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A simple general relation P = Q + R + 1 between the number P of kinds of species, the number Q of charge and elemental/core balances and the number R of independent equilibrium constants is deduced, and its validity is confirmed for non-redox and redox electrolytic systems, of different degree of complexity.

The quantitative, thermodynamic description of any electrolytic system requires prior information on 1) the species present in the system considered; 2) the equilibrium constants; 3) the balances. The balances and expressions for equilibrium constants interrelate molar concentrations of the species in the system. To do it, we should necessarily define these terms in an unambiguous manner. This possibility gives the Generalized Approach to Electrolytic Systems (GATES) [

This paper refers to batch and dynamic, multicomponent and mono- or multiphase closed systems with condensed (liquid, liquid + solid, liquid1 + liquid2, liquid1 + liquid2 + solid) phases, where quasistatic processes proceed under isobaric + isothermal conditions. The species forming solid phase are marked below in bold letters.

An example of a dynamic procedure is the titration, where V mL of titrant (T) is added into V_{0} mL of titrand (D). The D + T mixture thus formed involves the related species.

The issue raised in this paper concerns formulation of a general relation

between the number P of kinds of species, the number Q of charge and elemental/core balances and the number R of equilibrium constants, referred to redox and non-redox electrolytic systems, of different degree of complexity. As stated below, the terms: species, equilibrium constants and balances are presented in literature in ambiguous manner. These ambiguities can be avoided when the related terms are in accordance with the GATES principles.

Within GATES, the species are considered in their natural form, i.e., as hydrates _{2}O), or as _{s}) attached to (or involved in)_{3} and FeOOH are considered equivalently. However, one can find in litera-

ture [_{3})_{5}Al^{3+}, although soluble hydroxo-complex Al(OH)_{3} is unknown elsewhere in literature―probably for the same reasons as Fe(OH)_{3}; the ephemera/freak (Al(OH)_{3})_{5}Al^{3+} is not acceptable, although the precipitate Al(OH)_{3} is known.

Omission of the species

When collecting the species related to the system considered from the viewpoint of Equation (1), one can choose between the species containing different number of H_{2}O molecules involved in the related formula. This choice is made, however, in the context of accessible physico-chemical knowledge concerning the components considered. For boric acid, ref. [

_{1}, K_{2}, K_{3} for the reactions:_{4},_{3}BO_{3},

stituents by 2 is associated with an increase in the number of equilibrium constants by 2. This means that (P−2) = Q + (R−2) + 1 (or (P + 2) = Q + (R + 2) + 1) is equivalent to Equation (1). Factually, HB(OH)_{4} º H_{5}BO_{4} º H_{3}BO_{3}(H_{2}O) is equivalent to H_{3}BO_{3}. The inclusion/omission in considerations of m independent components is associated with inclusion/omission of m equilibrium constants; it means that (P ± m) = Q + (R ± m) + 1 Û Equation (1).

The value of any equilibrium constant formulated on the basis of mass action law depends on the reaction notation applied for this purpose; e.g., for a polyprotic acid H_{n}L, forming the species H_{j}L^{j−n} (j = 0, ×××, q), equal

numbers of dissociation constants (K_{i}, i = 1, ×××, q) or stability constants of the proto-complexes (_{i} and _{0}) and solubility products (K_{sp}) may be formulated in different manners, E_{01} for _{02 }for _{sp1} for Mg(OH)_{2} = Mg^{2+} + 2OH^{− }and K_{sp2} for Mg(OH)_{2}

+ 2H^{+} = Mg^{2+} + 2H_{2}O [_{02} = E_{01} − J∙pK_{W} (J = RT/F∙ln10); pK_{sp2} = pK_{sp1} − 2∙pK_{W}. In the context with known pK_{W}, pK_{sp1} and E_{01} values, E_{02} and K_{sp2} are not new/independent equilibrium constants. In the latter notation, suggested also in [_{2} [

The set of stability constants for tartrate ^{−} ions were not taken there into account−as elsewhere, for other Me-ions. Nevertheless, in the lack of other, competitive equilibrium data, the stability constants for

Concluding, any kind of complex species in involved with the related equilibrium constant, and inclusion/omission of independent complex(es) in calculations is involved with inclusion/omission of the related equilibrium constant(s); P and R are on opposite sides of equality sign in Equation (1). Inclusion of a species (or a set of species) in the related balances without prior physicochemical information related to its/their equilibrium constant(s) is pointless.

