The authors have declared that no competing interests exist.

The Generalized Approach To Electrolytic Systems (GATES) provides the best possible thermodynamic formulation of redox and non-redox, equilibrium and metastable, mono-, two- and three-phase systems, with all attainable/pre-selected physicochemical knowledge involved, without any simplifying assumptions made for calculation purposes, where different species may occur in batch or dynamic systems, of any degree of complexity. The Generalized Electron Balance (GEB) is the key concept completing the set of algebraic balances referred to redox systems, described according to GATES/GEB ⊂ GATES principles. The GEB, considered as the law of Nature, is fully compatible with charge and concentration balances, and relations for the corresponding equilibrium constants. Within GATES, the electrolytic systems are resolvable with use of MATLAB, or other iterative computer programs, if all necessary physicochemical knowledge is attainable.

The paper refers to fundamental/general regularities obligatory for electrolytic systems. The linear combination f_{12} = 2∙f(O) – f(H) of elemental balances: f_{1} = f(H) for Y_{1} = H and f_{2} = f(O) for Y_{2} = O, is the general, key relation distinguishing between electrolytic redox and non-redox systems in aqueous media _{12} is put in context with charge balance (f_{0} = ChB) and other, elemental and/or core balances f_{k} = f(Y_{k}) (k=3,…,K; Y_{k} ≠ H, O), related to the system in question. It is stated that f_{12} is (1^{o}) linearly independent on f_{0},f_{3},…,f_{K} for a redox system, or (2^{o}) linearly dependent on f_{0},f_{3},…,f_{K} when related to a non-redox system, and thus the independency/dependency property of f_{12} (3^{o}) provides a rigorous criterion distinguishing between redox and non-redox systems ^{o}) f_{12} is considered as the _{12} = ^{o}) f_{12} is fully compatible with f_{0},f_{3},…,f_{K}, and (6^{o}) f_{12} completes the set of K equations f_{0},f_{12},f_{3},…,f_{K}, necessary for thermodynamic solution of a redox system according to Generalized Approach to Redox Systems (GATES) ^{o}) The set of K–1 balances f_{0},f_{3},…,f_{K} is necessary for the solution of a non-redox system; (8^{o}) all relevant, available physicochemical knowledge can be applied for this formulation. Formulation of f_{12} within GATES/GEB ⊂ GATES (9^{o}) needs none prior knowledge of oxidation numbers (ONs) for all elements in components forming any redox system, and in the species of the system thus formed, where (10^{o}) oxidants and reductants are not indicated ^{o}) Stoichiometry, oxidation numbers, oxidants, reductants and equivalent mass (as GEM ^{o}) f_{12} is the hidden connection of physicochemical laws, and (13^{o}) f_{12} is the breakthrough in thermodynamic theory of electrolytic redox systems. Within GATES, f_{12} (14^{o}) indicates the unique, distinctive role of the two elements: Y_{1} = H and Y_{2} = O in mathematical description of electrolytic (redox and non-redox) systems of any degree of complexity, limited only by the quantitative, physicochemical knowledge related to the system in question, and GATES/GEB provides the best thermodynamic approach to electrolytic (static and dynamic) redox systems, of any degree of complexity. The GEB is perceived as a law of Nature _{12} formulation within GATES, designed primarily to aqueous media, can be extended on mixed-solvent media, where amphiprotic co-solvent(s) are involved

The GATES formulation is applicable for non-redox and redox systems of any degree of complexity. Applicability of GATES is limited only by the degree of physicochemical knowledge, expressed by a complete set of mutually independent equilibrium constants related to the system in question.

Static (batch) and dynamic electrolytic systems are resolvable within GATES, formulated (1992) by Michałowski

In this paper, we concern mainly some electrolytic redox systems, formulated according to GATES/GEB principles, with GEB discovered by Michałowski (1992, 2005) according to the Approaches (I, II) to GEB. The Approach I to GEB

All calculations made within GATES/GEB are related to condensed (liquid, or liquid+solid) phases separated from the environment by diathermal (freely permeable by heat) walls

Complex redox systems, where all types of elementary chemical reactions proceed simultaneously and/or sequentially, were also considered, see

The problem of linear dependency or independency of equations is of fundamental importance in GATES/GEB. Among others, it provides the new criterion distinguishing between redox and non-redox electrolytic systems. An independent equation obtained from charge and elemental and/or core balances related to electrolytic redox systems provides the independent equation known as Generalized Electron Balance (GEB).

