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Anna Maria Micha∤owska Kaczmarczyk^{1} and Tadeusz Micha∤owski*^{2}

^{1}Department of Oncology, The University Hospital in Cracow, Poland^{2}Department of Analytical Chemistry, Technical University of Cracow, Poland

**Received:** July 18,2018; **Published:** August 16,2018

***Corresponding author: ** Tadeusz Micha∤owski, Department of Analytical Chemistry, Technical University of Cracow, Cracow, Poland

**DOI:** 10.26717/BJSTR.2018.08.001591

The linear combination f_{12} = 2 f(O) - f(H) of elemental balances: f^{1} = f(H) for H and f_{2} = f(O) for O is the keystone in formulation of Generalized Electron Balance (GEB), which completes the set of equations necessary for resolution of an electrolytic redox system according to Approach II to GEB in aqueous media. The f does not involve the numbers of free and hydrating water molecules. The linear dependency or independency of f_{12} from charge balance (f_{0} = ChB) and other elemental/core balances f_{k} = f(Y_{k}) (k=3...,K) for elements/cores Y_{k} of the system is the general criterion distinguishing between non-redox and redox systems. This criterion is provided by linear combination where K* (≤K) is the number of electron-non-active elements (fans) of the system, and dk are the multipliers related to oxidation numbers (ONs) of the elements involved in Y_{k}. For non-redox systems, LC is transformed into identity, 0 = 0, whereas for redox systems, the LC and each linear combination are different from identity. The ONs, reductant, oxidant are derivative concepts in this formulation. All these interesting regularities are illustrated by simulated titration in a symproportionating redox system, resolved according to Generalized Approach to Electrolytic Systems (GATES) principles.

The Generalized Electron Balance (GEB) [1-6] concept, related to electrolytic redox systems, is put in context with the principle of conservation of all elements in electrolytic redox systems, in aqueous, non-aqueous or mixed-solvent media. Two equivalent approaches to GEB, termed as the Approaches I and II to GEB, were formulated. The Approach I, when perceived in convention of card game, is based on the common pool of electrons as money, introduced by electron-active elements, named as players; elec- tron-non-active elements are called there as fans [3]. The GEB, formulated according to Approach II, results from the linear combination f_{12} = 2-f(O) - f(H) of elemental balances: f_{1} = f(H) for H, and f_{2} = f(O) for O. For a redox system, the f is linearly independent on charge balance (f_{0} = ChB) and other elemental balances f_{k} = f(Y_{k}) (Y_{k} ≠ H, O; k=3...,K); all the balances are presented here as equations, fk = 0. For a non-redox system, f is linearly dependent on f_{0},f_{3}...,f_{K}. The redox system is described with use of K independent balances f_{0},f_{12},f_{3}...,f_{K}, whereas a non-redox system is described with use of K-1 independent balances f_{0},f_{3}...,f_{K}. This property of f_{12} provides the general criterion distinguishing between redox and non-redox systems of any degree of complexity, namely: equilibrium, non-equilibrium and metastable, mono- and polyphase, static and dynamic electrolytic systems, also in mixed-solvent media. The GEB enters, as GATES/GEB, the Generalized Approach to Electrolytic Systems (GATES), i.e. GATES/GEB ⊂ GATES.

The linear combination (1) is formulated, where all fans (K* elements) of the system are involved; K* = K for a non-redox system, K* < K for a redox system, where K-K* players are involved. When dk are equal to the oxidation numbers (ONs) of elements in the corresponding fans (k=1...,K) of a non-redox system, then LC is transformed into identity, 0 = 0. For a redox system, LC assumes the simpler form, where only the species and components related to players are involved. The f_{12} is considered as the primary form of GEB, f_{12} = pr-GEB. The f_{12}, LC (Eq. 1) and any other combination have full properties of GEB, although - for obvious reasons - the simplest/shortest form of GEB, involving the smallest number of components, is more desirable. The GEB concept is perceived as the law of the matter conservation, as the general law of Nature, as the hidden connection of physicochemical laws, and as a breakthrough in thermodynamic theory of electrolytic redox systems [3,7-15]. The GATES/GEB is perceived as an example of excellent paradigm, as the best possible thermodynamic approach to electrolytic redox systems [16].

