Research Article, J Chem Appl Chem Eng Vol: 1 Issue: 2

# Dynamic Buffer Capacities in Redox Systems

**Anna Maria MichaÅ�?�?owska-Kaczmarczyk ^{1}, Aneta Spórna- Kucab^{2} and Tadeusz MichaÅ�?�?owski^{2}^{*}**

^{1}Department of Oncology, The University Hospital in Cracow, 31-501 Cracow,

Poland

^{2}Department of Analytical Chemistry, Technical University of Cracow, 31-155

Cracow, Poland

***Corresponding Author :** **Tadeusz MichaÅ�?�?owski**

Faculty of Chemical Engineering and Technology, Cracow University of Technology, Warszawska 24, 31-155 Cracow, Poland

**Tel:** +48126282035

**E-mail**: michalot@o2.pl

**Received:** October 04, 2017 **Accepted:** October 19, 2017 **Published:** October 24, 2017

**Citation:** MichaÅ�?�?owska-Kaczmarczyk AM, Spórna-Kucab A, MichaÅ�?�?owski T (2017) Dynamic Buffer Capacities in Redox Systems . J Chem Appl Chem Eng 1:2 *doi:10.4172/2576-3954.1000107*

## Abstract

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.

### Keywords: Thermodynamics of electrolytic redox systems; Buffer capacity; GATES/GEB

## 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) âÂ�?�?¹ A(C_{0},V_{0}), denoted for brevity as B âÂ�?�?¹ A, is characterized by the fraction titrated [1-4]

(1)

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 [5-16], 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: (1°) monotonic pH = pH(Φ) and monotonic E = E(Φ) [18-20]; (2°) monotonic pH = pH(Φ) and non-monotonic E = E(Φ) [6]; (3°) non-monotonic pH = pH(Φ) and monotonic E = E(Φ) [5]; (4°) non-monotonic pH = pH(Φ), and nonmonotonic E = E(Φ) [7].

**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 âÂ�?�?¹ HIO and (S2) HCl âÂ�?�?¹ NaIO, characterized by monotonic changes of pH and E values during the related titrations (i.e., the case 1°). In both instances, the values: V_{0}=100, C_{0}=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 1.33âÂ�?�?Â�?�?10-3 mol/L. The related algorithms, prepared in MATLAB for S1 (NaOH âÂ�?�?¹ HIO) S2 (HCl âÂ�?�?¹ NaIO) system according to the GATES/GEB principles, are presented in Appendices 1 and 2.

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

1 | I_{2} + 2^{e–1} = 2I^{–1} (for dissolved I2) |
[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}·[I_{3}^{–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 | IO3–1 + 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 | H_{4}IO_{6}^{–1} = H^{+1} + H_{3}IO_{6}^{–2} |
[H^{+1}][H_{3}IO_{6}^{–2}] = K_{72}·[H_{4}IO_{6}^{–1}] |
pK_{72} = 3.3 |

10 | Cl_{2} + 2^{e–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_{11Cl}·[HClO] |
pK_{11Cl }= 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}]^{5} |
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}·[ClO_{4}^{–1}][H^{+1}]^{8}[e^{–1}]^{8} |
E_{013} = 1.38 V |

18 | 2ICl + 2e–1 = I2 + 2Cl–1 | [I_{2}][Cl^{–1}]^{2}= K_{e14}·[ICl]^{2}[e^{–1}]^{2} |
E_{014} = 1.105 V |

19 | I_{2}Cl^{–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}][O^{H-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 **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 [3,21-23]. 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)

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,|βV| and |η|, for βV (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 (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

(8)

has been suggested. From extension in Taylor series we have

where

(10)

From equations 7 and 9 we see that βV is the first approximation of B_{V}. 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 β^{E}_{V} and β^{E}_{V}, 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 1,2**, 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 âÂ�?�?¹ HIO; (S2) HCl âÂ�?�?¹ NaIO. The relations: (A) β_{V} vs. Φ, (B) βV vs. pH, (C) β_{V }vs. E and (D) E V β vs. Φ, (E) β^{E}_{V} vs. pH, (F) β^{E}_{V} vs. E are plotted in **Figures 3,4**.

## 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 [18-20]. 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: 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° - 4°) and exemplified in **Figures 7,8,9**. For comparison, in isohydric (acid-base) systems, the buffer capacity strives for infinity. In particular, it occurs in the titration HB (C,V) âÂ�?�?¹ 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}âÂ�?�?ª1, the isohydricity condition is expressed here by the MichaÅÂ�?�?owski formula [24-26].

**Figure 7:** Case (2°): (A) monotonic pH = pH(V) and (B) non-monotonic E = E(V) plots on the step 3 of the process presented in [6].

**Figure 8:** Case (3°): (A) non-monotonic pH = pH(Φ) and (B) monotonic E = E(Φ) functions for the system KBrO3 âÂ�?�?¹ NaBr presented in [5].

**Figure 9:** Case (4°): the (A) non-monotonic pH = pH(Φ) and (B) non-monotonic E = E(Φ) functions for the system HI âÂ�?�?¹ KIO_{3} presented in [7].

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.

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 2H_{2}O = O_{2(g)} + 4H^{+1} + 4e^{-1} at the outlet electrode in CE [36-39].

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

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