The terms: components of an electrolytic system and species in the system are distinguished, for balancing purposes. The components (solvent, solutes) are compounds forming a (static or dynamic) system, and the species X_{i}^{zi} are formed in the system thus obtained. We refer here to aqueous electrolytic systems, where the species X_{i}^{zi} exist as hydrates X_{i}^{zi}.n_{iw}, i=1,…, I; z_{i} = 0, ±1, ±2,…is a charge, expressed in elementary charge units, e = F/N_{A} (F _{A }– Avogadro’s number), n_{i} = n_{iW} = n_{i}H_{2}O ≥ 0 is a mean number of water (W=H_{2}O) molecules attached to X_{i}^{zi} ; the case n_{iW}=0 is then also admitted. The notation X_{i}^{zi}.n_{iw} presents the species as real entities in aqueous media. The n_{iW} values are virtually unknown – even n_{2W} for X_{2}^{z}^{2}. n_{2w} = H^{+}_{2w}

For some reasons, it is justifiable to start the balancing from the numbers of particular entities: N_{0j} – for components (j = 1,…,J) represented by molecules, 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.The mono- or two-phase electrolytic system thus obtained involve N_{1} molecules of H_{2}O and N_{i} species of i-th kind, X_{i}^{zi}.n_{iw} (i=2, 3,…,I), specified briefly as X_{i}^{zi} (N_{i}, n_{i}), where n_{i} ≡ n_{iW} ≡ n_{i}H_{2}O. For ordering purposes, we write: H^{+1} (N_{2}, n_{2}), OH^{-1} (N_{3}, n_{3}),… , i.e., z_{2} = 1, z_{3} = –1

Formation of linear combinations is applicable to check the linear dependency or independency of the balances. A very useful/effective manner for checking/stating the linear dependence of the balances is the transformation of an appropriate system of equations to the identity, 0 = 0

Within GATES, elemental and/or core balances are distinguished/formulated. The core is a cluster of elements of defined composition (expressed by chemical formula) and external charge; e.g., the core SO_{4}^{-2} enters the component FeSO_{4}∙xH_{2}O and the species: HSO_{4}^{-1}∙n_{5}H_{2}O, SO_{4}^{-2}∙n_{6}H_{2}O, FeSO_{4}∙n_{9}H_{2}O in aqueous (W=H_{2}O) solutions: D and D+T presented below. Molar concentrations ^{mol/L} of the species X_{i}^{zi}.n_{iw} is denoted as [X_{i}^{zi}], for brevity. Concentrations of all components and species be expressed in mol/L, and all volumes – in mL.

In further parts of this paper, we consider the D+T system, obtained after addition of V mL of T containing the reagent B (C M) into V_{0} mL of the solution containing the analyte A (C_{0} M). The number of mmoles of A equals n_{A} = C_{0}V_{0} and the number of mmoles of B, added up to a given point of titration, equals n_{B} = CV. The fraction = n_{B}/n_{A}, i.e.,

is termed as the fraction titrated. The variable introduces a kind of normalization (independence of V_{0} value) in graphical presentation of the results thus obtained. The will be also very useful in formulation of the equivalence mass according to generalized equivalent mass GEM

Modelling the electrolytic systems can be realized with use of an iterative computer program, with all attainable physicochemical knowledge involved in the algorithm applied for this purpose. The values of equilibrium constants, interrelating molar concentrations of particular species, are parameters of the corresponding balances.

For comparative purposes, we start from titrand D and titrant T as the static non-redox sub-systems participating the redox D+T system. We consider here:

1) D (V_{0}) composed of FeSO_{4}∙xH_{2}O (N_{04}) + H_{2}SO_{4} (N_{05}) + H_{2}O (N_{06}) + CO_{2} (N_{07});

2) T (V) composed of KMnO_{4} (N_{01}) + H_{2}O (N_{02}) + CO_{2} (N_{03}) ;

and

3) D+T (V_{0}+V) as the mixture of D and T, where the following species are formed:

H_{2}O (N_{1}); H^{+1} (N_{2}, n_{2}), OH^{-1} (N_{3}, n_{3}), HSO_{4}^{-1} (N_{4}, n_{4}), SO_{4}^{-2} (N_{5}, n_{5}), H_{2}CO_{3} (N_{6}, n_{6}), HCO_{3}^{-1} (N_{7}, n_{7}),

CO_{3}^{-2} (N_{8}, n_{8}), Fe^{+2} (N_{9}, n_{9}), FeOH^{+1} (N_{10}, n_{10}), FeSO_{4} (N_{11}, n_{11}), Fe^{+3} (N_{12}, n_{12}), FeOH^{+2} (N_{13}, n_{13}),

Fe(OH)_{2}^{+1} (N_{14}, n_{14}), Fe_{2}(OH)_{2}^{+4} (N_{15}, n_{15}), FeSO_{4}^{+1} (N_{16}, n_{16}), Fe(SO_{4})_{2}^{-1} (N_{17}, n_{17}), K^{+1} (N_{18}, n_{18}),

