Comparative Balancing of Non-Redox and Redox Electrolytic Systems and Its Consequences

doi:10.4236/ajac.2013.410A1006

Paper Menu >>

Journal Menu >>

American Journal of Analytical Chemistry, 2013, 4, 46-53

http://dx.doi.org/10.4236/ajac.2013.410A1006 Published Online October 2013 (http://www.scirp.org/journal/ajac)

Comparative Balancing of Non-Redox and Redox

Electrolytic Systems and Its Consequences

Anna Maria Michałowska-Kaczmarczyk1, Tadeusz Michałowski2*

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

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

Email: *michalot@o2.pl

Received May 31, 2013; revised July 1, 2013; accepted August 15, 2013

the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided

the original work is properly cited.

ABSTRACT

In this paper, it is proved that linear combination 2·f(O) − f(H) of elemental balances: f(O) for O and f(H) for H is line-

arly independent on charge and elemental/core balances for all redox systems of any degree of complexity; it is the

primary form of the Generalized Electron Balance (GEB),









-GEB =2OHprf f , considered as the Approach II

to GEB. The Approach II is equivalent to the Approach I based on the principle of common pool of electrons. Both Ap-

proaches are illustrated on an example of titration of acidified (H2SO4) solution of H2C2O4 with KMnO4. It is also stated,

on an example of titration of the same solution with NaOH, that 2·f(O) − f(H) is a linear combination of charge and

elemental/core balances, i.e. it is not an independent balance when related to the non-redox system. These properties of

2·f(O) − f(H) can be extended on redox and non-redox systems, of any degree of complexity, i.e. the linear independ-

ency/dependency of 2·f(O) − f(H) on other balances related to a system in question is a criterion distinguishing redox

and non-redox systems. The GEB completes the set of (charge and concentration) balances and a set of expressions for

independent equilibrium constants needed for modeling the related redox system.

Keywords: Electrolytic Systems; Redox Systems; GEB; GATES

1. Introduction

Before 1992, the principle of formulation of a complete

set of balances related to a redox system was unknown;

except k concentration balances and charge balance, the k

+ 2-th balance, completing the set of equations with k + 2

variables, was needed. All the trials [1-13] made after

1960s to find the missing equation were unsuccessful.

Those trials were slavishly related to the stoichiometric

reaction notations, involving only two pairs of species

participating in redox reaction; the species different from

those involved in the reaction notation were thus omitted

in considerations. What is more, the charge balance and

concentration balances for accompanying substances

were also omitted. Theoretical considerations were re-

lated to virtual cases, not to real, electrolytic redox sys-

tems. In conclusion, all authors of those papers did not

have any idea on how to resolve correctly the problem in

question. The principle of the formulation of the k + 2-th

equation, completing k + 1 equations composed of k

concentration balances and charge balance, was devised

in 1992 by Michałowski in a totally mature form, known

presently as the Approach I to Generalized Electron

Balance (GEB). This Approach is based on a card game

principle, with electron-active elements as “players”,

electron-non-active elements as “fans”, and electrons as

“money” (cash), constituting a common pool of electrons.

The common pool of electrons is ascribed to players,

whereas the fans’ cashes are untouched. The species in

the related systems are considered in their natural form,

i.e., as hydrates in aqueous media. For the species of i-th

kind, 2, we apply the notation

Xn





iii

Nn ,

where ni ≥ 0, Ni is a number of these entities. The Ap-

proach I was formulated by Michałowski within the Gen-

eralized Approach to Electrolytic Systems (GATES). The

GATES provides a comprehensive, compatible and con-

sistent knowledge on thermodynamics of electrolytic

redox and non-redox systems, referred to aqueous and

non-aqueous media. The formulation of GATES, with

GEB concept referred to redox systems, is denoted brief-

ly as GATES/GEB. The first works on GATES, with the

*Corresponding author.

A. M. MICHAŁOWSKA-KACZMARCZYK, T. MICHAŁOWSKI 47

Approach I to GEB, were published in 1994 [14-16], and

followed by other papers [17-23]. Later on, Michałowski

devised the Approach II to GEB.

The Approach II to GEB, presented first in [24] and

then in other issues [25-30], is based on a linear combi-

nation 2·f(O) − f(H) of the balances: f(O) for oxygen (O)

and f(H) for hydrogen (H) related to redox systems; the

balance 2·f(O) − f(H) is considered as the primary form

of GEB in redox systems







-GEB =2OHprf f .

The pr-GEB is linearly independent on charge balance

and other concentration balances in any electrolytic re-

dox system (aqueous and non-aqueous media), whereas

in any non-redox system the 2·f(O) − f(H) is dependent

on those balances. This property is thus the basis for a

distinction between redox and non-redox electrolytic sys-

tems. The Approach II is fully equivalent to the Appro-

ach I. However, the fundamental advantage of the Ap-

proach II (in context with the Approach I) to GEB is that

none prior knowledge on oxidation degrees of elements

in complex species of definite elemental composition and

charge is needed. This property of the Approach II to

GEB is advantageous/desired, inter alia, for redox sys-

tems where radical and ion-radical species are formed

[31-35]. The Approach I can be considered as a “short”

version of GEB applicable in the cases where calculation

of oxidation degrees of all elements in a system is easy.