We refer here to a dynamic D+T system (aqueous medium), composed of titrand D and titrant T; V mL of T is added into V_{0} mL of D according to titrimetric mode, and V0+V mL of D+T mixture is obtained at a given point of the titration T(V) D(V0), if the assumption of the volumes additivity is valid/tolerable. In this system, the species *X _{i}^{(zi)}*> exist as hydrates

It is advisable to start the balancing from the interrelations between numbers of particular entities: N_{0j} for components repre-sented by molecules composing D and T, and N_{i} - for species (ions and molecules) of i-th kind (i = 1...,I), where I is the number of kinds of the species *x ^{zi}_{i} n_{iW}* in D+T. The N

‘Electrolytic (redox and non-redox) systems are formulated according to the GATES principles [1], and the set of K+1 balances f_{0}, f_{1}, f_{2}, f_{3}..., f_{K} is thus obtained. The charged/ionic species ** X,o -n_{W}** of the system are involved in the charge balance (ChB)

applicable for static and dynamic systems. Note that 0- [X_{m}]= 0 for a species X_{m}. n_{mw} with zero charge (z_{m}=0), e.g., 0^ [H_{2}O] = 0 (z_{1}=0).

Free water particles (N_{1}), and water bound in the hydrates

Next, the linear combination

is obtained, where a_{ki} and b_{kj} in equations (4)-(6) are the numbers of atoms/cores y(k=1 _{k}(k=1.....k)in x^{zi}_{i} n_{iw} and in the j-th component of the system, resp. A core is perceived as non-transformable cluster of atoms, of the same elemental composition (expressed by chemical formula) and structure; the cores are involved only with fans. One can notice that: (a) N_{1} involved with H_{2}O as the species, (b) all N_{0i} related to water as the component of the system, and (c) all n_{i} = n_{iW} specified in f(H) and f(O), are cancelled within 2-f(O) - f(H) in aqueous media. The necessity of prior knowledge of n_{iW} values in the balancing procedure is thus avoided. All these regularities are valid for non-redox and redox systems. The elemental/core balances: f3..., f_{K}, interrelating the numbers of atoms/cores Y_{k} * H, O in components and species, are as follows

In an electrolytic system, with K elements/cores involved in it, where H and O do not participate as players, we arrange the elemental/core balances in the sequence f_{1}, f_{2}...,,,...,f_{K} , where K* is the number of fans. Then we formulate the linear combination

involving balances: f_{0}, f_{12}, f_{3}., f_{k}* for all fans. In particular, d1 = +1, d2 = -2.

In a non-redox system, we have K* = K fans, i.e., the number of players equals zero, K -K*= 0. In a redox system, we have K* fans, K* < K, i.e., the number of players equals K - K* (> 0). When the multipliers dk are equal to (or involved with) the oxidation num-bers (ONs) of the corresponding elements (k=1..., K) in fans of a non-redox system, then (Eq. 8) is transformed into identity, 0 = 0. This proves that f_{12} is not the independent equation in the set and then f^{o}, f^{3}....f^{K }is the
set of K-1 independent balances, composed of charge balance (f_{0}) and K-2 elemental/core balances f_{3},.,f_{K}. Referring now to a redox system, we state that the Eq. (8), i.e.,

involving the balances for all K* fans, with dk (k=1..., K*) equal to ONs of elements in fans, gives the balance involving only the com-ponents (N., N_{0}j) related to players in species and compounds, with coefficients equal to (or involved with) ONs of these elements. To facilitate the purposeful linear combination of the balances that enables to avoid possible/simple mistakes in the realization of the linear combination procedure, we apply the equivalent relations:

for elements with negative oxidation numbers, or

for elements with positive oxidation numbers, k 6 3., K; the change of places of numbers N_{i} for components and N_{0j} for species around equality sign is realized here.