MnO_{4}^{-1} (N_{19}, n_{19}), MnO_{4}^{-2} (N_{20}, n_{20}), Mn^{+3} (N_{21}, n_{21}), MnOH^{+2} (N_{22}, n_{22}), Mn^{+2} (N_{23}, n_{23}), MnOH^{+1} (N_{24}, n_{24}),

MnSO_{4} (N_{25}, n_{25}) .....(2)

The same notation (N_{0j}, N_{i}, n_{i}) will be applied, for brevity, also for D and T. The x value in FeSO_{4}∙xH_{2}O _{2} in water applied as solvent for preparation of D and T.

The complete set of equilibrium constants related to this D+T system is involved in the relations:

[H^{+1}][OH^{-1}] = 10^{-14.0}; [HSO_{4}^{-1}] = 10^{1.8}[H^{+1}][SO_{4}^{-2}]; [H_{2}CO_{3}] = 10^{16.4}[H^{+1}]^{2}[CO_{3}^{2}]; [HCO_{3}^{-1}] = 10^{10.1}[H^{+1}][CO_{3}^{-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}; [Fe_{2}(OH)_{2}^{+4}] = 10^{21.7}[Fe^{+3}]^{ 2}[OH^{-1}]^{2}; [FeSO_{4}] = 10^{2.3}[Fe^{+2}][SO_{4}^{-2}];

[FeSO_{4}^{+1}] = 10^{4.18}[Fe^{+3}][SO_{4}^{-2}]; [Fe(SO_{4})_{2}^{-1}] = 10^{7.4}[Fe^{+3}][SO_{4}^{-2}]^{ 2}; [MnO_{4}^{-1}] = [Mn^{+2}]∙10^{5A(E – 1.507) + 8pH};

[MnO_{4}^{-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}] ...(3)

The notation of balances, similar to one applied e.g. in

The dynamic system is realized according to titrimetric mode, where V mL of titrant T, added in successive portions into V_{0} mL of titrand D, and V_{0}+V mL of D+T mixture is obtained at this point of the titration, if the assumption of the volumes additivity is valid; D and T are the sub-systems of the D+T system.

We get here the balances:

_{0} = ChB

N_{2} – N_{3 }– N_{4} – 2N_{5}– N_{7} – 2N_{8} + 2N_{9} + N_{10} = 0

_{1} =

2N_{1} + N_{2}(1+2n_{2}) + N_{3}(1+2n_{3}) + N_{4}(1+2n_{4}) + 2N_{5}n_{5} + N_{6}(2+2n_{6}) + N_{7}(1+2n_{7}) + 2N_{8}n_{8} +

2N_{9}n_{9} + N_{10}(1+2n_{10}) + 2N_{11}n_{11} = 2xN_{04} + 2N_{05} + 2N_{06}

_{2} =

N_{1} + N_{2}n_{2} + N_{3}(1+n_{3}) + N_{4}(4+n_{4}) + N_{5}(4+n_{5}) + N_{6}(3+n_{6}) + N_{7}(3+n_{7}) + N_{8}(3+n_{8}) +

N_{9}n_{9} + N_{10}(1+n_{10}) + N_{11}(4+n_{11}) = (4+x)N_{04} + 4N_{05} + N_{06} + 2N_{07}

_{12} = 2

-N_{2} + N_{3} + 7N_{4} + 8N_{5} + 4N_{6} + 5N_{7} + 6N_{8} + N_{10} + 8N_{11} = 8N_{04} + 6N_{05} + 4N_{07}

–6_{3} = –6f(S)

6N_{04} + 6N_{05} = 6N_{4} + 6N_{5} + 6N_{11}

–2_{4} = –2f(Fe)

2N_{04} = 2N_{9} + 2N_{10} + 2N_{11}

–4_{5} = –4f(C)

4N_{07} = 4N_{6} + 4N_{7} + 4N_{8}

_{12} + f_{0}

6N_{4} + 6N_{5} + 4N_{6} + 4N_{7} + 4N_{8} + 2N_{9} + 2N_{10} + 8N_{11} = 8N_{04} + 6N_{05} + 4N_{07} ⇨

6(N_{4} + 6N_{5}) + 4(N_{6} + N_{7} + N_{8}) + 2(N_{9} + N_{10}) + 8N_{11} = (2+6)N_{04} + 6N_{05} + 4N_{07}

_{12} + _{0} – 6_{3} – 2_{4} – 4_{5} ⇨ ....(4)

(+1)⋅f(H) + (-2)⋅f(O) + (+6)⋅f(S) + (+2)⋅f(Fe) + (+4)⋅f(C) = ChB (4a)

0 = 0 ....(5)

Note that Fe(+2) species are not oxidized here.

