The Balance 2∙f (O) – f (H) as a Cornerstone in Formulation of Electrolytic Systems

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 V₀ mL of titrand D, and V₀ + 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 exist as hydrates zi = 0, ±1, ±2, … is a charge, expressed in terms of elementary charge units, e = F/NA (F = 96485 C∙mol⁻¹ – Faraday’s constant, NA = 6.022∙10²³ mol^-1 – Avogadro’s number), ni = n_iW ≥ 0 is a mean number of water (W=H₂O) molecules attached to ;the case niW = 0 is then also admitted. For ordering purposes we assume: = H⁺¹∙n_2W, = OH-1∙n_3W, i.e., z₂ = +1, z₃ = –1, … . Molar concentration of the species is denoted as ;for a static system (volume V₀ mL), we have

= 10³∙(N_i/N_A)/V₀       (2)

and for a dynamic D+T system

= 10³∙(N_i/N_A)/(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 (ions and molecules) of i-th kind (i = 1, …, I), where I is the number of kinds of the species in the D+T. The mono- or two-phase electrolytic D+T system thus obtained involves N1 molecules of H₂O and Ni species of i-th kind, (i = 2, 3, …, I), specified briefly as (N_i, n_i), where n_i ≡ n_iW ≡ niH₂O is the mean number of hydrating water molecules (W=H₂O) attached to The net charge of equals to the charge of z_i+ n_iW⋅0 = z_i. For ordering purposes, we write the sequence: H⁺¹ (N₂, n₂), OH^-1 (N₃, n₃), … , i.e., z₂ = 1, z₃ = –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 N₁ molecules of H₂O = ( H₂O, z₁=0) and N_i species of i-th kind, (i = 2, 3, …, I), denoted briefly as where n_i≡ n_iW ≡ niH₂O. The n_i = n_iW values are virtually unknown – even for = H^+1 1 in aqueous media, and depend on ionic strength (I) of the solution.

Electrolytic (redox and non-redox) systems are formulated according to the GATES principles ^{2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40}. For this purpose, the set of K+1 balances f₀, f₁, f₂, f₃, …, f_K is obtained. The charged/ionic species of the system are involved in the charge balance

       (3)

     (3a)

applicable for static and dynamic systems. Note that 0∙∙(x_m) = 0 for a species with zero charge (z_m=0), e.g., 0∙[H₂O] = 0 (z₁=0).

Free water particles (N₁), and water bound in the hydrates , are included in the balances: f₁ = f(H) and f₂ = f(O):

(4)

(5)

Next, the linear combination

  (6)

is obtained, where aki and bkj in equations 4-6 are the numbers of atoms/cores Y_k (k = 1,...,K) in , and in the j-th component of the system, resp. The core is perceived as a non-transformable cluster of atoms, of the same elemental composition (expressed by chemical formula) and structure; e.g. the set of species: HSO₄^-1, SO₄^-2 and Fe SO₄ contains SO₄^-2 as the core. One can notice that: (a) N₁ involved with H₂O 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: f₃, ..., 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 f₀, …, 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 ^{19, 41}, in accordance with Noether's conceptual approach to linear algebra ^{42, 43}. For this purpose, we consider first an example of titration in a redox system, where comproportionation ⁴⁴ reactions occur. The conclusions arising in the formulation of this system (denoted as the system S, for brevity) will then be generalized later in discussion.