We consider the T(V) D(V_{0}) titration, where V mL of C mol/L KIO3 is added into V_{0} mL of KI (C_{0}) + HCl (C_{01}); V mL of T is composed of N molecules of KIO_{3} and N_{02} molecules of H_{2}O, and V_{0} mL of D is composed of N_{03} molecules of KI, N_{04} molecules of HCl and N_{05} molecules of H_{2}O. In the D+T system thus formed we have the following species:

H2O (N_{1}), H+^{1} (N_{2}, n_{2}), OH^{-1} (N_{3}, n_{3}), K+1 (N_{4} n_{4}), I^{-1} (N_{5}, n_{5}), I_{3}^{-1} (N_{6} n_{6}) I_{2 } (N_{7} n_{7}), I_{2(s)} (N_{8} n_{8}), HI0(N_{9} n_{9}), IO-^{1} (N_{10}, n_{10}), HIO^{3} (N_{11} n_{11}), IO3^{-1} (N_{12}, n_{12}), H_{5}IO_{6} (N_{13}, n_{13}), H_{4}IO_{6}^{-1} (N_{14}, n_{14}), H_{3}IO_{6}^{-2} (N_{15}, n_{15}, Cl^{-1} (N_{16}, n_{16}), Cl_{2} (N_{17}, n_{17}), HClO (N_{18}, n_{18}), ClO^{-1} (N_{19}, n_{19},), HClO_{2} (N_{20}, n_{20}), ClO^{-1}_{2} (N_{21}, n_{21}), ClO_{2} (N_{22}, n_{22}), ClO^{-1}_{3} (N_{23}, n_{23}), ClO_{4}^{-1} (N_{24},n_{24}) I_{2}Cl^{-1} (N_{25},n_{25})ICl (N_{26},n_{26}) ICl_{2}^{-1} (N_{27},n_{27}) (11)

where I_{2(s)}is the solid (s) iodine, characterized by solubility s =1.3310^{-3} mol/L in aqueous media (25 ^{o}C).

For the system specified above, we formulate the charge (f_{0}) and elemental (f_{1}, f_{2}, f_{3}, f_{4}, f_{5}) balances:

The set of balances presented above was formulated according to Approach II to GEB, for the redox system with K - 5 elements: H, O, K, I, Cl involved. In accordance with the terminology applied in [3], we have their K* = 3 fans: H, O, K, and K - K* = 5 - 3 = 2 players: I and Cl. The balance f_{12} = 2-f(O) - f(H) (Eq. 18a) involves also the species: H+^{1}, OH^{-1} composed only of fans (H, O). These species are cancelled in the linear combination f12 + f0. On the other hand, the species formed by players and fans, where H and O are not involved, i.e., K^{+1}, I^{-1}, I3^{-1}, I _{2}, I _{2(s)}, Cl^{-1}, Cl_{2} , I_{2}Cl^{-1}, ICl and ICl ^{-1}, are also cancelled within f. The linear combination f + f_{0} - f_{3} = 0 (Eq. 19) composed of f_{0} and balances for fans: f_{1}, f_{2}, f_{3} includes only the species where players are involved. Note that the coefficients/multipliers for the balances: f_{1}, f_{2}, f_{3} are equal to ONs for the elements: H, O and K, respectively. The coefficients at concentrations in Eq. (22) are equal to the products of numbers of the player's atoms and the related ONs, see e.g., (-1).[I^{-1}] = ≡1-(-1)-[I^{-1}], 3-(-+)-[I3-1], 0-[I_{2}] = 2 0-[I_{2}], and 0-[I_{2}(s)] = 2-0-[I_{2}(s)]). In I_{2}Cl^{-1}, ICl and ICl_{2}^{-1}, where only players (I, Cl) are involved (Eq. 22), the coefficients are weighed sums of ONs of the players.