_{0} = ChB

N_{2} – N_{3 }– N_{4} – 2N_{5}– N_{7} – 2N_{8} + N_{18} – N_{19} = 0

_{1} =

2N_{1} + N_{2}(1+2n_{2}) + N_{3}(1+2n_{3}) + N_{6}(2+2n_{6}) + N_{7}(1+2n_{7}) + 2N_{8}n_{8} + 2N_{19}n_{19} = 2N_{02}

_{2} =

N_{1} + N_{2}n_{2} + N_{3}(1+n_{3}) + N_{6}(3+n_{6}) + N_{7}(3+n_{7}) + N_{8}(3+n_{8}) + N_{19}(4+n_{19}) = 4N_{01} + N_{02} + 2N_{03}

_{12} = 2_{2} - _{1}

-N_{2} + N_{3} + 4N_{6} + 5N_{7} + 6N_{8} + 8N_{19} = 8N_{01} + 4N_{03}

_{12} + f_{0}

4N_{6} + 4N_{7} + 4N_{8} + N_{18} + 7N_{19} = 8N_{01} + 4N_{03}

–7f_{3} = –7f(Mn)

7N_{01} = 7N_{19}

–4f_{4} = –4f(C)

4N_{03} = 4N_{6} + 4N_{7} + 4N_{8}

–f_{5} = –f(K)

N_{01} = N_{18}

_{12} + f_{0} – 7f_{3} – 4f_{4} – f_{5} ⇨ ....(6)

(+1)⋅f(H) + (–2)⋅f(O) + (+6)⋅f(S) + (+2)⋅f(Fe) + (+4)⋅f(C) = ChB (6a)

0 = 0.....(5)

Note that MnO_{4}^{-1} is not reduced here

We get here the balances:

_{0} = ChB

N_{2} – N_{3 }– N_{4} – 2N_{5} – N_{7} – 2N_{8} + 2N_{9} + N_{10} + 3N_{12} + 2N_{13} + N_{14} + 4N_{15} + N_{16} – N_{17} + N_{18} – N_{19} - 2N_{20} + 3N_{21} + 2N_{22} + 2N_{23} + N_{24} = 0....(7)

_{1} =

2N_{1} + N_{2}(1+2n_{2}) + N_{3}(1+2n_{3}) + N_{4}(1+2n_{4}) + 2N_{5}n_{5} + N_{6}(2+2n_{6}) + N_{7}(1+2n_{7}) + 2N_{8}n_{8} + 2N_{9}n_{9} + N_{10}(1+2n_{10}) + 2N_{11}n_{11} + 2N_{12}n_{12} + N_{13}(1+2n_{13}) + N_{14}(2+2n_{14}) + N_{15}(2+2n_{15}) + 2N_{16}n_{16} + 2N_{17}n_{17} + 2N_{18}n_{18} + 2N_{19}n_{19} + 2N_{20}n_{20} + 2N_{21}n_{21} + N_{22}(1+2n_{22}) + 2N_{23}n_{23} + N_{24}(1+2n_{24}) + 2N_{25}n_{25} = 2N_{02} + 2xN_{04} + 2N_{05} + 2N_{06}

_{2} =

N_{1} + N_{2}n_{2} + N_{3}(1+n_{3}) + N_{4}(4+n_{4}) + N_{5}(4+n_{5}) + N_{6}(3+n_{6}) + N_{7}(3+n_{7}) + N_{8}(3+n_{8}) + N_{9}n_{9} + N_{10}(1+n_{10}) + N_{11}(4+n_{11}) + N_{12}n_{12} + N_{13}(1+n_{13}) + N_{14}(2+n_{14}) + N_{15}(2+n_{15}) + N_{16}(4+n_{16}) + N_{17}(8+n_{17}) + n_{18}n_{18} + N_{19}(4+n_{19}) + N_{20}(4+n_{20}) +

N_{21}n_{21} + N_{22}(1+n_{22}) + N_{23}n_{23} + N_{24}(1+n_{24}) + N_{25}(4+n_{25}) = 4N_{01} + N_{02} + 2N_{03} + (4+x)N_{04} + 4N_{05} + N_{06} + 2N_{07}

f_{12} = 2

–N_{2} + N_{3} + 7N_{4} + 8N_{5} + 4N_{6} + 5N_{7} + 6N_{8} + N_{10} + 8N_{11} + N_{13} + 2N_{14} + 2N_{15} + 8N_{16} + 16N_{17} + 8N_{19} + 8N_{20} + N_{22} + N_{24} + 8N_{25} = 8N_{01} + 4N_{03} + 8N_{04} + 6N_{05} + 4N_{07} .....(8)

f_{12} + _{0}

6N_{4} + 6N_{5} + 4N_{6} + 4N_{7} + 4N_{8} + 2N_{9} + 2N_{10} + 8N_{11} + 3N_{12} + 3N_{13} + 3N_{14} + 6N_{15} + 9N_{16} + 15N_{17} + N_{18} + 7N_{19}

