Research Paper - Journal of Chemical Technology and Applications (2018) Volume 2, Issue 1

## A compact, algebraic formulation of disproportionation and symproportionation in Bromine systems.

- *Corresponding Author:
- Tadeusz Micha?owski

Department of Oncology The University Hospital in Cracow 31-501 Cracow, Poland

**Tel:**+212667845404

**E-mail:**[email protected]

**Accepted date:** July 03, 2018

**Citation: ***Michalowska-Kaczmarczyk AM, Michalowski T. A compact, algebraic formulation of disproportionation and symproportionation in Bromine systems. J Chem Tech App. 2018;2(2):1-14.*

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### Abstract

Numerous examples of disproportionation and symproportionation of bromine in dynamic redox systems are resolved according to GATES/GEB principles and represented graphically by the functions E = E(Φ) and pH = pH(Φ) of the fraction titrated Φ, and completed by dynamic speciation diagrams log zi ( ) i i log ??X ?? = ? Φ , for different species zi i X . The results of calculations can be considered from the viewpoint of relative efficiency of the competing reactions, on different steps of the titration procedure. The idea of linear combination of the balances related to electrolytic systems (aqueous media) is presented in context of GEB formulation GEB according to Approach II. Oxidation number, oxidant, reductant, are perceived as derivative (not primary) concepts within GATES/GEB. The equivalency of Approaches I and II to GEB is also proved.

### Keywords

Thermodynamics of electrolytic redox systems, GEB, GATES/GEB, disproportionation, symproportionation, Potentiometric titration.

**Notations:** GATES: Generalized Approach to Electrolytic Systems; GEB: Generalized Electron Balance; ON: Oxidation Number

### Introduction

Disproportionation [1] and symproportionation [2] are two mutually opposite phenomena classified as redox reactions. We will focus here our interest on aqueous solutions as electrolytic redox systems, of which physicochemical knowledge is relatively extensive.

Disproportionation is a special type of redox reaction, where an
element on an intermediate oxidation number (ON) in a species
is transformed – simultaneously – to the species with lower and
higher ONs of this element. It means that this element must be
able to form the species with at least three different oxidation
numbers (ONs). For example, bromine forms the species
with five ONs (–1, –1/3, 0, 1, 5) (**Figure 1**). In Br_{2} and BrO^{-1},
bromine has intermediate ONs: 0 and 1, resp. In particular, the
disproportionation of Br_{2}, affected by OH^{-1} ions, can be written
as follows [3]:

3Br_{2} + 6OH-1 = BrO_{3}-1 + 5Br^{-1} + 3H_{2}O (1a)

Br_{2} + 2OH-1 = BrO^{-1} + Br^{-1} + H_{2}O (1b)

etc. In the symproportionation, two reactants containing the
same element, here: Br with different ONs, react with formation
of the species on intermediate ONs of this element (**Figure 1**).
For example, symproportionation of BrO_{3}-1 and Br^{-1}, affected
by H^{+1} ions, can be written as follows:

BrO_{3}-1 + 5Br^{-1} + 6H^{+1} = 3Br_{2} + 3H_{2}O (2a)

BrO_{3}-1 + 8Br^{-1} + 6H^{+1} = 3Br_{3}-1 + 3H_{2}O 2(b)

etc.

*A remark*. The disproportionation reactions in biological
systems are termed as dismutation, when associated with
superoxide dismutases (SODs) – the enzymes catalysing a
dismutation of toxic superoxide (O_{2}^{−1}) radical [4]. In French,
the term dismutation refers also to non-biological systems [5].
Comproportionation [6] and synproportionation [7]), as the
synonyms of symproportionation, are also found in literature.

The disproportionation may be affected by an action of
the solvent, e.g. dilution with water, to which the bromine
compound at an intermediate oxidation state, e.g. HBrO, has
been introduced. The disproportionation effect can be greatly
enhanced by the action of an acid or base. In some instances, it
can also be stated that the disproportionating agent acts also as an
oxidant or reductant [3]. In a particular case, namely in reaction
Br_{2} + Br^{-1} = Br_{3}^{-1}, the symproportionation is indistinguishable
from the complexation effect.

The redox systems are formulated, from thermodynamic
viewpoint, according to Generalized Approach to Electrolytic
Systems (GATES) [8,9] principles, formulated by Michałowski.
For this purpose, the set of K algebraic equations,* f _{0},f_{12},f_{3},…,f_{K},* is formulated. It is composed of: charge balance (ChB,

*f*), the linear combination

_{0}*f*

_{12}= 2∙

*f*

_{2}–

*f*

_{1}, of elemental balances:

*f*

_{1}= f(H) for H and

*f*

_{2}=

*f*(O) for O, and K–2 elemental/core balances

*f*(Y

_{k}) (k=3,…,K) for elements/cores Y

_{k}(≠ H, O). The

*f*

_{12}is the primary form of the Generalized Electron Balance (GEB), discovered by Michałowski, and formulated as the Approach II to GEB [3,8-24]. The GATES related to redox systems will be denoted as GATES/GEB. The GATES is related to redox and non-redox systems, and then GATES/GEB ⊂ GATES.

Another option is the Approach I to GEB [25-28], discovered by Michałowski, and considered later as the ‘short’ version of GEB. The Approach I to GEB is based on a ‘card game’ principle, with electron-active elements as ‘players’, electronnon- active elements as ‘fans’, and electrons as ‘money’ [22]. The equivalency of Approaches I and II to GEB will be proved, and then the balances for GEB be formulated for different systems according to the Approach I.

All attainable physicochemical knowledge can be involved in further, numerical calculations, realized with use of an iterative computer program. The results of calculations are presented graphically and discussed. The GATES/GEB is perceived as the best tool for thermodynamic resolution of electrolytic redox systems, according to algebraic principles.