As results from the above, the formulation made according to Approach II to GEB does not require prior knowledge of oxidation numbers for all elements (players, fans) of the system in question. The roles of oxidants and reductants are not ascribed to components and species. Within the Approach II to GEB, the terms: oxidation number (ON), oxidant and reductant are derivative concepts.

In the D+T system considered above, calculation of ONs for all the elements is an easy task, however. Therefore, one can use here Approach I to GEB, known also as the 'short version’ of GEB [3]. In the Approach I to GEB, the players, and their ONs in the components and species are indicated.

Unlike the Approach II exemplified above, the Approach I to GEB needs prior knowledge of ONs for all (K = 5) elements in com-ponents and species of the system in question. In the D+T system, there are K* = 3 'fans', whereas iodine (I) and chlorine (Cl) are considered as players; KI and KIO_{3} are the carriers of iodine electrons, whereas HCl is the carrier of chlorine electrons. One molecule of KIO_{3} involves ZI^{-5} iodine electrons, one molecule of KI involves ZI+^{1} iodine electron, and one molecule of HCl involves ZCl+^{1} iodine electron. Then N molecules of KIO_{3} involve (ZI^{-5}) N iodine electrons, N_{03} molecules of KI involve (ZI+^{1}) -N_{03} iodine electrons, and N04 molecules of HCl involve (ZCl+1) -N_{04} chlorine electrons. Thus, the total number of the players' electrons introduced by the solutes in this system is (ZI^{-5}) N_{01} + (ZI+^{1}) N_{03} + (ZCl+_{1}) N_{04}. Referring to the species where the players are involved, we state that:

