^{1}

^{2}

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The Generalized Electron Balance (GEB), together with charge balance and concentration balances, completes the set of equations needed for resolution of electrolytic redox systems. The general formulae for GEB were obtained according to Approach II to GEB, i.e., on the basis of the equation 2?f(O) ? f(H) obtained from elemental balances: f(H) for H, and f(O) for O. Equivalency of the Approach II and the Approach I to GEB was proved for an aqueous solution and a binary-solvent system. On this basis, a compact form of GEB was derived.

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 [

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 [_{2}SO_{4}) solution of FeSO_{4} + H_{2}C_{2}O_{4} is titrated with KMnO_{4}, we have 41 species and 29 equilibrium constants, whereas in the system CuSO_{4} + H_{2}SO_{4} + NH_{3} + CH_{3}COOH + KI titrated with Na_{2}S_{2}O_{3}, we have 47 species interrelated in 35 equilibrium constants [_{3} + HCl + H_{2}SeO_{3} + HgCl_{2} titrated with ascorbic acid C_{6}H_{8}O_{6} we have 49 species and 39 equilibrium constants [

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

species _{i} be the number of _{i} is the external charge (q_{i} = z_{i}·e, z_{i} = 0,

±1, ±2, ···) of_{A}; F—Faraday’s constant, N_{A}—Avogadro’s

number. Let K be the number of different elements E(k) _{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

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

lecules attached to_{i} entities

we have H^{+1}(N_{2}, n_{2}); N_{2} ions _{2}(1 + 2n_{2}) atoms of H, and N_{2}n_{2} atoms of O. For

_{3}CN and B = C_{2}H_{5}OH as co-solvents, we apply the notation

_{i}_{A} ≥ 0 and n_{i}_{B} ≥ 0 are the mean numbers of A and B attached to

species are included in this notation); e.g. for _{7}, n_{7A}, n_{7B});

N_{7} entities of HBrO·n_{7A}CH_{3}CN·n_{7B}C_{2}H_{5}OH involve: N_{7}(1 + 3n_{7A} + 6n_{7B}) atoms of H, N_{7}(1 + n_{7B}) atoms of O, N_{7}(2n_{7A} + 2n_{7B}) atoms of C, and N_{7}n_{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 _{2}O, CH_{3}CN, C_{2}H_{5}OH) 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 H_{2}SO_{4} , HNO_{3} 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. [

N_{01} molecules of Br_{2} is mixed with N_{02} molecules of H_{2}O, and V mL of the solution is thus obtained. There are

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}), _{5}, n_{5}), HBrO (N_{6}, n_{6}),

BrO^{–1} (N_{7}, n_{7}), Br_{2} (N_{8}, n_{8}), _{9}, n_{9}), Br^{–1} (N_{10}, n_{10}), involved in the elemental balances:

f(H):

f(O):

f(Br):

Then we get 2∙f(O) - f(H)

Addition of (4) to charge balance

gives the equation

Subtraction of (6) from Z_{Br}∙ f(Br), where Z_{Br} = 35 is the atomic number for Br, gives

Applying the relations:

gives the equation [

obtained according to Approach I for C mol/L Br_{2}. It is assumed that Br_{2} does not react with H_{2}O, 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 (Br_{2}, H_{2}O). In the terminology relating to card games [_{7}H_{2}O is Br, whereas the elements: H, O are considered as “fans”. The species H^{+1}·n_{2}H_{2}O involves only “fans”. Equation (4a), obtained from 2·f(O) – f(H), involves concentrations of the species, composed only from “fans”: H and O.

V mL of the solution is obtained by introducing N_{01} molecules of Br_{2} into the mixture of N_{02} molecules of CH_{3}CN (=A) and N_{03} molecules of C_{2}H_{5}OH (=B). According to notation applied above, in the mixture thus formed we have the following species:

CH_{3}CN (N_{1}), C_{2}H_{5}OH (N_{2}), _{3}, n_{3A}, n_{3B}),

C_{2}H_{5}O^{–1} (N_{4}, n_{4A}, n_{4B}), HBrO_{3} (N_{5}, n_{5A}, n_{5B}),

_{6}, n_{6A}, n_{6B}), HBrO (N_{7}, n_{7A}, n_{7B}), BrO^{–1} (N_{8}, n_{8A}, n_{8B}),

Br_{2} (N_{9}, n_{9A}, n_{9B}), _{10}, n_{10A}, n_{10B}),

Br^{–1} (N_{11}, n_{11A}, n_{11B}) (11)

The N_{1} and N_{2} 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:

f(H) :

f(O) :

f(C) :

f(N) :

f(Br) :

From (12) and (13) we get

Addition of (15) to (17) gives

Addition of (18) to 2·f(C) (19)

gives

Subtraction of (19) from the charge balance (21)

gives the equation

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 [

where b_{k}—the pre-assumed numbers.

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 b_{k} 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 (Br_{2}, H_{2}O) does not contain, as components, the numbers: N_{02}, N_{1} and the n_{i} (i = 2, ···, 10) associated with the (undefined - except N_{02}) numbers of water molecules. The balance –(2·f(O) – f(H)) (and then the balance 2·f(O) – f(H)) formulated for the system (Br_{2}, CH_{3}CN, C_{2}H_{5}OH) contain the numbers involved with water molecules; the water molecules are cancelled completely after due combination of elemental balances for all “fans”. The difference

^{+1}] – [OH^{–1}] in aqueous media.

Let us assume that the aqueous system involves K elements 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:

i.e.

Denoting

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

we have

After a proper choice of b_{k} _{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.