The authors have declared that no competing interests exist.

The Generalized Electron Balance (GEB) concept, related to electrolytic redox systems, is considered according to principles of Generalized Approach to Electrolytic Systems (GATES). Two equivalent Approaches (I and II) to GEB are presented. The Approach I, when perceived in convention of the card game, is based on the common pool of electrons as money, introduced by electron-active elements, named as players; electron-non-active elements are called there as fans. The GEB obtained 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. Very important properties of f_{12} are presented here in details, and illustrated by a redox system where comproportion- ation reactions occur

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, can be formulated for electrolytic redox and non-redox systems, of any degree of complexity, where charge (f_{0} = ChB) and elemental/core balances f_{k} = f(Y_{k})for Y_{k} ≠ H, O (k≥3, …, K) are also formulated, according to Generalized approach to redox systems (GATES) principles. The set of K independent balances f_{0}, f_{12}, f_{3}, …, f_{K} is necessary for thermodynamic resolution of a redox system; all the balances are presented here in the form f_{k} = 0 (k = 0, …, K). A non-redox system is resolvable with use of K–1 independent balances f_{0}, f_{3}, …, f_{K}. In a non-redox system, f_{12} is the dependent balance, indicated by the linear combination _{1} = H, Y_{2} = O, and Y_{k} (k≥3, …, K). For a non-redox system, the LC is expressed by identity, 0 = 0, i.e., f_{12} is linearly dependent on f_{0}, f_{3}, …, f_{K}, and then _{12} is the new balance in the redox system. Therefore, the linear dependency/independency of f_{12} from f_{0}, f_{3}, …, f_{K} is the general criterion distinguishing between non-redox and redox systems, of any degree of complexity.

The Approach II to GEB (formulated 2005) was preceded by the Approach I to GEB (formulated 1992); both approaches are equivalent, i.e.,

Approach I to GEB ⟺ Approach II to GEB (1)

Formulation of GEB according to Approach I can be perceived in the convention of the card game, based on the common pool of electrons as money, introduced by electron-active elements, named as players; electron-non-active elements are called there as fans. In a redox system, the arranged set (f_{1}, f_{2}, …, f_{k}, f_{k+}_{1},, …, f_{K}) of K balances f_{1}, …, f_{K} involves K* balances f_{1}, f_{2}, …, f_{k}_{*}_{,} for fans, and K – K* balances f_{k}_{*}_{+1},…, f_{K} for players. In a non-redox system we have K* = K fans, i.e., the number of players is K – K* = 0 in there; H and O are considered here as fans.

The prior knowledge of ONs for all elements of the system in question is not needed in the Approach II to GEB, contrary to the formulation of GEB according to Approach I, where the knowledge of ONs for players is required.

All the regularities indicated above will be confirmed in an example presented below, and illustrated graphically. Further generalizing comments will also be presented.

In chemical laboratories, redox reactions are usually referred to processes occurred during redox titrations. Since redox reactions are so important class of chemical reactions, we should know their physicochemical nature thoroughly, from the titrimetric viewpoint.

Titration is considered as a kind of dynamic process, where V mL of titrant T is added, from the start, up to a given point of the titration, into V_{0} mL of titrand D, and V_{0} + V mL of D+T mixture is obtained, if the volumes additivity is valid/tolerable. The species formed in the D+T systems are involved in the related balances.

In aqueous media, the species ^{−1} – Faraday’s constant, NA = 6.022∙10^{23} mol^{-1} – Avogadro’s number), ni = n_{iW} ≥ 0 is a mean number of water (W=H_{2}O) molecules attached to ^{+1}∙n_{2W}, _{3W}, i.e., z_{2} = +1, z_{3} = –1, … . Molar concentration of the species _{0} mL), we have

^{= 103∙(Ni/NA)/V0 (2)}

and for a dynamic D+T system

^{= 103∙(Ni/NA)/(V0+V). (2a)}

It is advisable to start the balancing from the interrelations between numbers of particular entities: N_{0j} for components represented by molecules composing D and T, and N_{i} – for species _{2}O and Ni species of i-th kind, _{i}, n_{i}), where n_{i} ≡ n_{iW} ≡ niH_{2}O is the mean number of hydrating water molecules (W=H_{2}O) attached to _{i}+ n_{iW}⋅0 = z_{i}. For ordering purposes, we write the sequence: H^{+1} (N_{2}, n_{2}), OH^{-1} (N_{3}, n_{3}), … , i.e., z_{2} = 1, z_{3} = –1,.