**a) **N_{5} species I^{-1}-n_{5}H_{2}O involve (Z_{1}+1) -N_{s} iodine electrons;

**b) **N_{6} species I_{3}^{-1}-n_{6}H_{2}O involve (3Z_{1}+1) -N_{6} iodine electrons;

**c)** N_{7} species I_{2}-n_{7}H_{2}O involve 2Z_{1}N_{7} iodine electrons;

**d)** N_{8} species I_{2}(_{s}.n_{8}H_{2}O involve 2Z_{1}N_{8} iodine electrons;

**e) **N_{9} species HIO-n_{9}H_{2}O involve (Z_{1}-1) N iodine electrons;

**f)** N species IO^{-1}n_{10}H_{2}O involve (Z_{1}-1) W iodin electrons;

**g)** N_{11}pecies HIO_{3}.n_{11}H_{2}O involve (Z_{1}-5) -N iodine electrons;

**h)** N_{12} species IO3-1.n_{12}H_{2}O involve (Z_{:}-5) W iodine electrons;

**i)** N_{13} species H_{5}IO_{6}n_{13}H_{2}O involve (Z_{:}-7) .N_{13} iodine electrons;

**j)** N_{14} species H_{4}IO_{6}^{-1}.n_{14}H_{2}O involve (Z_{1}-7) .N_{14} iodine electrons;

**k)** N_{15} species H_{3}IO_{6}^{-2}.n_{15}H_{2}O involve (Z_{1}-7) .N_{1S} iodine electrons;

**l)** N_{16} species Cl^{-1}-n H_{2}O involve (Z_{Cl}+1) W chlorine electrons;

**m)** N_{17} species Cl_{2}-n_{17}H_{2}O involve 2Z_{C}l.N chlorine electrons;

**n)** N species HClO.n_{18}H_{2}O involve (Z_{Cl}-1) W chlorine electrons;

**o)** N_{19} species ClO^{-1}.n_{19}H_{2}O involve (Z_{Cl}-1) .N_{19} chlorine electrons;

**p)** N_{20} species HClO_{2}-n_{20}H_{2}O involve (Z_{Cl}-1) .N_{20} chlorine electrons;

**q)** N_{21} species ClO_{2}-1.n_{21}H_{2}O involve (Z_{Cl}-1) .N_{21} chlorine electrons;

**r)** N_{22} species ClO_{2}.n_{22}H_{2}O involve (Z_{Cl}-1) -N_{22} chlorine electrons;

**s)** N_{23} species ClO_{3}^{-1}-n_{23}H_{2}O involve (Z_{Cl}-1) W chlorine electrons;

**t)** N_{24} species ClO_{4}^{-1}.n_{24}H_{2}O involve (Z_{Cl}-1) W chlorine electrons;

**u)** N_{25} species I_{2}Cl^{-1}.n_{25}H_{2}O involve (2Zj+Z_{Cl}+1) .N_{25} io- dine+chlorine electrons;

**v)** N_{26} species ICl.n_{26}H_{2}O involve (Z,+Z_{Cl}) .N_{26} iodine+chlorine electrons;

**w)** N_{27} species ICl_{2}^{-1}.n_{27}H_{2}O involve (Zj+2Z_{Cl}+1) .N_{27} io- dine+chlorine electrons.

Approach II to GEB ↔ Approach I to GEB (24)

For more details - see [3] (pp. 41-43).

We can also formulate the balances for D and T, considered separately, as independent units. Applying the notation specified above, we have:

The relations 0 = 0, see (28a) and (31a), named as identities,mean here that:

**a)** f_{12} is linearly dependent on: f_{0}, f_{3}, f_{4} and f_{5} in Eq. (28), i.e., f_{12} = f_{3} - f_{4} - f_{5} - f_{0};?

**b)** f_{12} is linearly dependent on: f_{0}, f_{3} and f_{4} in Eq. (31), i.e., f_{12}
= f_{3 }+ 5f_{4} -f_{0}.

In other words, the f_{12} are not the independent equations in D and T, considered here as separate subsystems.

It can be added that IO_{3}^{-1} can be considered in T as the core, see Eq. (30a); IO_{3}^{-1} is not considered as the core in the D+T system, considered in section 3.

The computer program is analogous /like one presented elsewhere [14,18-20]. The values for parameters of the system were as follows: V_{0} = 10; C_{0} = 0.01; C_{01} = 0, 0.2 or 5.0; C = 0.1. The independent variables: x (1) - E, x(2) - pH, x(3) - pI, x(4) - pCl were applied for the set of equations: (23a) for GEB, (12a) for ChB, (16a) for iodide, (17a) for chloride, rewritten as follows:

As we see, the number of balances (equations 23b, 12b, 16b, 17b) related to the system in question is equal to the number of independent variables: x(1), x(2), x(3), x(4) (i.e., 4 = 4). The relation (15a) is considered as equality, not equation; at pre-assumed C_{0}, C and V_{0} values, it is simply a number (not variable!) at given V value; V is considered as parameter (not variable!) at defined step of calculation procedure, realized according to iterative computer program, e.g. MATLAB [1].

Some representative curves are plotted in Figures 1 - 5, as functions of the fraction titrated Φ [21] on the abscissa

The curve s in Figures 1a-1c are related to absence of HCl (C01 = 0) in D. The main products of the symproportio-nation reaction, indicated in Figure 1a, are formed in reactions:

**Figure 1: ** The plots for (1a) speciation curves, and (b) pH = pH(Φ), (1c) E = E(Φ) relationships, at C01 = 0 and (C0, C, V0) = (0.01, 0.1, 10).

only with very slight efficiencies, ca. 10-6 mol/L for I3-1. The (weakly) alkaline reaction of the mixture (Fig. 1b) is expected based on simplified form of ChB

