Some Regularities Involved with Oxidation Numbers Stated in Formulation of Redox Systems According to GATES/GEB Principles

Formulation of Generalized Electron Balance (GEB) for redox systems according to Approach II to GEB does not require the prior knowledge of oxidation numbers of all elements in components forming a system, and in the species of the system thus formed. This formulation is involved with linear combination of charge and elemental and/or core balances related to the system in question. The skillful choice of multipliers for the balances on the step of purposeful formulation of this linear combination allows for to find important regularities for electrolytic systems of different degree of complexity. These multipliers are related to the oxidation numbers of the elements; this regularity is important in the context of the fact that the oxidation number is the contractual concept. This property is valid for redox and non-redox systems. In this context, oxidation number is perceived as the derivative/redundant concept. The paper indicates the close relationships between different rules of conservation and indicates huge possibilities inherent in the generalized approach to electrolytic systems (GATES), and GATES/GEB in particular.

two-phase electrolytic system thus obtained involve N 1 molecules of H 2 O and N i species of i -th kind, X i zi n iw (i = 2, 3,…,I), denoted briefly as X i zi (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 ),… .
The GEB is formulated according to two equivalent approaches, named as the Approach I to GEB and the Approach II to GEB. The Approach I to GEB is formulated according to 'card game' principle [10] ; it is based on a common pool of electrons introduced into the system by electron-active elements [10][11][12][13][14][15] . The Approach II to GEB is formulated from linear combination 2•f(O) -f(H) of elemental balances: f(H) for H, and f(O). Both Approaches (I and II) to GEB are equivalent.
‫؞‬ Approach II to GEB <=> Approach I to GEB (1) All the inferences made within GATES/GEB are based on firm, algebraic foundations. The approach proposed allows understanding far better all physicochemical phenomena occurring in the system in question and improving 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.
The Approach I to GEB needs prior knowledge of oxidation numbers. The Approach II to GEB does not require any prior knowledge of the oxidation numbers of elements in the components and in the species. Because the 'oxidation number' is essentially the contractual concept, it is a fact of capital importance, particularly in relation to organic species (molecules, ions, radicals and ion-radicals) of any degree of complexity. However, when the oxidation numbers are easily determined, the Approach I to GEB, known as the 'short' version of GEB, can be applied. The roles of oxidants and reductants are not assigned a priori to individual components within the Approaches I and II to GEB; GATES/GEB provides full 'democracy' in this regard.
The principle of GEB formulation, discovered by Michałowski (1992) as the Approach I to GEB, and 2006 as the Approach II to GEB) was unknown in earlier literature. The GEB is considered as a new law of the matter conservation related to electrolytic redox systems, as a Law of Nature [9,10] .
The importance of GATES/GEB in area of electrolytic redox systems is unquestionable. Then the main purpose of the present paper is to familiarize it to a wider community. It will also be indicated, in a simple mathematical manner, the fundamental criterion distinguishing between non-redox and redox systems. Contrary to appearances, this criterion is not immediately associated with free electrons in electrolytic system.

Formulation of balances for the
The charge balance (f 0 = ChB) is then derivable from balances for electrons and protons. It is nothing strange because the external charge of a species is a simple sum of charges brought by nuclear protons and orbital electrons.
In particular, from Equations (3)  Note that z i ≠ 0 are multipliers for concentrations of the corresponding ions in the ChB.

Formulation of balances for the System 2
In the System 2 we have the set (2) of the species identical as in the System 1. Applying similar notation, we assume that V 0 mL of D is composed of HBrO (N 01 molecules) + H 2 O (N 02 molecules) and V mL of T is composed of NaOH (N 03 molecules) + H 2 O (N 04 molecules). The numbers N 01 and N 02 of the molecules composing these systems and the numbers N i of the species in the System 2 are different than that in the System 1, in principle. The f 0 = ChB in the System 2 is identical with Eq. (9a), and -f 4 = -f(Na) is identical with Eq. (14). Then after formulation of f 1 and f 2 we have here, by turns, The balances in terms of molar concentrations For the System 1, from Eqs. (3) -(5), (13), (14) and ( Eqs. (13a) and (14a) are termed as concentration balances, obtained from elemental balances (13) and (14), and Eq. (16a) is the shortest form of GEB. The relation (14a) can be immediately introduced into Eq. (9a); then we get Eqs. (9c), (13a) and (16a) form the complete set of balances related to the System 1.
We can also refer to static systems, formed by (a) C 0 mol/L Br 2 and (b) C 0 mol/L HBrO. These solutions are identical with the titrand D in the related Systems 1 and 2. The balances for these static systems are obtained assuming V = 0 in Eqs. (9c), (13a) and (14a), or in Eqs. (9c),(19a) and (20a), resp. Note that Br 2 and HBrO do not oxidize water molecules, i.e., products of H 2 O oxidation do not exist there as species.