In the notation applied here, N_{0j} (j =1, 2, …, J) is the number of molecules of the component of j-th kind, including water, forming D in static (V=0), or D and T in a dynamic D+T system. The D+T system thus obtained involves N_{1} molecules of H_{2}O = ( _{2}O, z_{1}=0) and N_{i} species of i-th kind, (i = 2, 3, …, I), denoted briefly as _{i}≡ n_{iW} ≡ niH_{2}O. The n_{i} = n_{iW} values are virtually unknown – even for ^{+1}

Electrolytic (redox and non-redox) systems are formulated according to the GATES principles _{0}, f_{1}, f_{2}, f_{3}, …, f_{K} is obtained. The charged/ionic species of the system are involved in the charge balance

applicable for static and dynamic systems. Note that 0∙∙(x_{m}) = 0 for a species _{m}=0), e.g., 0∙[H_{2}O] = 0 (z_{1}=0).

Free water particles (N_{1}), and water bound in the hydrates _{1} = f(H) and f_{2} = f(O):

(4)

(5)

Next, the linear combination

is obtained, where aki and bkj in equations 4-6 are the numbers of atoms/cores Y_{k} (k = 1,...,K) in _{4}^{-1}, SO_{4}^{-2} and Fe SO_{4} contains SO_{4}^{-2} as the core. One can notice that: (a) N_{1} involved with H_{2}O as the species, (b) all N0i related to water as the component of the system, and (c) all ni = niW specified in f(H) and f(O) are cancelled within 2∙f(O) – f(H) in aqueous media. The necessity of prior knowledge of niW values in the balancing procedure is thus avoided. All these regularities are valid for non-redox and redox systems.

The elemental/core balances: f_{3}, ..., f_{K}, interrelating the numbers of atoms/cores Y_{k} ≠ H, O in components and species, are as follows

fk = f(Y_{k}) =

(k = 3, ..., K) (7)

All the balances f_{0}, …, f_{K} are presented here are as equations f_{k} = 0, see equations 3–7.

In further parts of this paper, we concern on the procedure known in elementary algebra as linear combination

_{3}_{2}_{4}_{01}

Let us consider the system, where V_{0} mL of Dis composed of NaBr (N01 molecules) + H_{2} SO_{4} (N_{02} molecules) + H_{2}O (N_{03} molecules), and V mL of T is composed of KBrO_{3 }(N_{04} molecules) + H_{2}O (N_{05} molecules). The D+T mixture thus formed involves the following species:

H_{2}O (N_{1}), H^{+1} (N_{2},n_{2}), OH^{-1} (N_{3},n_{3}), HBrO_{3} (N_{4},n_{4}),s, BrO_{3}^{-1} (N_{5},n_{5}), HBrO (N_{6},n_{6}), BrO^{-1} (N_{7},n_{7}),

Br_{2} (N_{8},n_{8}), Br_{3}^{-1} (N_{9},n_{9}), Br^{-1} (N_{10},n_{10}), Na+1 (N_{11},n_{11}) , K^{+1} (N_{12},n_{12}),

HSO_{4}^{-1} (N_{13},n_{13}), SO_{4}^{-2} (N_{14},n_{14}).

Denoting C_{0}V_{0} = 10^{3}∙N_{01}/N_{A}, C_{01}V_{0} = 10^{3}∙N_{02}/N_{A}, CV = 10^{3}∙N_{04}/N_{A}, we formulate the fraction titrated

Φ =

It provides a kind of normalization in the related graphs, i.e., independency on V0 value. The balances are as follows:

f_{0} = ChB :

N_{2} – N_{3} – N_{5} – N_{7} – N_{9} – N_{10} + N_{11} + N_{12} –

N_{13} – 2N_{14} = 0 ⟹ (9)

[H^{+1}] – [OH^{-1}] – [BrO_{3}^{-1}] – [BrO^{-1}] – [Br3^{-1}] – [Br^{-1}] + [Na^{+1}] + [K^{+1}]

