American Journal of Analytical Chemistry, 2013, 4, 4653 http://dx.doi.org/10.4236/ajac.2013.410A1006 Published Online October 2013 (http://www.scirp.org/journal/ajac) Comparative Balancing of NonRedox and Redox Electrolytic Systems and Its Consequences Anna Maria MichałowskaKaczmarczyk1, 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 MichałowskaKaczmarczyk, 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 nonredox system. These properties of 2·f(O) − f(H) can be extended on redox and nonredox 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 nonredox 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 + 2th balance, completing the set of equations with k + 2 variables, was needed. All the trials [113] 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 + 2th 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 electronactive elements as “players”, electronnonactive 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 ith kind, 2, we apply the notation HO i z ii Xn , i z 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 nonredox systems, referred to aqueous and nonaqueous 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. MICHAŁOWSKAKACZMARCZYK, T. MICHAŁOWSKI 47 Approach I to GEB, were published in 1994 [1416], and followed by other papers [1723]. Later on, Michałowski devised the Approach II to GEB. The Approach II to GEB, presented first in [24] and then in other issues [2530], 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 prGEB is linearly independent on charge balance and other concentration balances in any electrolytic re dox system (aqueous and nonaqueous media), whereas in any nonredox system the 2·f(O) − f(H) is dependent on those balances. This property is thus the basis for a distinction between redox and nonredox 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 ionradical species are formed [3135]. 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 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 , 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. MICHAŁOWSKAKACZMARCZYK, T. MICHAŁOWSKI 48 pendency of elementary and core () balances relating to nonredox 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 nonredox electrolytic systems of any degree of complexity. 2.1. A NonRedox 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), 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 ( 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. MICHAŁOWSKAKACZMARCZYK, 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 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 rGEB – 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 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. MICHAŁOWSKAKACZMARCZYK, 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 [3739] val ues, are involved with nonredox: 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 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. MICHAŁOWSKAKACZMARCZYK, 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 OxidationReduc Journal of Chemical Education, Vol. 37, No. 7, 1960, pp. 364366. 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 nonredox 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 (nonredox 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 nonredox 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 nonredox 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 ionradicals 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 nonredox or redox systems of any degree of complexity. These properties can also be extended on mixedsolvent 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 [2428]. 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, nonequilibrium, mono and polyphase 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. MICHAŁOWSKAKACZMARCZYK, 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. 397405. http://dx.doi.org/10.1016/S00032670(00)884056 [3] J. A. Goldman, “The Equivalence Po dox Titrations,” Analytica Chimica Ac int Potential in Re ta, Vol. 33, 1965, pp. 217218. http://dx.doi.org/10.1016/S00032670(01)848777 [4] J. A. Goldman, “A General Equation for the Description of Redox Titration Curves,” Journal of Electroanalytical Chemistry, Vol. 11, No. 4, 1966, pp. 255261. http://dx.doi.org/10.1016/00220728(66)800906 [5] J. A. Goldman, “Further Considerations on Redox Titra tion Equations,” Journal of Electroanalytical Chemistry Vol. 11, No. 6, 1966, pp. 416424. , http://dx.doi.org/10.1016/00220728(66)800104 [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. 373383. http://dx.doi.org/10.1016/00220728(67)800184 [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. 12, 1968, pp. 4145. http://dx.doi.org/10.1016/S00220728(68)801585 [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. 205214. http://dx.doi.org/10.1016/S00220728(68)801196 [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. 557562. http://dx.doi.org/10.1016/00399140(78)801465 [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. 125129. http://dx.doi.org/10.1016/S00032670(00)848199 [11] R. de Levie, “A Simple Expression for the Redox Titr tion Curve,” Journal of Electroanalytical Chemistry a , Vol. 323, No. 12, 1992, pp. 347355. http://dx.doi.org/10.1016/00220728(92)80022V [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. 560562. http://dx.doi.org/10.1021/ed071p560 [15] T. Michałowski and A. Lesiak, “AcidBase Titration Cur ves in Disproportionating Redox Systems,” Journal of Chemical Education, Vol. 71, No. 8, 1994, pp. 632636. 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. 667685. [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/S00032670(01)011722 [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. 12411253. 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. 470472. 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. 172183. 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. 78, 2007, pp. 625640. [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. 216. http://dx.doi.org/10.1080/10408340903001292 [27] T. Michałowski, A. Pietrzyk, M. PonikvarSvet and M. hemis Rymanowski, “Critical Reviews in Analytical C try,” Vol. 40, No. 1, 2010, pp. 1729. 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 NonEqui icha łowski, Ed., Applications of MATLAB in Science and En gineering, InTech, 2011, pp. 135. http://www.intechopen.com/books/show/title/applications ofmatlabinscienceandengineering [29] T. Michałowski, M. PonikvarSvet, 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. 436446. http://dx.doi.org/10.1007/s1095301298156 Copyright © 2013 SciRes. AJAC
A. M. MICHAŁOWSKAKACZMARCZYK, 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. MichałowskaKacz marczyk and A. G. Asuero, “New Tre Electrolytic Redox Systems,” Electrochimica Acta, Vol. 109, 2013, pp. 519531. [31] B. Nemzer, Z. Pietrzkowski, A. Spórna, P. Stalica, W. Thresher, T. Michałowski and S. Wybraniec, “Betalainic and Nutritional Profiles of PigmentEnriched Root (Beta vulgaris L.) Dried Extracts,” Food 22, No. 2, 2013, pp. 353363. http://dx.doi.org/10.1007/s1006801300887 [35] S. Wybraniec, K. Starzak, A. Skopińska, B. Nemzer, Z. Pietrzkowski and T. Michalowski, “Studies on NonEn zymatic Oxidation Mechanism and Decarboxylated Betanins, Journal of Agricu , Vol. 127, No. 1, 2011, pp. 4253. 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. 96129622. 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. 1216312170. 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, “GuideBook for Chemist (in Russian),” Vol. 3, Khimia, Moscow, 1964. [40] T. Michałowski, A. M. MichałowskaKaczmarczyk and M. Toporek, “Formulation guishing between NonRedox 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).
