The mathematical description of electrolytic redox systems in aqueous media is realizable with use of the set of equations composed of Generalized Electron Balance (GEB), charge balance, and elemental balances, related to elements E(i) different from H and O [1] -[4] . Ultimately, all the balances are expressed in terms of molar concentrations. The GEB concept can be extended on non-aqueous and mixed-solvent media (binary-solvent systems, in particular), where amphiprotic solvents/co-solvents are involved [5] . The charge balance is based on the principle of electroneutrality of the solution, whereas elemental balances express the preservation of particular elements in the closed system, separated from its environment by diathermal walls; it enables any process in the system to proceed under isothermal conditions. It will also be assumed that none nuclear transformations occur for all elements of the system.

Concentrations of the species in the appropriate equations are also involved in dependencies resulting from the expressions for the equilibrium constants. Because the equilibrium constants are dependent (among others) on the temperature, the assumption concerning the possibility of carrying out the process under isothermal conditions is advisable.

In the papers issued previously (and ones cited mainly in [1] [3] ), two approaches to GEB, named as Approach I and Approach II, were introduced. The Approach I (named as “short” version of GEB) is based on the principle of common pool of electrons, and applicable in the cases where oxidation numbers for all elements in the redox system considered are known beforehand. Combination 2f(O)-f(H) of elemental balances: f(H) for H and f(O) for O is the quintessence of the Approach II to GEB. The fundamental advantage of the Approach II (in context with the Approach I) to GEB is that none prior knowledge on oxidation degrees of elements in complex species of definite elemental composition and charge is needed. Also the terms: “oxidizer” and “reductant” are not applicable as starting terms in considerations on redox systems. These facts are of capital importance, when redox equilibria in the systems with complex organic species, involving radicals and ion-radicals, are considered. Both approaches were realized in numerous examples, where results of calculations obtained according to iterative computer programs [4] [6] were illustrated graphically. In the algorithms applied in calculations, all attainable physicochemical knowledge was involved. This knowledge is of qualitative (particular species) and quantitative (equilibrium constants) nature. For example, in the system where the acidified (H₂SO₄) solution of FeSO₄ + H₂C₂O₄ is titrated with KMnO₄, we have 41 species and 29 equilibrium constants, whereas in the system CuSO₄ + H₂SO₄ + NH₃ + CH₃COOH + KI titrated with Na₂S₂O₃, we have 47 species interrelated in 35 equilibrium constants [6] . In the system KIO₃ + HCl + H₂SeO₃ + HgCl₂ titrated with ascorbic acid C₆H₈O₆ we have 49 species and 39 equilibrium constants [4] [7] [8] . These numbers give an idea of the complexity of relevant systems. On the other hand, the number of these species makes notation of the appropriate balances lengthy, i.e., they take up many space in the text. The present paper is an attempt of generalizing the notation. This allows the synthesis of certain problems associated with the description of the system using a mathematical formalism.

Let an electrolytic system, of volume V [mL], be formed by mixing J different components, ,

and N_0j be the number of entities (uncharged molecules) of. In this mixture we have I different kinds of

species and N_i be the number of entities, where z_i is the external charge (q_i = z_i·e, z_i = 0,

±1, ±2, ···) of, expressed in elementary charge units, e = F/N_A; F—Faraday’s constant, N_A—Avogadro’s

number. Let K be the number of different elements E(k) composing the system, and m_jk be the

number of atoms of the k-th element in Y_0j , and n_ik be the number of atoms of the k-th element in the species.

All the species in the system are considered in their natural form, i.e. as solvates. So, in aqueous solutions

the species exist as hydrates, in principle, i.e., where n_i ≥ 0 is the mean number of water mo-

lecules attached to. For N_i 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 n_iA ≥ 0 and n_iB ≥ 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, n_7A, n_7B);

N₇ entities of HBrO·n_7ACH₃CN·n_7BC₂H₅OH involve: N₇(1 + 3n_7A + 6n_7B) atoms of H, N₇(1 + n_7B) atoms of O, N₇(2n_7A + 2n_7B) atoms of C, and N₇n_7A 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 n_i or n_Ai and n_Bi 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. [9] .

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

i.e.

Denoting

from Equations. (26), (8), (27), after addition of charge balance (28) [12] [13]

we have

After a proper choice of b_k and cancellations, Equation (29) does not involve “fans”; it is the simplest form of GEB. The n_ik and m_jk values should involve particular elements in the corresponding solvates, considered as the species, see Sections 3.1 and 3.2.

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.