The Approach II to GEB can be formulated for static

(batch) and dynamic (titration) systems. In any titration,

a titrant (T) is added into titrand (solution titrated, D) and

a D + T system is thus formed. It is assumed that the D +

T system and the subsystems (D and T) are closed (sub)

systems, separated from the environment by diathermal

walls; any flow of a matter between the environment and

the system and its subsystems is not possible. The dia-

thermal walls enable the related process (i.e., titration) to

be carried out under isotheral conditions; constancy of

temperature is one of the conditions assuring constancy

of the related equilibrium constants values. Such titration

is an approximation of a real titration.

The GEB is principally formulated for the systems

where none nuclear transformations (involved with





−,



+ radioactive decay, electron capture, and



radiation) of

elements occur, although its extension on such systems is

also possible. In this paper, the systems with stable iso-

topes are considered.

From a mathematical viewpoint, the GEB formulated

for any (static or dynamic) redox system is the equation

linearly independent on charge and concentration bal-

ances. The GEB is fully compatible with these balances

related to the system in question. This compatibility, re-

sulting from the “common root” principle, is directly

seen from the viewpoint of the Approach II to GEB.

In compliance with the thesis put in the title of this

paper, we refer first to the concept of linear dependency

or independency of linear algebraic equations, known

from elementary algebra [36].

2. Linear Dependency of Balances

In algebra, a system of linear equations is a collection of

two or more linear equations, involving the same set of

variables in all the equations. The set of m linear equa-

tions with k variables represented by the vector





xxx

has the form (T—transposition sign)

11 112211

21 122222

112 2

mm mnn

axax b

axaxax b

axa xaxb









m



where aij are numerical coefficients; some of the aij

values can be equal zero. To facilitate further reasoning,

let us consider, for simplicity, the system of m = 3

equations with n = 2 variables,



xx, i.e.

a11x1 + a12x2 = b1 (1)

a21x1 + a22x2 = b2 (2)

a31x1 + a32x2 = b3 (3)

where (1) and (2) are independent equations, i.e.

D = a11a22 − a12a21

is ≠ 0. For example, let us take the equations:

x1 + x2 = 3 (a11 = a12 = 1) (1a)

x1 − x2 = 1 (a21 = 1, a22 = −1) (2a)

x1 − 7x2 = −5 (3a)

On the basis of (1a), (2a), from (4) we get





D1111 20



 ,

and then x1 = 2, x2 = 1. Multiplication of (1a) by −3, (2a)

by 4 and addition of the sides of the resulting equations

gives (3a), i.e., (3a) is a linear combination of (1a) and

(2a). Then (3a) is not an independent equation in this

collection of equations, i.e., it depends linearly on (1a)

and (2a).

In this paper, we apply more convenient (shorter) pro-

cedure to check the dependency of linear equations. This

procedure is based on obtaining the identity, equivalent

to the statement that 0 = 0. For this purpose, we multiply

(1a) by 3, (2a) by −4 and add the related sum to (3a)

3x1 + 3x2 = 9 (1b)

−4x1 + 4x2 = −4 (2b)

x1 − 7x2 = −5 (3b)

Adding the Equations: (1b), (2b), (3b), we get the

identity, 0 = 0 (see also Appendix).

On this simple principle, we prove the linear de-

A. M. MICHAŁOWSKA-KACZMARCZYK, T. MICHAŁOWSKI

pendency of elementary and core () balances relating to

non-redox electrolytic systems, of different degree of

complexity. On this basis, we put forward a thesis that it

is a property of general nature, applicable to non-redox

electrolytic systems of any degree of complexity.

2.1. A Non-Redox System (System A)

V mL of T containing C mol/L NaOH + C2 mol/L CO2 is

added into V0 mL of D containing H2C2O4 (C0) + H2SO4

(C0z1) + C02 mol/L CO2. V0 mL of D is composed of N10

molecules of H2C2O4·2H2O + N20 molecules of CO2 +

N30 molecules of H2SO4 + N40 molecules of H2O and V

mL of T is composed of N50 molecules of NaOH + N60

molecules of CO2 + N70 molecules of H2O. The follow-

ing species:

H2O (N1); H+ (N2, n2), OH− (N3, n3), 4 (N4, n4),

(N5, n5); Na+ (N6, n6), H2C2O4 (N7, n7),

HSO

SO 

HC O



(N8, n8), (N9, n9); H2CO3 (N11, n11), 3

COHCO



(N12, n12), (N13, n13) are formed in the system in

question. Note, for example, that N10 molecules of

H2C2O4·2H2O introduce 6N10 atoms of H, 6N10 atoms of

O and 2N10 atoms of C; N8 ions of 24

·n8H2O con-

tain N8 (1 + 2n8) atoms of H, N8 (4 + n8) atoms of O and

2N8 atoms of C. On this basis, one can formulate the ba-

lances:

