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

# Dynamic Buffer Capacities in Redox Systems

**Anna Maria MichaÅowska-Kaczmarczyk ^{1}, Aneta Spórna- Kucab^{2} and Tadeusz MichaÅowski^{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 MichaÅowski**

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:** MichaÅowska-Kaczmarczyk AM, Spórna-Kucab A, MichaÅowski 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) âÂ¹ A(C_{0},V_{0}), denoted for brevity as B âÂ¹ 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 âÂ¹ HIO and (S2) HCl âÂ¹ 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 1.33âÂÂ10-3 mol/L. The related algorithms, prepared in MATLAB for S1 (NaOH âÂ¹ HIO) S2 (HCl âÂ¹ 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 âÂ¹ HIO; (S2) HCl âÂ¹ 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) âÂ¹ 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}âÂª1, the isohydricity condition is expressed here by the MichaÅÂowski 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 âÂ¹ 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 âÂ¹ 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.

## References

- MichaÅÂowski T (2010) The generalized approach to electrolytic systems: I. Physicochemical and analytical implications. Crit Rev Anal Chem 40: 2-16.
- MichaÅÂowski T, A Pietrzyk, M Ponikvar-Svet, M Rymanowski (2010) The generalized approach to electrolytic systems: II. The generalized equivalent mass (GEM) concept. Crit Rev Anal Chem 40: 17-29.
- Asuero AG, MichaÅÂowski T (2011) Comprehensive formulation of titration curves referred to complex acid-base systems and its analytical implications. Crit Rev Anal Chem 41: 151-187.
- MichaÅÂowski T, Asuero AG, Ponikvar-Svet M, Toporek M, Pietrzyk A, et al. (2012) Liebig–Denigès Method of Cyanide Determination: A Comparative Study of Two Approaches. J Solution Chem 41: 1224-1239.
- MichaÅÂowska-Kaczmarczyk AM, Asuero AG , Toporek M, MichaÅÂowski T (2015) “Why not stoichiometry” versus “Stoichiometry – why not?” Part II. GATES in context with redox systems. Crit Rev Anal Chem 45: 240-268.
- MichaÅÂowska-Kaczmarczyk AM, MichaÅÂowski T, Toporek M, Asuero AG (2015) Why not stoichiometry” versus “Stoichiometry – why not?” Part III, Extension of GATES/GEB on complex dynamic redox systems. Crit Rev Anal Chem 45: 348-366.
- MichaÅÂowski T, Toporek M, MichaÅÂowska-Kaczmarczyk AM, Asuero AG (2013) New trends in studies on electrolytic redox systems. Electrochimica Acta 109: 519-531.
- MichaÅÂowski T, MichaÅÂowska-Kaczmarczyk AM, Toporek M (2013) Formulation of general criterion distinguishing between non-redox and redox systems. Electrochimica Acta 112: 199-211.
- MichaÅÂowska-Kaczmarczyk AM, Toporek M, MichaÅÂowski T (2015) Speciation diagrams in dynamic Iodide + Dichromate system. Electrochimica Acta 155: 217-227.
- Toporek M, MichaÅÂowska-Kaczmarczyk AM, MichaÅÂowski T (2015) Symproportionation versus disproportionation in bromine redox systems. Electrochimica Acta 171: 176-187.
- MichaÅÂowski T (2011) Application of GATES and MATLAB for resolution of equilibrium, metastable and non-equilibrium electrolytic systems, In: Applications of MATLAB in science and engineering. MichaÅÂowski T (edt) InTech, Rijeka, Croatia, 1-34.
- MichaÅÂowski T, Pilarski B, Asuero AG , MichaÅÂowska-Kaczmarczyk AM (2014) Modeling of acid-base properties in binary-solvent systems In: Handbook of solvents. George Wypych (edt), (1stedn) ChemTec Publishing, Toronto, 623-648.
- MichaÅÂowska-Kaczmarczyk AM, Spórna-Kucab A, MichaÅÂowski T (2017) Generalized electron balance (GEB) as the law of nature in electrolytic redox systems, In: Redox: principles and advanced applications, MA Ali Khalid (edt), InTech, Rijeka, Croatia, 10-55.
- MichaÅÂowska-Kaczmarczyk AM, Spórna-Kucab A , MichaÅÂowski T (2017) Principles of titrimetric analyses according to GATES, In: Advances in titration techniques Vu Dang Hoang (Edt), InTech, Rijeka, Croatia, 133-171.
- MichaÅÂowska-Kaczmarczyk AM, Spórna-Kucab A , MichaÅÂowski T (2017) Principles of titrimetric analyses according to GATES, In: Advances in titration techniques Vu Dang Hoang (Edt), InTech, Rijeka, Croatia, 133-171.