obtained from Eq. 12a, after omission of charged chlorine species/complexes. Moreover, [I2] < s, and then [I2(s)] = 0. Note that H3IO6-2 is the minor species in Eq. (37); its graph lies far below the plane of Fig. 1c. At Φ=2, we have pH(Φ=2) = 8.796 (Figure 1b); it results from reactions (33), (34), (36), where protons are consumed. The solution of KI is neutral, pH(⒃=0) = 7.00 in Figure 1b. Greater (but narrow) changes in pH and E values comprise, generally, the initial fragments of the respective curves in Figures 1b, c. The plots in Figures 2 & 3 were obtained at C = 0.2. The curves in Figures 2a, b is characterized by break points, whereas the curve in Figure 2c has inflection points. To explain the essence of these interest-
ing changes, we will use speciation diagram (Fig. 3). The analysis of the course of the speciation curves will allow to formulate the reactions occurring at stages of the titration. A slope of the curve in Fig. 2a indicates greater consumption of protons at Φ < 0.5 than at O > 0.5. From Fig. 2b, we see that more Cl^{-1} ions are consumed within O ∈ <0.2, 0.5> than outside this Φ-interval. Moreover, the Cl^{-1} consumption within Φ ∈ <0, 0.2> and Φ ∈ <0.5, 2> proceeds in a similar degree. Within Φ ∈ <0, 0.2>, the main product (among I_{2}Cl^{-1}, ICl, ICl_{2}^{-1}) where Cl^{-1} is involved, is I_{2}Cl^{-1}. The reaction where I_{2}Cl^{-1} is formed can be written as follows (Figure 3):

**Figure 2: ** The plots of (2a) pH = pH (Φ), (2b) log [Cl-1] = φ(Φ) and (2c) E = E (Φ) relationships, at (C0, C01, C, V0) = (0.01, 0.2, 0.1,10).

IO_{3}^{-1} + 5I^{-1} + 3Cl^{-1} + 6H^{+1} = 3I_{2}Cl^{-1} + 3H_{2}O

For O ∈ < 0.2, 0.5 >, concentration of I_{2}Cl^{-1} decreases mono- tonically while concentrations of ICl_{2}^{-1} and ICl increased; one can write the reactions:

where Cl^{-1} is consumed in greater degree than within O ∈ <0, 0.2>, 6 >, compare Equations 37, 38 with Eq. 36; ICl_{2}^{-1} predominates there significantly over ICl, see Figure 3. Within O ∈ <0, 0.5>, other iodine species: I_{2}, I , and I_{3}^{-1} are also formed

The number of protons consumed in reactions (36) - (39) is the same: 6. It explains the right course of the line in Fig. 2a within Φ ∈ <0, 0.5>. A decrease in concentration of I_{2}Cl^{-1} on account of formation of ICl_{2}^{-1} and ICl can be expressed by reactions: I_{2}Cl^{-1} + Cl^{-1} = I^{-1} + ICl_{2}^{-1} and I_{2}Cl^{-1} = I^{-1} + ICl (where protons are not involved), obtained from linear combination of equations (36) - (38), respectively.

A precise location of inflection points on the E - E (Φ) curve (Figure 2c) is facilitated by derivative curves (Figures 4a & 4b). We see that the inflection points on the curve E - E(Φ) (Figure 2c) fit the related stoichiometric points exactly (Fig. 4a) or approximately (Figure 4b) - see location of break-points in Figures 2a, 2b and stoi- chiometries of reactions: 1:5 (in Eq. 36) versus 1:2 (in Equations 37, 38).

**Figure 4: ** The ΔE/ΔΦ = (Ej+1–Ej)/(Φj+1–Φj) vs. (Φj+1+Φj)/2 relationships in the vicinity of Φ=0.2 (4a) and Φ=0.5 (4b), at (C0, C01, V0, C) = (0.01,0.2,100,0.1).

The D and T, when considered separately, are non-redox-sys- tems; ONs for I are: -1 in KI, and +5 in KIO_{3}. The -1 is the lowest, boundary value of ON for iodine. On the other hand, +7 (not +5) is the highest, boundary value for iodine. However, the species: H_{5}IO_{6}, H_{4}IO_{6}'^{1} and H_{3}IO_{6}'^{2}, represented by the numbers: N_{13}, N_{14} and N_{15}, were not involved in Eq. (30). This omission resulted from the fact that at E ≈ 0.5 V and pH ≈ 7 (Figures 1b, c), concentrations of the iodine (+7) species are extraordinarily small. It testifies because of the assumption that iodine (+5) species: IO_{3}^{_1} and HIO_{3} do not disproportionate, practically, in KIO_{3} solution, and the simplified balance (30) is valid. The iodine (+7) species were involved in the balances related to D+T system, see section 3.

