## Some Regularities Involved with Oxidation Numbers Stated in Formulation of Redox Systems According to GATES/GEB Principles

Anna Maria Michalowska-Kaczmarczyk

^{1}, Aneta Sporna-Kucab

^{2}, Tadeusz Michalowski

^{2*}

## Affiliation

^{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 Michalowski, Department of Analytical Chemistry, Technical University of Cracow, 31-155 Cracow, Poland, michalot@o2.pl

## Citation

Tadeusz Michalowski, T., et al. Some Regularities Involved with Oxidation Numbers Stated In Formulation of Redox Systems According to GATES/GEB Principles. (2017) J Anal Bioanal Sep Tech 2(2): 102- 110.

## Copy rights

© 2017 Michalowski, T. This is an Open access article distributed under the terms of Creative Commons Attribution 4.0 International License.

## Keywords

Electrolytic systems; Redox systems; GEB; GATES; Oxidation number

##### Abstract

Formulation of Generalized Electron Balance (GEB) for redox systems according to Approach II to GEB does not require the prior knowledge of oxidation numbers of all elements in components forming a system, and in the species of the system thus formed. This formulation is involved with linear combination of charge and elemental and/or core balances related to the system in question. The skillful choice of multipliers for the balances on the step of purposeful formulation of this linear combination allows for to find important regularities for electrolytic systems of different degree of complexity. These multipliers are related to the oxidation numbers of the elements; this regularity is important in the context of the fact that the oxidation number is the contractual concept. This property is valid for redox and non-redox systems. In this context, oxidation number is perceived as the derivative/redundant concept. The paper indicates the close relationships between different rules of conservation and indicates huge possibilities inherent in the generalized approach to electrolytic systems (GATES), and GATES/GEB in particular.

##### Introduction

The paper refers critically to some fundamental concepts, known to a wide community of chemists from early stages of education. It concerns the terms: oxidation state/number; oxidant and reductant; equivalent weight. All those concepts, defended by IUPAC, raise many reservations and controversies, expressed in our review papers issued in recent years[1-5], and discussed elsewhere[6-8]. All these terms were introduced/considered in context of stoichiometry, i.e., the concept straight from the 18^{th} century[3]. The factual place of these terms is indicated here by the mathematical formulation of redox systems, realized according to GATES/GEB principles, i.e., based on the Generalized Approach to Electrolytic Systems (GATES)[9], with Generalized Electron Balance (GEB) formulated according to Approach II to GEB[1-5,9-15].

Within GATES/GEB, the species in electrolytic systems are considered in their natural form, particularly as hydrates X_{i}^{zi}.n_{iw} in aqueous solutions[10], where z_{i}(z_{i}= 0, ± 1, ± 2,…) is the charge of X_{i}^{zi}, expressed in elementary charge unit e = F/N_{A} (F – Faraday constant, N_{A} – Avogadro’s constant), and n_{iW} ( ≥ 0) is the mean number of water (W = H_{2}O) molecules attached to X_{i}^{zi}. The known chemical formulas of the X_{i}^{zi} of the and their respective external charges provide the information necessary/sufficient to formulate the respective balances[4].

The terms: components and species are distinguished. In the notation applied here, N_{0j} (j = 1,2,…,J) is the number of molecules of components of j^{-th} kind composing the static or dynamic D+T system, whereby the D and T are composed separately, from defined components, including water. The mono- or two-phase electrolytic system thus obtained involve N_{1} molecules of H_{2}O and Ni species of i^{-th} kind, X_{i}^{zi}n_{iw} (i = 2, 3,…,I), denoted briefly as X_{i}^{zi} (N_{i}, n_{i}), where n_{i} ≡ n_{iz} ≡ n_{i}H_{2}O; then we have: H^{+1} (N_{2},n_{2}), OH^{-1} (N_{3},n_{3}),… .

The GEB is formulated according to two equivalent approaches, named as the Approach I to GEB and the Approach II to GEB. The Approach I to GEB is formulated according to ‘card game’ principle[10]; it is based on a common pool of electrons introduced into the system by electron-active elements[10-15]. The Approach II to GEB is formulated from linear combination 2.f(O) – f(H) of elemental balances: f(H) for H, and f(O). Both Approaches (I and II) to GEB are equivalent.