Relations between concentrations of the species
From the interrelations obtained on the basis of expressions for equilibrium data [16] collected in Table 1

Graphical presentation
Formulation of the static system with Br 2 (C 0 ) was presented first in [17] , whereas the dynamic Systems: 1 and 2 were presented in [18] . The pH = pH (Φ) and E = E (Φ) relationships for the Systems 1 and 2 are plotted in Figures 1A, B, where the fraction titrated   -1 ]. The 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 .
In C 0 = 0.01 mol/L HBrO, more than 90% HBrO disproportionates according to the reaction 5HBrO = BrO 3  The main task of titration is the estimation of the equivalent volume, V eq , corresponding to the volume V = V eq of T, where the fraction titrated (Eq. (22)) assumes the value C . V eq Φ eq = ---------- The V e [mL] value is the volume of T consumed up to the end (e) point, where the titration is terminated (ended). The V e value is usually determined in visual titration, when a pre-assumed color (or color change) of D+T mixture is obtained. In a visual acid-base titration, pH e value corresponds to the volume V e [mL] of T added from the start for the titration and C . V e Φ e = ---------(28) C 0 . V 0 is the Φ-value related to the end point. From Eqs. (25) and (27), one obtains: (29) This does not mean that we may choose between Eqs. (29a) and (29b), to calculate m A . Namely, Eq. (29a) cannot be applied for the evaluation of m A : V e is known, but Φ e unknown. Calculation of Φ e needs prior knowledge of C 0 value; e.g., for the titration system NaOH(C,V) → HCl(C 0 ,V 0 ) we have [2] Φ e = C/C 0 . (C 0 -α e ) / (C + α e ) (30) where: α = [H +1 ] -[OH -1 ] = 10 -pH -10 pH-14 , and α e = α(pH e ). However, C 0 is unknown before the titration; otherwise, the titration would be purposeless. The pH e value in visual titration is known only approximately. Also Eq. (29b) is useless: the 'round' Φ eq value is known exactly, but V eq is unknown; V e (not V eq ) is determined in visual titrations. Because Eqs: (29a) and (29b) appear to be useless, the third, approximate formula for m A , has to be applied [19] , namely: (31) where Φ eq is put for Φ e in Eq. 29a, and R A eq = M A /Φ eq (32) is named as the equivalent mass. The relative error in accuracy, resulting from this substitution, equals to δ = (m A ' -m A )/m A = m A '/m A -1 = V e /V eq -1 = Φ e /Φ eq -1 (33) In particular, for the titration of V 0 = 100 mL of C 0 = 0.01 mol/L A = Br 2 (MBr 2 = 159.808 g/mol) with V mL of C = 0.1 mol/L NaOH we have Φ eq = 2 (see Table 2). At pH = pH e = 9.413 we have Φ e = 2.003, and then δ = 2.003/2 -1 = 0.15%. Note that this result was obtained on the basis of thermodynamic knowledge (GATES/GEB), not from invalid, stoichiometric considerations.

Equations and equalities
Among the concentration balances one can distinguish equations and equalities. An equality, represented by the relation (14a), involves only one species, whereas an equation involves more species, see e.g. Eq. (13a) or Eq. (16a). Concentrations of the species in the balances are involved in expression(s) for the corresponding equilibrium constant(s), exemplified by the relations presented in Table 1. In the equality (14a), the value for [Na +1 ] is a number for the pre-assumed C and V 0 values, at given V-value; as such, it can enter immediately the related ChB, see Eq. (9c). Then (14a) is not considered as equation, when the number of equations is compared with the number of independent variables. where: logK e1 = 6AE 01 , logK ei = 2AE 0i (i = 2,3,4); A = 16.9; pK 51 = -logK 51 , pK 11 = -logK 11 , pK W = -logK W

Number of independent equations and dependency/independency property
In the System 1, three equations: (9c), (13a) and (16a) form a set of independent [20,21] (16a),(17a)), pH to ChB (Eq. (9c)), and pBr -to the concentration balance for Br (Eq. (13a)). The volume V of the titrant (T) added is considered as a parameter, not variable; the V value is changed, in steps, in the calculation procedure, realized according to iterative computer program, here: MATLAB.
The linear dependence of algebraic equations is usually checked using the Gaussian elimination method [22] or the

