**Anna Maria MichaÃ
Âowska- Kaczmarczyk ^{1}, Aneta Spórna-Kucab^{2} and Tadeusz MichaÃ
Âowski^{2*}**

^{1}Department of Oncology, The University Hospital in Cracow, Cracow, Poland

^{2}Department of Analytical Chemistry, Technical University of Cracow, Cracow, Poland

- *Corresponding Author:
- Tadeusz MichaÃ
Âowski

Department of Analytical Chemistry

Technical University of Cracow

24, 31-155 Cracow, Poland.

**Tel:**+48 12 628 20 00

**E-mail:**[email protected]

**Received Date:** September 21, 2017;** Accepted Date:** October 04, 2017;** Published Date:**October 07, 2017

**Citation:**MichaÃ
Âowska-Kaczmarczyk AM, Spórna-Kucab A, MichaÃ
Âowski T (2017) Dynamic Buffer Capacities in Redox Systems. Biochem Mol Biol J. Vol.3 No. 3:11 DOI: 10.21767/2471-8084.100039

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. and pH vs. Φ relationships were considered. The systems were modeled according to GATES/GEB principles.

Thermodynamics of electrolytic redox systems; Buffer capacity; GATES/GEB.

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 V0 mL of titrand D, containing a substance A (C_{0} mol/L). The advance of a titration B(C,V) ⇒ A(C0,V0), denoted for brevity as B ⇒ A is characterized by the fraction titrated [1-4]

(1)

That introduces a kind of normalization (independence on V0 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 [5-16], and in references to other authors' papers cited therein.

According to earlier conviction expressed by Gran [17], all titration curves: pH=pH(Φ) and E=E(Φ), were perceived as monotonic; that generalizing statement is not true [7], however. According to contemporary knowledge, full diversity in this regard is stated, namely: (1o) monotonic pH=pH(Φ) and monotonic E=E(Φ) [18- 20]; (2o) monotonic pH=pH(Φ) and non-monotonic E=E(Φ) [6]; (3o) non-monotonic pH=pH(Φ) and monotonic E=E(Φ) [5]; (4o) non-monotonic pH=pH(Φ), and non-monotonic E=E(Φ) [7].

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 1o). In both instances, the values: V0=100, C0=0.01, and C=0.1 were assumed. The set of equilibrium data [18-20] applied in calculations, presented in **Table 1**, is completed by the solubility of solid iodine, I_{2(s)}, in water, equal 1.33Ã¢ÂÂ10-3 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.

No. | Reaction | Equilibrium equation | Equilibrium data |
---|---|---|---|

1 | I_{2} + 2e^{–1}=2I^{–1} (for dissolved I_{2} ) |
[I^{–1}]^{2}=K_{e1}·[I^{2}][e^{–1}]^{2} |
E_{01}=0.621 V |

2 | I_{3}^{–1} + 2e^{–1}=3I^{–1} |
[I^{–1}]^{3}=K_{e2}·[I3^{–1}][e^{–1}]^{2} |
E_{02}=0.545 V |

3 | IO^{–1} + H_{2}O + 2e^{–1}=I^{–1} + 2OH^{–1} |
[I^{–1}][OH^{–1}]^{2}=K_{e3}·[IO^{–1}][e^{–1}]^{2} |
E_{03}=0.49 V |

4 | IO_{3}^{–1} + 6H^{+1} + 6e^{–1}=I^{–1} + 3H_{2}O |
[I^{–1}]=K_{e4}·[IO_{3}^{–1}][H^{+1}]6[e^{–1}]^{6} |
E_{04}=1.08 V |

5 | H_{5}IO_{6} + 7H^{+1} + 8e^{–1}=I^{–1} + 6H_{2}O |
[I^{–1}]=K_{e5}·[H_{5}IO_{6}][H^{+1}]7[e^{–1}]^{8} |
E_{05}=1.24 V |

6 | H_{3}IO_{6}^{–2} + 3H_{2}O + 8e^{–1}=I^{–1} + 9OH^{–1} |
[I^{–1}][OH^{–1}]9=K_{e6}·[H_{3}IO_{6}–2][e^{–1}]^{8} |
E_{06}=0.37 V |

7 | HIO=H^{+1} + IO^{–1} |
[H^{+1}][IO^{–1}]=K_{11I}·[HIO] |
pK_{11I}=10.6 |

8 | HIO_{3}=H^{+1} + IO_{3}^{–1} |
[H^{+1}][IO_{3}^{–1}]=K_{51I}·[HIO_{3}] |
pK_{51I}=0.79 |

9 | H4IO_{6}^{–1}=H^{+1} + H3IO_{6}^{–2} |
[H^{+1}][H_{3}IO_{6}^{–2}]=K_{72}·[H_{4}IO_{6}^{–1}] |
pK_{72}=3.3 |

10 | Cl_{2} + 2e^{–1}=2Cl^{–1} |
[Cl^{–1}]^{2}=K_{e7}·[Cl^{2}][e^{–1}]^{2} |
E_{07}=1.359 V |

11 | ClO^{–1} + H_{2}O + 2e^{–1}=Cl^{–1} + 2OH^{–1} |
[Cl^{–1}][OH^{–1}]^{2}=K_{e8}·[ClO^{–1}][e^{–1}]^{2} |
E_{08}=0.88 V |

12 | ClO_{2}^{–1} + 2H_{2}O + 4e^{–1}=Cl^{–1} + 4OH^{–1} |
[Cl^{–1}][OH^{–1}]^{4}=K_{e9}·[ClO_{2}^{–1}][e^{–1}]^{4} |
E_{09}=0.77 V |