– [HSO_{4}^{-1}] – 2[SO_{4}^{-2}] = 0 (9a)

f_{1} = f(H) :

2N_{1} + N_{2}(1+2n_{2}) + N_{3}(1+2n_{3}) + N_{4}(1+2n_{4}) + 2N_{5}n_{5} + N_{6}(1+2n_{6}) + 2N_{7}n_{7} + 2N_{8}n_{8} + 2N_{9}n_{9} + 2N_{10}n_{10} + 2N_{11}n_{11} + 2N_{12}n_{12} + N_{13}(1+2n_{13}) + 2N_{14}n_{14} = 2N_{02} + 2N_{03} + 2N_{05}

f_{2} = f(O) :

N_{1} + N_{2}(1+n_{2}) + N_{3}(1+n_{3}) + N_{4}(3+n_{4}) +

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

N_{9}n_{9} + N_{10}n_{10} + N_{11}n_{11} + N_{12}n_{12} + N_{13}(4+n_{13}) +

N_{14}(4+n_{14}) = 4N_{02} + N_{03} + 3N_{04} + N_{05}

–f_{3 }= –f(Na) :

N_{01} = N_{11} ⟹ [Na^{+1}] = C_{0}V_{0}/(V_{0}+V) (10)

–f4 = –f(K) :

N_{04} = N_{12} ⟹ [K^{+1}] = CV/(V_{0}+V) (11)

–f_{5} = –f(SO_{4}) ⟺ –f5 = –f(S): (12)

N_{02} = N_{13} + N_{14} ⟹ (12a)

[H SO_{4}^{-1}] + [SO_{4}^{-2}] = C_{01}V_{0}/(V_{0}+V) (12b)

f_{6} = f(Br) : N_{4} + N_{5} + N_{6} + N_{7} + 2N_{8} + 3N_{9} + N_{10}

= N_{01} + N_{04} ⟹ (13)

[HBrO_{3}] + [BrO_{3}^{-1}] + [HBrO] + [BrO^{-1}] +2[Br_{2}] +

3[Br_{3}^{-1}] + [Br^{-1}] = (C_{0}V_{0} + CV)/(V_{0}+V) (13a)

f_{12} = 2 f(O) – f(H)

– N_{2} + N_{3} + 5N_{4} + 6N_{5} + N_{6} + 2N_{7} + 7N_{13} + 8N_{14}

= 6N_{02} + 6N_{04} (14)

– [H^{+1}] + [OH^{-1}] + 5[HBrO_{3}] + 6[BrO3^{-1}] + [HBrO] + 2[BrO^{-1}] + 7[HSO_{4}^{-1}] + 8[SO_{4}^{-2}]

= 6(C_{01}V_{0} + CV)/(V_{0}+V) (14a)

f_{12} + f_{0} – f_{3} – f_{4} – 6∙f_{5} = 0 ⟺ (+1)f1 +

(–2)f_{2 }+ (+1)f_{3} + (+1)f_{4} + (+6)f_{5} – f_{0} = 0 ⟺

(+1)f(H) + (–2)f(O) + (+1)f(Na) + (+1)f(K) +

(+6)f(S) – ChB = 0 (15)

5(N_{4}+N_{5}) + 1(N_{6}+N_{7}) – N_{9} – N_{10}

= – N_{01} + 5N_{04} ⟹ (15a)

5([HBrO_{3}] + [BrO_{3}^{-1}]) + [HBrO] + [BrO^{-1}] –

[Br3^{-1}] – [Br^{-1}] =

(+5)([HBrO_{3}]+[BrO_{3}^{-1}]) + (+1)([HBrO]+[BrO^{-1}]) + 2∙0∙[Br_{2}] + 3∙_{3}^{-1}] + (–1)[Br^{-1}]

= (–1)∙

Z_{Br}∙f6 – (f_{12} + f_{0} – f_{3} – f_{4} – 6∙f_{5})

(Z_{Br}–5)(N_{4}+N_{5}) + (Z_{Br}–1)(N_{6}+N_{7}) + 2Z_{Br}N_{8} + (3Z_{Br}+1)N_{9} + (Z_{Br}+1)N_{10}

= (Z_{Br}+1)N_{01} + (Z_{Br}–5)N_{04} ⟹ (16)