+ 6N_{20} + 3N_{21} + 3N_{22} + 2N_{23} + 2N_{24} + 8N_{25} = 8N_{01} + 4N_{03} + 8N_{04} + 6N_{05} + 4N_{07}

–6_{3} = –6f(S)

6N_{04} + 6N_{05} = 6N_{4} + 6N_{5} + 6N_{11} + 6N_{16} + 12N_{17} + 6N_{25} .....(9)

–4_{4} = –4f(C)

4N_{03} + 4N_{07} = 4N_{6} + 4N_{7} + 4N_{8} .....(10)

–_{5} = –f(K)

N_{01} = N_{18} .....(11)

Then we have:

A = f_{12} + _{0} – 6_{3} – 4_{4} – _{5} .....(12)

2N_{9} + 2N_{10} + 2N_{11} + 3N_{12} + 3N_{13} + 3N_{14} + 6N_{15} + 3N_{16} + 3N_{17} + 7N_{19} + 6N_{20} + 3N_{21} + 3N_{22} + 2N_{23} + 2N_{24} + 2N_{25}

= 7N_{01} + 2N_{04} ⇨

2(N_{9}+N_{10}+N_{11}) + 3(N_{12}+N_{13}+N_{14} +2N_{15}+N_{16}+N_{17}) + 7N_{19} + 6N_{20} + 3(N_{21}+N_{22}) + 2(N_{23}+N_{24}+N_{25})

= 7N_{01} + 2N_{04} .....(13)

_{6} = f(Fe)

(N_{9}+N_{10}+N_{11}) + (N_{12}+N_{13}+N_{14}+2N_{15}+N_{16}+N_{17}) = N_{04} .....(14)

_{7} = f(Mn)

N_{19} + N_{20} + (N_{21}+N_{22}) + (N_{23}+N_{24}+N_{25}) = N_{01} .....(15)

The atomic numbers for Fe and Mn are: Z_{Fe} = 26 and Z_{Mn} = 25, respectively. Then we get

Z_{Fe}∙f_{6} + Z_{Mn}∙f_{7} – (f_{12} + _{0} – 6_{3} – 4_{4} – _{5})

(Z_{Fe}–2)(N_{9}+N_{10}+N_{11}) + (Z_{Fe}–3)(N_{12}+N_{13}+N_{14}+2N_{15}+N_{16}+N_{17}) + (Z_{Mn}–7)N_{19} + (Z_{Mn}–6)N_{20} +

(Z_{Mn}–3)(N_{21}+N_{22}) + (Z_{Mn}–2)(N_{23}+N_{24}+N_{25}) = (Z_{Fe}–2)N_{04} + (Z_{Mn}–7)N_{01} ....(16)

Another combination of the equations is also noteworthy. Namely, after addition of the balances

– (f_{12} + _{0} – 6_{3} – 4_{4} – _{5}) = –A.....(12a)

7N_{01} + 2N_{04} = 2(N_{9}+N_{10}+N_{11}) + 3(N_{12}+N_{13}+N_{14} +2N_{15}+N_{16}+N_{17}) + 7N_{19} + 6N_{20} + 3(N_{21}+N_{22}) + 2(N_{23}+N_{24}+N_{25})

3_{6} = 3f(Fe)

3(N_{9}+N_{10}+N_{11}) + 3(N_{12}+N_{13}+N_{14}+2N_{15}+N_{16}+N_{17}) = 3N_{04}

2_{7} = 2f(Mn)

2N_{19} + 2N_{20} + 2(N_{21}+N_{22}) + 2(N_{23}+N_{24}+N_{25}) = 2N_{01}

After cancelling similar components in the sum thus obtained and further rearrangements, we get

N_{9}+N_{10}+N_{11} – (5N_{19}+4N_{20}+N_{21}+N_{22}) = N_{04} – 5N_{01} ....(17)

It is the shortest (in terms of the number of its constituents) among the equivalent forms of GEB, obtained from the linear combination