**Figure 5: ** Speciation diagrams for mixed species: I2Cl^{-1}, ICl, IC12^{-1} at (C_{0}, C, V_{0}) = (0.01, 0.1, 10), and: (5a) C_{01} = 0.2, (5b) C_{0}1 = 5.0.

In the model applied for the D+T system, the iodine and chlorine species with different ONs were involved a priori. However, the Cl^{-1} ions introduced by HCl (C_{01}) practically do not participate as a reducing agent in this system; they act, however, as a complexing agent for iodine, depending on the concentration of C , compare Figures 3 & 5. Analogous situation was stated in the D+T system HCl NaIO, considered in [12, 18-20]; HCl acts in this system (first of all) as the component providing H+^{1} ions. Nota bene, the D+T system HCl NaIO is a simple example of electrolytic system, where all four types of chemical reactions, i.e. redox, acid-base, complexation (I_{2}Cl'^{1}, ICl, ICl_{2}^{-1}) and precipitation (I.) occur simultaneously.

Referring to the main problem raised in the title of this paper, we summarize and generalize the remarks stated above. From K+1 starting balances: f_{0}, f_{1}, f_{2}, f_{3}, ..., f_{K}, after linear combination f - 2-f_{2}-f_{1} we obtain the set of K balances: f_{0}, f , f_{3}, ... , f_{K}. In the linear combination procedure, the identity 0 = 0, obtained from linear combination of balances, indicates that the equations involved in the combination are linearly dependent, i.e., they are not linearly independent. In a non-redox system, f is the dependent balance; we have their K-1 independent balances: f_{0}, f_{3},..., f_{K}. In other words, f_{12} is not a new, independent balance in non-redox systems; f is then omitted in formulation of any non-redox system. In a redox system, any linear combination of the balances f_{0}, f_{1}, f_{2}, f_{3}... , f_{K} is different from (is not reducible to) the identity, 0=0, i.e., f is the independent balance. Then in a redox system we have K independent balances: f_{0}, f_{12}, f_{3}, ..., f_{K}.

The difference between redox and non-redox systems is then ‘coded’ in f . This unique, general property explains clearly why the elemental balances: f and f_{2} are not formulated for non-redox systems, of any degree of complexity. In conclusion, the linear independency/dependency of f - 2-f(O) - f(H) from other balances: f_{0},f_{3},...,f_{K} is the general criterion distinguishing between redox and non-redox systems [13,22-24], also in mixed-solvent media [2,25]; the proper linear combination, with d_{k} equal to ON’s, is the way towards the simplest/shortest form of GEB; for a non-redox system, it is the way towards identity 0 = 0. These remarks coincide well with the quotation ascribed to Einstein: ‘Everything should be made as simple as possible, but not simpler’. Transformation of a set of equations into the identity, 0 - 0, is a relatively simple/efficient/ useful operation when applied to the system of algebraic equations related to non-redox electrolytic systems [23].

In a redox system, f is considered as the primary form of Generalized Electron Balance (GEB), f = pr-GEB. The linear combination with balances for K* elec- tron-non-active elements (fans) has full properties of GEB. Any linear combination of the LC with K - K* balances for electron-active elements (players) can also be made. Other linear combinations of the K balances for redox systems, although acceptable (from an accounting point of view), devoid of motivation. Two equivalent (Eq. 24) approaches to GEB, known as the Approach I to GEB [26-34] and Approach II to GEB were formulated. The equivalence (24) is valid for redox systems of any degree of complexity, where one, two or more electron-active elements are involved. Formulation of GEB according to Approach II has - undoubtedly - the cognitive advantages, even in the cases when the oxidation numbers for all elements in the system are known beforehand.