**Thermodynamic modelling of redox systems**

Modelling of electrolytic redox systems according to GATES/
GEB principles is based on general laws of elements and charge
preservation, related to closed systems composed of condensed
phases, separated from the environment by diathermal walls. In
further discussion, we refer to redox systems formed in aqueous
media, where the species are perceived in their natural form,
as hydrates , where z_{i} (z_{i} = 0, ±1, ±2,…) is the external charge of expressed in elementary charge unit e = F/N_{A} (F –
Faraday constant, N_{A} – Avogadro’s constant), and niW (≥ 0) is the
mean number of water (W = H_{2}O) molecules attached to

**Components and species:** The terms: components and species
are distinguished. Components form a system, the species are
present in the system thus formed. A static system is obtained
after disposable mixing the components: H_{2}O as solvent, and
solute(s). A dynamic D+T system is a result of addition of
titrant T into titrand D, in consecutive portions. The D and T
are composed separately before the titration, where the D+T
mixture is formed; the D and T are subsystems of the D+T
system. A volume V mL of T is added into V_{0} mL of D, up to a
given point of the titration, and V_{0}+V mL of D+T mixture is thus
obtained, if the assumption of the volumes additivity is valid/
tolerable. In the notation applied here, N_{0j} (j=1,2,…,J) is the
number of molecules of the component of j-th kind, including
water, forming D and T in dynamic D+T system. The D+T
system thus obtained involves N_{1} molecules of H_{2}O ( = H_{2}O,
z_{1}=0) and Ni species of i-th kind, (i=2,3,…,I), denoted
briefly as (N_{i} ,n_{i}) where n_{i}≡n_{iW}≡n_{i}H_{2}O.

The known chemical formulas of and their respective charges provide the information necessary/sufficient to formulate the respective balances, for elements or cores. A core is a cluster of elements with defined composition, structure and external charge that remains unchanged in a system considered.

**Formulation of balances: general remarks: **It is advisable
to start the balancing from the interrelations between numbers
of particular entities: N_{0j} for components represented by
molecules composing D and T, and N_{i} – for the species (ions
and molecules) of i-th kind (i = 1,…,I), where I is the number of
kinds of the species in D+T. The mono- or two-phase
electrolytic D+T system thus obtained involve N_{1} molecules of
H_{2}O and Ni species of i-th kind, (i=2,3,…,I), specified
briefly as (N_{i} ,n_{i}) ,where n_{i} ≡n_{i}W ≡n_{i}H_{2}O is the mean number
of hydrating water molecules (W=H_{2}O) attached to .The net charge of equals to the charge of , i.e., z_{i} + n_{iW}∙0 = z_{i}.For ordering purposes, we write the sequence: H^{+1} (N_{2}, n_{2}), OH^{-1} (N_{3}, n_{3}), … , i.e., z_{2} = 1, z_{3} = –1, … . The 's, with different
numbers of H_{2}O molecules involved in , e.g. {H^{+1}, H_{3}O^{+1},H_{9}O_{4}^{+1}}, {H_{4}IO_{6}^{-1}, IO_{4}^{-1}} are considered equivalently, i.e., as the
same species in this medium. The charge of a species , expressed in elementary charge units, results from the numbers
of protons in nuclei, and orbital electrons in atoms composing
the species.

Presentation of the species in natural forms in aqueous media,
i.e., as , has several advantages. This way, after linear
combinations of the related balances, one can discover some
regularities hidden earlier by notation of the species in the
form This notation can be extended on electrolytic systems
in mixed-solvent A_{s} (s=1,…,S) media, where mixed solvates are assumed, and is the mean numbers
of A_{s} (s=1,…,S) molecules attached to [15,21,29]. In other
instances (reaction notation), the common/simpler notation of the species, e.g. HSO_{4}^{-1}∙ n_{4}H_{2}O as HSO_{4}^{-1}, will be practiced. Molar concentrations [mol/L] of the species be
denoted as , for brevity.

The notation of the species is useful on the step of
formulation of the related balances: charge balance (*f _{0}* = ChB) and elemental balances,

*f*

_{k}=

*f*(Y

_{k}), in the system where K elements Y

_{k}(k=1,…,K) are involved. The ChB expresses the electroneutrality of the electrolytic system, whereas the

*f*(Y

_{k}), (k=1,…,K) express the conservation of all the elements in the closed system, chosen for modelling purposes. For simplicity/ uniformity of notation, we assume the sequence:

*f*=

_{1}*f*(H),

*f*=

_{2}*f*(O),… ,

*f*

_{k}= f(Y

_{k}).

The charged/ionic species of the system, i.e., the species
with z_{i} ≠ 0 (z_{i} > 0 for cations, z_{i} < 0 for anions), are involved in
the charge balance

(3)

Free water particles, and water bound in the hydrates are included in the balances: *f _{1}* =

*f*(H),

*f*= f(O):

_{2}(4)

(5)

Then the balance

(6)

is formulated.

The elemental balances: *f _{3},...,f_{K}* , interrelating the numbers of
atoms Y

_{k}≠ H, O in components and species, are as follows

(7)

where a_{ki} and b_{kj} in equations 4, 5 and 7 are the numbers of
element Y_{k} (k=1,...,K) in , and in the j-th component of
the system, resp.

The linear combination

(8)

involves K balances: *f _{0}, f_{12}, f_{3} ,…,f_{K}*. In particular, d

_{1}= +1, d

_{2}= –2. As will be indicated below, when the multipliers dk are equal to (or involved with) the oxidation numbers (ON’s) of the corresponding elements E

_{k}(k=1,…,K) in a redox system, we get the simplest (most desired) form of the related linear combination (eq. 8), as will be explained on the example presented in section 3.5.

In eq. 6 and then in eq. 8, the terms involved with water, i.e., N_{1},
N_{0j} (for j related to H_{2}O as the component), and all n_{i} = n_{iW} are
not involved. The necessity of prior knowledge of niW values in
the balancing is thus avoided. The n_{i} = n_{iW} = niH_{2}O values are
virtually unknown – even for X_{2}^{z2} = H^{+1} [30] in aqueous media,
and depend on ionic strength (I) of the solution.

For a redox system, (*f _{0}, f_{12}, f_{3} ,…,f_{K}*) is the set of K independent,
algebraic equations. The

*f*is the primary form of Generalized Electron Balance (GEB),

_{12}*f*= pr-GEB [12,17]. All the balances thus obtained are expressed in terms of concentrations, see

_{12}**Table 1**in context of

**Table 2**. The charge balance has there the form

1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|

System | D (V_{0}) |
T (V) | GEB | Charge balance | Concentration balances |

S1 | Br_{2} (C_{0}) |
NaOH (C) | P_{1Br} = 2Z_{Br}?β_{0} |
α ? P_{2Br} + β = 0 |
P_{3Br} = 2?β_{0} |

S2 | HBrO (C_{0}) |
NaOH (C) | P_{1Br} = (Z_{Br}?1)?β_{0} |
α ? P_{2Br} + β = 0 |
P_{3Br} = β_{0} |

S3 | NaBr (C_{0}) |
KBrO_{3} (C) |
P_{1Br} = (Z_{Br}+1)?β_{0} + (Z_{Br}?5)?β |
α ? P_{2Br} + β_{0} + β = 0 |
P_{3Br} = β_{0} + β |