– (f_{12} + _{0} – 6_{3} – 4_{4} – _{5}) + 3_{6} + 2_{7} ⇨ – f_{0} – (2f_{2} – f_{1}) + 6f_{3} + 4f_{4} + f_{5} + 3f_{6} + 2f_{7} ⇨

– f_{0} + f_{1} + (–2)f_{2} + 6f_{3} + 4f_{4} + f_{5} + 3f_{6} + 2f_{7} ⇨ 1⋅f_{1} + (–2)⋅f_{2} + 6⋅f_{3} + 4⋅f_{4} + 1⋅f_{5} + 3⋅f_{6} + 2⋅f_{7} = f_{0} ⇨ ....(18)

(+1)⋅f(H) + (–2)⋅f(O) + (+6)⋅f(S) + (+4)⋅f(C) + (+1)⋅f(Na) + (+3)⋅f(Fe) + (+2)⋅f(Mn) = f_{0} (18a)

The f_{12} is considered as the _{12} =

When formulating the balances f_{1} and f_{2}, it is possible to take into account the formation of water clusters(H_{2}O)_{𝞴} (N_{1}_{𝞴}, 𝞴 =1, 2,...) in aqueous solutions

f_{1} = f(H):

2⋅ + N_{2} (1 + 2n_{2}) + N_{3}(1 + 2n_{3}) + ...

f_{2} = f(O) :

_{2} (1 + n_{2}) + N_{3}(1 + n_{3}) + ...

we have:

2f_{2} – f_{1} :

– N_{2} + N_{3} + ...

i.e., all components related to the clusters are cancelled. The f_{12} and all linear combinations of f_{12} with f_{0} and other balances do not include the terms: N_{1}, n_{iW}, x, N_{02}, N_{06} involved with water.

The coefficients/multipliers at the corresponding balances f_{k} = f(Y_{k}) in Equations 4a, 6a, 9a are equal to oxidation numbers (ON’s) of elements in the corresponding species, see also

Applying the Relations:

[X_{i}^{zi }]∙(V_{0}+V) = 10^{3}∙N_{i}/N_{A} CV = 10^{3}∙N_{01}/N_{A}, C_{0}V_{0} = 10^{3}∙N_{05}/N_{A},

C_{1}V = 10^{3}∙N_{04}/N_{A} , C_{01}V_{0} = 10^{3}∙N_{06}/N_{A} , C_{02}V_{0} = 10^{3}∙N_{08}/N_{A} ....(19)

in the balances derived above, we have the optional/equivalent equations for GEB :

[Fe^{+2}] + [FeOH^{+1}] + [FeSO_{4}] – (5[MnO_{4}^{-1}] + 4[MnO_{4}^{-2}] + [Mn^{+3}] + [MnOH^{+2}])

– (C_{0}V_{0} – 5CV)/(V_{0}+V) = 0 ....(17a)

[H^{+1}] – [OH^{-1}] – [HSO_{4}^{-1}] – 2[SO_{4}^{-2}] – [HCO_{3}^{-1}] – 2[CO_{3}^{-2}] + 2[Fe^{+2}] + [FeOH^{+1}] + 3[Fe^{+3}] + 2[FeOH^{+2}] + [Fe(OH)_{2}^{+1}]+ 4[Fe_{2}(OH)_{2}^{+4}] + [FeSO_{4}^{+1}] – [Fe(SO_{4})_{2}^{-1}] + [K^{+1}] – [MnO_{4}^{-1}] – 2[MnO_{4}^{-2}] +

3[Mn^{+3}] + 2[MnOH^{+2}] + 2[Mn^{+2}] + [MnOH^{+1}] = 0....(7a)

[HSO_{4}^{-1}] + [SO_{4}^{-2}] + [FeSO_{4}] + [FeSO_{4}^{+1}] + 2[Fe(SO_{4})_{2}^{-1}] + [MnSO_{4}] – (C_{0}+C_{01})V_{0}/(V_{0}+V) = 0..(9a)

[H_{2}CO_{3}] + [HCO_{3}^{-1}] + [CO_{3}^{-2}] – (C_{02}V_{0}+C_{1}V)/(V_{0}+V) = 0...(10a)

[K^{+1}] = CV/(V_{0}+V) ....(11a)

[Fe^{+2}] + [FeOH^{+1}] + [FeSO_{4}] + [Fe^{+3}] + [FeOH^{+2}] + [Fe(OH)_{2}^{+1}] + 2[Fe_{2}(OH)_{2}^{+4}] +

[FeSO_{4}^{+1}] + [Fe(SO_{4})_{2}^{-1}] – C_{0}V_{0}/(V_{0}+V) = 0....(14a)