CO 

HC O

f(H)





12 23344

5566 7788

99111112121313

103040 5070

2NN 1 2nN 1 2nN 1 2n

2Nn2NnN22nN12n

2N nN22nN12n2Nn

6N 2N 2N N 2N



 

(4)

f(O)















12233 4455

66 778899

111112121313

10 20 304050 6070

NnN1nN4nN 4 n

NnN4nN4nN4n

N3n N3nN3n

6N 2N4NNN2NN



  



(5)

2f(O) − f(H)

23 4 5 7 89

1213102030 5060

– NN7N8N6N7N8N4N

5N 6N6N 4N6NN4N

  

   

4N20 + 4N60 =

N50 = N6 (11)

(6)

Addition of (6) to charge balance (7), 6·f(SO4) (8),

6·f(C2O4) (9), 4·f(CO3) (10) and f(Na) (11)

N2 − N3 − N4 − 2N5 + N6 − N8 − 2N9 − N12 − 2N13 = 0

(7)

6N30 = 6N4 + 6N5 (8)

6N10 = 6N7 + 6N8 + 6N9 (9)

4N11 + 4N12 + 4N13 (10)

gives the identity, 0 = 0.

2.2. A Redox System (System B)

V mL of T containing C mol/L KMnO4 + C2 mol/L CO2

is added into V0 mL of D containing H2C2O4 (C0) +

H2SO4 (C01) + C02 mol/L CO2. V0 mL of D is composed

of N10 molecules of H2C2O4·2H2O + N20 molecules of

CO2 + N30 molecules of H2SO4 + N40 molecules of H2O

and V mL of T is composed of N50 molecules of KMnO4

+ N6 molecules o

0f CO2 + N70 molecules of H2O. The

llowing species:

H2O (N1); H+ (N2, n2), OH- (N3, n3), 4

HSO



(

N, n ),

4 4



(N5K+ (N6, n6), H2C2O4 (N7, n7), , n);

524

HC O



(N8, n8), 24



(N9, n9); H2CO3 (N11, n11), 3

HCO



(N12, n12), 2



(N13, n13); 4

MnO (N14, n14), 2

MnO



(N15, n15), Mn3+ (N16, n16), MnOH2+ (N17, n17), Mn2+ (N18,

n18), MnOH+ (N19, n19), MnSO4 (N21, n21) are formed in

the system in question. Before addition of the first por-

tion of the titrant (T), the solution is heated up to ca. 80˚C;

it is a metastable system at room temperature [26,28].

This system will be considered from the viewpoints of

Approaches I and II to GEB. In the Approach I, Mn and

C are considered as players whose electrons are balanced.

The players are involved with fans in hydrates or other

·n

)N21 electrons. Then the following bal-

ance is valid

complexes.

2.2.1. Approach I to GEB

Denoting atomic numbers of Mn and C by ZMn = 25 and

ZC = 6, we state that N10 molecules of KMnO4 introduce

N50(ZMn − 7) electrons, whereas the carbon components

introduce 2(ZC − 3)N10 + (ZC − 4)N20 + (ZC − 4)N60 elec-

trons to the common pool. In the resulting D + T system

(solution), N7 entities H2C2O4H2O involve 2 (ZC − 3)

N7 electrons, N8 entities 24

HC O8H2O involve 2 (ZC −

3) N8 electrons... N13 entities 3

CO ·n8H2O involve (ZC

− 4) N13 electrons, N14 entities 4

MnO·n14H2O involve

(ZMn – 7)N14 electrons, ... N21 entities MnSO4·n21H2O

involve (ZMn – 2

2







 





 

 

C789C11121

Mn14Mn15Mn16 17

18 192110

20 6050

2Z3 NNNZ4NNN

Z7Z6 Z3NN

2N N N23N

4N N7N

Mn C

CMn

 

  

 



(12)

Applying the relations:

The term “core”, when referred to the species in electrolytic systems,

means a common group of elements of the same composition and

structure; e.g., C2O4 is a common core for different oxalate clusters:

H2C2O4, and ; SO4 is a common core for different

sulfate clusters: , and MnSO4.