S4 | NaBr (C_{0}) + H_{2}SO_{4} (C_{01}) |
KBrO_{3} (C) |
P_{1Br} = (Z_{Br}+1)?β_{0} + (Z_{Br}?5)?β |
α ? P_{2Br} ? P_{2S} + β_{0} + β = 0 |
P_{3Br} = β_{0} + β, P_{3S} = β_{0}1 |

S5 | NaBr (C_{0}) |
NaBrO (C) | P_{1Br} = (Z_{Br}+1)?β_{0} + (Z_{Br}?1)?β |
α ? P_{2Br} + β_{0} + β = 0 |
P_{3Br} = β_{0} + β |

S6 | NaBr (C_{0}) +H_{2}SO_{4} (C_{01}) |
NaBrO (C) | P_{1Br} = (Z_{Br}+1)?β_{0} + (Z_{Br}?1)?β |
α ? P_{2Br} ? P_{2S} + β_{0} + β = 0 |
P_{3Br} = β_{0} + β, P_{3S} = β_{0}1 |

S7 | NaBr (C_{0}) |
Br_{2} (C) |
P_{1Br} = (Z_{Br}+1)?β_{0} + 2Z_{Br}?β |
α ? P_{2Br} + β_{0} = 0 |
P_{3Br} = β_{0} + 2?β |

S8 | NaBr (C_{0}) +H_{2}SO_{4} (C_{01}) |
Br_{2} (C) |
P_{1Br} = (Z_{Br}+1)?β_{0} + 2Z_{Br}?β |
α ? P_{2Br} ? P_{2S} + β_{0} = 0 |
P_{3Br} = β_{0} + 2?β, P_{3S} = β_{0}1 |

S9 | NaBr (C_{0}) |
KBrO_{3} (C) |
P_{1Br} = (Z_{Br}+1)?β_{0} + (Z_{Br}?5)?β |
α ? P_{2Br} + β_{0} + β = 0 |
P_{3Br} = β_{0} + β |

S10 | NaBr (C_{0}) + Br_{2} (C_{01}) |
KBrO_{3} (C) |
P_{1Br} = (Z_{Br}+1)?β_{0} + 2Z_{Br}?β_{0}1 + (Z_{Br}?5)?β |
α ? P_{2Br} + β_{0} + β = 0 |
P_{3Br} = β_{0} + 2?β_{0}1 + β |

S11 | NaBr (C_{0}) + Br_{2} (C_{01}) + H_{2}SO_{4} (C_{02}) |
KBrO_{3} (C) |
P_{1Br} = (Z_{Br}+1)?β_{0} + 2Z_{Br}?β_{0}1 + (Z_{Br}?5)?β |
α ? P_{2Br} ? P_{2S} + β_{0} + β = 0 |
P_{3Br} = β_{0} + 2?β_{0}1 + β, P_{3S} = β_{0}2 |

**Table 1: ** Composition of titrand D and titrant T in the systems S1,?,S11.

System S1 | System S2 | ||||
---|---|---|---|---|---|

NaOH → Br_{2} |
NaOH → HBrO | ||||

? | pH | E | ? | pH | E |

1,995 | 6,666 | 1,0491 | 0,995 | 6,347 | 1,0720 |

1,996 | 6,728 | 1,0455 | 0,996 | 6,411 | 1,0681 |

1,997 | 6,811 | 1,0406 | 0,997 | 6,498 | 1,0630 |

1,998 | 6,933 | 1,0334 | 0,998 | 6,625 | 1,0555 |

1,999 | 7,161 | 1,0199 | 0,999 | 6,866 | 1,0412 |

2,000 | 8,143 | 0,9619 | 1,000 | 8,102 | 0,9682 |

2,001 | 8,966 | 0,9132 | 1,001 | 9,002 | 0,9150 |

2,002 | 9,244 | 0,8968 | 1,002 | 9,281 | 0,8985 |

2,003 | 9,413 | 0,8868 | 1,003 | 9,450 | 0,8885 |

2,004 | 9,534 | 0,8797 | 1,004 | 9,571 | 0,8814 |

2,005 | 9,628 | 0,8741 | 1,005 | 9,666 | 0,8758 |

**Table 2: ** (S1,S2). The sets of (?, ph, E) values taken from the vicinity of the equivalence points, at (C_{0},V_{0},C) = (0.01,100,0.1).

(3a)

Note that [ X_{m}] = 0 for a species m mw X_{m}.n_{mw} with zero charge
(z_{m}=0), e.g., 0∙[H_{2}O] = 0 (z_{1}=0).

The term charge balance (ChB) is used for both forms of this
relation, e.g., for ChB expressed by equations: 3 (in terms
of N_{i}, N_{0j}), and 3a (in terms of concentrations); it is done in
accordance with the ‘Ockham razor’ principle. This conceptual
‘abuse’ should not lead to ambiguities, in the right context. In
addition, the term GEB will be applied both to *f _{12}*, and to the linear combinations, expressed by eq. 8 (in terms of N

_{i}, N

_{0j}), and to the related balances written in terms of concentrations. The elemental/core balances expressed in terms of concentrations are named as concentration balances, for Y

_{k}≠ H, O (k=3,…,K). The balances expressed in terms of concentrations are compatible with expressions for equilibrium constants, specified in

**Table 3**. Consequently, the system of balances related to a redox system consists of three types of balances, expressed in terms of concentrations: GEB, ChB, and K–2 concentration balances for the related elements/cores Y

_{k}≠ H, O; k= 3,…,K. For modelling purposes, the balances are related to the closed system, separated from the environment by diathermal walls [22].

1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|

System: | ? | pBrO_{3} |
pBrO | pBr | Δ1 = pBrO ? pBrO_{3} |
Δ2 = pBr ? pBrO_{3} |

S1 | 1.5 | 2.669 | 6.857 | 1.976 | 4.188 | 0.693 |

2.5 | 2.574 | 6.758 | 1.875 | 4.184 | 0.699 | |

S2 | 1.5 | 2.538 | 6.988 | 2.237 | 4.450 | 0.301 |

2.5 | 2.574 | 7.024 | 2.273 | 4.450 | 0.301 |

**Table 3: ** Supplementary computational data for the systems S1 and S2.

As indicated elsewhere [31], for a non-redox system, the
equation (eq. 8) is transformed into identity,
0 = 0, and then *f _{12}* is not an independent equation. Consequently,
in non-redox systems,

*f*is the equation dependent on

_{12}*f*, and . In other words,

_{0},f_{3},…,f_{K}*f*is the set of K – 1 independent equations for a non-redox system. For non-redox systems,

_{0},f_{3},…,f_{K}*f*is transformed into charge balance (eq. 3a) and K – 2 concentration balances for Y

_{0},f_{3},…,f_{K}_{k}≠ H, O. In other words, f1, f2 and f12 are not formulated for non-redox systems. The dependency or independency of f12 from the balances

*f*is the general criterion distinguishing between non-redox and redox systems [11].