Formulating the balances for electrolytic systems is also ambiguous. First, the most natural form of balance for ions is the charge balance

expressing the principle of electroneutrality of the solution [

Within GATES, the term “core balance” is also applied; e.g., _{4} (complex) in the balance

related to C mol/L FeSO_{4} solution. Generally, a core is considered as a cluster of elements with defined composition (expressed by chemical formula) and charge [_{4} (C) + H_{2}S (C_{1}), where none synproportionation [

where [FeS] is the concentration of the precipitate FeS; Equation (4) can be presented as the sum of core (for ^{2−}) balances: Equation (3), and [FeS] + [H_{2}S] + [HS^{−}] + [S^{2−}] = C_{1}. For the system H_{2}C_{2}O_{4} (C_{1}) + Na_{2}CO_{3} (C_{2}), the elemental balance for C, i.e.,

can be presented as the linear combination of core (for

However, when V mL of C mol/L KMnO_{4} is added into V_{0} mL of C_{0} mol/L of acidified (H_{2}SO_{4}) solution of H_{2}C_{2}O_{4} (C_{0}), the elemental balance for carbon C (in closed system)

cannot be presented as the (combined) sum of core balances for

The balances are usually formulated for aqueous solutions. For beginners, it may seem to be strange that the concentration balances were not applied for hydrogen (H) and oxygen (O), i.e., for the basic elements, introduced mainly by the solvent (H_{2}O) in aqueous media. The latter issue was not known before formulation (2006) [_{i} = n_{iW} of coordinating water molecules at the species_{iW} for H^{+} in aqueous media is practically unknown [_{p} ¹ H, O) elemental/core balances, all coordination num-

bers

The balance 2∙f(O) − f(H), when referred to redox systems of any degree of complexity, is the balance independent on charge and elemental/core balances for Y_{p} ¹ H, O, contrary to non-redox systems [

For comparison, earlier (i.e., after 1960s) approaches to dynamic redox systems were based on primitive formulations, with stoichiometric reactions involved. The roles of oxidizers and reducers were ascribed to indicated species, and homogeneous, non-homogeneous and symmetrical reactions were distinguished. Irrespectively of the complexity of the redox system considered, only two pairs {(Ox_{i}, Red_{i}) çi = 1, 2} were involved in two concentration balances. Charge balance and concentration balances for other elements were not formulated. Only two equilibrium constants, namely “formal” potentials related to the (Ox_{i}, Red_{i}) pairs were used and applied to formulate the equilibrium constant for the redox reaction notation considered. These approaches were extensively criticized, mainly in [

Example 1. For aqueous solution of the mixture KCl (C_{1}) + NaNO_{3} (C_{2}) + NaCl (C_{3}) + HCl (C_{4}) + KNO_{3}(C_{5}) + HNO_{3} (C_{6}) we have: P = 7 (H_{2}O, H^{+}, OH^{−}, K^{+}, Na^{+}, Cl^{−},^{+}] = C_{1}+C_{5}, [Na^{+}] = C_{2}+C_{3}, [Cl^{−}] = C_{1}+C_{4}, _{W}); then 7 = 1 + 5 + 1.

Example 2. For NaCl (excess) + H_{2}O system: P = 6 (H_{2}O, H^{+}, OH^{−}, Na^{+}, Cl^{−}, NaCl_{(s)}); Q = 3 (charge, Na, Cl); R = 2 (K_{W}, s); then 6 = 3 + 2 + 1.

NaCl is usually considered as wholly dissociated in diluted aqueous solutions. However, in the saturated solution, not dissociated forms of this salt are admitted, [NaCl] > 0, i.e., NaCl can be considered as a soluble complex [_{sp} for M_{n}L_{u} by the relation s^{n+u} = n^{n}u^{u}∙K_{sp}, as is usually practiced, e.g. in [

Example 3. CaCO_{3} + H_{2}O system: P = 11 (H_{2}O, H^{+}, OH^{−}, Ca^{2+}, CaOH^{+}, _{3}, CaCO_{3}_{(s)} H_{2}CO_{3},

Example 4. AgCl + H_{2}O + NH_{3} system: P = 16 (H_{2}O, H^{+}, OH^{−}, Ag^{+}, AgOH, _{(s)}, Cl^{−}, _{3}), Q = 4, R = 11; then 16 = 4 + 11 + 1.

Example 5. AgCl + H_{2}O + NH_{3} (excess) system: P = 15 (H_{2}O, H^{+}, OH^{−}, Ag^{+}, AgOH, ^{−}, _{3}); Q = 4, R = 10; then 15 = 4 + 10 + 1.

Example 6. HgCl_{2} + H_{2}O + KI : P = 18 (H_{2}O, H^{+}, OH^{−}, Hg^{2+}, HgOH^{+}, Hg(OH)_{2}, HgCl^{+}, HgCl_{2}, ^{+}, HgI_{2}, _{2}_{(s)}, I^{−}, Cl^{−}, K^{+} ), Q = 5, R = 12; then 18 = 5 + 12 + 1.