HC O2

CO

2

SO 

HSO

A. M. MICHAŁOWSKA-KACZMARCZYK, T. MICHAŁOWSKI 49



Zi 33

i0iA00 10A

50 A01020 A

160A

XVV10NN, CV10NN,

CV10NN,CV10NN,

CV 10 NN

  



 



(13)

in (12), we get the equation















 







 









C22424 24

C2333

Mn4 Mn4

Mn 4

C00C 0202

Mn 0

2Z3 HCOHCOCO

Z4H COHCOCO

Z7MnOZ6MnO

Z3MnMnOH

Z2 MnMnOHMnSO

2Z3CVZ4C VCV

Z7CVVV

















 

 

 

 

 

 

 

 

 

 

 





(14)

2.2.2. Approach II to GEB

We formulate the balances:

f(H)







12 2334 4

5566 7788

99111112121313

14 141515161617171818

1919212110304070

2NN 1 2nN 1 2nN 1 2n

2N n2NnN22nN12n

2N nN22nN12n2Nn

2N n2N n2N nN12n2N n

12n2N n6N2N2N2N



 



  

 

(15)

f(O)













 



1223 34455

66 778899

1111 12121313

1414151516 161717

18 1819192121

10 20 3040 50 6070

NNn N1nN4nN4n

NnN4 nN 4 nN4 n

N3n N3nN3n

+N4 nN4 nNnN1n

NnN1 nN4n

6N2N 4N N 4N2N N



  

 







(16)

 

2345789

12131415 17 1921

1020 30 5060

2OH r-GEB

–

N7N8N6N7N8N4N

5N 6N 8N8NNN 8N

6N 4N6N 8N4N

ff p

  

  



(17)

Addition of (17) to charge balance (18), 6·f(SO4) (19)

and f(K) (20)

234 568 912

141516171819

N N2N N N2NN2N

N2N3N2N2NN 0

 

 (18)

6N30 = 6N4 + 6N5 + 6N21 (19)

N50 = N6 (20)

gives













789 111213

141516 17181921

1020 6050

6N N N4NNN

7N6N3NN2NNN

6N4 NN7N

 

 

 



(21)

Subtraction of (21) from ZMn·f(Mn) + ZC·f(C), i.e.









C789 1112 13Mn14

Mn15Mn 16 17Mn 18 1921

C10C2060Mn 50

Z2N2N2NN NNZ N

ZN ZNNZNNN

Z2N ZNNZN



 





gives







 







 



C789C1112

Mn14Mn15Mn16 17

Mn181921C10

C2060Mn50

Z32N2N2NZ4NNN

Z7N Z6N Z3NN

Z2NNN Z32N

Z4N NZ7N



 

 

 

(22)

Introducing the relations (13) to (22), we get the equa-

tion identical with (14); it proves the equivalency of the

Approaches I and II to GEB. Equation (14) is considered

as GEB for the related system. However, the equation











 

4422

24242 33

344

4000202

01 00

HOH7HSO8SO6HCO

7HCO8CO4HCO 5HCO

6CO8MnO 8MnOMnOH

MnOH8MnSO6CV 4CVCV

6CV8CVVV



 

 





 



 



 



 





 



 

(23)

obtained after application of the relations (13) and

01 030A

CV10NN in (17) and any other combination

of (17) with other (charge and/or elemental/core) bal-

ances provides an equivalent form of GEB. For example,

subtraction of (ZC − 4)·f(C) + (ZMn − 2)·f(Mn) from (12)

and application of (13) gives the simplest (in terms of the

number of components) form of GEB











22424 24

23 2

00 0

2HCO HCOCO

5 MnO4 MnOMnMnOH

2C V5CVVV















 





 

 

(24)

which is one of the equivalent forms of GEB, referring to

the system in question. As we see, application to the re-

dox system of the same procedure, i.e., linear combina-

tion of the corresponding set of equations, does not led to

the identity, but a new balance, independent from charge

and concentration balances, is obtained.

A. M. MICHAŁOWSKA-KACZMARCZYK, T. MICHAŁOWSKI

3. Completing the Set of Balances

s completed







(25)

and concentration balances:

The GEB, expressed e.g. by Equation (24), i

by charge (see Equation (18))

HOHHSO



 

 

24 2433

2SO K

HCO2 COHCO2 CO

MnO2 MnO+3Mn+2 MnOH

+2 MnMnOH0









 



 



 



 



 

 



 

f(C)











22424242 3

33 0002020

2HCO2 HCO2COHCO

HCOCO2CV+CV+CVV+V















(26)

f(Mn)





23 2

MnO Mn MnOH Mn

MnOHMnSOCV VV

 



 



 







(27)

f(S) (or f(SO4))

MnO 











44 40100

HSO SO







 MnSOC VVV





 

 (28)

The relation





KCVVV







 (see Eq

(2ered as a concentratio

d that the species with different

4. Completing the Set of Equilibrium

Th, charge and concentration) balances is

uation

0)) is not considn balance; at de-

fined C, V and V0 values, it enters as a number in the

charge balance (25).

It should be notice

res, namely C2O4 and CO3, are involved in the same

balance (26). It results from the fact that oxalate species

are transformed in the System B into carbonate species,

i.e., a change in the core occurs. Such a transformation

does not occur in the System A, where the balances for

oxalates and carbonates can be written as separate equa-

tions.