- MichaÅÂowska-Kaczmarczyk AM, Spórna-Kucab A , MichaÅÂowski T (2017) A distinguishing feature of the balance 2âÂÂf(O) – f(H) in electrolytic systems. The reference to titrimetric methods of analysis, In: Advances in titration techniques, Vu Dang Hoang (edt), InTech, Rijeka, Croatia, 174-207.
- MichaÅÂowska-Kaczmarczyk AM, Spórna-Kucab A , MichaÅÂowski T (2017) Some remarks on solubility products and solubility concepts, In: Descriptive inorganic chemistry. Researches of metal compounds T Akitsu (edt), InTech, Rijeka, Croatia, 93-134.
- Gran G (1988) Equivalence volumes in potentiometric titrations. Anal Chim Acta 206: 111-123.
- Meija J, MichaÅÂowska-Kaczmarczyk AM, MichaÅÂowski T (2017) Redox titration challenge. Anal Bioanal Chem 409: 11-13.
- MichaÅÂowski T, MichaÅÂowska-Kaczmarczyk AM, Meija J (2017) Solution of redox titration challenge. Anal Bioanal Chem 409: 4113-4115.
- Toporek M, MichaÅÂowska-Kaczmarczyk AM, MichaÅÂowski T (2014) Disproportionation reactions of HIO and NaIO in static and dynamic systems. Am J Analyt Chem 5: 1046-1056.
- MichaÅÂowska-Kaczmarczyk AM, MichaÅÂowski T (2015) Dynamic buffer capacity in acid-base systems. J Solution Chem 44: 1256-1266.
- MichaÅÂowska-Kaczmarczyk AM, MichaÅÂowski T, Asuero AG (2015) Formulation of dynamic buffer capacity for phytic acid.Am J Analyt Chem 2: 5-9.
- MichaÅÂowski T, Asuero AG (2012) New approaches in modelling the carbonate alkalinity and total alkalinity. Crit Rev Anal Chem 42: 220-244.
- MichaÅÂowski T, Pilarski B, Asuero AG, Dobkowska A (2010) A new sensitive method of dissociation constants determination based on the isohydric solutions principle. Talanta 82: 1965-1973.
- MichaÅÂowski T, Pilarski B, Asuero AG, Dobkowska A, Wybraniec S (2011) Determination of dissociation parameters of weak acids in different media according to the isohydric method. Talanta 86: 447-451.
- MichaÅÂowski T, Asuero AG (2012) Formulation of the system of isohydric solutions. J Analyt Sci Methods Instrumentation 2: 1-4.
- Bard AJ, Inzelt G, Scholz F (edts) (2012) Electrochemical dictionary (2nd edtn), Springer-Verlag Berlin Heidelberg, Germany, 87.
- de Levie R (1999) Redox bufferstrength.J Chem Educ 76: 574-577.
- MichaÅÂowska-Kaczmarczyk AM, Asuero AG, MichaÅÂowski T (2015) “Why not stoichiometry” versus “Stoichiometry – why not?” Part I. General context. Crit Rev Anal Chem 45: 166-188.
- MichaÅÂowski T, Baterowicz A, Madej A, Kochana J (2001) Extended Gran method and its applicability for simultaneous determination of Fe(II) and Fe(III). Anal Chim Acta 442: 287-293.
- MichaÅÂowski T, Toporek M, Rymanowski M (2005) Overview on the Gran and other linearization methods applied in titrimetric analyses. Talanta 65: 1241-1253.
- MichaÅÂowski T, Kupiec K, Rymanowski M (2008) Numerical analysis of the Gran methods. A comparative study. Anal Chim Acta 606: 172-183.
- Ponikvar M, MichaÅÂowski T, Kupiec K, Wybraniec S, Rymanowski M (2008) Experimental verification of the modified Gran methods applicable to redox systems. Anal Chim Acta 628: 181-189.
- MichaÅÂowski T, Toporek M, Rymanowski M (2007) pH-Static titration: A quasistatic approach. J Chem Educ 84: 142-150.
- Pilarski B, Dobkowska A, Foks H, MichaÅÂowski T (2010) Modelling of acid–base equilibria in binary-solvent systems: A comparative study. Talanta 80: 1073-1080.
- Smith AD, Moini M (2001) Control of electrochemical reactions at the capillary electrophoresis outlet/electrospray emitter electrode under CE/ESI-MS through the application of redox buffers. Anal Chem 73: 240-246.
- Moini M, Cao P, Bard AJ (1999) Hydroquinone as a buffer additive for suppression of bubbles formed by electrochemical oxidation of the CE buffer at the outlet electrode in capillary electrophoresis/electrospray ionization-mass spectrometry. Anal Chem 71: 1658-1661.
- Van Berkel GJ, Kertesz V (2001) Redox buffering in an electrospray ion source using a copper capillary emitter. J Mass Spectrom 36: 1125-1132.
- Shintani H (1997) Handbook of capillary electrophoresis applications. Polensky J (edt) Blackie Academic & Professional, London.