_{0},f_{3},…,f_{K}Formulation of the proper (i.e., with dk equal to ON’s) linear
combinations is applicable to check the linear dependency or
independency of the balances. This way is realized a very useful/
effective manner for checking/stating the linear dependence
of the balances: *f _{0}, f_{12}, f_{3} ,…,f_{K}* related to non-redox systems,
named as the transformation of the linear combination (6) to the
identity, 0 = 0 [12]. For this purpose, in all instances, we try to
obtain the simplest form of the linear combination (6).

Let us repeat: For a redox system, the proper linear combination
(6), with d_{k} equal to ON’s, is the way towards the simplest/
shortest form of GEB; for a non-redox system, it is the way
towards identity, 0 = 0.

To avoid possible/simple mistakes in the realization of the linear combination procedure, we apply the equivalent relations:

(9)

for elements with negative oxidation numbers, or

(10)

for elements with positive oxidation numbers, k ∈ 3,…,K. In
this notation, *f _{k}* will be essentially treated not as the algebraic
expression on the left side of the equation fk = 0, but as an
equation that can be expressed in alternative forms presented
above.

Note, for example, that . The change of places of numbers N_{i} for components and N_{0j} for species (see equations 9, 10) facilitates the purposeful
linear combination of the balances, and enables to avoid simple
mistakes in this operation.

Concentrations of solutes in D and T of D+T redox systems S1 –
S11 considered in this paper are specified in **Table 1** (columns 2,
3). The set of balances for a particular redox system, specified in
rows of **Table 1**, consists of: generalized electron balance (GEB,
column 4), charge balance (ChB, column 5) and concentration
balance(s) (column 6). The symbols used in columns 4,5,6 of **Table 1** and further notations are as follows:

C_{0} – concentration of analyte (A) in D, C_{01} – concentration of
H_{2}SO_{4} in D, C – concentration of reagent (B) in T;

u = V/(V_{0}+V); u_{0} = V_{0}/(V_{0}+V);

β = C∙u, β_{0}= C0∙u_{0}, β_{01} = C_{01}∙u_{0} ;

– fraction titrated ;

Atomic number Z_{Br}=35 for Br;

Molar concentration of the species in the D+T mixture is involved in the relation

;

α = [H^{+1}] – [OH^{-1}] = 10^{-pH} – 10^{pH-14} ;

ϑ_{0} = RT/F∙lN_{10} – Nernstian slope; ϑ_{0} = 0.05916 V at T = 298 K;A = 1/0.05916 = 16.9;

P_{1Br} = (Z_{Br}–5)([HBrO_{3}]+[BrO_{3}^{-1}]) + (Z_{B}r–1)([HBrO]+[BrO^{-1}]) + 2Z_{Br}[Br_{2}] + (3Z_{Br}+1)[Br_{3}^{-1}] + (Z_{Br}+1)[Br^{-1}];

P_{2Br} = [BrO_{3}^{-1}] + [BrO^{-1}] + [Br_{3}^{-1}] + [Br^{-1}];

P_{3Br} = [HBrO_{3}] + [BrO_{3}^{-1}] + [HBrO] + [BrO^{-1}] + 2[Br_{2}] + 3[Br_{3}^{-1}]+ [B_{r}^{-1}];

P_{2S} = [HSO_{4}^{-1}] + 2[SO_{4}^{-2}] ; P_{3S} = [HSO_{4}^{-1}] + [SO_{4}^{-2}].

All concentrations of components and species involved with
notation applied in **Table 1** are expressed in mol/L, and all
volumes – in mL. In this notation, common segments of the
related balances are distinguished; it allows to simplify the
extensive formulation, and may be helpful in construction of
the appropriate algorithms. Numerical values of equilibrium
constants needed/used in the calculations are involved in the
relations:

[H^{+1}] = 10^{-pH} ; [OH^{-1}] = 10^{pH-14} ; [HSO_{4}^{-1}] = 10^{1.8-pH}[SO_{4}^{-2}]; n b

[Br_{2}] = 10^{2A(E-1.087)-2pBr} ; [Br_{3}^{-1}] = 10^{2A(E-1.05)-2pBr} ; [BrO^{-1}] = 10^{2A(E-0.76)-pBr+2pH-28} ;

[HBrO] = 10^{8.6-pH∙}[BrO^{-1}] ; [BrO_{3}^{-1}] = 10^{6A(E-1.45)-pBr+6pH} ; [HBrO_{3}] = 10^{0.7-pH∙}[BrO_{3}^{-1}];

As stated above, formulation of GEB can be realized according to Approaches I or II to GEB; both Approaches are equivalent, i.e.,

Approach I to GEB Approach II to GEB (11)

**A remark: **The GEB concept is quite different from the ‘electron
balancing’ procedure applied in laying the redox reactions
according to stoichiometric ‘rules’, criticized extensively in
[8,18,19,32-34]; it is also different from the term ‘electron
balance’ applied for description of: microbial metabolism
[35], electrons in light-emitting diodes [36], etc., and… from
‘electronic balance’, as the device for mass measurement [37].

**Formulation of the system S1**

**Preliminary data: **Formulation of GEB according to both
Approaches (I and II) to GEB will be exemplified first by the
system S1 in **Table 1**, formed from D and T, considered as
subsystems of the D+T system. 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
the D+T system we have I = 11 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}) (12)

Note, for example, that N_{4} molecules of HBrO_{3}∙n_{4}H_{2}O (z_{4}=0)
involve: N_{4}(1+2n_{4}) atoms of H, N_{4}(3+n_{4} ) atoms of O, and N_{4} atoms of Br.

The species formed from Br_{2} and H_{2}O in D, and then present in
D+T, resulted from hydrolytic disproportionation: 3Br_{2} + 3H_{2}O
= BrO_{3}^{-1} + 5Br^{-1} + 6H^{+1}; 3Br_{2} + 3H_{2}O = HBrO_{3} + 5Br^{-1} + 5H^{+1};
Br_{2} + H_{2}O = HBrO + Br^{-1} + H^{+1}; Br_{2} + H_{2}O = BrO^{-1} + Br^{-1} + 2H^{+1}.
The symproportionation reaction Br_{2} + Br^{-1} = Br_{3} -1 can be also
perceived as complexation. The ONs for H, O and Na are not
changed, i.e., these elements are not oxidized or reduced.