Constants

e set of (GEB

completed by interrelations between concentrations of

the species entering these balances. These interrelations

are expressed by a set of expressions for independent

equilibrium constants. These equilibrium constants, ob-

tained from tables of equilibrium constants [37-39] val-

ues, are involved with non-redox:



H14

OH 10

 







HOH+OHpK14 ,





and redox reactions:

where:





22424 1

pH 1.25

24 224

HCO=HHCO pK1.25

HC O10H C O















2424 2

2pH4.27

24 24

HC O=HCOpK4.27

CO 10HCO

 

 



 



 





233 1

pH 6.3

HCOH +HCOpK6.3

HCO=10H CO







 





332

2pH10.1

HCOHCO pK 10.1

CO 10HCO

 

 

 

 



 



442

2pH1.8

HSOHSOpK 1.8

SO 10HSO



 

 

 



 



32OH

2pH+0.23

MnOHMnOHlog K14.2

MnOH 10Mn

 



 

 



 



pH 10.62

MnOHMnOHlog K3.4

MnOH 10Mn

 

 

 



 





441

2.28 22

MnSOMnSO logK2.28

MnSO 10MnSO



 









5AE 1.5078pH

MnO8H5eMn4HOE1.507 V

 

 



 







MnO Mn





440

E 0.56

MnOeMnOE0.56 V

MnOMnO 10

 





 

 



 



E 1.509

32A

MneMnE1.509 V

MnMn 10

 





 















E0.396pH

23224 20

0.5

23 224

2H CO +2H+2e=HCO+2H OE0.386

HCOHCO 10





HlogH,



AFRTln10











, F = Fara-

day constant, T = temperature (K); A

16.92able (qualitative and quantita-

tive) kned in these balances and equi-

species, e.g.

nstant, R = gas co

at 298 K. All attain

owledge is involv

ants. Othe

librium constr



n SOi





complexes, are unknown in literature, see [17,28]. The

requirement involved with independent set of equilib-

rium constants prevents receiving of contradictory equa-

tions.

It is assumed that the solution in the redox system is

sufficiently acidified (H2SO 4) to prevent formation of

MnO2, i.e. , the system is homogeneous (solution) during

the titration, and thus MnO is not involved in the bal-

ce (27).

A. M. MICHAŁOWSKA-KACZMARCZYK, T. MICHAŁOWSKI 51

5. Prevention of Contradictory Equations

As were stated above, GEB is fully compatible with

charge and concentration balances. The balances should

be arranged rightly; it means that all known species fr

the system in question should be included properly a

ue of a defined equilibrium con-

consequently in the related balances. All more complex

species should be involved in expressions for the related

equilibrium constants.

Tables of the equilibrium constants usually contain re-

dundant and incompatible equilibrium constants values.

The equilibrium constants were determined experimen-

tally, then the quoted val

ant may be different in different tables. For example,

the pK1 value found in for acetic acid is 4.65 [37] (at I =

0.1 mol/L, without indication of temperature) and 4.76

[38] (at 25˚C, without indication of ionic strength, I).

However, some hidden divergence in the equilibrium

data is usually encountered in the same book. For exam-

ple, in [38] we have the following values of standard

potentials: E01 = 0.907 V for 22

linear comb

f(H) for H and f(O) for O were

redox systems (aqueous media),

System B, respectively. It was

t independent

cording to Approach II.

x sys-

tems.

stems, where complex organic species are con-

s (e.g. binary-

referring to

nd concentration balances,

environment. The calcula-

tio

e Equivalence Point Potential in Oxidation-Reduc-

Journal of Chemical Education, Vol. 37,

No. 7, 1960, pp. 364-366.

http://dx.doi.org/10.1021/ed037p364

Hg2e Hg

 

 , E02 =

0.850 V for Hg2+ + 2e− = Hg, and E03 = 0.792 V for

Hg2e 2Hg



 . Then we get E03’ = 2E02 − E

01 =

0.793  E

03. Far greater discrepancies between equilib-

rium data at the apparent abhysico-

chemical data may be quoted profusely. Referring to the

in this paper, one can find E01 = 1.69

V for 4

MnO + 4H+ + 3e− = MnO2 + 2H2O, E02 = 2.26

V for 2

MnO  + 4H+ + 2e− = MnO2 + 2H2O, and E03 =

0.56 V for 2

MnO eMnO

 

 ; we state that E03’ = E02

− E01 = 0.57  E03.

6. Final Comments

The properties of the ination 2·f(O) − f(H) of

elemental balances:

undance of the p

systems presented

referred to non-redox and

denoted as System A and

assumed that the composition of titrand (D) is the same,

but the titrants applied were different: NaOH in the

System A, and KMnO4 in the System B.

In the System A (non-redox system), it was stated that

2·f(O) − f(H) is the linear combination of charge and

other elemental or core balances: f(Na), f(SO4), f(C2O4)

and f(CO3), i. e . 2·f(O) − f(H) are no

lances in this system.