(ZBr–5)([HBrO_{3}]+[BrO_{3}^{-1}]) + (ZBr^{–1}(^{HBrO} +[BrO^{-1}]) + 2ZBr[Br_{2}] + (3ZBr+1)[Br_{3}^{-1}] + (ZBr+1)[Br^{-1}] =

(ZBr+1)

Other linear combinations are also possible. From the linear combination (f_{12} + f_{0} – f_{3} – f_{4} – 6∙f_{5} + f_{6})/2 we get the shortest (involving the smallest number of components) form of GEB for the system S:

3(N_{4}+N_{5}) + (N_{6}+N_{7}) + N_{8} + N_{9} = 3N_{04} (17)

3([HBrO_{3}] + [BrO_{3}^{-1}]) + ([HBrO] + [BrO^{-1}]) + [Br_{2}] +

[Br_{3}^{-1}] = 3∙

Note that the coefficients in eq. 15c are equal tothe oxidation numbers (ONs) of the elements considered as ‘fans’ in the system S, i.e., H, O, Na, K, S. Eq. 15b obtained from the combination 15 interrelates components and species formed by Br, as the player.

Other linear combinations a_{1}∙f_{0} + a_{2}∙f_{12} + _{0}, f_{12}, f_{3}, …, f_{6} are also acceptable, ak ∈ ℝ.

Equations 14a, 15b, 16a and 17a are alternative/equivalent equations for GEB related to the system S. One of the equations for GEB, together with equations 9a, 12b and 13a, form a complete set of equations related to the system S. The relations 10 and 11, considered as equalities (not equations), can be immediately introduced into eq. 9a as numbers.

In eq. 12b, the SO_{4}^{-2} can be perceived as the core. However, because the system S has no other competing sulfate forms, the choice between f(S) and f(SO_{4}) (eq. 12) is irrelevant.

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

for D :

f_{0} = ChB : N_{2} – N_{3} – N_{10} + N_{11} – N_{13} – 2N_{14} = 0

f_{1} = f(H) : 2N_{1} + N_{2}(1+2n_{2}) + N_{3}(1+2n_{3}) +

2N_{11}n_{11} + N_{13}(1+2n_{13}) + 2N_{14}n14 = 2N_{02} + 2N_{03}

f_{2} = f(O) : N_{1} + N_{2}n_{2} + N_{3}(1+n_{3}) + N_{11}n_{11} +

N_{13}(4+n_{13}) + N_{14}(4+n_{14}) = 4N_{02} + N_{03}

–f_{3} = –f(Na) : N_{01} = N_{11}

–f_{5} = –f(SO_{4}) : N_{02} = N_{13} + N_{14}

f_{6} = f(Br) : N_{10} = N_{01}

and then:

f_{12} = 2∙f_{2} – f_{1} : – N_{2} + N_{3} + 7N_{13} + 8N_{14} = 6N_{02}

f_{12} + f_{0} – f_{3} – 6f_{5} + f_{6} = 0 ⟺ (+1)f(H) + (–2)f(O) + (+1)f(Na) + (+6)f(S) + (–1)f(Br) – ChB = 0 (18)

0 = 0 (18a)

for T :

f_{0} = ChB : N_{2} – N_{3} – N_{5} + N_{12} = 0

f_{1} = f(H) : 2N_{1} + N_{2}(1+2n_{2}) + N_{3}(1+2n_{3}) +

N_{4}(1+2n_{4}) + 2N_{5}n_{5} + 2N_{12}n_{12} = 2N_{05}

f_{2} = f(O) : N_{1} + N_{2}n_{2} + N_{3}(1+n_{3}) + N_{4}(3+n_{4}) +

N_{5}(3+n_{5}) + N_{12}n_{12} = 3N_{04} + N_{05}

–f_{4} = –f(K) :N_{04} = N_{12}

–f_{6} = –f(Br) :N_{04} = N_{4} + N_{5}

and then:

f_{12} = 2∙f_{2} – f_{1} : – N_{2} + N_{3} + 5N_{4} + 6N_{5} + 8N_{14} = 6N_{04}

f_{12} + f_{0} – f_{4} – 5f_{6} = 0 ⟺ (+1)f(H) + (–2)f(O) + (+1)f(K) + (+5)f(Br) – ChB = 0 (19)