System S: KBrO₃ (C) ⇒NaBr (C0) + H₂SO₄ (C₀₁)

Let us consider the system, where V₀ mL of Dis composed of NaBr (N01 molecules) + H₂ SO₄ (N₀₂ molecules) + H₂O (N₀₃ molecules), and V mL of T is composed of KBrO₃ (N₀₄ molecules) + H₂O (N₀₅ molecules). The D+T mixture thus formed involves the following species:

H₂O (N₁), H⁺¹ (N₂,n₂), OH^-1 (N₃,n₃), HBrO₃ (N₄,n₄),s, BrO₃^-1 (N₅,n₅), HBrO (N₆,n₆), BrO^-1 (N₇,n₇),

Br₂ (N₈,n₈), Br₃^-1 (N₉,n₉), Br^-1 (N₁₀,n₁₀), Na+1 (N₁₁,n₁₁) , K⁺¹ (N₁₂,n₁₂),

HSO₄^-1 (N₁₃,n₁₃), SO₄^-2 (N₁₄,n₁₄).

Denoting C₀V₀ = 10³∙N₀₁/N_A, C₀₁V₀ = 10³∙N₀₂/N_A, CV = 10³∙N₀₄/N_A, we formulate the fraction titrated ^{16, 25, 45}

Φ =      (8)

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

f₀ = ChB :

N₂ – N₃ – N₅ – N₇ – N₉ – N₁₀ + N₁₁ + N₁₂ –

N₁₃ – 2N₁₄ = 0                         ⟹        (9)

[H⁺¹] – [OH^-1] – [BrO₃^-1] – [BrO^-1] – [Br3^-1] – [Br^-1] + [Na⁺¹] + [K⁺¹]

– [HSO₄^-1] – 2[SO₄^-2] = 0      (9a)

f₁ = f(H) :

2N₁ + N₂(1+2n₂) + N₃(1+2n₃) + N₄(1+2n₄) + 2N₅n₅ + N₆(1+2n₆) + 2N₇n₇ + 2N₈n₈ + 2N₉n₉ + 2N₁₀n₁₀ + 2N₁₁n₁₁ + 2N₁₂n₁₂ + N₁₃(1+2n₁₃) + 2N₁₄n₁₄ = 2N₀₂ + 2N₀₃ + 2N₀₅

f₂ = f(O) :

N₁ + N₂(1+n₂) + N₃(1+n₃) + N₄(3+n₄) +

N₅(3+n₅) + N₆(1+n₆) + N₇(1+n₇) + N₈n₈ +

N₉n₉ + N₁₀n₁₀ + N₁₁n₁₁ + N₁₂n₁₂ + N₁₃(4+n₁₃) +

N₁₄(4+n₁₄) = 4N₀₂ + N₀₃ + 3N₀₄ + N₀₅

–f₃ = –f(Na) :

N₀₁ = N₁₁               ⟹           [Na⁺¹] = C₀V₀/(V₀+V)       (10)

–f4 = –f(K) :

N₀₄ = N₁₂               ⟹           [K⁺¹] = CV/(V₀+V)     (11)

–f₅ = –f(SO₄)               ⟺                –f5 = –f(S):      (12)

N₀₂ = N₁₃ + N₁₄     ⟹   (12a)

[H SO₄^-1] + [SO₄^-2] = C₀₁V₀/(V₀+V)   (12b)

f₆ = f(Br) : N₄ + N₅ + N₆ + N₇ + 2N₈ + 3N₉ + N₁₀ 

= N₀₁ + N₀₄              ⟹     (13)

[HBrO₃] + [BrO₃^-1] + [HBrO] + [BrO^-1] +2[Br₂] +

 3[Br₃^-1] + [Br^-1] = (C₀V₀ + CV)/(V₀+V)   (13a)

f₁₂ = 2 f(O) – f(H) 

– N₂ + N₃ + 5N₄ + 6N₅ + N₆ + 2N₇ + 7N₁₃ + 8N₁₄

 = 6N₀₂ + 6N₀₄     (14)

– [H⁺¹] + [OH^-1] + 5[HBrO₃] + 6[BrO3^-1] + [HBrO] + 2[BrO^-1] + 7[HSO₄^-1] + 8[SO₄^-2]

= 6(C₀₁V₀ + CV)/(V₀+V)   (14a)

f₁₂ + f₀ – f₃ – f₄ – 6∙f₅ = 0            ⟺           (+1)f1 +

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

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

(+6)f(S) – ChB = 0     (15)