∴ Approach II to GEB <=> Approach I to GEB (1)

All the inferences made within GATES/GEB are based on firm, algebraic foundations. The approach proposed allows understanding far better all physicochemical phenomena occurring in the system in question and improving some methods of analysis. All the facts testify very well about the potency of simulated calculations made, according to GATES, on the basis of all attainable physicochemical knowledge.

The Approach I to GEB needs prior knowledge of oxidation numbers. The Approach II to GEB does not require any prior knowledge of the oxidation numbers of elements in the components and in the species. Because the ‘oxidation number’ is essentially the contractual concept, it is a fact of capital importance, particularly in relation to organic species (molecules, ions, radicals and ion-radicals) of any degree of complexity. However, when the oxidation numbers are easily determined, the Approach I to GEB, known as the ‘short’ version of GEB, can be applied. The roles of oxidants and reductants are not assigned a priori to individual components within the Approaches I and II to GEB; GATES/GEB provides full ‘democracy’ in this regard.

The principle of GEB formulation, discovered by Michalowski (1992) as the Approach I to GEB, and 2006 as the Approach II to GEB) was unknown in earlier literature. The GEB is considered as a new law of the matter conservation related to electrolytic redox systems, as a Law of Nature[9,10].

The importance of GATES/GEB in area of electrolytic redox systems is unquestionable. Then the main purpose of the present paper is to familiarize it to a wider community. It will also be indicated, in a simple mathematical manner, the fundamental criterion distinguishing between non-redox and redox systems. Contrary to appearances, this criterion is not immediately associated with free electrons in electrolytic system.

In this paper we consider first a relatively simple dynamic redox system, where V mL of C mol/L NaOH as titrant T is added; up to a given point of the titration, into V_{0} mL of C_{0} mol/L Br_{2} as titrant D and the D+T mixture with volume V_{0}+V is thus obtained.

V_{0} mL of D is composed of Br_{2} (N_{01} molecules)+H_{2}O (N_{02} molecules), and V mL of T is composed of NaOH (N_{03} molecules)+H_{2}O (N_{04} molecules). In V_{0}+V mL of D+T mixture we have the following species:

H_{2}O (N_{1}), H^{+1} (N_{2},n_{2}), OH^{-1} (N_{3},n_{3}), HBrO_{3} (N_{4},n_{4}), BrO_{3}^{-1} (N_{5},n_{5}), HBrO (N_{6},n_{6}), BrO^{-1} (N_{7},n_{7}), Br_{2} (N_{8},n_{8}), Br_{3}^{-1} (N_{9},n_{9}), Br^{-1} (N_{10},n_{10}), Na^{+1} (N_{11},n_{11}). (2)

In the D+T mixture, the molar concentrations of the species are as follows:

[X_{i}^{iz}] . (V_{0}+V) = 10^{3}.N_{i}/N_{A} (i = 2,…,11) (3)

And molar concentrations of the solutes are equal to

C_{0}.V_{0} = 10^{3}.N_{01}/N_{A} (4)

C.V = 10^{3}.N_{03}/N_{A} (5)

At V = 0, from Eq. (3) we have

[X_{i}^{zi}] .V_{0} = 10^{3}.N_{i}/N_{A} (i = 2,…,11) (6)

The system considered above will be denoted as T => D or, more exactly, as

System-1: NaOH (C, V) => Br_{2} (C_{0}, V_{0})

At V = 0, the D+T is limited to D, i.e., C_{0}mol/L Br_{2} solution. Analogously, we apply the notation

System-2: NaOH (C, V) => HBrO (C_{0}, V_{0})

for the second system, considered in further parts of this paper. At V = 0, the System 2 is limited to C_{0} mol/L HBrO solution. The C_{0} mol/L Br_{2} and C_{0} mol/L HBrO are considered as static systems, obtained after disposable mixing of the related components.