0 = 0 (19a)

The D and T, considered separately, form non-redox-systems; ONs for Br are: –1 in NaBr, and +5 in KBrO_{3}, i.e., there are the boundary values of ONs in bromine redox systems; Br(+7) species are omitted in considerations

The relations 0 = 0, named as identities, mean here that:

f_{12} is linearly dependent on: f_{0}, f_{3}, f_{5} and f_{6} in eq. 18, i.e., f_{12} = f_{3} + 6f_{5} – f_{6} – f_{0};

f_{12} is linearly dependent on: f_{0}, f_{4} and f_{6} in eq. 19, i.e., f_{12} = f_{4} + 5f_{6} –f_{0}.

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

Unlike the Approach II exemplified above, the Approach I to GEB needs prior knowledge of ONs for all elements in components and species of the system in question. In the system S, there are K* = 5 ‘fans’, whereas bromine (Br) is considered as the ‘player, K – K* = 6 – 5 = 1 is here the number of players.

In the system S, bromine (as NaBr and KBrO_{3}) is the carrier/distributor of the player electrons. One atom of Br has Z_{Br} bromine electrons, and then one molecule of NaBr has Z_{Br} +1 bromine electrons, one molecule of KBrO_{3} has Z_{Br}–5 bromine electrons; then N_{01} molecules of NaBr involve (Z_{Br} + 1)∙N_{01} bromine electrons, N_{04} molecules of KBrO_{3} involve (Z_{Br} –5)N04 bromine electrons. Thus, the total number of bromine electrons introduced by NaBr and KBrO_{3} is (Z_{Br} + 1)∙N_{01} + (Z_{Br} – 5)N_{04}. On this basis, we state that

N_{4} species HBrO_{3}∙n_{4}H_{2}O involve (Z_{Br} – 5)∙N_{4 }bromine electrons;

N_{5} species BrO_{3}^{-1}∙n_{5}H_{2}O involve (Z_{Br}–5)∙N_{5} bromine electrons;

N_{6} species HBrO∙n_{6}H_{2}O involve (Z_{Br}–1)∙N_{6} bromine

electrons;

N_{7} species BrO^{-1}∙n_{7}H_{2}O involve (Z_{Br}–1)∙N_{7} bromine

electrons;

N_{8} species Br_{2}∙n_{8}H_{2}O involve 2Z_{Br}∙N_{8} bromine

electrons;

N_{9} species Br_{3}^{-1}∙n_{9}H_{2}O involve (3Z_{Br}+1)∙N_{9} bromine

electrons;

N_{10} species Br^{-1}∙n_{10}H_{2}O involve (Z_{Br}+1)∙N_{10} bromine electrons.

The balance for the bromine electrons is then expressed by eq. 16 and then by eq. 16a. This confirms the equivalency of the Approaches I and II to GEB (eq. 1).

As stated above, the Approach I to GEB is compared to the ‘card game’

function F = Function_NaBr_H_{2} SO_{4}_KBrO3(x)

%NaBr (C0) H_{2} SO_{4} (C_{01}) V_{0} KBrO_{3} (C) V

global V Vmin Vstep Vmax V0 C C0 C01 fi H OH pH E Kw pKw A

global Br Br2 Br3 HBrO BrO HBrO3 BrO3 Na K H SO_{4} SO_{4}

global logBr logBr2 logBr3 logHBrO logBrO logHBrO3 logBrO3

global logNa logK logH SO4 log SO4

E=x(1);

pH=x(2);

Br=10.^-x(3);

SO_{4}=10.^-x(4);

H=10.^-pH;

pKw=14;

Kw=10.^-14;

OH=Kw./H;

A=16.9;

ZBr=35;

BrO3=Br.*10.^(6.*A.*(E-1.45)+6.*pH);

HBrO3=10.^0.7.*H.*BrO3;

BrO=Br.*10.^(2.*A.*(E-0.76)+2.*pH-2.*pKw);

HBrO=10.^8.6.*H.*BrO;

Br3=Br.^3.*10.^(2.*A.*(E-1.05));

Br2=Br.^2.*10.^(2.*A.*(E-1.087));

Na=C0.*V0./(V0+V);

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

%Charge balance

F=[(H-OH-BrO3-BrO-Br3-Br+Na+K- HSO4-2.* SO4);

%Concentration balance of Br

(HBrO3+BrO3+HBrO+BrO+2.*Br2+3.*Br3+

Br-(C0.*V0.+C.*V)/(V0+V));

%Concentration balance for SO4

(H SO4^{+}^{2}.* SO4-C01.*V0/(V0+V));

%GEB

((ZBr-5).*(HBrO3+BrO3)+(ZBr-1).*(HBrO+BrO)+2.*ZBr.*Br2...