In the System B (redox system) 2·f(O) − f(H) is inde-

pendent on those balances. For redox systems, 2·f(O) −

f(H) is the primary form of Generalized Electron Bal-

ance (GEB), obtained ac

These properties are valid for non-redox and redox

systems of any degree of complexity. The linear depend-

ency/independency property of 2·f(O) − f(H) is then a

new criterion distinguishing non-redox and redo

The Approach II is equivalent to the Approach I; the

latter is based on the common pool of the electron prin-

ciple. However, the Approach II offers special advan-

tages, of capital importance, particularly when referred to

redox sy

dered. Namely, the knowledge of oxidation numbers of

all elements and structure of the elements in these species

is not needed. A known composition of all the species,

expressed by their formula and external charge, provides

information sufficient to formulate the related GEB. In

particular, the Approach II is convenient in formulation

of GEB for the systems where radicals and ion-radicals

are formed. The Approach I to GEB, considered as a

“short” version of GEB, is applicable for the systems

where oxidation degree can easily be calculated for all

elements in the redox system in question.

The properties of 2·f(O) − f(H) can be extended on

non-redox or redox systems of any degree of complexity.

These properties can also be extended on mixed-solvent

systems, where polar protic (e.g. H2O, CH3OH) or apro-

tic (e.g. (CH3)2SO) solvents are involved.

The terms: “oxidant” and “reductant” are not neces-

sary (not applicable) in considerations on a redox system

[24-28]. Different species are considered in their natural

form, i.e. as solvates, e.g. hydrates in aqueous media;

although the systems with mixed solvent

lvent systems) were also considered [40].

The Approach II shows that GEB is based on reliable

principles of the matter conservation, and—in this re-

gard—it is equally robust as equations for charge and

concentration balances. From this viewpoint, GEB is

considered as a relatively new law of Nature,

ectrolytic redox systems of any degree of complexity,

namely equilibrium, metastable, non-equilibrium, mono-

and poly-phase systems [28].

GEB completes the set of charge and concentration

balances and a complete set of independent expressions

for equilibrium constants, needed for quantitative de-

scription of redox system.

In summary, GEB, charge a

gether with the set of independent equilibrium con-

stants, provide the numerical algorithm, implemented to

software packages that support advanced programming,

such as MATLAB computing

n procedure, based on iterative computer programs

[28], enables the desired relationships to be plotted gra-

phically.

REFERENCES

[1] A. J. Bard and S. H. Simonsen, “The General Equation

for th

tion Titrations,”

A. M. MICHAŁOWSKA-KACZMARCZYK, T. MICHAŁOWSKI

[2] E. Bishop, “Srations in Analyti-

cal Chemistry.alculation of Data

ome Theoretical Conside

Part VI. The Precise C

for Redox Titration Curves,” Analytica Chimica Acta,

Vol. 26, 1962, pp. 397-405.

http://dx.doi.org/10.1016/S0003-2670(00)88405-6

[3] J. A. Goldman, “The Equivalence Po

dox Titrations,” Analytica Chimica Ac

int Potential in Re-

ta, Vol. 33, 1965, pp.

217-218.

http://dx.doi.org/10.1016/S0003-2670(01)84877-7

[4] J. A. Goldman, “A General Equation for the Description

of Redox Titration Curves,” Journal of Electroanalytical

Chemistry, Vol. 11, No. 4, 1966, pp. 255-261.

http://dx.doi.org/10.1016/0022-0728(66)80090-6

[5] J. A. Goldman, “Further Considerations on Redox Titra-

tion Equations,” Journal of Electroanalytical Chemistry

Vol. 11, No. 6, 1966, pp. 416-424.

http://dx.doi.org/10.1016/0022-0728(66)80010-4

[6] J. A. Goldman, “The Locations of Inflection Points on Ti-

tration Curves for Symmetrical Redox Reactions

nal of Electroanalytical Chemistry, Vol. 14, No. 4, 1967,

,” Jour-

pp. 373-383.

http://dx.doi.org/10.1016/0022-0728(67)80018-4

[7] J. A. Goldman, “Redox Equilibria. V. The Locations of

Inflection Points on Titration Curves for Homo

geneous

Reactions,” Journal of Electroanalytical Chemistry, Vol.

18, No. 1-2, 1968, pp. 41-45.

http://dx.doi.org/10.1016/S0022-0728(68)80158-5

[8] J. A. Goldman, “Redox Equilibria. Part VI. General Titra-

tion Curve Equation for Homogeneous and Sym

metrical

Redox Reactions,” Journal of Electroanalytical Chemis-

try, Vol. 19, No. 3, 1968, pp. 205-214.

http://dx.doi.org/10.1016/S0022-0728(68)80119-6

[9] A. Meretoja, O. Lukkari and E. Hakoila, “Redox

tions—II. Location of Inflection Points on Titratio

Titra-

n Cur-

ves for Homogeneous Redox Reactions,” Talanta, Vol.