Example 7. HgCl_{2} + H_{2}O + KI (excess) : P = 17 (H_{2}O, H^{+}, OH^{−}, Hg^{2+}, HgOH^{+}, Hg(OH)_{2}, HgCl^{+}, HgCl_{2}, ^{+}, HgI_{2}, ^{−}, Cl^{−}, K^{+}), Q = 5, R = 11 ; then 17 = 5 + 11 + 1.

Example 8. Struvite (pr1 = MgNH_{4}PO_{4}) is introduced into V mL of aqueous solution with dissolved CO_{2}. As results from detailed calculations made in [_{sp1}) for pr1 is attained, the solubility product (K_{sp2}) for pr2 = Mg_{3}(PO_{4})_{2} is crossed. At (pC_{0}, pC_{CO2}, pC_{b}) = (3, 4, ∞), the process pr1 à pr2 leads to total depletion of pr1; the solubility product K_{sp1} is not attained (q_{1} < 1). At (pC_{0}, pC_{CO2}, pC_{b}) = (2, 4, ∞), K_{sp2} for pr2 is attained at ppr1 = 2.013 and pr2 precipitates according to reaction 3pr1 = pr2 + _{3} up to ppr1 = 2.362, where the solubility product for pr1 is crossed and the dissolution process pr1 à pr2 is terminated. At equilibrium, the solid phase consists of the two species: pr1 and pr2, and the expressions for K_{sp1} and K_{sp2} are valid. Then at (pC_{0}, pC_{CO2}, pC_{b}) = (3, 4, ∞) we have: P = 23, Q = 5, R = 17; then 23 = 17 + 5 + 1. At (pC_{0}, pC_{CO2}, pC_{b}) = (2, 4, ∞) we have: P = 24, Q = 5, R = 18; then 24 = 18 + 5 + 1.

Example 9. For the Liebig-Denigès titration, described in [

Example 10. V_{0} mL of ZnSO_{4} (C_{0}) + NH_{3 }(C_{01}) + NH_{4}Cl (C_{02}) + NaH_{2}In = C_{20}H_{12}N_{3}O_{7}SNa (erio T) solution is titrated with V mL of C mol/L EDTA = C_{10}H_{14}N_{2}O_{8}Na_{2}. On the basis of the data presented in [

Example 11. For aqueous solution of C mol/L Br_{2} we have [_{2}O, H^{+} , OH^{−}, e^{−}, HBrO_{3}, ^{−}, Br_{2}, ^{−}); Q = 3 (^{−}]) + 2Z[Br_{2}] + (Z+1)[Br^{−}] = 2ZC, where Z = 35-atomic number for Br; [H^{+}] − [OH^{−}] −^{−}] −^{−}] = 0; [HBrO_{3}] + ^{−}] + 2[Br_{2}] + ^{−}] = 2C ); R = 7 (K_{W}, ^{8.6}∙[H^{+}][BrO^{−}] = [HBrO], ^{−}]∙10^{6A(E−1.45)+6pH} , [BrO^{−}] = [Br^{−}]∙10^{2A(E−0.76)+2pH−28}, [Br_{2}] = [Br^{−}]^{2}∙10^{2A(E−1.087)}, ^{−}]^{3}∙10^{2A(E−1.05)}, where A = 16.92 at 25 ^{o}C; then 11 = 3 + 7 + 1.

Example 11a. For C mol/L NaBrO, we have 12 = 4 + 7 + 1.

Example 12. V mL of KMnO_{4} (C) + CO_{2} (C_{1}) as T is added into V_{0} mL of D FeSO_{4} (C_{01}) + H_{2}C_{2}O_{4} (C_{02}) + H_{2}SO_{4} (C_{03}) + CO_{2} (C_{04}). This system involves P = 39 species: H_{2}O, H^{+}, OH^{−}, e^{−}, K^{+}, _{2}C_{2}O_{4},

_{2}CO_{3}, ^{3}^{+}, MnOH^{2+}, ^{2+}, MnOH^{+}, MnSO_{4}, MnC_{2}O_{4}, ^{2+}, FeOH^{+}, FeSO_{4}, ^{3+}, FeOH^{2+}, _{03} value is sufficiently high to prevent precipitation of MnO_{2}, FeC_{2}O_{4} and MnC_{2}O_{4}, i.e. the solubility products of these precipitates are not crossed [

Three electron-active elements: Fe, C and Mn are involved in this system. Denoting atomic numbers: Z_{C} = 6 for C, Z_{Mn} = 25 for Mn, Z_{Fe} = 26 for Fe, the resulting GEB is written according to Approach I to GEB as follows:

The equation for charge balance (Equation (7)) and equations for concentration balances for Fe (Equation (8)), Mn (Equation (9)), C (Equation 10) and SO_{4} (Equation (11)), and K (Equation (12)) are as follows:

The relationships between concentrations of the species in Equations (6)-(12) are formulated on the basis of equilibrium constants, involved in R = 31 relations for: K_{W} and:

then we have 39 = 7 + 31 + 1.