In all instances, the balances are ultimately expressed in terms of molar concentrations [mol/L] of compounds and species, like the expressions for equilibrium constants. The related algorithm, with all attainable/complete physicochemical/thermodynamic knowledge involved in it, is resolvable with use of an iterative computer program, and then presented graphically. The Approach II to GEB does not require any prior knowledge of the oxidation numbers (ONs) of elements in the components and in the species; it is a fact of capital importance, particularly in relation to organic species (molecules, ions, radicals and ion-radicals), of any degree of complexity, also in two-phase systems [15,27,35]. Note that the oxidation number was essentially the contractual concept [36-38], based on arbitrary assumption that even the small difference in electro-negativities of atoms makes ionic bond between the atoms. However, the 'clean' ionic bonding does not exist, i.e., polar covalent bonds are formed. Even for CsF, i.e., for the compound with the greatest electronegativity difference 4.0 - 0.7 = 3.3 in the Pauling scale, we have 92% ionic bond, i.e., no compound can be 100% ionic. From the viewpoint of the Approach II to GEB, the ONs values are defined as multipliers of the corresponding linear combinations (Eq. 8a).

Some kinetic effects involved with organic compounds can also be tested [39]. The dynamic buffer capacities [40] in the form analogous to dynamic buffer capacity in acid-base systems [41,42] were also formulated. From the GATES/GEB [1,3-8,10-16,43-46] viewpoint, the terms: oxidation number, oxidant and reductant, equivalent mass [8] are derivative terms [47], not necessary (not applicable) for considerations on a redox system. Within the Approaches I and II to GEB, the roles/terms of oxidants and reductants are not assigned a priori to individual components; GATES/GEB provides full 'democracy' in this regard. If the oxidation numbers are easily determined, the Approach I to GEB, known as the 'short' version of GEB, can be applied. The Generalized Electron Balance (GEB) completing the set of equations necessary for resolution of electrolytic redox systems of any degree of complexity is a general law of Nature [3], one of the laws of the matter conservation. The GEB, together with linearly independent set of charge and concentration/core balances, completed by a set of independent equilibrium constants, provides a complete set of equations used for thermodynamic description of a redox system taken for quantitative considerations.

Within GATES, we refer to closed electrolytic systems, separated from their surroundings by diathermal (freely permeable by heat) walls; a matter cannot be exchanged by these walls. A proper heat exchange through diathermal walls enables the temperature T of the system to be kept constant, i.e., processes in the closed system can occur under isothermal conditions. Constancy of temperature, T = const, is one of the assumptions needed to secure constant values of the related equilibrium constants, defined based on mass action law. In principle, the diathermal walls separate condensed, e.g., liquid, liquid+liquid [27,48] or liquid solid [1,6,11,18-20,35,39] phases from its gaseous environment. An open chemical system approximates the closed system - provided that the matter (e.g., H_{2}O, CO_{2}, O_{2}) exchange between the system and its surroundings can be neglected, within a relatively short period of time assumed to carry out the dynamic process, e.g. titration. 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.

All the inferences made within GATES/GEB are based on firmly established physical, physicochemical and algebraic foundations. Consequently, it allows to understand far better the 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, based on all attainable and preselected physicochemical knowledge involved. Contrary to appearances, established by the current paradigm, the criterion distinguishing non-redox and redox systems is not immediately associated with free electrons in the related system; it provides a kind of uniformity in the formulas derived for this purpose. This fact, especially the simple calculations [11] of free electron concentrations in redox systems, deny the unique role of free electrons in elementary redox reactions, as described in half- or partial reactions, where the species of the same element with different oxidation numbers are involved. On the other hand, it points to the unique role of H and O in redox systems, suggested in earlier theoretical/ hypothetical/qualitative considerations on these systems. Let's repeat: the GEB concept, valid for redox electrolytic systems, is the emanation of balances for H and O, referred to aqueous media.

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