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Dynamic Buffer Capacities in Redox Systems

Anna Maria Michałowska- Kaczmarczyk1, Aneta Spórna-Kucab2 and Tadeusz Michałowski2*

1Department of Oncology, The University Hospital in Cracow, Cracow, Poland

2Department of Analytical Chemistry, Technical University of Cracow, Cracow, Poland

*Corresponding Author:
Tadeusz Michałowski
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:Michałowska-Kaczmarczyk AM, Spórna-Kucab A, Michałowski T (2017) Dynamic Buffer Capacities in Redox Systems. Biochem Mol Biol J. Vol.3 No. 3:11 DOI: 10.21767/2471-8084.100039

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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 (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]

Equation (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, I2(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.

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 βV and BV

Dynamic buffer capacity was referred previously only to acid base equilibria in non-redox systems [3,21-23]. However, the dynamic (βV) and windowed (BV) buffer capacities can be also related to acid-base equilibria in redox systems. The βV is formulated as follows [3,21].

Equation (2)


Equation (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

Equation (4)

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

Equation (5)


Equation (6)

is the sharpness index on the titration curve. For comparative purposes, the absolute values,|βV| and |η|, for βV (Equations 1 and 5) and η (Equation 6) are considered. At C0/C << 1 and small Φ value, from Equation 3 we get Equation

The βV 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 (βV is a non–linear function of pH). For this purpose, the ‘windowed’ buffer capacity, BV, defined by the formula [3,21].

Equation (7)



has been suggested. From extension in Taylor series we have



Equation (10)

From Equations 7 and 9 we see that βV 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 capacitiesEquation andEquation

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

Equation (11)

Equation (12)

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


Equation (13)


Equation (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: βV vs. Φ, βV vs. pH, βV vs. E, and βEV vs. Φ, βEV vs. pH, βEV vs. E for the systems: (S1) NaOH ⇒ HIO; (S2) HCl ⇒ NaIO. The relations: (A) βV vs. Φ, (B) βV vs. pH, (C) βV vs. E and (D) βEV vs. Φ, (E) βEV vs. pH, (F) βEV vs. E are plotted in Figures 3 and 4.


Figure 3:The relations: legend


Figure 4: The relations: legend


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, I2(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 (I2(s)) is precipitated and soluble complexes: I2Cl-1, ICl and ICl2-1 are formed, see Figure 6A. Note that I2(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.


The redox buffer capacity concepts: βV and βEV 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 (2o–4o) 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 4KW/C2<<1, the isohydricity condition is expressed here by the Michałowski formula Equation [24-26].


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


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


Figure 9: Case (4o): 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 Michałowski [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 Michałowska-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=O2(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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