5(N₄+N₅) + 1(N₆+N₇) – N₉ – N₁₀

= – N₀₁ + 5N₀₄                         ⟹   (15a)

5([HBrO₃] + [BrO₃^-1]) + [HBrO] + [BrO^-1] –

[Br3^-1] – [Br^-1] = + 5∙   (15b)

(+5)([HBrO₃]+[BrO₃^-1]) + (+1)([HBrO]+[BrO^-1]) + 2∙0∙[Br₂] + 3∙∙[Br₃^-1] + (–1)[Br^-1]

= (–1)∙ + (+5)∙ (15c)

Z_Br∙f6 – (f₁₂ + f₀ – f₃ – f₄ – 6∙f₅)

(Z_Br–5)(N₄+N₅) + (Z_Br–1)(N₆+N₇) + 2Z_BrN₈ + (3Z_Br+1)N₉ + (Z_Br+1)N₁₀

= (Z_Br+1)N₀₁ + (Z_Br–5)N₀₄       ⟹     (16)

(ZBr–5)([HBrO₃]+[BrO₃^-1]) + (ZBr^–1(^HBrO +[BrO^-1]) + 2ZBr[Br₂] + (3ZBr+1)[Br₃^-1] + (ZBr+1)[Br^-1] =

(ZBr+1) + (ZBr–5)∙   (16a)

Other linear combinations are also possible. From the linear combination (f₁₂ + f₀ – f₃ – f₄ – 6∙f₅ + f₆)/2 we get the shortest (involving the smallest number of components) form of GEB for the system S:

3(N₄+N₅) + (N₆+N₇) + N₈ + N₉ = 3N₀₄    (17)

3([HBrO₃] + [BrO₃^-1]) + ([HBrO] + [BrO^-1]) + [Br₂] +

[Br₃^-1] = 3∙   (17a)

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 a₁∙f₀ + a₂∙f₁₂ + of the balances: f₀, f₁₂, f₃, …, f₆ 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 SO₄^-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(SO₄) (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 :

f₀ = ChB :       N₂ – N₃ – N₁₀ + N₁₁ – N₁₃ – 2N₁₄ = 0

f₁ = f(H) :      2N₁ + N₂(1+2n₂) + N₃(1+2n₃) +

2N₁₁n₁₁ + N₁₃(1+2n₁₃) + 2N₁₄n14 = 2N₀₂ + 2N₀₃

f₂ = f(O) :       N₁ + N₂n₂ + N₃(1+n₃) + N₁₁n₁₁ +

N₁₃(4+n₁₃) + N₁₄(4+n₁₄) = 4N₀₂ + N₀₃

–f₃ = –f(Na) :      N₀₁ = N₁₁

–f₅ = –f(SO₄) :  N₀₂ = N₁₃ + N₁₄

f₆ = f(Br) :     N₁₀ = N₀₁

and then:

f₁₂ = 2∙f₂ – f₁ :  – N₂ + N₃ + 7N₁₃ + 8N₁₄ = 6N₀₂

f₁₂ + f₀ – f₃ – 6f₅ + f₆ = 0     ⟺     (+1)f(H) + (–2)f(O) + (+1)f(Na) + (+6)f(S) + (–1)f(Br) – ChB = 0      (18)

0 = 0   (18a)

for T :

f₀ = ChB :     N₂ – N₃ – N₅ + N₁₂ = 0

f₁ = f(H) :    2N₁ + N₂(1+2n₂) + N₃(1+2n₃) +

N₄(1+2n₄) + 2N₅n₅ + 2N₁₂n₁₂ = 2N₀₅

f₂ = f(O) :    N₁ + N₂n₂ + N₃(1+n₃) + N₄(3+n₄) +

N₅(3+n₅) + N₁₂n₁₂ = 3N₀₄ + N₀₅

–f₄ = –f(K) :N₀₄ = N₁₂

–f₆ = –f(Br) :N₀₄ = N₄ + N₅

and then:

f₁₂ = 2∙f₂ – f₁ :      – N₂ + N₃ + 5N₄ + 6N₅ + 8N₁₄ = 6N₀₄

f₁₂ + f₀ – f₄ – 5f₆ = 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 KBrO₃, i.e., there are the boundary values of ONs in bromine redox systems; Br(+7) species are omitted in considerations ²⁴.

