.^-x(6);

H=10.^-pH;

pKw=14;

Kw=10.^-14;

OH=Kw./H;

A=16.9;

Mn7O4=Mn2.*10.^(5.*A.*(E-1.507)+8.*pH);

Mn6O4=Mn2.*10.^(4.*A.*(E-1.743)+8.*pH);

Mn3=Mn2.*10.^(A.*(E-1.509));

Fe3=Fe2.*10.^(A.*(E-0.771));

HSO4=10.^1.8.*H.*SO4;

CO3=10.^-16.4*H^-2*H2CO3;

HCO3=10.^10.1*H*CO3;

Fe2OH=10.^4.5.*Fe2.*OH;

Fe2SO4=10.^2.3.*Fe2.*SO4;

Fe3OH=10.^11.0.*Fe3.*OH;

Fe3OH2=10.^21.7.*Fe3.*OH.^2;

Fe32OH2=10.^25.1.*Fe3.^2.*OH.^2;

Fe3SO4=10.^4.18.*Fe3.*SO4;

Fe3SO42=10.^7.4.*Fe3.*SO4.^2;

Mn2OH=10.^3.4.*Mn2.*OH;

Mn2SO4=10.^2.28.*Mn2.*SO4;

Mn3OH=10.^14.2.*Mn3.*OH;

K=C.*V./(V0+V);

%Charge balance

F=[(H-OH -HSO4-2.*SO4-HCO3-2.*CO3-+2.*Fe2+Fe2OH... +3.*Fe3+2.*Fe3OH+Fe3OH2+4.*Fe32OH2+Fe3SO4- Fe3SO42...

+K-Mn7O4-2.*Mn6O4+3*Mn3+2.*Mn3OH+2.*Mn2+Mn2 OH);

%Concentration balance of Mn

(Mn7O4+Mn6O4+Mn3+Mn3OH+Mn2+Mn2OH+Mn2 SO4-C.*V./(V0+V));

%Concentration balance of Fe

(Fe2+Fe2OH+Fe2SO4+Fe3+Fe3OH+Fe3OH2+2.*Fe32 OH2...+Fe3SO4+Fe3SO42-C0.*V0./(V0+V));

%Concentration balance of SO4

(HSO4+SO4+Mn2SO4+Fe2SO4+Fe3SO4+2.*Fe3SO42- (C0+Ca).*V0./(V0+V));

%Concentration balance of CO3

(H2CO3+HCO3+CO3-(C02.*V0+C1.*V)/(V0+V));

%Electron balance

(Fe2+Fe2OH+Fe2SO4-(5.*Mn7O4+4.*Mn6O4+ Mn3+Mn3OH)... -(C0.*V0.-5.*C.*V.)/(V0+V))];

logMn2=log10(Mn2);

logMn2OH=log10(Mn2OH);

logMn2SO4=log10(Mn2SO4);
logMn3=log10(Mn3);
logMn3OH=log10(Mn3OH);
logMn6O4=log10(Mn6O4);
logMn7O4=log10(Mn7O4);
logFe2=log10(Fe2);
logFe2OH=log10(Fe2OH);
logFe2SO4=log10(Fe2SO4);
logFe3=log10(Fe3);
logFe3OH=log10(Fe3OH);
logFe3OH2=log10(Fe3OH2);
logFe32OH2=log10(Fe32OH2);
logFe3SO4=log10(Fe3SO4);
logFe3SO42=log10(Fe3SO42);
logHSO4=log10(HSO4);
logSO4=log10(SO4);
logH2CO3=log10(H2CO3);
logHCO3=log10(HCO3);
logCO3=log10(CO3);
logK=log10(K);

The results of calculations made at V₀=100, C₀=0.01, C=0.02, C₀₁=0.5, C₀₂=C₁ = 0 are plotted in Figures 1 and 2. The jump of E on the curve in Fig. 1a occurs at Φ = Φ_eq = 0.2, i.e., at the equivalent (eq) point where C∙V_eq = 0.2·C0∙V0. Relatively small pH changes (Figure 1b) result from high buffer capacity of the titrand D [41-46]. From Figure 2a we see that [Fe⁺³] << [FeSO₄ ⁺¹] << [Fe(SO₄)₂ ^-1]. Note that MnOH+2 and Mn+3 (not MnO4 -1) ions are the predominating manganese species immediately after crossing the related equivalence point (Figure 2b), Some points from the vicinity of equivalence point are presented in Table 1.