**Approach I to GEB in S1: **The Approach I to GEB needs prior
knowledge of ON’s for all elements in components and species
of the system in question. In S1, one element (Br) is considered *a priori *as the only one electron-active element (player); K*=1
is here the number of players.

In the system S1, bromine (as Br_{2}) is the carrier/distributor of
its own, bromine electrons. One atom of Br has Z_{Br} bromine
electrons, and then one molecule of Br_{2} has 2Z_{Br} bromine
electrons, i.e., N0_{1} molecules of Br_{2} involve 2Z_{Br}∙N0_{1} bromine
electrons. The oxidation degree x of an atom in a simple species,
such as ones formed here by bromine, is calculated on the basis
of known oxidation degrees: +1 for H, and –2 for O, and external
charge of this species. We have, by turns, the relations: 1∙1 + 1∙x
+ 3∙(–2) = 0 → x= 5 for HBrO_{3}; 1∙x + 3∙(–2) = –1 → x = 5 for
BrO_{3}^{-1}; 1∙1 + 1∙x + 1∙(–2) = 0 → x = 1 for HBrO;…; 3∙x = –1 →
x = –1/3; 1∙x = –1 → x = –1 for Br^{-1}.

The oxidation degree is the net charge resulting from the
presence of charge carriers, inherently involved in an atom:
protons in nuclei and orbital electrons, expressed in elementary
charge units as: +1 for protons, and –1 for electrons. The number
y of bromine electrons in one molecule of HBrO_{3} is calculated
from the formula: Z_{Br}∙(+1) + y∙(–1) = 5, i.e., bromine involves
y = Z Br–5 bromine electrons, etc. On this basis, we state that
[22,38]:

N_{4} species HBrO_{3}∙N_{4}H_{2}O involve (Z_{Br}–5)∙N_{4} bromine electrons;

N_{5} species BrO_{3}^{-1}∙N_{5}H_{2}O involve (Z_{Br}–5)∙N_{5} bromine electrons;

N_{6} species HBrO∙N_{6}H_{2}O involve (Z_{Br}–1)∙N_{6} bromine electrons;

N_{7} species BrO^{-1}∙N_{7}H_{2}O involve (Z_{Br}–1)∙N_{7} bromine electrons;

N_{8} species Br_{2}∙N_{8}H_{2}O involve 2Z_{Br}∙N_{8} bromine electrons;

N_{9} species Br_{3} -1∙N_{9}H_{2}O involve (3Z_{Br}+1)∙N_{9} bromine electrons;

N_{10} species Br^{-1}∙N_{10}H_{2}O involve (Z_{Br}+1)∙N_{10} bromine electrons;

The N0_{1} molecules of Br_{2} involved 2Z_{Br}∙N0_{1} bromine electrons.
These (bromine) electrons were dissipated between different
bromine species, indicated above. The balance for the bromine
electrons is then as follows:

2Z_{Br}∙N0_{1} = (Z_{Br}–5)∙N_{4} + (Z_{Br}–5)∙N_{5} + (Z_{Br}–1)∙N_{6} + (Z_{Br}–1)∙N_{7} +
2Z_{Br}∙N_{8} + (3Z_{Br}+1)∙N_{9} + (Z_{Br}+1)∙N_{10} (13)

Applying in (13) the relations:

and

(14)

we get

Eq. 13a is the GEB for S1, obtained according to the Approach
I to GEB. Applying the notation from **Table 2**, from eq. 13a we
get the balance P_{1Br} = 2Z_{Br}∙β_{0}, identical with the one specified in
column 4 for S1.

**Approach II to GEB in S1: **The balances related to the D+T
mixture are as follows:

*f*_{0} = ChB

N_{2} – N_{3} – N_{5} – N_{7} – N_{9} – N_{10} + N_{11} = 0 (15)

*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_{11}n_{11} = 2N_{02} + N_{03} + 2N_{04} (16)

*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_{11}n_{11}

= N_{02} + N_{03} + N_{04} (17)

– *f*_{3} = – *f*(Na)

N_{03} = N_{11} (18)

*f*_{4} = *f*(Br)

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

From Equations 16 and 17 we have

*f*_{12} = 2∙*f*_{2}– *f*_{1} :

– N_{2}+
N_{3} + 5N_{4} + 6N_{5} + N_{6} + 2N_{7} = N_{03} (20)

From equations 15, 18 and 20 we have

*f*_{1}_{2} + *f*_{0} – *f*_{3} :

5N_{4} + 5N_{5} + N_{6} + N_{7} – N_{9} – N_{10} = 0 (21)

Applying the atomic number Z_{Br} for Br, from Equations 19 and
21 we obtain the equation

Z_{Br}∙*f*_{4} – (*f*_{12} + *f*_{0} – *f*_{3}) :

(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}

identical with eq. 13. This way, the equivalency of the Approaches I and II (eq. 11) is proved for the system S1.

In the balance *f*_{12} = 2∙*f*_{2} – *f*_{1} = 2∙*f*(O) – *f*(H) (eq. 20), the numbers
of water molecules: N_{1}, n_{iW}, and those N_{0j} related to H_{2}O as
component/solvent (here: N_{02} and N_{04}) used for preparation of
D and T are cancelled. In other systems, also hydrating water
molecules introduced by some components (e.g. CuSO_{4}∙5H_{2}O
in [20]), are also cancelled within *f*_{12}.

**Charge and concentration balances for S1:** From equations
14, 15, 18, 19 and **Tables 1** and **2** we have

[H^{+1}] – [OH^{-1}] – [BrO_{3}^{-1}] – [BrO^{-1}] – [Br_{3}^{-1}] – [Br^{-1}] + [Na^{+1}] = 0

(15a)

(18a)

Eq. 19a is the concentration balance for Br, see column 6 in
**Table 1** for S1. Note that α, P_{2Br} and β are considered as segments
of eq. 15a, applied also in balances related to other systems
specified in **Table 1**.

**Other/equivalent forms of GEB in S1:** Note that the GEB for
the system S1, obtained from *f*_{12} (eq. 20), has the form

From eq. 21 we have

(21a)

Other combinations of f12 (eq. 20) with *f*_{0}, *f*_{3} and *f*_{4} (equations
15,18,19) have also full properties of GEB for the system S1.
Among others, we have

(22)

Equations 21a and 22a can be perceived as the steps towards the shortest (involving the smallest number of terms) form of GEB in S1. Summarizing, the equations 13a, 20a, 21a and 22a are the equivalent forms of GEB in the system S1.