+(3.*ZBr+1).*Br3+(ZBr+1).*Br...

-((ZBr+1).*C0.*V0+(ZBr-5).*C.*V)./(V0+V))];

logBr=log10(Br);

logBr2=log10(Br2);

logBr3=log10(Br3);

logHBrO=log10(HBrO);

logBrO=log10(BrO);

logHBrO3=log10(HBrO3);

logBrO3=log10(BrO3);

logH SO4=log10(H SO4);

log SO4=log10(SO4);

logNa=log10(Na);

logK=log10(K);

In the algorithm, prepared according to MATLAB computational software, the potential E ^{V} was expressed in SHE scale ^{+1}], , pBr = – log[Br^{-1}], V_{0} = 100, C_{0} = 0.01, C = 0.1, C_{01} = 0, 0.01 or 0.1. The equilibrium constants related to this system were cited in ref.

The results of calculations are presented graphically (

At C_{01} = 0, comproportionation practically does not occur (IIIa); concentration of HBrO, as the major product formed in the comprortionation reaction

BrO3^{-1} + 2Br^{-1} + 3H^{+1} = 3HBrO sss (20)

is ca. 10^{-6} mol/L. The potential E increases monotonically (

At C_{01} = 0.01 (

BrO_{3}^{-1} + 5Br^{-1} + 6H^{+1} = 3Br_{2} + 3H_{2}O (21)

For Φ > 0.2, an increase of efficiency of the competing reaction 21 is noted. A growth of C01 value causes a small extension of the potential range in the jump region, on the side of higher E-values ((

Concluding remarks

The linear combination

involves K balances: f_{0}, f_{12}, f_{3},…,f_{K} where d_{1} = +1, d_{2} = –2. All the balances are presented here as equations, f_{k} = 0.

In a non-redox system, we have K fans, i.e., the number of players equals zero. 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 numbers (ONs) of the corresponding elements (k = 1, …, K) in a non-redox system, then eq. 22 is transformed into identity, 0 = 0. This proves that f_{12} is not the independent equation in the f_{0}, f_{12}, f_{3}, …, f_{K}, and _{0}, 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 arrange the elemental/core balances in the sequence f_{1}, f_{2}, …, f_{K}_{*}_{,} f_{K}_{*}_{+}_{1} , f_{K} , and then formulate the equation

involving the balances for K* electron-non-active elements, compare with equations 15, 18, 19.

If d_{k} (k=1,…,K*) are equal to (or involved with) ONs of electron-non-active elements, then the resulting balance involves only the components (N_{i}, N_{0j}) related to electron-active species/compounds, with coefficients equal to (or involved with) ONs of these element. More precisely, dk are the products of the number λ_{k} of defined atoms in the species and the ON value ωk, i.e. _{3}^{-1} in eq. 15c, where –1 = 3_{2} in the eq. 15c, where 0 = 2∙0, i.e., the coefficient d_{9} = –1 at [Br_{3}^{-1}] in eq. 15b is involved with (not equal to) the oxidation number ω9 = _{3}^{-1} (λ_{9} = 3).

The equivalent relations were applied:

f_{k} =

(24)

for elements with negative oxidation numbers, or

-f_{k} =

(25)

for elements with positive oxidation numbers, k ∈ 3, …,K. The change of places of numbers Ni for components and N0j for species at the equality sign in relations 24, 25 were made in order to avoid possible/simple mistakes in the realization of the linear combination procedure. This facilitates the purposeful linear combination of the balances, and enables to avoid simple mistakes in this operation. Note, for example, that f_{4} = f(Na) ⟺ f(Na) = f_{4} ⟺ –f_{4} = –f(Na).