Formulation of balances for the System-1

Charge balance

Denoting atomic numbers: Z_{H} for H, Z_{0} for O, Z_{Br} for Br, Z_{Na} for Na we have the balances:

• For nuclear protons

(2Z_{H}+Z_{0})N_{1}+Z_{H}+n_{2}(2Z_{H}+Z_{0}))N_{2}+Z_{H}+Z_{0}+n_{3}(2Z_{H}+Z_{0}))N_{3}+Z_{H}+Z_{Br}+3Z_{0}+n_{4}(2Z_{H}+Z_{0}))N_{4}+Z_{Br}+3Z_{0}+n_{5}(2Z_{H}+Z_{0}))N_{5}

+Z_{H}+Z_{Br}+Z_{0}+n_{6}(2Z_{H}+Z_{0}))N_{6}+(Z_{Br}+Z_{0}+n_{7}(2Z_{H}+Z_{0}))N_{7}+(2Z_{Br}+n_{8}(2Z_{H}+Z_{0}))N_{8}+(3Z_{Br}+n_{9}(2Z_{H}+Z_{0}))N_{9}+(Z_{Br}+n_{10}(2Z_{H}+Z_{0}))N_{10}+(Z_{Na}+n_{11}(2Z_{H}+Z_{0}))N_{11 =} 2Z_{Br}N_{01}+ (2Z_{H}+Z_{0})N_{02}+ (Z_{Na}+Z_{0}+Z_{H})N_{03}+ (2Z_{H}+Z_{0})N_{04} (7)

• For orbital electrons

(2Z_{H}+Z_{0})N_{1}+Z_{H}^{1}+n_{2}(2Z_{H}+Z_{0}))N_{2}+Z_{H}^{+1}+Z_{0}+n_{3}(2Z_{H}+Z_{0}))N_{3}+Z_{H}+Z_{Br}+3ZO+n_{4}(2Z_{H}+Z_{0}))N_{4}+Z_{Br}+3Z_{0}+1+n_{5}(2Z_{H}+Z_{0}))N_{5}

+Z_{H}+Z_{Br}+Z_{0}+n_{6}(2Z_{H}+Z_{0}))N_{6}+(Z_{Br}+Z_{0}+1+n_{7}(2Z_{H}+Z_{0}))N_{7}+(2Z_{Br}+n_{8}(2Z_{H}+Z_{0}))N_{8}+(3Z_{Br}+1+n_{9}(2Z_{H}+Z_{0}))N_{9}+(Z_{Br}+1+n_{10}(2Z_{Br}+Z_{0}))N_{10}+(Z_{Na}–1+n_{11}(2Z_{H}+Z_{0}))N_{11} = 2Z_{Br}N_{01}+(2Z_{H}+Z_{0})N_{02} (8)

Subtraction of Equation-(8) from Equation-(7) gives the charge balance (ChB)

f_{0} = ChB : N_{2} – N_{3} – N_{5} – N_{7} – N_{9} – N_{10}+N_{11} = 0 ↔

(+1)N_{2}+(–1)N_{3}+(–1)N_{5}+(–1)N_{7}+(–1)N_{9}+(–1)N_{10}+(+1)N_{11} = 0 (9)

The charge balance (f_{0} = ChB) is then derivable from balances for electrons and protons. It is nothing strange because the external charge of a species is a simple sum of charges brought by nuclear protons and orbital electrons.

In particular, from Equations (3) and (9) we have

[H^{+1}] – [OH^{-1}] – [BrO_{3}^{-1}] – [BrO^{-1}] – [Br_{3}^{-1}] – [Br^{-1}]+[Na^{+1}] = 0 → (9a)

(+1).[H^{+1}]+(–1).[OH^{-1}]+(–1).[BrO_{3}^{-1}]+(–1).[BrO^{-1}]+(–1).[Br_{3}^{-1}]+(–1).[Br^{-1}]+(+1).[Na^{+1}] = 0 (9b)

Generally, ChB is expressed by equation

∑_{i}^{zi} . N_{i} = 0 ↔ ∑_{i}^{zi} . [X_{i}^{zi} } = 0 10

Note that z_{i} ≠ 0 are multipliers for concentrations of the corresponding ions in the ChB.