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

f₁₂ is linearly dependent on: f₀, f₃, f₅ and f₆ in eq. 18, i.e., f₁₂ = f₃ + 6f₅ – f₆ – f₀;

f₁₂ is linearly dependent on: f₀, f₄ and f₆ in eq. 19, i.e., f₁₂ = f₄ + 5f₆ –f₀.

In other words, the f₁₂ 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 KBrO₃) 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 KBrO₃ has Z_Br–5 bromine electrons; then N₀₁ molecules of NaBr involve (Z_Br + 1)∙N₀₁ bromine electrons, N₀₄ molecules of KBrO₃ involve (Z_Br –5)N04 bromine electrons. Thus, the total number of bromine electrons introduced by NaBr and KBrO₃ is (Z_Br + 1)∙N₀₁ + (Z_Br – 5)N₀₄. On this basis, we state that ^{2, 27}:

N₄ species HBrO₃∙n₄H₂O involve (Z_Br – 5)∙N₄ bromine electrons;

N₅ species BrO₃^-1∙n₅H₂O involve (Z_Br–5)∙N₅ bromine electrons;

N₆ species HBrO∙n₆H₂O involve (Z_Br–1)∙N₆ bromine

electrons;

N₇ species BrO^-1∙n₇H₂O involve (Z_Br–1)∙N₇ bromine

electrons;

N₈ species Br₂∙n₈H₂O involve 2Z_Br∙N₈ bromine

electrons;

N₉ species Br₃^-1∙n₉H₂O involve (3Z_Br+1)∙N₉ bromine

electrons;

N₁₀ species Br^-1∙n₁₀H₂O involve (Z_Br+1)∙N₁₀ 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’ ²⁷ (pp. 41-43), and – nominally – all electrons of the players are involved in the balance 16a. Following this line of reasoning, it can be also stated that the card players do not engage, as a rule, all their cash resources in the game. What's more – the ‘debt of honour’ principle can be applied ²⁷ (p. 43). Simply, on the ground of linear combination, in eq. 16a one can replace ZBr for Br by ζBr ≠ ZBr; in particular, one can apply ζBr = 0, see equations 15a and 15b in context with equations 16 and 16a.

function F = Function_NaBr_H₂ SO₄_KBrO3(x)

%NaBr (C0) H₂ SO₄ (C₀₁) V₀ KBrO₃ (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 SO₄ SO₄

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);

SO₄=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 ⁴⁶, pH = – log[H⁺¹], , pBr = – log[Br^-1], V₀ = 100, C₀ = 0.01, C = 0.1, C₀₁ = 0, 0.01 or 0.1. The equilibrium constants related to this system were cited in ref. ².

The results of calculations are presented graphically (Figure 1), as the graphs: (a) E = E(Φ), (b) pH = pH(Φ) and (c) speciation curves [ ] = φi(Φ), with the fraction titrated Φ (eq. 8) on the abscissas. The impact of growth in C01 concentration is illustrated here.

At C₀₁ = 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 (Figure. Ia), whereas pH first increases, passes through maximum and then decreases, see IIa). The relevant E and pH changes are small (Figures Ia, IIa). Binding the H+1 ions in reaction 21 causes a weakly alkaline reaction (Figure. IIa).