25, No. 10, 1978, pp. 557-562.

http://dx.doi.org/10.1016/0039-9140(78)80146-5

[10] J. Stur, M. Bos and W. E. van der Linden, “A Generalized

Approach for the Calculation and Automation of Pot

en-

tiometric Titrations Part 2. Redox Titrations,” Analytica

Chimica Acta, Vol. 158, 1984, pp. 125-129.

http://dx.doi.org/10.1016/S0003-2670(00)84819-9

[11] R. de Levie, “A Simple Expression for the Redox Titr

tion Curve,” Journal of Electroanalytical Chemistry

, Vol.

323, No. 1-2, 1992, pp. 347-355.

http://dx.doi.org/10.1016/0022-0728(92)80022-V

[12] R. de Levie, “Advanced Excel for Scientific Data Analy-

sis,” 2nd Edition, Oxford University Press, New

2008.

York

l E for

[13] G. Raj, “Advanced Physical Chemistry,” 35th Edition,

GOEL Publ. House, 2009.

[14] T. Michałowski, “Calculation of pH and Potentia

Bromine Aqueous Solutions,” Journal of Chemical Edu-

cation, Vol. 71, No. 7, 1994, pp. 560-562.

http://dx.doi.org/10.1021/ed071p560

[15] T. Michałowski and A. Lesiak, “Acid-Base Titration Cur-

ves in Disproportionating Redox Systems,” Journal of

Chemical Education, Vol. 71, No. 8, 1994, pp. 632-636.

http://dx.doi.org/10.1021/ed071p632

[16] T. Michałowski and A. Lesiak, “Formulation of General-

ized Equations for Redox Titration Curves,”

lityczna, Vol. 39, No. 4, 1994, pp. 623

Chemia Ana-

-637.

nalytical Chemistry

d J. Kochana,

-293.

[17] T. Michałowski, N. Wajda and D. Janecki, “A Unified

Quantitative Approach to Electrolytic Systems,” Chemia

Analityczna, Vol. 41, No. 4, 1996, pp. 667-685.

[18] T. Michałowski, “Calculations in A

with Elements of Computer Programming,” 2001.

http://suw.biblos.pk.edu.pl/resourceDetails&rId=3974

[19] T. Michałowski, A. Baterowicz, A. Madej an

“An Extended Gran Method and Its Applicability for Si-

multaneous Determination of Fe(II) and Fe(III),” Analy-

tica Chimica Acta, Vol. 442, No. 2, 2001, pp. 287

http://dx.doi.org/10.1016/S0003-2670(01)01172-2

[20] T. Michałowski, M. Toporek and M. Rymanowski, “

view on the Gran and Other Linearization Methods A

Over-

plied in Titrimetric Analyses,” Talanta, Vol. 65, No. 5,

2005, pp. 1241-1253.

http://dx.doi.org/10.1016/j.talanta.2004.08.053

[21] T. Michałowski, M. Rymanowski and A. Pietrzyk, “Non

typical Brønsted Acids and Bases,” Journal of Ch

emical

Education, Vol. 82, No. 3, 2005, pp. 470-472.

http://dx.doi.org/10.1021/ed082p470

[22] T. Michałowski, K. Kupiec and M. Rymanowski, “Nu-

merical Analysis of the

Study,” Analytica Chimica Acta, Vol. 606, No

Gran Methods: A Comparative

. 2, 2008,

pp. 172-183. http://dx.doi.org/10.1016/j.aca.2007.11.020

[23] M. Ponikvar, T. Michałowski, K. Kupiec, S. Wybraniec

and M. Rymanowski, “Experimental Verificati

Modified Gran Methods Applicable t

on of the

o Redox Systems,”

Analytica Chimica Acta, Vol. 628, No. 2, 2008, pp. 181-

189. http://dx.doi.org/10.1016/j.aca.2008.09.012

[24] T. Michałowski and A. Pietrzyk, “Complementarity of

Physical and Chemical Laws of Conservation in Aspect

pli-

of Electrolytic Systems (in Polish),” Wiadomości Chemic-

zne, Vol. 61, No. 7-8, 2007, pp. 625-640.

[25] http://www.chemia.uj.edu.pl/~ictchem/book.html

[26] T. Michałowski, “The Generalized Approach to Electro-

lytic Systems: I. Physicochemical and Analytical Im

cations,” Critical Reviews in Analytical Chemistry, Vol.

40, No. 1, 2010, pp. 2-16.

http://dx.doi.org/10.1080/10408340903001292

[27] T. Michałowski, A. Pietrzyk, M. Ponikvar-Svet and M.

hemis- Rymanowski, “Critical Reviews in Analytical C

try,” Vol. 40, No. 1, 2010, pp. 17-29.

http://dx.doi.org/10.1080/10408340903001292

[28] T. Michałowski, “Application of GATES and MATLAB

for Resolution of Equilibriu

librium Electrolytic Systems,” Chapter 1, In: T. M

m, Metastable and Non-Equi-

icha-

łowski, Ed., Applications of MATLAB in Science and En-

gineering, InTech, 2011, pp. 1-35.

http://www.intechopen.com/books/show/title/applications

-of-matlab-in-science-and-engineering

[29] T. Michałowski, M. Ponikvar-Svet, A. G. Asuero and K.