Elemental balances

f_{1} = f(H) : 2N_{1}+N_{2}(1+2n_{2})+N_{3}(1+2n_{3})+N_{4}(1+2n_{4})+2N_{5}n_{5}+N_{6}(1+2n_{6})+2N_{7}n_{7}+2N_{8}n_{8}+2N_{9}n_{9}+2N_{10}n_{10} = 2N_{02 } (11)

f_{2} = f(O) : N_{1}+N_{2}n_{2}+N_{3}(1+n_{3})+N_{4}(3+n_{4})+N_{5}(3+n_{5})+N_{6}(1+n_{6})+N_{7}(1+n_{7})+N_{8}n_{8}+N_{9}n_{9}+N_{10}n_{10} = N_{02} (12)

f_{3} = f(Br) : N_{4}+N_{5}+N_{6}+N_{7}+2N_{8}+3N_{9}+N_{10} = 2N_{01} (13)

–f_{4} = –f(Na) : N_{03} = N_{11} (14)

Linear combinations of the balances

From Eqs. (11) and (12) we have

f_{12} = 2.f_{2} – f_{1}: – N_{2}+N_{3}+5N_{4}+6N_{5}+N_{6}+2N_{7} = 0 (15)

From Eqs. (15) and (9)

f_{12}+f_{0} – f_{4}: 5N_{4}+5N_{5}+N_{6}+N_{7} – N_{9} – N_{10} = 0 (16)

From Eqs. (13) and (16)

Z_{Br}.f_{3} – (f_{12}+ f_{0} – f_{4}): (Z_{Br}-5)(N_{4}+N_{5})+(Z_{Br}–1)(N_{6}+N_{7})+2Z_{Br}N_{8}+(3Z_{Br}+1)N_{9}+(Z_{Br}+1)N_{10} = 2Z_{Br}N_{01} (17)

Formulation of balances for the System 2

In the System 2 we have the set (2) of the species identical as in the System 1. Applying similar notation, we assume that V_{0} mL of D is composed of HBrO (N_{01} molecules)+H_{2}O (N_{02} molecules) and V mL of T is composed of NaOH (N_{03} molecules)+H_{2}O (N_{04} molecules). The numbers N_{01} and N_{02} of the molecules composing these systems and the numbers N_{i} of the species in the System 2 are different than that in the System 1, in principle. The f_{0} = ChB in the System 2 is identical with Eq. (9a), and –f_{4} = –f(Na) is identical with Eq. (14). Then after formulation of f_{1} and f_{2} we have here, by turns,

f_{12} = 2.f_{2} – f_{1} :– N_{2}+N_{3}+5N_{4}+6N_{5}+N_{6}+2N_{7} = N_{01}+N_{03} (18)

f_{3} = f (Br) : N_{4}+N_{5}+ N_{6}+N_{7}+2N_{8}+3N_{9}+N_{10} = N_{01} (19)

f_{12}+f_{0}– f_{4} : 5N_{4}+5N_{5}+N_{6}+N_{7} – N_{9} – N_{10} = N_{01} (20)

Z_{Br}.f_{3} – (f_{12}+f_{0}–f_{4}): (Z_{Br}–5)(N_{4}+N_{5})+(Z_{Br}–1)(N_{6}+N_{7})+2Z_{Br}N_{8}+ (3Z_{Br}+1)N_{9}+(Z_{Br}+1)N_{10} = (Z_{Br}–1)N_{01} (21)

The balances in terms of molar concentrations

For the System 1, from Eqs. (3) – (5), (13), (14) and (16), we have:

([HBrO_{3}]+[BrO_{3}^{-1}])+([HBrO]+[BrO^{-1}])+2[Br_{2}]+3[Br_{3}^{-1}]+[Br^{-1}] = 2.C_{0}V_{0}/ (V_{0}+V) (13a)

[Na^{+1}] = CV/(V_{0}+V) (14a)

5([HBrO_{3}]+[BrO_{3}^{-1}])+([HBrO]+[BrO^{-1}]) – [Br_{3}^{-1}] – [Br^{-1}] = 0 (16a)

Eqs. (13a) and (14a) are termed as concentration balances, obtained from elemental balances (13) and (14), and Eq. (16a) is the shortest form of GEB. The relation (14a) can be immediately introduced into Eq. (9a); then we get

[H^{+1}] – [OH^{-1}] – [BrO_{3}^{-1}] – [BrO^{-1}] – [Br_{3}^{-1}] – [Br^{-1}]+CV/ (V_{0}+V) = 0 (9c)

Eqs. (9c), (13a) and (16a) form the complete set of balances related to the System 1.