[MnO_{4}^{-1}] + [MnO_{4}^{-2}] + [Mn^{+3}] + [MnOH^{+2}] + [Mn^{+2}] + [MnOH^{+1}] + [MnSO_{4}] – CV/(V_{0}+V) = 0..(15a)

The simplest form of GEB, expressed by Eq. (17a), is chosen here (arbitrarily) for calculation purposes - for obvious reasons _{12} = pr-GEB (Eq. 8) is as follows

– [ H^{+1}] + [OH^{-1}] + 7[HSO_{4}^{-1}] + 8[SO_{4}^{-2}] + 4[H_{2}CO_{3}] + 5[HCO_{3}^{-1}] + 6[CO_{3}^{-2}] + [FeOH^{+1}] +

8[FeSO_{4}] + [FeOH^{+2}] + 2[Fe(OH)_{2}^{+1}] + 2[Fe_{2}(OH)_{2}^{+4}] + 8[FeSO_{4}^{+1}] + 16[Fe(SO_{4})_{2}^{-1}] +

8[MnO_{4}^{-1}] + 8[MnO_{4}^{-2}] + [MnOH^{+2}] + [MnOH^{+1}] + 8[MnSO_{4}]

– (8CV + 4C_{1}V + 8C_{0}V_{0} + 6C_{01}V_{0} + 4C_{02}V_{0})/(V_{0}+V) = 0 ...(8a)

The GEB, obtained from Eq. 16,

(Z_{Fe}–2)([Fe^{+2}]+[FeOH^{+1}]+[FeSO_{4}]) + (Z_{Fe}–3)([Fe^{+3}]+[FeOH^{+2}]+[Fe(OH)_{2}^{+1}]+2[Fe_{2}(OH)_{2}^{+4}]+

[FeSO_{4}^{+1}]+[Fe(SO_{4})_{2}^{-1}]) + (Z_{Mn}–7)[MnO_{4}^{-1}] + (Z_{Mn}–6)[MnO_{4}^{-2}] + (Z_{Mn}–3)([Mn^{+3}]+[MnOH^{+2}]) +

(Z_{Mn}–2)([Mn^{+2}]+[MnOH^{+1}]+[MnSO_{4}]) – ((Z_{Fe}–2)C_{0}V_{0} + (Z_{Mn}–7)CV)/(V_{0}+V) = 0...(16a)

is identical to that obtained directly from Approach I to GEB, compare with

2([Fe^{+2}]+[FeOH^{+1}]+[FeSO_{4}]) + 3([Fe^{+3}]+[FeOH^{+2}]+[Fe(OH)_{2}^{+1}]+2[Fe_{2}(OH)_{2}^{+4}]+[FeSO_{4}^{+1}] +

[Fe(SO_{4})_{2}^{-1}]) + 7[MnO_{4}^{-1}] + 6[MnO_{4}^{-2}] + 3([Mn^{+3}]+[MnOH^{+2}]) +

2([Mn^{+2}]+[MnOH^{+1}]+[MnSO_{4}]) – (2C_{0}V_{0 }+ 7CV)/(V_{0}+V) = 0....(13a)

Eq. 13a is identical with the one obtained from Eq.16a under assumption that Z_{Fe} = 0, Z_{Mn} = 0 (compare with the ‘debt of honor’ principle/idea suggested in

The set of 6 independent equations: 7a, 9a, 10a, 14a, 15a, 17a is then distinguished. The number of the equations is equal to the number of 6 independent variables, considered as components of the vector:

_{1},…,x_{6}]^{T} = (E,pH,pMn2,pFe2,pSO4,pH2CO3)^{T}

where pMn2 = – log[Mn^{+2}], pFe2 = – log[Fe^{+2}], pSO4 = – log[SO_{4}^{-2}], pH2CO3 = – log[H_{2}CO_{3}]. The pX_{i} = - log [X_{i}^{zi}] refer to mutually independent species X_{i}^{zi}.n_{iw}. Volume V of the titrant T added is the parameter (not variable) of the system, at defined point of the titration. The relation (11a) is considered as the equality (not equation!); [K^{+1}] can enter immediately Eq. 7a, like a number at defined point of the calculation procedure, defined here by C, V_{0} and V values.

