157-9818-82c1af6d01b5.png width=45 height=43.125 />. For Ni entities we apply the notation, e.g. for
we have H+1(N2, n2); N2 ions involve: N2(1 + 2n2) atoms of H, and N2n2 atoms of O. For
in binary-solvent medium, composed of A = CH3CN and B = C2H5OH as co-solvents, we apply the notation
, where niA ≥ 0 and niB ≥ 0 are the mean numbers of A and B attached to (not solvated species are included in this notation); e.g. for we have HBrO (N7, n7A, n7B);
N7 entities of HBrO·n7ACH3CN·n7BC2H5OH involve: N7(1 + 3n7A + 6n7B) atoms of H, N7(1 + n7B) atoms of O, N7(2n7A + 2n7B) atoms of C, and N7n7A atoms of N.
The notation can be extended on other co-solvents A, B or more complex systems, with co-solvents A, B, C, ··· included. In all instances, the (external) charge of is introduced by; the solvating molecules (e.g., H2O, CH3CN, C2H5OH) are neutral.
Referring again to ni or nAi and nBi values, one should also take into account the fact that the solvents are not always the dominant components of an electrolytic system. In some instances, e.g. concentrated H2SO4 , HNO3 or HCl solutions, the roles of solvent and solute may be interchanged. The hydration number of individual species varies with the concentration of a suitable, aqueous solution. It should be noted that these values are factually unknown and vary with concentration of solutes. It suffice to say that even the hydration number of H+1 ion in aqueous solutions is not clearly specified, see e.g.  .
3.1. The (Br2, H2O) System
N01 molecules of Br2 is mixed with N02 molecules of H2O, and V mL of the solution is thus obtained. There are the following species: H2O (N1), H+1 (N2, n2), OH–1 (N3, n3), HBrO3 (N4, n4), (N5, n5), HBrO (N6, n6)BrO–1 (N7, n7), Br2 (N8, n8), (N9, n9), Br–1 (N10, n10), involved in the elemental balances:
Then we get 2∙f(O) - f(H)
Addition of (4) to charge balance
gives the equation
Subtraction of (6) from ZBr∙ f(Br), where ZBr = 35 is the atomic number for Br, gives
Applying the relations:
gives the equation  
obtained according to Approach I for C mol/L Br2. It is assumed that Br2 does not react with H2O, i.e., none products of this (virtual, not real) reaction are formed, i.e., application of (8) and (9) to (3) – (6) gives the balances, expressed in terms of molar concentrations:
The Br is the only one electron-active element in the system (Br2, H2O). In the terminology relating to card games  , applied in the Approach I, the redox systems have electron-active elements called as “players” and electron-non-active elements, named as “fans”. In this context, the electrons are seen as “money”. The “player” in HBrO∙n7H2O is Br, whereas the elements: H, O are considered as “fans”. The species H+1·n2H2O involves only “fans”. Equation (4a), obtained from 2·f(O) – f(H), involves concentrations of the species, composed only from “fans”: H and O.
3.2. The (Br2, CH3CN, C2H5OH) System
V mL of the solution is obtained by introducing N01 molecules of Br2 into the mixture of N02 molecules of CH3CN (=A) and N03 molecules of C2H5OH (=B). According to notation applied above, in the mixture thus formed we have the following species:
CH3CN (N1), C2H5OH (N2), (N3, n3A, n3B)C2H5O–1 (N4, n4A, n4B), HBrO3 (N5, n5A, n5B),
(N6, n6A, n6B), HBrO (N7, n7A, n7B), BrO–1 (N8, n8A, n8B)Br2 (N9, n9A, n9B), (N10, n10A, n10B)Br–1 (N11, n11A, n11B) (11)
The N1 and N2 in (11) refer to the numbers of molecules of the co-solvents A and B not involved in the related solvates. On this basis, we formulate the elemental balances:
From (12) and (13) we get
Addition of (15) to (17) gives
Addition of (18) to 2·f(C) (19)
Subtraction of (19) from the charge balance (21)
gives the equation equivalent to Equation (6); then we get Equations (6a) and (10).
Note that the procedure involved with multiplication, e.g. 2·f(O), 2·f(C), f(N) = 1·f(N), and then addition/ subtraction of the corresponding equations is a realization of linear combination . Generally, a linear combination of equations has the form
where bk—the pre-assumed numbers.
3.3. Comparison of (Br2, H2O) and (Br2, CH3CN, C2H5OH) Systems
From linear combination of the charge balance and elemental balances related to “fans”: H, O, N and C, we obtain the simplest/shortest form of GEB, expressed by Equation (6a); the bk values (Equation (22)) are properly chosen for this purpose. Equation (10) is the more extended, but equivalent to (6a), form of GEB, obtained for C mol/L according to Approach I. We see that the form of GEB does not depend on the solvent composition—assuming that the solvent does not form other (new) species with a solute.
It should be noted that the balance 2·f(O) – f(H) obtained for the system (Br2, H2O) does not contain, as components, the numbers: N02, N1 and the ni (i = 2, ···, 10) associated with the (undefined - except N02) numbers of water molecules. The balance –(2·f(O) – f(H)) (and then the balance 2·f(O) – f(H)) formulated for the system (Br2, CH3CN, C2H5OH) contain the numbers involved with water molecules; the water molecules are cancelled completely after due combination of elemental balances for all “fans”. The difference
plays the role of [H+1] – [OH–1] in aqueous media.
4. Some Generalizing Remarks
Let us assume that the aqueous system involves K elements E(k),; P elements are considered as “players” and then F = K – P elements are treated as “fans”; the elemental balance related to the element E(k) will be denoted by f(E(k)).
A special role among the elements related to aqueous systems play H and O; the related balances are: f(E(1)) = f(H), and f(E(2)) = f(O). The balances for successive “fans” are denoted as f(E(3)), ···, f(E(K–P)), whereas f(E(K–P+1)), ···, f(E(K)) are formulated for “players”. Applying the notations specified above, we have:
On this basis we formulate the balance 2·f(O) - f(H)
The elemental balances f(E(3)), ···, f(E(K–P)) are multiplied by the corresponding numbers: and the linear combination
from Equations. (26), (8), (27), after addition of charge balance (28)  
After a proper choice of bk and cancellations, Equation (29) does not involve “fans”; it is the simplest form of GEB. The nik and mjk values should involve particular elements in the corresponding solvates, considered as the species, see Sections 3.1 and 3.2.
5. Final Comments
In the article it is proved that the Generalized Electron Balance (GEB), referred to a redox electrolytic system (aqueous media), is derivable from the equation 2·f(O) – f(H) resulting from comparison of elemental balances: f(H) for H and f(O) for O. This approach, named as the Approach II, is equivalent to the Approach I, based on a common pool of electrons brought by elements forming the system, of any degree of complexity.
The GEB is ultimately expressed in terms of molar concentrations, as charge and concentrations balances, and the expressions for equilibrium constants. Contrary to a redox system, the equation 2·f(O) – f(H) related to a non-redox system of any degree of complexity, is linearly independent on charge and elemental balances, related to elements ≠ H, O. This property, valid for the systems of any degree of complexity, distinguishes between redox and non-redox systems. The GEB is perceived as a rule of a matter conservation, related to electrolytic redox systems.
The terms: Generalized Electron Balance (GEB) and Generalized Approach to Electrolytic Systems (GATES) are still unknown to a wider community. This article aims to fill this gap. Therefore, this article present a concise description of redox systems, in the context of linear transformations of algebraic equations.