Dynamic Buffer Capacities in Redox Systems

The buffer capacity concept is extended on dynamic redox systems, realized according to titrimetric mode, where changes in pH are accompanied by changes in potential E values; it is the basic novelty of this paper. Two examples of monotonic course of the related curves of potential E vs. Φ and pH vs. Φ relationships were considered. The systems were modeled according to GATES/GEB principles.


Introduction
The buffer capacity concept is usually referred to as a measure of resistance of a solution (D) on pH change, affected by an acid or base, added as a titrant T, i.e., according to titrimetric mode; in this case, D is termed as titrand.
The titration is a dynamic procedure, where V mL of titrant T, containing a reagent B (C mol/L), is added into V 0 mL of titrand D, containing a substance A (C 0 mol/L). The advance of a titration B(C,V) ⟹ A(C 0 ,V 0 ), denoted for brevity as B ⟹ A, is characterized by the fraction titrated [1][2][3][4] that introduces a kind of normalization (independence on V 0 value) for titration curves, expressed by pH = pH(Φ), and E = E(Φ) for potential E [V] expressed in SHE scale. The redox systems with one, two or more electron-active elements are modeled according to principles of Generalized Approach to Electrolytic Systems with Generalized Electron Balance involved (GATES/GEB), described in details in [5][6][7][8][9][10][11][12][13][14][15][16], and in references to other authors' papers cited therein.

Examples of titration curves pH = pH(Φ) and E = E(Φ) in redox systems
In this paper, we refer to the disproportionating systems: (S1) NaOH ⟹ HIO and (S2) HCl ⟹ NaIO, characterized by monotonic changes of pH and E values during the related titrations (i.e., the case 1 o ). In both instances, the values: V 0 =100, C 0 =0.01, and C=0.1 were assumed. The set of equilibrium data [18][19][20] applied in calculations, presented in Table 1, is completed by the solubility of solid iodine, I 2(s) , in water, equal 1.33•10 -3 mol/L. The related algorithms, prepared in MATLAB for S1 (NaOH ⟹ HIO) S2 (HCl ⟹ NaIO) system according to the GATES/GEB principles, are presented in Appendices 1 and 2.
The titration curves: pH = pH(Φ) and E = E(Φ) presented in Figure 1 and Figure 2 are the basis to formulation of dynamic buffer capacities in the systems S1 and S2.

Dynamic acid-base buffer capacities β V and B V
Dynamic buffer capacity was referred previously only to acid-base equilibria in non-redox systems [3,[21][22][23]. However, the dynamic (β V ) and windowed (B V ) buffer capacities can be also related to acid-base equilibria in redox systems. The β V is formulated as follows [3,21] is the current concentration of B in D+T mixture, at any point of the titration. In the simplest case, D is a solution of one substance A (C 0 mol/L), and then equation 3 can be rewritten as follows where Φ is the fraction titrated (equation 1). Then we get where is the sharpness index on the titration curve. For comparative purposes, the absolute values,β V  and η, for β V (equations 1,5) and η (equation 6) are considered. At C 0 /C << 1 and small Φ value, from equation 3 we get The β V value is the point-assessment and then cannot be used in the case of finite pH-changes (∆pH) corresponding to an addition of a finite volume of titrant (β V is a non-linear function of pH). For this purpose, the 'windowed' buffer capacity, B V , defined by the formula [3,21] has been suggested. From extension in Taylor series we have From equations 7 and 9 we see that β V is the first approximation of B V . One should take here into account that finite changes (∆pH) in pH, e.g. ∆pH = 1, are involved with addition of a finite volume of a reagent endowed with acid-base properties, here: base NaOH, of a finite concentration, C.

Dynamic redox buffer capacities
In similar manner, one can formulate dynamic buffer capacities E V β and E V B , involved with infinitesimal and finite changes of potential E values: where c is defined by equation 2, and then we have

Graphical presentation of dynamic buffer capacities in redox systems
Referring to dynamic redox systems represented by titration curves presented in Figures 1,2

Discussion
Disproportionation of the solutes considered (HIO or NaIO) in D occurs directly after introducing them into pure water. The disproportionation is intensified, by greater pH changes, after addition of the respective titrants: NaOH (in S1) or HCl (in S2), and the monotonic changes of E = E(Φ) and pH = pH(Φ) occur in all instances.
All attainable equilibrium data related to these systems are included in the algorithms implemented in the MATLAB computer program (Appendices 1 and 2). In all instances, the system of equations was composed of: generalized electron balance (GEB), charge balance (ChB) and concentration balances for particular elements ≠ H,O.
In the system S1, the precipitate of solid iodine, I 2(s) , is formed ( Figure 5). In the (relatively simple) redox system S2, we have all four basic kinds of reactions; except redox and acid-base reactions, the solid iodine (I 2(s) ) is precipitated and soluble complexes: I 2 Cl -1 , ICl and ICl 2 -1 are formed ( Figure 6A). Note that I 2(s) + I -1 = I 3 -1 is also the complexation reaction.   In the system S2, all oxidized forms of Cl -1 were involved, i.e. the oxidation of Cl -1 ions was thus pre-assumed. This way, full "democracy" was assumed, with no simplifications [18][19][20]. However, from the calculations we see that HCl acts primarily as a disproportionating, and not as reducing agent. The oxidation of Cl -1 occurred here only in an insignificant degree ( Figure 6B); the main product of the oxidation was Cl 2 , whose concentration was on the level ca. 10 -16 -10 -17 mol/L.

Final comments
The redox buffer capacity concepts: V β and E V β can be principally related to monotonic functions. This concept looks awkwardly for non-monotonic functions pH = pH(Φ) and/or E = E(Φ) specified above (2 o -4 o ) and exemplified in Figures 7,8,9. For comparison, in isohydric (acid-base) systems, the buffer capacity strives for infinity.
The formula for the buffer capacity, suggested in [27] after [28], is not correct. Moreover, it involves formal potential value, perceived as a kind of conditional equilibrium constant idea, put in (apparent) analogy with the simplest static acid-base buffer capacity, see criticizing remarks in [29]; it is not adaptable for real redox systems.
In Baicu et al. [40], a nice proposal of "slyke", as the name for (acid-base, pH) buffer capacity unit, has been raised.