Option 1. To indicate a complexation effect of sulphate ions, introduced by the H₂SO₄ solution, we compare the plots of E = E(Φ) curves: (1) the curve from Fig. 1a with one (2) obtained after omission of the sulfate complexes (FeSO₄, FeSO₄⁺¹, Fe(SO₄)₂^-1, MnSO₄) from the algorithm in section 4.5. The related plots are compared in Figure 3.

Option 2. The sulphate complexes formed by Mn⁺³ ions are unknown in literature, although on the basis of analogy with other trivalent ions (e.g., Fe⁺³, Al⁺³) it could be expected that the stability constants Ki for virtual Mn(SO₄)i ^+3-2i complexes, [Mn(SO₄)_i^+3-2i] = K_i·[Mn⁺³] [SO₄ ^-2]ⁱ, may have significant values. However, a numerical analysis of the data obtained for the pre-assumed stability constants Ki of sulphate complexes (Figure 4) with the curve obtained experimentally [14] has revealed that the Mn(SO₄)_i ^+3-2i complexes – if they exist – are relatively weak [6].

The quantitative, algebraic description of any electrolytic system according to GATES principles is based on electro neutrality rule, and on rules of conservation of particular elements in the systems, where none radioactive transformations occur [2]. The electro-neutrality rule is expressed by charge balance (f₀ = ChB). The conservation of particular elements is expressed in terms of elemental and core balances f_k = f(Y_k), where Y_k (k=1,2,…,K) is an element or core. For ordering purposes, it is assumed that Y₁ = H, Y₂ = O, and the linear combination f₁₂ = 2·f₂ – f₁ = 2∙f(O) – f(H) is formulated. In redox systems, K* electron-non-active elements/cores (‘fans’), and K–K* electron-active elements (‘players’) are distinguished. In all examples presented here, H and O did not participate in redox systems as players. The set of K balances (f₀, f₁₂, f₃,…, f_k) is needed for mathematical formulation of a redox system (i.e., at K* < K), whereas the K–1 balances f₀, f₃,…, f_k are needed for resolution of a non-redox system, where K* = K and f₁₂ is the balance linearly dependent on f₀, f₃,…,f_k. The linear independency or dependency of f12 within the balances f₀, f₁₂, f₃,…,f_k is then the general, dichotomous criterion distinguishing between redox and non-redox systems. The linear combination * $${\displaystyle \sum {}_{k=1}^{K*}}{d}_k.f_k-f_0$$ (Eq. 3) applied to a non-redox system (1º) gives the identity, 0 = 0, for a non-redox system, or (2 º) does not give the identity for a redox system, also after any linear combination with K - K* balances for the players in this system. These regularities are valid for coefficients dk equal to oxidation numbers (ONs) of the elements in the corresponding balances f_k = f(Y_k) for elements/cores Yk related to fans.

The f12 and any linear combination of f₁₂ with f₀, f₃,…, f_k, have full properties of Generalized Electron Balance (GEB), completing the set of K balances, f₀,f₁₂,f₃,…,f_k, needed for resolution of a redox system, of any degree of complexity. The supreme role of this independency/dependency criterion, put also in context with calculation of ONs, is of great importance, in context with the contractual nature of the ON concept known from the literature issued hitherto [47-49]. These regularities are the clear confirmation of the Emmy Noether’s general theorem [50] applied to conservation laws of a physical/electrolytic system, expressed in terms of algebraic equations, where GEB is perceived as the Law of Nature [15], as the hidden connection of physicochemical laws, and as the breakthrough in thermodynamic theory of electrolytic redox systems. Resolution of several redox systems, according to principles of Generalized Approach to Electrolytic Systems (GATES), is presented. The GATES/ GEB is the best thermodynamic approach of electrolytic redox systems, of any degree of complexity. The Generalized Equivalent Mass (GEM) concept [14], fully compatible with the GATES principles, was introduced. The mathematical models, applicable for handling the results obtained from potentiometric titrations in redox systems, were also presented. The chapter offers – undoubtedly – the best possible ways to resolution of the issues raised. Formulation of an electrolytic redox system specified in section 3.1.3 illustrates the huge possibilities/advantages inherent in GATES/GEB.