**Computer program for the system S1:** Some of the balances
specified above are involved in the set of independent balances
applied in the computer program.

function F = NaOH_Br2(x)

global V C0 V0 C yy

E = x(1);

pH = x(2);

pBr = x(3);

H = 10^(-pH);

Kw = 10^-14;

pKw = 14;

OH = Kw/H;

A = 16.9;

Br = 10^-pBr;

ZBr = 35;

Br2=Br^2*10^(2*A*(E-1.087));

Br3=Br^3*10^(2*A*(E-1.05));

BrO=Br*10^(2*A*(E-0.76)+2*pH-2*pKw);

BrO3=Br*10^(6*A*(E-1.45)+6*pH);

HBrO = 10^8.6*H*BrO;

HBrO3=10^0.7*H*BrO3;

Na=C*V/(V0+V);

F = [%Charge balance

(H-OH -Br-Br3-BrO-BrO3+Na);

%Concentration balance for Br

(Br+3*Br3+2*Br2+HBrO+BrO+HBrO3+BrO3-2*C0*V0/ (V0+V));

%Electron balance

( ( Z B r + 1 ) * B r + ( 3 * Z B r + 1 ) * B r 3 + 2 * Z B r * B r 2 + ( Z B r - 1)*(HBrO+BrO)...

+(ZBr-5)*(HBrO3+BrO3)-2*ZBr*C0*V0/(V0+V))];

yy(1)=log10(Br);

yy(2)=log10(Br3);

yy(3)=log10(Br2);

yy(4)=log10(HBrO);

yy(5)=log10(BrO);

yy(6)=log10(HBrO3);

yy(7)=log10(BrO3);

yy(8)=log10(Na);

end

The calculation procedure is realized according to an iterative computer program, here: MATLAB [8]. The volume V of the titrant (T) added is the parameter. In this program, the set of 3 independent variables, forming a (transposed) vector is considered. The number of the (independent, ‘homogeneous’) variables is equal to the number of equations; this ensures a unique solution of the equations related to the system S1, at the pre-set C0, C and V0 values, and the V-value at which the calculations are realized, at defined step of the calculation procedure. The set of equations {15a, 19a, 13a}, involving GEB, obtained according to Approach I to GEB, was applied there. The set of independent equations: {15a, 19a, 20a}, {15a, 19a, 21a} or {15a, 19a, 22a} can be chosen, optionally, for this purpose. The complete set of interrelations between concentrations of the species in the balances, taken from Table 3, is applied.

x^{T} = (x(1), x(2), x(3)) = (E, pH, pBr) (23)

The ‘homogeneity’ of the variables in (23) results from the fact
that all them are found in the exponents of the power for 10 in:
[e^{-1}] = 10^{-A∙E}, [H^{+1}] = 10^{-pH}, [Br^{-1}] = 10^{-pBr}, where A = 16.9 at *T* =
298 K (**Table 2**).

In the computer program, two measurable variables: potential
E and pH are involved in the set (23). The E values are related
here to the normal hydrogen electrode (NHE) scale. In all cases
presented in this paper, the curves E = E(Φ), pH = pH(Φ) and
speciation diagrams with the curves zi
log [X_{i}^{zi}] vs. Φ are plotted.

**Equations and equalities in S1: **Among the concentration balances for the systems specified in **Table 1**, one can distinguish
equations and equalities. In the system S1, we have an equality,
represented by the balance 18a, which involves only one species
(here: Na^{+1}). In the equality 18a, the value for [Na^{+1}] is a number
(not variable) for the pre-assumed C and V0 values, at given
V-value; as such, it can enter immediately the related ChB,
column 5 for S1 in **Table 1**. Other balances, here: 13a, 15a and
19a, involve more species, and then are classified as equations.
Then (18a) is not considered as equation, if the number of
equations be compared with the number of independent
variables, here: 3 = 3.

**Oxidation number, oxidant and reductant as the redundant
terms in S1:** The GEB related to the system S1, and expressed
by eq. 21a, obtained according to Approach II to GEB, can be
rewritten as follows:

(21b)

As we see, the balance (21b), obtained from *f*_{12} + *f*_{0} – *f*_{3}, i.e., from
the linear combination of *f*_{0} and balances for ‘fans’ (H, O, Na),
has the oxidation numbers (ON) of Br in its species equal to (or
involved with) the coefficient/multiplier at the concentration of
the Br-species, and at concentration of the Br-component (here:
Br_{2}). If the species or component involves more Br-atoms, e.g.,
Br_{3}^{-1}, then we have , i.e. the product of the number of
Br-atoms, and the ON for Br in Br_{3}^{-1}.

This regularity can be extended on other redox systems. Concluding, the formulation of GEB according to Approach II to GEB

• needs none prior knowledge of ON’s for elements participating the redox system; it means that ON is the derivative concept within GATES/GEB;

• the terms: oxidant and reductant (as distinguisher, attribute,
*differentia specifica*) are not assigned a *priori* to individual
components and species of the redox system; there is simply
no need for this, i.e., full ‘democracy’ in this respect is
assumed.

In this context, the linear combination *f*_{12} + *f*_{0} – *f*_{3} and the
resulting eq. 21b exemplify the ‘purposeful’ linear combination.

**Completeness/redundancy/compatibility of equilibrium
constants:** The preparatory step in the formulation of redox
systems according to GATES/GEB principles involves
gathering of the corresponding equilibrium data, i.e., the
standard potentials E0’s, and other equilibrium constants
(**Table 3**). The set of equilibrium constants should be complete,
as far as possible. The point is that these sets of data, often
presented in the corresponding tables of equilibrium constants,
are usually incomplete and/or refer to different equations of
the related reactions. The related equilibrium constants can be
obtained from other equilibrium constants, as were shown in
[12,28], where the problem of redundancy and compatibility
was considered in context with the system S1. The problem of
redundancy is involved with seemingly excessive number of physicochemical data, collected from various thematic studies,
or different works.

**Disproportionation in dynamic bromine systems**

In algorithms applied for all dynamic systems S1 – S6,
specified in **Table 1** and presented below, it is assumed that V_{0}
= 100, C_{0} = 0.01, C = 0.1. All the systems will be illustrated
graphically, on the graphs (a) E = E(Φ), (b) pH = pH(Φ) and
(c) log [X_{i}^{zi}] =ϕ (Φ) plotted as the functions of the fraction titrated

(24)

It provides a kind of normalization in the related graphs, i.e.,
independency on V_{0} value. In principle, C_{0} is related to an
analyte (A), and C – to a reagent B for this analyte (**Tables 1** and **2**). Some dynamic systems are presented in extended, graphical
forms.