Starting from K+1 balances: f_{0}, f_{1, }f_{2}, f_{3}, …, f_{K},, after formulation of the linear combination f_{12} we obtain the set of K balances: f_{0}, f_{12}, f_{3}, …, f_{K},. In a non-redox system, f_{12} is the dependent balance; we have there 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; it is then omitted in formulation of any non-redox system. The identity 0 = 0 for the linear combinations indicates that the equations f_{0}, f_{12}, f_{3}, …, f_{K}, are linearly dependent for non-redox systems. In a redox system, f_{12} is the independent balance (i.e., different from the identity, 0=0); then we have K independent balances: f_{0}, f_{12}, f_{3}, …, f_{K}, that will be rearranged – optionally – as the set (f_{12}, f_{0}, f_{3}, …, f_{K}) involved with GEB, ChB, and f(Y_{k}) (k = 3, …, K), respectively. The number of elemental/core balances, both in non-redox and redox systems, equals K–2. Then:

• For a non-redox system, a proper linear combination of f_{12} with f_{0}, f_{3}, …, f_{K} is reducible to identity 0 = 0, i.e., f_{12} is linearly dependent on f_{0}, f_{3}, … , f_{K} (equations 24,25).

• For a redox system, any linear combination of f_{12} with f_{0}, f_{3}, …, f_{K} is not reducible to identity 0 = 0, i.e., f_{12} is linearly independent on f_{0}, f_{3}, …, f_{K}.

• In conclusion, the linear independency/dependency of f_{12} = 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; the proper linear combination LC with d_{k} equal to ON’s (see eq. 15), is the way towards the simplest/shortest form of GEB; the shortest form (eq. 17a) was obtained after further combination with the balance for Br (player). For a non-redox system, the linear combination indicated it is the way towards identity 0 = 0 (equations 18a, 19a).

• f_{12} = 2∙f(O) – f(H) is the primary form of Generalized Electron Balance (GEB), f_{12} = prGEB, completing the set of K balances f_{0}, f_{12}, f_{3}, …, f_{K} necessary for resolution of redox systems of any degree of complexity.

• Any linear combination of f_{12} with the balances f_{0}, f_{3}, …, f_{K} has full properties of GEB related to the redox system considered, i.e., all them are equivalent forms of GEB.

• The Approach II to GEB does not indicate oxidants and reductants, i.e., oxidized and reduced forms in the system in question.

• The prior knowledge of oxidation numbers (ONs) for all elements of the system is not required; this fact is of capital importance when redox equilibria are involved, e.g., with complex organic species; the known composition of a species, expressed by its formula, together with external charge of this species, provides information sufficient to formulate the related balances.

• When the oxidation numbers of all elements of a system are known beforehand, the GEB can be formulated according to Approach I to GEB, known also as the 'short version' of GEB; the GEB obtained according to Approach I to GEB involves all electron-active elements – as components and species – of the system tested.

• Both Approaches (I and II) to GEB are equivalent (eq. 1).

where all ‘fans’ (K* elements or cores) 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 there simpler/simplest 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 and any other combination

• The criterion distinguishing between non-redox and redox systems is valid for redox systems of any degree of complexity. We can check it also on more complex redox systems, of any degree of complexity, where two or more electron-active elements as ‘players’, are involved. The relation (12) was also confirmed for electrolytic systems in binary and (generally) mixed-solvent As (s = 1, …, S) media

are assumed, and niAs≥0 is the mean numbers of as molecules attached to

• The formulation of GEB according to Approach II is relatively/extraordinarily simple, although receiving the shorter equation for GEB, when using the linear combinations of pr-GEB with other balances, requires implementation of the time-consuming, preparatory activities. However, the 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.

The GEB concept, discovered by Michałowski as the Approaches I (1992,

The Approach II to GEB does not require any prior knowledge of the oxidation numbers 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. Note that the ‘oxidation number’ was essentially the contractual concept

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, on the basis of 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 kindof uniformity in the formulas derived for this purpose. This fact, especially the simple calculations (

D – titrand (solution titrated),

GATES – Generalized Approach to Electrolytic Systems,

GEB – Generalized Electron Balance,

ON – oxidation number,

T – titrant,

V – volume (mL) of T,

V0 – volume (mL) of D.