Kupiec, “Thermodynamic and Kinetic Effects Involved in

the pH Titration of As(III) with Iodine in a Buffered

Malonate System,” Journal of Solution Chemistry, Vol.

41, No. 3, 2012, pp. 436-446.

http://dx.doi.org/10.1007/s10953-012-9815-6

A. M. MICHAŁOWSKA-KACZMARCZYK, T. MICHAŁOWSKI

nds in Studies on

Red Beet

Chemistry

ects of Metal Cations on Be

Organic Solutions,” Food Science and

tanin Stability in Aqueous-

Biotechnology, Vol.

[30] T. Michałowski, M. Toporek, A. M. Michałowska-Kacz-

marczyk and A. G. Asuero, “New Tre

Electrolytic Redox Systems,” Electrochimica Acta, Vol.

109, 2013, pp. 519-531.

[31] B. Nemzer, Z. Pietrzkowski, A. Spórna, P. Stalica, W.

Thresher, T. Michałowski and S. Wybraniec, “Betalainic

and Nutritional Profiles of Pigment-Enriched

Root (Beta vulgaris L.) Dried Extracts,” Food

22, No. 2, 2013, pp. 353-363.

http://dx.doi.org/10.1007/s10068-013-0088-7

[35] S. Wybraniec, K. Starzak, A. Skopińska, B. Nemzer, Z.

Pietrzkowski and T. Michalowski, “Studies on Non-En-

zymatic Oxidation Mechanism

and Decarboxylated Betanins, Journal of Agricu

Vol. 127, No. 1, 2011, pp. 42-53.

http://dx.doi.org/10.1016/j.foodchem.2010.12.081

[32] S. Wybraniec and T. Michałowski, “New Pathways of

Betanidin and Betanin Enzymatic Oxidation,” Journal of

Agricultural and Food Chemistry, Vol. 59, No. 17, 2011,

pp. 9612-9622. http://dx.doi.org/10.1021/jf2020107

[33] S. Wybraniec, P. Stalica, A. Spórna, B. Nemzer, Z. Pie-

trzkowski and T. Michałowski, “Antioxidant Activity of

Betanidin: Electrochemical Study in Aqueous

Journal of Agricultural and Food Chemistry, Vol. 5

Media,”

9, No.

22, 2011, pp. 12163-12170.

http://dx.doi.org/10.1021/jf2024769

[34] S. Wybraniec, K. Starzak, A. Skopińska, M. Szaleniec, J.

Słupski, K. Mitka, P. Kowalski and T. Michałowski, “Eff-

in Neobetanin, Betanin

ltural and

of General Criterion Distin-

ppendix (a More Sophisticated Presentation of the Linear Dependency)

Let us take the set of linear equations:

ax + ax + ax = b

co quations, i.e.,

Food Chemistry, Vol. 61, No. 26, 2013, pp. 6465–6476.

[36] http://en.wikipedia.org/wiki/System_of_linear_equations

[37] J. Inczedy, “Analytical Applications of Complex Equilib-

ria,” Horwood, Chichester, 1976.

[38] Yu. Lurie, “Handbook of Analytical Chemistry,” Mir

Publishers, Moscow, 1975.

[39] B. P. Nikolsky, “Guide-Book for Chemist (in Russian),”

Vol. 3, Khimia, Moscow, 1964.

[40] T. Michałowski, A. M. Michałowska-Kaczmarczyk and

M. Toporek, “Formulation

guishing between Non-Redox and Redox Systems,” Elec-

trochimica Acta.

11 112 213 31

a21x1 + a22x2 + a23x3 = b2

mpleted by linear combination of these e

 













111 112213322112222322221 1322331122

caxax axcaxax acaxcacaxcbcb

31 1122111 12

xcac axca .

Applying matrix algebra we see that the determnt ina

11 1213

21 2223

1112 21

ca c a

1122 221132 23

aaa

cac acac a





has zero value

1112 131112 13

12122 23 221222312

11 121321 2223

aaaaaa

caaacaaacc

aaa aaa

 D 0

irrespectively on the c1 and c2 values; at D = 0, calculation of x1, x2 and x3 is impossible. On the other hand, even a small

change in the third equation makes a system of equations contradictory. For example, tang the system of equations:

(1a), (2a) (see text) and (



) x1 − 7x2 = −5.01, (β) 1.01x1 − 7x2 = −5, (



) x1 − 7.01x2 = −5 instead of (3a) we get the con-

tradictions: −5  −5.01 in (



) and (



), and −4.98  −5 in (β).