All the regularities specified above are valid for redox and non-redox systems of any degree of complexity, also for the multi-solvent systems [51-54].

The key role of H and O (not free electrons, e^-1) in redox systems, inherent in f₁₂, was stressed in [23]. Contrary to appearances, established by the current paradigm “obligatory” till now, the criterion distinguishing non-redox and redox systems is not immediately associated with free electrons in the system. The new/fundamental/practical criterion involved with f₁₂ = 2∙f(O) - f(H) and its properties, unknown in earlier literature, provides a kind of uniformity in the formulas derived for this purpose. This fact, especially the simple calculations of free electron concentrations in redox systems [23], deny the unique role of free electrons in redox reactions. On the other hand, it points to the decisive role of H and O in redox systems [2], suggested elsewhere, in earlier theoretical/hypothetical considerations on these systems. Here is the hidden simplicity, which had to be discovered, as the Approach II to GEB. The author ™ contends that the discovery of the Approach II GEB would most likely be impossible without the prior discovery of the Approach I to GEB.

The GEB concept, valid for electrolytic redox systems, is the emanation of balances for H and O, referred to aqueous media. GEB is compatible with other (charge and concentration) balances and enables to resolve the electrolytic (mono- or/and two-phase) redox systems of any degree of complexity, within the scope of GATES, perceived as the thermodynamic approach to equilibrium and metastable systems, where all necessary physicochemical knowledge on the systems tested is involved. The GATES is perceived as the unrivalled tool applicable, among others: (a) to mathematical modelling of thermodynamic behavior of the systems, (b) in choice of optimal a priori conditions of chemical analyses, and (c) in gaining chemical information invisible in real experiments, in general.

GATES/GEB is a counter-proposal in relation to earlier IUPAC decisions, presented in three subsequent editions of the Orange Book, and based on the reaction stoichiometry; that viewpoint was criticized unequivocally/exhaustively/ convincingly, especially in a series of authors’ articles cited herein. It was demonstrated, on examples of redox systems of different complexity, that stoichiometry is a secondary/ derivative/ “fragile” concept, from the viewpoint of GATES, and GATES/ GEB, in particular.

Conservation laws of physics are very closely related to the symmetry of physical laws under various transformations. The nature of these connections is an intriguing physical problem. The theory of these connections, as it appears in classical physics, constitutes one of the most beautiful aspects of mathematical physics. It confirms a general theorem of Emmy Noether which states that symmetries and conservation laws of a physical system correspond to each other [50]. The Noether's conceptual approach to algebra led to a body of principles unifying algebra, geometry, linear algebra, topology, and logic. The theory of this connection constitutes one of the most beautiful chapters of mathematical physics.

Concluding, GATES is the overall, thermodynamic approach to redox and non-redox, static and dynamic, single and multiphase equilibrium, metastable and nonequilibrium electrolytic systems, of any degree of complexity. Possibilities of GATES/GEB are far greater than ones offered by the actual physicochemical knowledge related to the system in question.

Stoichiometry, oxidation number, equivalent mass, order of reaction, equivalent mass [14], etc. are derivative (not primary!) concepts within GATES. The Equilibrium Law (EL), based on the Gibbs function and the Lagrange multipliers idea [15], can be put instead of Mass Action Law (MAL), based on a stoichiometric reaction notation, and other principles. Equilibrium, kinetic and metastable systems are distinguished. Within GATES, thermodynamics of electrolytic systems is based on purely algebraic principles; the stoichiometry is considered here only as a kind of “dummy” [22,23,55].

Summarizing, this paper offers the best possible ways to resolution of the issues raised.