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
Copyright © 2013 Anna Maria Michałowska-Kaczmarczyk, Tadeusz Michałowski. This is an open access article distributed under
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
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
HO
i
z
ii
Xn
,
i
z
iii
X
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.
C
opyright © 2013 SciRes. AJAC
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
12
,
T
n
x
xxx
ax
has the form (T—transposition sign)
11 112211
21 122222
112 2
+=
=
=
nn
nn
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,
T
12
,
x
xx, 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-
Copyright © 2013 SciRes. AJAC
A. M. MICHAŁOWSKA-KACZMARCZYK, T. MICHAŁOWSKI
48
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
2
4
SO
24
HC O
(N8, n8), (N9, n9); H2CO3 (N11, n11), 3
2
24
COHCO
(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:
2
3
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
  
   
11
4N20 + 4N60 =
N50 = N6 (11)
fo
(6)
Addition of (6) to charge balance (7), 6·f(SO4) (8),
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
(
2
4
SO
N, n ),
4 4
(N5K+ (N6, n6), H2C2O4 (N7, n7), , n);
524
HC O
(N8, n8), 24
CO
2
(N9, n9); H2CO3 (N11, n11), 3
HCO
(N12, n12), 2
3
CO
(N13, n13); 4
MnO (N14, n14), 2
4
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
·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
7
2
2
 


 
 
C789C11121
Mn14Mn15Mn16 17
18 192110
20 6050
2Z3 NNNZ4NNN
Z7Z6 Z3NN
2N N N23N
4N N7N
Mn C
CMn
NN
ZZ
ZZ
 
  
 

3
(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.
24
HC O2
24
CO
4
2
4
SO
HSO
Copyright © 2013 SciRes. AJAC
A. M. MICHAŁOWSKA-KACZMARCZYK, T. MICHAŁOWSKI 49

Zi 33
i0iA00 10A
33
50 A01020 A
3
160A
XVV10NN, CV10NN,
CV10NN,CV10NN,
CV 10 NN
  

 

(13)
in (12), we get the equation





 





 


2
C22424 24
2
C2333
2
Mn4 Mn4
32
Mn
2
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
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


  
 


11
(16)
 
2345789
12131415 17 1921
1020 30 5060
2OH r-GEB
N
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
13
N
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

 
 
 
13
(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



 
2
4422
2
24242 33
22
344
4000202
01 00
HOH7HSO8SO6HCO
7HCO8CO4HCO 5HCO
6CO8MnO 8MnOMnOH
MnOH8MnSO6CV 4CVCV
6CV8CVVV

 
 

 

 

 

 


 

 
4
2
(23)
obtained after application of the relations (13) and
3
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



2
22424 24
23 2
44
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.
Copyright © 2013 SciRes. AJAC
A. M. MICHAŁOWSKA-KACZMARCZYK, T. MICHAŁOWSKI
50
3. Completing the Set of Balances
s completed
2
(25)
and concentration balances:
The GEB, expressed e.g. by Equation (24), i
by charge (see Equation (18))
HOHHSO

 
 
2
44
22
24 2433
23
44
2
2SO K
HCO2 COHCO2 CO
MnO2 MnO+3Mn+2 MnOH
+2 MnMnOH0




 
 

 

 

 
 

 
f(C)

2
22424242 3
2
33 0002020
2HCO2 HCO2COHCO
HCOCO2CV+CV+CVV+V








(26)
f(Mn)


23 2
44
40
2
MnO Mn MnOH Mn
MnOHMnSOCV VV
 
 

 



(27)
f(S) (or f(SO4))
MnO 


2
44 40100
HSO SO

 MnSOC VVV
 
(28)
The relation
0
KCVVV


 (see Eq
(2ered as a concentratio
d that the species with different
co
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:

p
H14
2w
OH 10
 


HOH+OHpK14 ,


3
H
and redox reactions:
0
where:


22424 1
pH 1.25
24 224
HCO=HHCO pK1.25
HC O10H C O







2
2424 2
2pH4.27
24 24
HC O=HCOpK4.27
CO 10HCO
 
 

 

 


233 1
pH 6.3
32
HCOH +HCOpK6.3
HCO=10H CO






2
332
2pH10.1
33
HCOHCO pK 10.1
CO 10HCO
 
 
 
 

 

2
442
2pH1.8
44
HSOHSOpK 1.8
SO 10HSO

 
 
 

 

32OH
1
2pH+0.23
MnOHMnOHlog K14.2
MnOH 10Mn
 

 
 

 

2O
1
pH 10.62
MnOHMnOHlog K3.4
MnOH 10Mn
 
 
 
 

 


22
441
2.28 22
44
MnSOMnSO logK2.28
MnSO 10MnSO


 





2
42
5AE 1.5078pH
2
MnO8H5eMn4HOE1.507 V
10
 
 

 

4
MnO Mn



2
440
E 0.56
2A
44
MnOeMnOE0.56 V
MnOMnO 10
 


 
 

 


32
0
E 1.509
32A
MneMnE1.509 V
MnMn 10
 


 








E0.396pH
A
23224 20
0.5
23 224
2H CO +2H+2e=HCO+2H OE0.386
HCOHCO 10


p
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

32
4
n SOi
i
M
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-
an
2
ce (27).
Copyright © 2013 SciRes. AJAC
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
om
nd
ue of a defined equilibrium con-
st
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
2
linear comb
f(H) for H and f(O) for O were
redox systems (aqueous media),
System B, respectively. It was
t independent
ba
cording to Approach II.
x sys-
tems.
stems, where complex organic species are con-
si
s (e.g. binary-
so
referring to
el
nd concentration balances,
to
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
2
2
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
4
MnO + 4H+ + 2e = MnO2 + 2H2O, and E03 =
0.56 V for 2
44
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,”
Copyright © 2013 SciRes. AJAC
A. M. MICHAŁOWSKA-KACZMARCZYK, T. MICHAŁOWSKI
52
[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
a-
, 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-
p-
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
Copyright © 2013 SciRes. AJAC
A. M. MICHAŁOWSKA-KACZMARCZYK, T. MICHAŁOWSKI
Copyright © 2013 SciRes. AJAC
53
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.
A
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
aaa
cac acac a

D
has zero value
1112 131112 13
12122 23 221222312
11 121321 2223
00
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-
ki
tradictions: 5 5.01 in (
) and (
), and 4.98 5 in (β).