At C₀₁ = 0.01 (Figures Ib, IIb, IIIb) and 0.1 (Figures Ic, IIc, IIIc), the stoichiometry 1 : 5, i.e., Φ eq = 0.2, see ^{16, 28}, resulting from the shape of the related graphs, is expressed by reaction

BrO₃^-1 + 5Br^-1 + 6H⁺¹ = 3Br₂ + 3H₂O    (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 ((Figure 1, column I). With an increase of the C01 value, the graphs of pH vs. Φ resemble two almost straight line segments intersecting at Φ eq = 0.2 ((Figure 2, column II). However, the pH-ranges covered by the titration curves are gradually narrowed; it is an effect in growth of dynamic buffer capacity of the related redox systems ³².

Figure 1.The functions: E = E(Φ) (column I), pH = pH(Φ) (column II) and speciation diagrams (column III), at (C0,V0,C) = (0.01,100,0.1), and three different C01 values, indicated at the graphs.

Concluding remarks

The linear combination

     ⟺

                          ⟺

                 ⟺

(22)

involves K balances: f₀, f₁₂, f₃,…,f_K where d₁ = +1, d₂ = –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 f₁₂ is not the independent equation in the f₀, f₁₂, f₃, …, f_K, and Then f₀, f₃, …, f_K is the set of K–1 independent balances, composed of charge balance (f₀) and K–2 elemental/core balances f₃, …, f_K.

Referring now to a redox system, we arrange the elemental/core balances in the sequence f₁, f₂, …, f_K*, f_K*+1 , f_K , and then formulate the equation

(23)

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. (k = 1,…,K*). It is clearly visible in the case of Br₃^-1 in eq. 15c, where –1 = 3, and (less visible) in the case of Br₂ in the eq. 15c, where 0 = 2∙0, i.e., the coefficient d₉ = –1 at [Br₃^-1] in eq. 15b is involved with (not equal to) the oxidation number ω9 = of bromine in Br₃^-1 (λ₉ = 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 f₄ = f(Na)    ⟺ f(Na) = f₄    ⟺    –f₄ = –f(Na).

Starting from K+1 balances: f₀, f_1, f₂, f₃, …, f_K,, after formulation of the linear combination f₁₂ we obtain the set of K balances: f₀, f₁₂, f₃, …, f_K,. In a non-redox system, f₁₂ is the dependent balance; we have there K–1 independent balances: f₀, f₃, …, f_K. In other words, f₁₂ 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 f₀, f₁₂, f₃, …, f_K, are linearly dependent for non-redox systems. In a redox system, f₁₂ is the independent balance (i.e., different from the identity, 0=0); then we have K independent balances: f₀, f₁₂, f₃, …, f_K, that will be rearranged – optionally – as the set (f₁₂, f₀, f₃, …, 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 f₁₂ with f₀, f₃, …, f_K is reducible to identity 0 = 0, i.e., f₁₂ is linearly dependent on f₀, f₃, … , f_K (equations 24,25).

• For a redox system, any linear combination of f₁₂ with f₀, f₃, …, f_K is not reducible to identity 0 = 0, i.e., f₁₂ is linearly independent on f₀, f₃, …, f_K.

• In conclusion, the linear independency/dependency of f₁₂ = 2∙f(O) – f(H) from other balances: f₀, f₃, …, 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).

• f₁₂ = 2∙f(O) – f(H) is the primary form of Generalized Electron Balance (GEB), f₁₂ = prGEB, completing the set of K balances f₀, f₁₂, f₃, …, f_K necessary for resolution of redox systems of any degree of complexity.

• Any linear combination of f₁₂ with the balances f₀, f₃, …, 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 f₁₂ is considered as the primary form of GEB, f₁₂ = pr-GEB. The f₁₂, LC and any other combination , where ak ∈ ℝ, have full properties of GEB, although the simplest/shortest form of GEB, involving the smallest number of components, is more desirable.

• 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 ^{47, 48, 49}, where mixed solvates

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.

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.