Generalized Electron Balance for Dynamic Redox Systems in Mixed-Solvent Media

A complex example of electrolytic redox system involving 47 species, 3 electron-active elements and five (3 amphiprotic + 2 aprotic) co-solvents, is presented. Mixed solvates of the species thus formed are admitted in the system considered. It is proved that the Generalized Electron Balance (GEB) in its simplest form obtained according to the Approach II to GEB is identical with the one obtained for aqueous media and binary-solvent system, and is equivalent to the Approach I to GEB.


Introduction
Motto: "Everything should be made as simple as possible, but not simpler" [1].
In the previous issues [2]- [8] and in earlier papers cited therein, the concept of Generalized Electron Balance (GEB), completing the set of compatible equations necessary for quantitative/mathematical solution of electrolytic redox systems, was introduced as two alternative options, named as Approach I and Approach II to GEB.In both Approaches it is assumed/admitted, that all the species i z i X exist in an electrolytic system in their natural form, i.e., as solvates.In particular, there are hydrates X n n ⋅ in binary-solvent media (W, A), (W, B) or (A, B) [9]- [11], respectively.The values of n Wi , n Ai and n Bi , considered as mean numbers of W, A and B attached to i z i X , are unknown, in principle, and vary with the co-solvent(s) composition, and solute(s) concentration.
In this paper, we refer also to more complex media with the mixture of co-solvents: W, A, B, E and F. We assume that the co-solvents are mutually miscible and at least one of the co-solvents has amphiprotic properties [5].Eeach of the co-solvents has potential/real solvating properties, i.e., , , , , 0 X other than those formed in (known from) aqueous media.In other words, W, A, B, E, F enter (potentially) the solvating sphere of i z i X .On this basis, the elemental balances f(E(i)) for particular elements E(i) are formulated.For ordering purposes, we denote: We apply also the balances f(C(Y)) for the cores C(Y), Y = A, B, E, F. The "core" is a cluster of elements of the same composition, structure and charge, that does not undergo a change in the system in question; e.g., CH 3 OH,

Formulation of Balances
Let us consider a system obtained after addition of V mL of titrant (T) containing I 2 (C) + KI (C 1 ) + CO 2 (C 2 ) in A + B + E + F into V 0 mL of titrand (D) containing KBrO 3 (C 0 ) + HCl (C 01 ) + CO 2 (C 02 ) in W + A + E; all concentrations are expressed in mol/L.The volume V 0 mL of D is composed of N 01 molecules of KBrO 3 , N 02 molecules of HCl, N 03 molecules of CO 2 , N 04 molecules of W, N 05 molecules of A, and N 06 molecules of E; V mL of T is composed of N 07 molecules of I 2 , N 08 molecules of KI, N 09 molecules of CO 2 and N 011 molecules of A, N 012 molecules of B, N 013 molecules of E, and N 014 molecules of F.
We assume that the solutes composing D and T were introduced in single solvents or mixtures of solvents.In ca.V 0 + V mL of a D + T mixture thus obtained, we have the following species (all changes in oxidation degrees are admitted). ) )     (1) )       1) and ( 2) we obtain ) As we see, Equation (3) does not involve the terms N 1 , N 04 , and {n iW } related to water.To cancel the terms involved with A, B, E and F, we add Equation (3) to the core balances ( 4) -( 7 7) and charge balance (8).Further simplification gives addition of the balance for K (9), and of the core balance 4•f(CO 3 ) (10): • 3•f(CH 3 CN) • Charge balance • f(K) • 4•f(CO As the result of this addition, considered as a kind of linear combination [4], we obtain the simplest form of GEB, expressed in terms of numbers of entities: Applying the relations: (where N A -Avogadro's constant), from Equations ( 11), (12) we have Elemental balances for electro-active elements ("players") are as follows: f(Br) After subtracting (13) from (17), we get the equation for GEB, identical with one obtained according to Approach I to GEB The balance (18) is equivalent to the balance (13).
A remark is needed in relation to the charge balance.Rewriting Equation ( 8) in terms of concentrations (see Equation ( 12)), we have As we see, Equation (19) involves the ionic species related to amphiprotic co-solvents.However, in accordance with the remarks presented in [5], the solvates of pairs of ions: (

Final Comments
The complex redox system in a mixture with five solvents is considered.The discussion can be extended on mixtures with S solvents, 1 , , S A A  , where at least one of the co-solvents has amphiprotic properties.In such systems, the solvates 1, , j S =  ; Equation (13) is the simplest/shortest form of GEB obtained for the redox system considered according to Approach II to GEB.Equation (13) was obtained from linear combination of the related balance 2•f(O) -f(H) (Equation ( 3)) with: charge balance (Equation ( 8)), elemental balances for other "fans" (C, K) (Equations ( 9), ( 10)), and core balances (Equations ( 4) -( 7)) related to organic solvents in this system.This GEB does not involve the species composed only of "fans": H, O, C, K.In particular, it does not contain the components explicitly related to the solvent species.The paper is an illustration of the compact formulation of redox systems according to GATES/ GEB Principles, presented in Ref. [5].
Note that-at the start-the Approach II does not distinguish between "fans" and "players"; the terms "fans" and "players" are used here only for the needs of the Approach I to GEB.In further parts of this text, the "players" (the electro-active elements) are distinguished later only to indicate the equivalency of the Approaches I and II.
N i entities of these species in the related system.In further part of the paper, we assume W = H 2 O, A = CH 3 OH, B = C 2 H 5 OH, E = (CH 3 ) 2 SO, F = CH 3 CN; W, A, B have amphiprotic properties, and E, F-have not.In particular, N 15 ions 4 6 15 15 15 15 15 N 15 n 15E atoms of S, and N 15 n 15F atoms of N. It is assumed that the solvents do not form-with solvates-the species denotes i z i (  )