V mL of KMnO_{4} (C = 0.02 M) is added into V_{0} = 100 mL of FeSO_{4} (C_{0} = 0.01 M) + H_{2}SO_{4} (C_{01} = 1.0 M). The simplest form of GEB is there as follows

[Fe^{+}^{2}]+[FeOH^{+1}]+[FeSO_{4}] – (5[MnO_{4}^{–1}]+4[MnO_{4}^{–2}]+[Mn^{+3}]+[MnOH^{+2}]) = (C_{0}V_{0} – 5CV)/(V_{0}+V)...(23)

The presence of H_{2}SO_{4} in D affects the related titration curve 1 in _{4}, FeSO_{4}^{+1}, Fe(SO_{4})_{2}^{-1}, MnSO_{4} were (intentionally) omitted in the related balances, formulated according to GATES/GEB principles. The relations: [FeSO_{4}^{+}^{1}]+[Fe(SO_{4})_{2}^{-1}] >> [Fe^{+3}] and [FeSO_{4}] > [Fe^{+2}] are valid in the whole Φ-range, and [Mn^{+3}]+[MnOH^{+2}] > [MnO_{4}^{-1}] at Φ > 0.2, see

All calculations were realized with use of MATLAB iterative computer program _{e}, E_{e}) points from the close vicinity of equivalence (eq) point are collected in

Φ_{e} |
E_{e}^{V} |

0.198 | 0.701 |

0.199 | 0.719 |

0.1998 | 0.661 |

0.1999 | 0.778 |

0.2 | 1.034 |

0.2001 | 1.365 |

0.2002 | 1.382 |

0.202 | 1.442 |

for _{e} = _{eq} = 0.2, one gets = 0.

The Fe^{+2}, Fe^{+3} and Mn^{+2} ions form sulphate complexes, but sulphate complexes Mn(SO_{4})_{i}^{+3-2i} formed by Mn^{+3} ions are unknown in literature. However, the comparison of the curves obtained for pre-assumed stability constants of Mn(SO_{4})_{i}^{+3-2i} complexes presented in

Redox systems are the most important and the most complex electrolytic systems. From thermodynamic viewpoint, the complexity of a redox system is expressed by the number and diversity of equilibrium constants involved in this system. The diversity is involved with the number of species, different types of reactions occurring between the species, and the number of elements involved in these species. The best thermodynamic approach to electrolytic systems is realizable according to the GATES principles, with none relevance to the stoichiometry of a chemical reaction, resulting from IUPAC recommendations

The correct formulation of dynamic redox systems was unknown in earlier (< 1994) literature, see _{2} with NaOH

GATES and GATES/GEB in particular, relies on the fact that the chemistry involved is predictable on the basis of knowledge of credible physicochemical data related to the species involved in the system in question.

The GEB plays a distinguishing role in redox systems; it is fully compatible with charge and concentration balances and, therefore, completes the set of equations necessary for thermodynamic solution of redox systems, of any degree of complexity, assuming that all relevant physicochemical knowledge is available. The results obtained according to GATES principles are also applicable for analytical purposes, especially in context with GEM

The identity (0 = 0) procedure applied to linear combination of equations relating to non-redox systems(see Eq. 5 in D and T), is far more convenient/efficient than the procedure based directly on the matrix calculations

In D and D+T, the core balance for _{4}) is identical with elemental balance

If oxidation numbers for some elements constituting the system under consideration are unknown beforehand, the Approach II to GEB must be applied, because the prior knowledge of oxidation numbers is not needed there. Oxidation number is the redundant concept within GATES/GEB. Note that the oxidation number is the contractual term, and sometimes encounters fundamental difficulties, or atoms in organic compounds and species, with radicals and ion-radicals involved. It is one of the paramount advantages of the Approach II to GEB. 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. Moreover, within the Approaches I and II to GEB, the roles of oxidants and reductants are not ascribed

These basic properties of the balance 2∙

Further (loose) remarks concerning the words: card game, players, fans, money, debt of honor. Pure mathematics is, in its way, the poetry of logical ideas. One seeks the most general ideas of operation which will bring together in simple, logical and unified form the largest possible circle of formal relationships. In this effort toward logical beauty, new formulas are discovered, necessary for the deeper penetration into the laws of nature.

The art of reasoning is nothing more than a well-ordered language. The Renaissance posed the question of the criterion of determining whether the sign actually means what it meant, and the decision was provided by similarity. With the advent of the classical age, the question arose: how and how is the sign related to what it means? Classicism resolved the problem by offering an analysis of the representation, and the present day an analysis of the sense of meaning.

According to Aristotle, creating a good metaphor is tantamount to seeing similarities in dissimilar things. Metaphors can be alive or dead; living metaphors are revealing. Nevertheless, the best determinant of living metaphors is their uniqueness - which makes them so unusual and desired by their recipients. The paraphrase of such metaphors cannot exhaust the meaning that is the vehicle of innovation.

Pure mathematics is, in its way, the poetry of logical ideas. One seeks the most general ideas of operation which will bring together in simple, logical and unified form the largest possible circle of formal relationships. In this effort toward logical beauty spiritual formulas are discovered necessary for the deeper penetration into the laws of Nature (A. Einstein).