13 | HClO=H^{+1}` + ClO^{–1} |
[H^{+1}][ClO^{–1}]=K_{11}Cl·[HClO] |
p_{K11Cl}=7.3 |

14 | HClO_{2} + 3H^{+1} + 4e^{–1}=Cl^{–1} + 2H_{2}O |
[Cl^{–1}]=K_{e10}·[HClO_{2}][H^{+1}]^{3}[e^{–1}]^{4} |
E_{010}=1.56 V |

15 | ClO_{2} + 4H^{+1} + 5e^{–1}=Cl^{–1} + 4H_{2}O |
[Cl^{–1}]=K_{e11}·[ClO_{2}][H^{+1}]^{4}[e^{–1}]^{4} |
E_{011}=1.50 V |

16 | ClO_{3}^{–1} + 6H^{+1} + 6e^{–1}=Cl^{–1} + 3H_{2}O |
[Cl^{–1}]=K_{e12}·[ClO_{3}^{–1}][H^{+1}]^{6}[e^{–1}]^{6} |
E_{012}=1.45 V |

17 | ClO_{4}^{–1} + 8H^{+1} + 8e^{–1}=Cl^{–1} + 4H_{2}O |
[Cl^{–1}]=K_{e13}·[ClO4–1][H^{+1}]^{8}[e^{–1}]^{8} |
E_{013}=1.38 V |

18 | 2ICl + 2e^{–1}=I_{2} + 2Cl^{–1} |
[I_{2} ][Cl^{–1}]^{2}=K_{e14}·[ICl]^{2}[e^{–1}]^{2} |
E_{014}=1.105 V |

19 | I2Cl^{–1}=I_{2} + Cl^{–1} |
[I_{2} ][Cl^{–1}]=K_{1}·[I_{2}Cl^{–1}] |
logK_{1}=0.2 |

20 | ICl_{2}^{–1}=ICl + Cl^{–1} |
[ICl][Cl^{–1}]=K_{2}·[ICl_{2}^{–1}] |
logK_{2}=2.2 |

21 | H_{2}O=H^{+1} + OH^{-1} |
[H^{+1}][OH^{-1}]=K_{W} |
pK_{W}=14.0 |

**Table 1:** Physicochemical data related to the systems S1 and S2.

The titration curves: pH=pH(Φ) and E=E(Φ) presented in **Figures 1** and **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 [3,21-23]. 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].

(2)

where

(3)

It 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

(4)

where Φ is the fraction titrated (Equation 1). Then we get

(5)

where

(6)

is the sharpness index on the titration curve. For comparative purposes, the absolute values,|β_{V}| and |η|, for β_{V} (Equations 1 and 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 (ΔpH) 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].

(7)

where

has been suggested. From extension in Taylor series we have

where

(10)

From Equations 7 and 9 we see that β_{V} is the first approximation of BV. One should take here into account that finite changes (ΔpH) in pH, e.g. ΔpH=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** **and**

In similar manner, one can formulate dynamic buffer capacities and , involved with infinitesimal and finite changes of potential E values:

(11)

(12)

Where c is defined by Equation 2, and then we have

(13)

where

(14)

**Graphical presentation of dynamic buffer capacities in redox systems**

Referring to dynamic redox systems represented by titration curves presented in **Figures 1** and **2**, we plot the relationships: β_{V} vs. Φ, β_{V} vs. pH, β_{V} vs. E, and β^{E}_{V} vs. Φ, β^{E}_{V} vs. pH, β^{E}_{V} vs. E for the systems: (S1) NaOH ⇒ HIO; (S2) HCl ⇒ NaIO. The relations: (A) β_{V} vs. Φ, (B) β_{V} vs. pH, (C) β_{V} vs. E and (D) β^{E}_{V} vs. Φ, (E) β^{E}_{V} vs. pH, (F) β^{E}_{V} vs. E are plotted in **Figures 3 **and **4**.

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 (see 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, see **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, see **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 [18-20]. 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 Cl2, whose concentration was on the level ca. 10-16-10-17 mol/L.

The redox buffer capacity concepts: β_{V} and β^{E}_{V} 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 7-9** presented in Appendix 3. For comparison, in isohydric (acid-base) systems, the buffer capacity strives for infinity. In particular, it occurs in the titration HB (C,V) ⇒ HL (C_{0},V_{0}), where HB is a strong monoprotic acid HB and HL is a weak monoprotic acid characterized by the dissociation constant K_{1}=[H^{+1}][L^{-1}]/[HL]; at 4K_{W}/C^{2}<<1, the isohydricity condition is expressed here by the MichaÃ
Âowski formula [24-26].

The formula for the buffer capacity, suggested by Bard et al. [27] after Levie [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 the study by MichaÃ Âowska-Kaczmarczyk et al. [29]; it is not adaptable for real redox systems.

Buffered solutions are commonly applied in different procedures involved with classical (titrimetric, gravimetric) and instrumental analyses [30-33]. There are in close relevance to isohydric solutions [24-26] and pH-static titration [4,34], and titration in binary-solvent systems [12,35]. Buffering property is usually referred to an action of an external agent (mainly: strong acid, HB, or strong base, MOH) inducing pH change, ΔpH, of the solution. Redox buffer capacity is also involved with the problem of interfacing in CE-MS analysis, and bubbles formation in reaction 2 H_{2}O=O_{2}_{(g)} + 4H^{+1} + 4e^{-1} at the outlet electrode in CE [36-39].

In the paper, a nice proposal of “*slyke*”, as the name for (acid- base, pH) buffer capacity unit, has been raised [40].

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