Biochemistry & Molecular Biology Journal Open Access

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 V0 mL of titrand D, containing a substance A (C₀ 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].

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 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 1.33âˆÂââ€Å¾Â¢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.

No.	Reaction	Equilibrium equation	Equilibrium data
1	I2 + 2e–1=2I–1 (for dissolved I2 )	[I–1]2=Ke1·[I­2][e–1]2	E01=0.621 V
2	I3–1 + 2e–1=3I–1	[I–1]3=Ke2·[I­3–1][e–1]2	E02=0.545 V
3	IO–1 + H2O + 2e–1=I–1 + 2OH–1	[I–1][OH–1]2=Ke3·[IO–1][e–1]2	E03=0.49 V
4	IO3–1 + 6H+1 + 6e–1=I–1 + 3H2O	[I–1]=Ke4·[IO3–1][H+1]6[e–1]6	E04=1.08 V
5	H5IO6 + 7H+1 + 8e–1=I–1 + 6H2O	[I–1]=Ke5·[H5IO6][H+1]7[e–1]8	E05=1.24 V
6	H3IO6–2 + 3H2O + 8e–1=I–1 + 9OH–1	[I–1][OH–1]9=Ke6·[H3IO6–2][e–1]8	E06=0.37 V
7	HIO=H+1 + IO–1	[H+1][IO–1]=K11I·[HIO]	pK11I=10.6
8	HIO3=H+1 + IO3–1	[H+1][IO3–1]=K51I·[HIO3]	pK51I=0.79
9	H4IO6–1=H+1 + H3IO6–2	[H+1][H3IO6–2]=K72·[H4IO6–1]	pK72=3.3
10	Cl2 + 2e–1=2Cl–1	[Cl–1]2=Ke7·[Cl­2][e–1]2	E07=1.359 V
11	ClO–1 + H2O + 2e–1=Cl–1 + 2OH–1	[Cl–1][OH–1]2=Ke8·[ClO–1][e–1]2	E08=0.88 V
12	ClO2–1 + 2H2O + 4e–1=Cl–1 + 4OH–1	[Cl–1][OH–1]4=Ke9·[ClO2–1][e–1]4	E09=0.77 V
13	HClO=H+1` + ClO–1	[H+1][ClO–1]=K11Cl·[HClO]	pK11Cl=7.3
14	HClO2 + 3H+1 + 4e–1=Cl–1 + 2H2O	[Cl–1]=Ke10·[HClO2][H+1]3[e–1]4	E010=1.56 V
15	ClO2 + 4H+1 + 5e–1=Cl–1 + 4H2O	[Cl–1]=Ke11·[ClO2][H+1]4[e–1]4	E011=1.50 V
16	ClO3–1 + 6H+1 + 6e–1=Cl–1 + 3H2O	[Cl–1]=Ke12·[ClO3–1][H+1]6[e–1]6	E012=1.45 V
17	ClO4–1 + 8H+1 + 8e–1=Cl–1 + 4H2O	[Cl–1]=Ke13·[ClO4–1][H+1]8[e–1]8	E013=1.38 V
18	2ICl + 2e–1=I2 + 2Cl–1	[I2 ][Cl–1]2=Ke14·[ICl]2[e–1]2	E014=1.105 V
19	I2Cl–1=I2 + Cl–1	[I2 ][Cl–1]=K1·[I2Cl–1]	logK1=0.2
20	ICl2–1=ICl + Cl–1	[ICl][Cl–1]=K2·[ICl2–1]	logK2=2.2
21	H2O=H+1 + OH-1	[H+1][OH-1]=KW	pKW=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.

Figure 1: (A) pH=pH(Φ) and (B) E=E(Φ) relationships plotted for the system NaOH ⇒ HIO.

Figure 2: (A) pH=pH(Φ) and (B) E=E(Φ) relationships plotted for the system HCl ⇒ NaIO.

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 (C₀ 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 C₀/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.

Figure 3:The relations:

Figure 4: The relations:

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 (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: I₂Cl^-1, ICl and ICl₂^-1 are formed, see Figure 6A. Note that I_2(s) + I^-1=I₃^-1 is also the complexation reaction.

Figure 5: Speciation diagram for the system (S1) NaOH ⇒ HIO.

Figure 6: Speciation diagram for the system (S2) HCl ⇒ NaIO: (A) for iodine species; (B) for oxidized forms of chlorine species.

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.

Conclusion

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 (C₀,V₀), where HB is a strong monoprotic acid HB and HL is a weak monoprotic acid characterized by the dissociation constant K₁=[H⁺¹][L^-1]/[HL]; at 4K_W/C²<<1, the isohydricity condition is expressed here by the Michaowski formula [24-26].

Figure 7: Case (2^o): (A) monotonic pH=pH(V) and (B) nonmonotonic E=E(V) plots on the step 3 of the process presented in the study by Michaowska-Kaczmarczyk et al. [6].

Figure 8: Case (3^o): (A) non-monotonic pH=pH(Φ) and (B) monotonic E=E(Φ) functions for the system KBrO₃ ⇒ NaBr presented in study by Michaowska- Kaczmarczyk et al. [5].

Figure 9: Case (4^o): the (A) non-monotonic pH=pH(Φ) and (B) non-monotonic E=E(Φ) functions for the system HI ⇒ KIO₃ presented in the study by Michaowski [7].

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 Michaowska-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 H₂O=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].