Oxidation number, oxidant and reductant as the redundant terms
In formulation of GEB according to the Approach II to GEB, the prior knowledge of oxidation numbers is not needed; it is the paramount advantage of the Approach II to GEB, particularly when applied to complex organic species, with radicals and ion-radicals involved. It should be noticed that the oxidation number, representing the degree of oxidation of an element in a compound or a species, is a contractual concept. None prior knowledge, except composition and external charge of the species formed in the system is needed. The equivalency of the Approaches I and II to GEB means that the equation obtained by a suitable linear combination of pr-GEB with charge balance and other elemental/core balances becomes identical with the one obtained directly from the Approach I to GEB. The GEB related to the System 1 with Br 2 (C 0 ) and expressed by Eq. (16a), obtained according to Approach II to GEB, can be rewritten as follows: (20b) Analogous balances are obtained for static systems, with C 0 mol/L (a) Br 2 , (b) HBrO. As we see, the coefficients at the concentrations of the corresponding species and at the concentrations of the related components are equal to the oxidation numbers of Br as an electron-active component in the related Systems (1 and 2). It is a general property, valid also in the systems where two or more electron-active elements are involved.
We refer here to titration of V 0 mL of the titrand D containing FeSO 4 (C 0 ) + H 2 SO 4 (C 01 ) with V mL of titrant T, containing KMnO 4 (C), added up to a given point of the titration. In this case, the linear combination of electron-non-active elements ('fans'), expressed as follows [28] where γ = 1 if precipitation of MnO 2 in this system is admitted (e.g. at too small C 01 value), or γ = 0 if it does not occur. The simplest form of GEB is obtained there from linear combination of the above balance (34) with the balances for the electron-active elements ('players'). At γ = 0 we have [29][30][31][32] (+1) As results from the examples presented above, (1) Formulation of GEB according to Approach II to GEB needs none prior knowledge of the oxidation states (oxidation numbers) of all elements participating the System 1 or 2, i.e., Br, H, O; it means that the oxidation state (oxidation number) is the derivative concept within GATES/GEB; (2) The terms: oxidant and reductant are not distinguished, i.e., not ascribed a priori to particular components and species in electrolytic systems, considered according to GATES/GEB principles, with GEB obtained according to Approaches I and II to GEB; full 'democracy' in this respect is assumed; (3) In the balance obtained from f 12 + f 0 , the oxidation numbers in particular Br-species are equal to (or involved with) the coefficient/multiplier at the concentration of the corresponding species with electron-active element (here: Br), and in the com-ponent (Br 2 or HBrO) or bromine species. If the species or component involves more atoms of an electron-active element, then the coefficient is equal to the product of the related numbers; e.g.

Concluding remarks The unique property of 2•f(O) -f(H)
In all instances it can be stated that -for all redox systems -any linear combination of f 12 = 2•f(O) -f(H) with f 0 = ChB and other elemental/core [20] balances related to a redox system of any degree of complexity does not give the identity, see e.g. Eqs.
The redox systems are 'coded' in the respective balances: f(H) and f(O). The balance 2•f(O) -f(H) is linearly independent on charge and concentration balances related to redox systems, and it is dependent on those balances when related to non-redox systems. The independency or dependency property is then the general criterion distinguishing between redox and non-redox systems [11][12][13][14][15][16][17][18][19][20] , of any degree of complexity, also in mixed-solvent media [25][26][27] . This unique, general property explains clearly why the elemental balances: f(H) and f(O) are not formulated in any non-redox system. This regularity applies to non-redox systems of any degree of complexity, both for systems with aqueous solutions, as well as with amphiprotic solvents or a mixture of solvents, including at least one amphiprotic co-solvent [21] . Contrary to appearances, established by the current paradigm, this criterion distinguishing non-redox and redox systems and unknown in earlier literature, is not immediately/explicitly associated with free electrons in the system. It 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 [5] , deny the unique role of free electrons in redox reactions, expressed 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 [33][34][35][36][37][38][39][40][41] , suggested in earlier theoretical/hypothetical considerations on these systems. This issue will be developed in a separate work.
Before 1992, the basic property of the balance 2•f(O)f(H) for redox systems was unknown at all in scientific world, and the linear independency/dependency of 2•f(O) -f(H) as the fundamental/practical criterion distinguishing redox/non-redox systems of any degree of complexity was also unknown.
Here is the hidden simplicity, which had to be discovered, as the Approach II to GEB. One of the authors™ 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 Generalized Electron Balance (GEB) concept, valid for redox electrolytic 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 Generalized Approach To Electrolytic Systems (GATES), perceived as the thermody-namic approach to equilibrium and metastable systems, where all necessary physicochemical knowledge on the systems tested is involved. The advantages of GATES are illustrated on examples of redox and non-redox analytical systems. 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. Furthermore, GATES is the basis for Generalized Equivalence Mass (GEM) concept, considered as the advantageous alternative against the obligatory Equivalence Mass (EM) concept suggested by IUPAC. According to GATES, any chemical reaction notation is only a basis to formulate the related expression for equilibrium constant according to mass action law.
GATES/GEB allows tracking not only the individual titrations, but also the processes composed of several steps, as shown in the example of iodometric determination of copper [5] . The entire analytical process is partitioned here into 4 stages: two preparatory stages, with non-redox reactions, and two further stages in which redox reactions occurred. The majority of the dynamic systems, realized according to titrimetric mode, are perceived also as an important tool in the classical chemical analysis.
GATES/GEB is a counter-proposal in relation to earlier IUPAC decisions, presented in three subsequent editions of the Orange Book, and based on reaction stoichiometry; that viewpoint was criticized unequivocally/exhaustively/convincingly, especially in a series of review articles [2][3][4][5] . It was demonstrated, on examples of redox systems of different complexity, that stoichiometry is a secondary/derivative concept from the viewpoint of GATES, and GATES/GEB, in particular.
Concluding, GATES is the overall, thermodynamic approach to redox and non-redox, static and dynamic, single and multiphase equilibrium, metastable and non-equilibrium 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.