Example 13. V_{0} mL of D containing KIO_{3} (C_{0}), HCl (C_{01}), H_{2}SeO_{3} (C_{02}) and HgCl_{2} (C_{03}) is titrated with V mL of ascorbic acid C_{6}H_{8}O_{6} (C). Within defined volume interval of ascorbic acid solution added, solid iodine, I_{2} (solubility s = 1.33 ´ 10^{−3} mol/L at 20˚C) is formed. We have there P = 46 species: H_{2}O, H^{+}, OH^{−}, e^{−}, K^{+}, C_{6}H_{8}O_{6}, _{6}H_{6}O_{6}, I_{2}, I_{2}, I^{−}, ^{−}, HIO_{3}, _{5}IO_{6}, _{2}, Cl^{−}, HClO, ClO^{−}, HClO_{2}, _{2}, _{2}Cl^{−}, ICl, _{2}SeO_{3}, ^{2+}, HgOH^{+}, Hg(OH)_{2}, HgCl^{+}, HgCl_{2}, ^{+}, HgI_{2},

and R = 37 equilibrium constants: K_{W} and ones involved in the relations:

where: A = 16.92 (at 298 K), pCl = −log[Cl^{−}], pI = −log[I^{−}]; the solubility s of I_{2} is the 37th equilibrium constant. Within F range where I_{2} is the equilibrium solid phase, we have 46 = 8 + 37 + 1; if not (i.e. a = 0 in Equations (17) and (18)), we have 45 = 8 + 36 + 1.

Example 14. V mL NaOH (C) + CO_{2} (C_{1}) is added into V_{0} mL of I_{2} (C_{0}) + KI (C_{01}) + CO_{2} (C_{02}) + W_{(o)} mL of CCl_{4} [_{2}O, H^{+}, OH^{−}, e^{−}, I^{−}, _{2}, I_{2}_{(s)}, I_{2(o)}, HIO, IO^{−}, HIO_{3}, _{5}IO_{6}, _{2}CO_{3}, _{4}), Q = 5 (GEB, charge, I, C, CCl_{4}), R = 14; then 20 = 5 + 14 + 1.

The equilibrium data on electrolytic systems refer almost exclusively to aqueous solutions. The relevant data for the systems with non-aqueous solvents are fragmentary and relate almost exclusively to acid-base equilibria. In the binary-solvent acid-base systems, the pK_{i} values for dissociation constants K_{i} are replaced by dissociation parameters, pK_{i}(x), as functions of the mole fraction x of one of the co-solvents in the related mixtures [

Generalizing, any electrolytic non-redox systems is characterized by one charge balance, and one or more concentration balances. The concentration balances are obtained from elemental or core balances, related to elements Y_{i} ¹ H, O.

Any redox system is characterized by one charge balance, one electron balance, named as generalized electron balance (GEB), and one or more concentration balances for elements/cores Y_{i} ¹ H, O. The GEB can be formulated according to the Approach I to GEB or according to the Approach II to GEB. The Approach I, named also as a “short” version of GEB, is realized under assumption that the oxidation numbers for all elements of the system considered are known beforehand. The Approach II to GEB is formulated on the basis of linear combination 2∙f(O) − f(H) of elemental balances: f(H) for H and f(O) for O. Any linear combination of 2∙f(O) − f(H) with charge and elemental/core balances has full properties of GEB. The GEB is expressed in terms of concentrations of the related species, as charge and concentration balances.

Equation (1) was checked in on numerous examples, related to electrolytic systems of different degree of complexity, and concluding formula was obtained in deductive manner. The non-redox and redox systems are distinguished on the basis of properties of the equation 2∙f(O) − f(H) [

In the examples, well, moderately, or sparingly soluble species are involved in batch or dynamic (titration) systems. An action of some components on the phase composition is also considered. The effect of an excess of the corresponding regent on phase composition is also taken into account. Electron is considered among the species in redox systems. GEB is considered as a one of the balances needed for mathematical description of redox systems, as fully compatible with charge and concentration balances related to the system in question.

Equation (1) deduced in this paper is in close relevance to GATES [

Omission/introduction of a species in the related balance(s) is involved with omission/introduction of the related equilibrium constant. Therefore, the Equation (1) is valid after such operations, (P ± m) = Q + (R ± m) + 1 Û P = Q + R + 1.

Anna MariaMichałowska-Kaczmarczyk,TadeuszMichałowski, (2015) General Relation Valid for Electrolytic Systems. Journal of Analytical Sciences, Methods and Instrumentation,05,74-85. doi: 10.4236/jasmi.2015.54009