**Systems S1 : NaOH ⇒ Br _{2} and S2 : NaOH ⇒ HBrO:** The
curves are presented in (

**Figures 2A-C**) for S1, and 3a,b,c for S2. The points (Φ,E,pH) from the vicinity of the related equivalence points on the titration curves: E = E(Φ) and pH = pH(Φ) are collected in

**Table 2**. Moreover, from the result files we have the set of (Φ,pBrO

_{3},pBrO,pBr) values collected in

**Table 3**, where:

pBrO_{3} = –log[BrO_{3}^{-1}], pBrO = –log[BrO^{-1}], pBr = –log[Br^{-1}]

On this basis, we can compare two main competing reactions: (1a) and (1b) in the system S1, and:

3HBrO + 3OH^{-1}= BrO_{3}^{-1} + 2Br^{-1} + 3H_{2}O and (25)

HBrO + OH^{-1} = BrO^{-1} + H_{2}O (26)

in the system S2. Note that BrO-1 is the main competing product
relative to BrO_{3}^{-1}, both in S1 and S2 (**Figures 2C,3C**), at a due
excess of NaOH. On this basis, we find the relative efficiencies
equal to 1041 (column 6) : 10^{4.188} at Φ = 1.5 and 10^{4.184} at Φ =
2.5 for reactions (1a), (1b); and 10^{4.450} at Φ =1.5 and 2.5 for
reactions (25) and (26). Note that (26) is the dissociation (not
disproportionation) reaction. Stoichiometries of reactions (1a)
and (1b) are the same for the competing pairs of reactions: 3:6
= 1:2 for reactions (1a), (1b), and 3:3 = 1:1 for reactions (25)
and (26). Moreover, we find the ratio [BrO_{3}^{-1}]/[Br^{-1}] equal to
10^{42} = 10^{0.699} = 5 = 5:1 for S1 at Φ = 2.5 (i.e. at the excess of
NaOH), and close to 5 at the point where 1.5/2 = 75% of Br_{2} is
already consumed. The stoichiometry of products of reaction
(25) is confirmed by the ratio [Br^{-1}]/[BrO_{3}^{-1}] = 10^{0.301} = 2 = 2 : 1
(**Table 3**). This confirms the reaction 1a, and testifies against the
reaction 1b (of the same stoichiometry!), commonly met (‘given
to believe’) in literature and elsewhere, e.g. [38].

**Symproportionation in dynamic bromine systems**

**System 3 : kbro _{3} ⇒ nabr:** In this case, symproportionation
practically does not occur (

**Figure 4C**); concentration of HBrO, as the major product formed in the symproportionation reaction

BrO_{3}^{-1} + 2Br^{-1} + 3H^{+1} = 3HBrO (27)

is ca. 10^{-6} mol/L. The potential E increases monotonically (**Figure 4A**), whereas pH first increases, passes through maximum and
then decreases (**Figure 4B**). The relevant pH and E changes are small. Binding the H^{+1} ions in reaction (27) causes a weakly
alkaline reaction (**Figure 4B**).

**Table 2** (S1,S2). The sets of (Φ, pH, E) values taken from the
vicinity of the equivalence points, at (C_{0},V_{0},C) = (0.01,100,0.1).

**System S4 : KBrO _{3} ⇒ NaBr + H_{2}SO_{4}: **The stoichiometry 1
: 5, i.e., Φeq = 0.2, stated for C

_{01}values indicated at the curves plotted in F

**igure 5**(column 5a), results from reaction (2a). At Φ > 0.2, an increase of efficiency of the competing reaction (27) is noted. A growth of C01 value causes a small extension of the potential range in the jump region, on the side of higher E-values (

**Figure 5**, column 5a). With an increase of the C

_{01}value, the graphs of pH vs. Φ resemble two almost straight line segments intersecting at Φ

_{eq}= 0.2 (

**Figure 5**, column 5b).

However, the pH-ranges covered by the titration curves are
gradually narrowed (**Figure 5**, column 5b).

**System S5 : NaBrO ⇒ NaBr:** The basic reaction in this
system (**Figure 6B**) results from the relation

[HBrO_{3}]+[HBrO]+2[Br_{2}]+ 2[Br_{3}^{-1}] = [OH^{-1}] – [H^{+1}] > 0

obtained for NaIO solution from combination (addition) of
charge and concentration balances (**Table 1**):

α – P_{2Br} + β_{0} + β = 0 and P_{3Br} = β0 + β. We get: – α = P_{3Br} – P_{2Br} = [HBrO] + 2[Br_{2}] + 2[Br_{3}^{-1}] > 0,

i.e., – α > 0 ⇒ α < 0 ⇒ [OH^{-1}] > [H^{+1}]. The disproportionation reactions

3HBrO = BrO_{3}^{-1} + 2Br^{-1} + 3H^{+1} (28)

3BrO^{-1} = BrO_{3}^{-1} + 2Br^{-1} (29)

occur in a small degree only in the initial step of the titration
(**Figure 6C**), and then a basicity resulting from growth of
NaBrO concentration in the system prevails over the growth
in H+1 concentration resulting from reaction 28, i.e., dpH/dΦ
> 0 (**Figure 6B**). Buffer capacity of NaBr solution is very low
and even small changes in acidity cause substantial pH changes.
Potential E passes through maximum, and then decreases
(**Figure 6A**); this results from changes in [BrO_{3}^{-1}] and pH. The (small) growth in Br_{}^{-1} concentration, d[Br^{-1}]/dΦ > 0 (**Figure 6B**), resulting from reactions 28 and 29, overcomes the dilution
effect in D+T, affected by T addition.

**System S6: NaBrO ⇒ NaBr + H _{2}SO_{4}:** Symproportionation
occurs here mainly according to the scheme

HBrO + Br_{}^{-1} + H^{+1} = Br_{2} + H_{2}O (30)

(**Figure 7C**), stoichiometry 1:1. At Φ_{eq} = 1, there is a jump of
the potential E (**Figure 7A**) and slightly marked fracture on the
curve pH = pH(Φ) (**Figure 7B**). For Φ > 1, the excess of HBrO
in the titrant disproportionates gradually according to schemes:
5HBrO = BrO_{3}^{-1} + 2Br_{2} + H^{+1} + 2H_{2}O and 5HBrO = HBrO_{3} + 2Br_{2} + 2H_{2}O.