References

1. Michałowski T (2010) the generalized approach to electrolytic systems: I. physicochemical and analytical implications.Crit Rev Anal Chem 40: 2-16.
2. Michałowski T, Pietrzyk A, Ponikvar-Svet M, Rymanowski M (2010) The generalized approach to electrolytic systems: II. The generalized equivalent mass (GEM) concept.Crit Rev Anal Chem 40: 17-29.
3. Asuero AG, Michałowski T (2011) Comprehensive formulation of titration curves referred to complex acid-base systems and its analytical implications.Crit Rev Anal Chem 41: 51-187.
4. Michałowski TT, Asuero AG, Ponikvar-Svet M, Toporek M, Pietrzyk A, et al. (2012) Principles of computer programming applied to simulated pH-static titration of cyanide according to a modified Liebig-Denigès method. J Solution Chem41: 1224-1239.
5. Michałowska-Kaczmarczyk AM, Asuero AG, Toporek M, Michałowski T (2015) “Why not stoichiometry” versus “Stoichiometry-why not?” Part II. GATES in context with redox systems.Crit Rev Anal Chem 45: 240-268.
6. Michałowska-Kaczmarczyk AM, Michałowski 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.
7. Michałowski T, Toporek M, Michałowska-Kaczmarczyk AM, Asuero AG (2013) New trends in studies on electrolytic redox systems.Electrochimica Acta 109: 519-531.
8. Michałowski T, Michałowska-Kaczmarczyk AM, Toporek M(2013) Formulation of general criterion distinguishing between non-redox and redox systems.Electrochimica Acta 112: 199-211.
9. Michałowska-Kaczmarczyk AM, Toporek M, Michałowski T (2015) Speciation diagrams in dynamic iodide+dichromate system.Electrochimica Acta 155: 217-227.
10. Toporek M, Michałowska-Kaczmarczyk AM, Michałowski T (2015) Symproportionation versus disproportionation in bromine redox systems.Electrochimica Acta 171: 176-187.
11. https://www.intechopen.com/books/show/title/applications-of-matlab-in-science-and-engineering.
12. https://www.sciencedirect.com/science/book/9781895198645.
13. https://cdn.intechopen.com/pdfs-wm/56043.pdf .
14. https://cdn.intechopen.com/pdfs-wm/55881.pdf .
15. https://cdn.intechopen.com/pdfs-wm/55742.pdf .
16. https://cdn.intechopen.com/pdfs-wm/55440.pdf .
17. Gran G (1988) Equivalence volumes in potentiometric titrations.AnalyticaChimica Acta 206: 111-123.
18. Meija J, Michałowska-Kaczmarczyk AM, Michałowski T (2017) Redox titration challenge. AnalBioanalChem 409: 11-13.
19. Michałowski T, Michałowska-Kaczmarczyk AM, Meija J (2017) Solution of redox titration challenge. Anal BioanalChem 409: 4113-4115.
20. Toporek M, Michałowska-Kaczmarczyk AM, Michałowski T (2014) Disproportionation reactions of HIO and NaIO in static and dynamic systems. Am J Anal Chem 5: 1046-1056.
21. Michałowska-Kaczmarczyk AM, Michałowski T (2015) Dynamic buffer capacity in acid-base systems. JSolution Chem 44: 1256-1266.
22. Michałowska-Kaczmarczyk AM, Michałowski T, Asuero AG(2015) Formulation of dynamic buffer capacity for phytic acid.Am J ChemAppl 2: 5-9.
23. Michałowski T, Asuero AG (2012) New approaches in modelling the carbonate alkalinity and total alkalinity.Crit Rev Anal Chem 42: 220-244.
24. Michałowski 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.
25. Michałowski 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.
26. Michałowski T, Asuero AG (2012) Formulation of the system of isohydric solutions.J Anal Sci 2: 1-4.
27. BardAJ, Inzelt G, Scholz F (2012) Electrochemical dictionary, (2nd edn), Springer-Verlag Berlin Heidelberg, Germany.pp. 87.
28. Levie R (1999) Redox buffer strength. J Chem Edu 76: 574-577.
29. Michałowska-Kaczmarczyk AM, Asuero AG, Michałowski T (2015) “Why not stoichiometry” versus “Stoichiometry-why not?” Part I. General context.Crit Rev Anal Chem 45: 166-188.
30. Michałowski T, Baterowicz A, Madej A, Kochana J (2001) An extended Gran method and its applicability for simultaneous determination of Fe(II) and Fe(III).AnalyticaChimica Acta 442: 287-293.
31. Michałowski T, Toporek M,Rymanowski M (2005), Overview on the Gran and other linearization methods applied in titrimetric analyses.Talanta 65: 1241-1253.
32. Michałowski T, Kupiec K,Rymanowski M (2008) Numerical analysis of the Gran methods. A comparative study.AnalyticaChimica Acta 606: 172-183.
33. Ponikvar M, Michałowski T, Kupiec K,Wybraniec S, Rymanowski M (2008) Experimental verification of the modified Gran methods applicable to redox systems.AnalyticaChimica Acta 628: 181-189.
34. Michałowski T, Toporek M, Rymanowski M (2007) pH-static titration: A quasistatic approach. J ChemEduc 84: 142-150.
35. Pilarski B,Dobkowska A, Foks H, Michałowski T (2010) Modelling of acid-base equilibria in binary-solvent systems: A comparative study.Talanta 80: 1073-1080.
36. 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
37. 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.
38. Berkel V, Kertesz V (2001), Redox buffering in an electrospray ion source using a copper capillary emitter. J Mass Spectrom 36: 1125-1132.
39. Shintani H, Polensky J (1997) Handbook of capillary electrophoresis applications, Blackie Academic and Professional, London, UK.
40. Baicu SC, Taylor MJ (2002) Acid-base buffering in organ preservation solutions as a function of temperature: New parameters for comparing buffer capacity and efficiency.Cryobiol 45: 33-48.