For the System 2,from Eqs. (3) – (5), (19) and (20), we obtain the balances:

([HBrO_{3}]+[BrO_{3}^{-1}])+([HBrO]+[BrO^{-1}])+2[Br_{2}]+3[Br_{3}^{-1}]+[Br^{-1}] = C_{0}V_{0}/ (V_{0}+V) (19a)

5([HBrO_{3}]+[BrO_{3}^{-1}])+([HBrO]+[BrO^{-1}]) – [Br_{3}^{-1}] – [Br^{-1}] = C_{0}V_{0}/ (V_{0}+V) (20a)

completed by the balance (9c).

We can also refer to static systems, formed by (a) C_{0} mol/L Br_{2} and (b) C_{0} mol/L HBrO. These solutions are identical with the titrand D in the related Systems 1 and 2. The balances for these static systems are obtained assuming V = 0 in Eqs. (9c), (13a) and (14a), or in Eqs. (9c),(19a) and (20a), resp.

Note that Br_{2} and HBrO do not oxidize water molecules, i.e., products of H_{2}O oxidation do not exist there as species.

Relations between concentrations of the species

From the interrelations obtained on the basis of expressions for equilibrium data[16] collected in Table 1 we have:

[H^{+1}] = 10^{-pH}; [OH^{-1}] = 10^{pH-14}; [BrO_{3}^{-1}] = 10^{6A(E-1.45)-pBr+6pH}; [BrO^{-1}] = 10^{2A(E-0.76)-pBr+2pH-28}; [Br_{2}] = 10^{2A(E-1.087)-2pBr}; [Br_{3}^{-1}] = 10^{2A(E-1.05)-2pBr}; [HBrO_{3}] = 10^{0.7-pH}.[BrO_{3}^{-1}]; [HBrO] = 10^{8.6-pH}.[BrO^{-1}]

Where the uniformly defined scalar variables: E, pH and pBr, forming a vector x = (E, pH, pBr)T, are involved:

A.E = –log[e^{-1}], pH = –log[H^{+1}], pBr = –log[Br^{-1}]

All the variables are in the exponents of the power for 10 in: [e^{-1}] = 10^{-A.E}, [H^{+1}] = 10^{-pH}, [Br^{-1}] = 10^{-pBr}, where 1/A = RT/F.ln10; A = 16.9 at T = 298 K. The number of the (independent) variables equals to the number of equations, 3 = 3; this ensures a unique solution of the equations related to the Systems 1 and 2, at the preset C_{0}, C and V_{0} values, and the V-value at which the calculations are realized, at defined step of the calculation procedure, according to iterative computer program presented in Appendix.

Graphical presentation

Formulation of the static system with Br_{2} (C_{0}) was presented first in[17], whereas the dynamic Systems: 1 and 2 were presented in[18]. The pH = pH (Φ) and E = E (Φ) relationships for the Systems 1 and 2 are plotted in Figures 1A, B, where the fraction titrated

C . V

Φ = ---------- (22)

C_{0}. V_{0}

introduces a kind of normalization(independence on V_{0} value) for the related titration curves. The computer program related to the System 1 is presented in Appendix. Speciation diagrams for the Systems 1 and 2 are presented in Figures 1C, D.

The Br_{2} solution is acidic, as results e.g. from the ChB (Eq. 9c) at V = 0: at [Na^{+1}] = 0 (Φ = 0) we have: [H^{+1}] – [OH^{-1}] = [BrO_{3}^{-1}]+[BrO^{-1}]+[Br_{3}^{-1}]+[Br^{-1}] > 0, i.e. [H^{+1}] > [OH^{-1}]. The Br_{2} is an acid with a strength comparable to that of acetic acid; at C_{0} = 0.01, pH equals: 3.40 for Br_{2}, and 3.325 for CH_{3}COOH (pK1 = 4.65). Disproportionation of Br_{2}occurs initially to a small extent (several %), according to the scheme Br_{2}+OH^{-1} = HBrO+Br^{-1}.