**System S7 : Br2 ⇒ NaBr: **After the titrant addition, the
concentration of Br_{3}^{-1} increases in the reaction

Br_{2} + Br^{-1} = Br_{3}^{-1} (31)

(Figure 8C); it affects the monotonic E-growth, (**Figure 8A**). Reaction (31) can be considered both as symproportionation
(0, –1) → (–1/3), and as the complex formation. The Br_{2} solution
acts as a weak acid and then pH decreases (**Figure 8B**); this
property of Br_{2} results immediately from the charge balance: α = [H+1] – [OH^{-1}] = [BrO_{3}^{-1}]+[BrO^{-1}]+[Br_{3}^{-1}]+[Br^{-1}] > 0 for Br_{2}.
The disproportionation Br_{2} + H_{2}O = HBrO + Br^{-1} + H+1 occurs
to a small extent; the ratio [HBrO]/[Br_{2}] equals: 0.0113 at
Φ=0.5; 0.0085 at Φ=1, 0.0068 at Φ=2. Other disproportionation
products are formed with much lesser efficiency.

**System S8: Br _{2} ⇒ NaBr + H_{2}SO_{4}:** Disproportionation of Br
occurs here in very small degree (

**Figure 9C**), smaller than in S7 (

**Figure 8C**). The potential changes (

**Figure 9A**) are very similar to those in Figure 8A. The pH changes are indicated in

**Figure 9B**.

**System S9: KBrO _{3} ⇒ NaBr:** In this case, symproportionation
practically does not occur (

**Figure 10C**); concentration of HBrO, as the major product formed in the symprortionation reaction

BrO_{3}^{-1} + 2Br^{-1} + 3H^{+1} = 3HBrO (32)

is ca. 10^{-6} mol/L. The potential E increases monotonically
(**Figure 10A**), whereas pH first increases, passes through
maximum and then decreases (**Figure 10B**). The relevant pH
and E changes are small. Binding the H^{+1} ions in reaction (32)
causes a weakly alkaline reaction (**Figure 10B**).

**System S10 : KBrO _{3} ⇒ NaBr + Br_{2}:** The symproportionation
effect (

**Figure 11C**) is greater here than in the system 13 (

**Figure 11C**), but is also small; concentration of HBrO, as the major product of the symproportionation (Eq. 32) is lower than 10

^{-3}mol/L (

**Figure 11C**). The increase in [HBrO], can be accounted on symproportionation resulting from a weakly acidic solution, caused by the presence of Br

_{2}, which disproportionates partially according to the scheme

Br_{2} + H_{2}O = HBrO + Br^{-1} + H^{+1} (11)

compare with **Figure 10C**. The pH values (**Figure 11B**) cover
acidic range, and E covers greater E values (**Figure 11A**) than
those in **Figure 10A**.

**System S11: KBrO _{3} ⇒ NaBr + Br_{2} + H_{2}SO_{4}:**

The related graphs plotted at different concentrations C0_{2} of
H_{2}SO_{4} are shown in **Figure 12**. From a comparison of the graphs
in the related columns of **Figures 7** and **12** it follows that the
presence of Br_{2} in D affects the related graphs. However, the
position of inflection points in **Figures 7A** and **12A**, and breaking
points in **Figures 7B** and **12B** are the same, in principle.

**Final comments**

The paper presents dynamic redox systems, with bromine
species on different ONs involved. The systems were tested in simulation procedures, realized according to GATES/GEB
principles. The results of calculations made with use of iterative
computer programs, were presented graphically. On the basis
of speciation curves, the reactions occurred in the systems can
be formulated, together with their relative efficiencies. Among
others, the effects resulting from presence of H_{2}SO_{4} in the
titrand, are considered and illustrated graphically

The Generalized approach to electrolytic systems (GATES) with the Generalized electron balance (GEB) involved and termed therefore as GATES/GEB, is adaptable for resolution of thermodynamic (equilibrium and metastable) redox systems of any degree of complexity; none simplifying assumptions are needed. Application of GATES provides the reference levels for real analytical systems. The GATES makes possible to exhibit some important details, of qualitative and quantitative nature, invisible in real experiment, e.g. speciation.

Contrary to appearances, the available physicochemical knowledge on the thermodynamic properties of basic species formed by halogens: chlorine, bromine and iodine in aqueous media, raises fundamental doubts, both of qualitative, and quantitative nature.

The knowledge of equilibrium constants, collected in the past/
distant times, for decades, is not substantially supplemented and verified in contemporary times. Frankly, the physicochemical
analysis of electrolytic systems is not currently one of the top
issues raised in scientific research. The quantitative data published
in literature are closely related to the quality of mathematical
models applied to their determination in electrochemical
research, with the main emphasis put on potentiometry. The
stoichiometry concept, based on the chemical reaction notation
principle, and especially its use and abuse, have been criticized
repeatedly by the author, especially in the works [8,19,20,32-34] issued in recent years. Stoichiometry cannot be perceived
as a true mathematics consequently inherent within chemistry.
Additionally, significant uncertainties arise in the context of
instability of the relevant compounds in aqueous solutions,
raised e.g., under the links [39,40], and in the literature cited
therein. In particular, the instability of some compounds after
their introduction into aqueous media is explained rightly by
their disproportionation. However, the disproportionation
scheme suggested this way (i.e., a *priori*) is inconsistent with
the results of calculations carried out on the basis of the physical
laws of elements conservation and equilibrium constants values.

The *f*_{12} , and any linear combination of *f*_{12} with *f*_{0},*f*_{3},…,*f*_{K} ,
have full properties of Generalized Electron Balance (GEB),
completing the set of K balances, *f*_{0},*f*_{12},*f*_{3},…,*f*_{K}, needed for
resolution of a redox system, of any degree of complexity. The
K–1 balances *f*_{0},*f*_{3},…,*f*_{K} are needed for resolution of a redox
system, of any degree of complexity. The linear independency/
dependency of *f*_{0},*f*_{12},*f*_{3},…,*f*K is then the general criterion
distinguishing between redox and non-redox systems. The
supreme role of this independency/dependency criterion, put
also in context with calculation of ONs, is of great importance,
in context with the contractual nature of the ON concept [41,42],
known from the literature issued hitherto. These regularities
are the clear confirmation of the Emmy Noether’s general
theorem [43-50] applied to conservation laws of a physical/
electrolytic system, expressed in terms of algebraic equations,
where GEB is perceived as the Law of Nature, as the hidden
connection of physicochemical laws, and as the breakthrough in
thermodynamic theory of electrolytic redox systems.

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