In the System 2, disproportionation of HBrO affected by NaOH (C) added according to titrimetric mode is presented in Fig. 1D [18]. The [Br^{-1}]/ [BrO_{3}^{-1}] ratio equals: 10^{-2.2553}/10^{-2.5563} at Φ = 2.0; 10^{-2.2730}/10^{-2.5740} at Φ = 2.5, i.e. 10^{0.3010} = 2 = 2:1, corresponding to the stoichiometric ratio of products of this reaction. As results from Fig. 1D, the disproportionation of HBrO, at an excess of NaOH added, occurs mainly according to the scheme 3HBrO+3OH^{-1} = 2Br^{-1}+BrO_{3}^{-1}+3H_{2}O (stoichiometry 3:3 = 1:1), resulting from half-reactions: HBrO+2e^{-1}+H^{+1} = Br^{-1}+H_{2}O, HBrO – 4e^{-1}+2H_{2}O = BrO_{3}^{-1}+5H^{+1}, and 3H^{+1}+3OH^{-1} = 3H_{2}O. The (Φ,pH,E) values from the close vicinity of the corresponding equivalence (eq) points on the curves in Figs. 1A,B are collected in Table 2.

In C_{0} = 0.01 mol/L HBrO, more than 90% HBrO disproportionates according to the reaction 5HBrO = BrO_{3}^{-1}+2Br_{2}+2H_{2}O+H^{+1}; at V = 0 we have: [Br_{2}] = 10^{-2.4406}, [BrO_{3}^{-1}] = 10^{-2.7442}, i.e. [Br_{2}]/ [BrO_{3}^{-1}] = 10^{0.3036} ≈ 2, which confirms this stoichiometry of the reaction. The H^{+1} ions formed in this reaction acidify the solution significantly: at C_{0} = 0.01 and V = 0 we have pH = 2.74, although HBrO itself is a relatively weak acid.

Figure-1. The relationships: (A) pH = pH(Φ) and (B) E = E(Φ) for the Systems 1 and 2, and the related speciation diagrams for the Systems: 1 (C) and 2 (D), at V_{0} = 100, C_{0} = 0.01, C = 0.1.

The main task of titration is the estimation of the equivalent volume, V_{eq}, corresponding to the volume V ≡ V_{eq} of T, where the fraction titrated (Eq. (22)) assumes the value

C . V_{eq}

Φ_{eq}= ------------- (23)

C_{0}. V_{0}

in contradistinction to visual titrations, where the end volume V_{e} V_{eq} is registered, all instrumental titrations aim, in principle, to obtain the V_{eq} value on the basis of experimental data {(V_{j}, y_{j}) | j = 1,…,N}, where y = pH, E for potentiometric methods of analysis. Referring to Eq. (22), we have

C_{0}.V_{0} = 10^{3 . }m_{A}/M_{A} (24)

where m_{A} [g] and M_{A} [g/mol] denote mass and molar mass of analyte (A), respectively. From Eqs. (22) and (24), we get

m_{A} = 10^{-3}. C . M_{A}. V/Φ (25)

The value of the fraction V/Φ in Eq. (25), obtained from Eq. (22),

V/Φ = C_{0}. V_{0}/C (26)

is constant during the titration. Particularly, at the end (e) and equivalent (eq) points we have

V/Φ = V_{e}/Φ_{e} = V_{eq}/Φ_{eq} (27)

The V_{e} [mL] value is the volume of T consumed up to the end (e) point, where the titration is terminated (ended). The V_{e} value is usually determined in visual titration, when a pre-assumed color (or color change) of D+T mixture is obtained. In a visual acid-base titration, pH_{e} value corresponds to the volume V_{e} [mL] of T added from the start for the titration and

C . V_{e}

Φ_{e}= ------------ (28)

C_{0}. V_{0}

is the Φ-value related to the end point. From Eqs. (25) and (27), one obtains:

(a) m_{A} = 10^{-3}.C.V_{e}.M_{A}/Φ_{e} and (b) m_{A} = 10^{-3}.C.V_{eq}.M_{A}/Φ_{eq} (29)

This does not mean that we may choose between Eqs. (29a) and (29b), to calculate m_{A}. Namely, Eq. (29a) cannot be applied for the evaluation of m_{A}: V_{e} is known, but Φ_{e} unknown. Calculation of Φ_{e} needs prior knowledge of C_{0} value; e.g., for the titration system NaOH(C,V) → HCl(C_{0},V_{0}) we have [2]

Φ_{e}= C/C_{0}. (C_{0} - α_{e}) / (C+α_{e}) &n