American Journal of Analytical Chemistry
Vol.05 No.17(2014), Article ID:52615,4 pages
10.4236/ajac.2014.517134

Linear Dependence of Balances for Non-Redox Electrolytic Systems

Anna M. Michałowska-Kaczmarczyk1, Tadeusz Michałowski2*

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

2Faculty of Engineering and Chemical Technology, Technical University of Cracow, Cracow, Poland

Email: *michalot@o2.pl

Copyright © 2014 by authors and Scientific Research Publishing Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY).

http://creativecommons.org/licenses/by/4.0/

Received 12 October 2014; revised 29 November 2014; accepted 11 December 2014

ABSTRACT

Two complex dynamic non-redox systems are considered as examples, providing interdependent linear equations. A simple and efficient linear combination method that leads the system of equations to the identity, 0 = 0, is used for this purpose. These examples are clear confirmations of the general property differentiating non-redox and redox electrolytic systems. This property is involved with linear dependence or independence of on charge and elemental/ core balances for elements/cores ¹ H, O, where and are elemental balances for H and O, respectively.

Keywords:

Titration, Liebig-Denigès Method, Complexonometric Detemination of Zinc

1. Introduction

In numerous examples of electrolytic redox systems presented in our previous papers [1] -[4] , it was found that the linear combination of elemental balances: for H and for O jest is linearly independent on charge and elemental/core balances for elements/cores different from H and O. This linear independence proves that is a new equation, considered as the starting form of the electron balance (GEB) related to the redox system in question. In this way it has been shown, inter alia, that the simplest form of GEB related to a redox system in mixed-solvent media is the same, regardless the solvent composition―assuming that the solutes do not form with solvents other species (except those known from aqueous media) besides solvates [5] -[7] .

For non-redox systems, it was stated that the linear combination is linearly dependent on charge and elemental/core balances for elements/cores ≠ H, O. This can easily be stated by transformation of the linear combination of the relevant equations to the identity, 0 = 0. In this work, this kind of transformation will be applied to two relatively complex systems involved with 1) determination of cyanide according to Liebig- Denigès method [8] and 2) complexometric titration of zinc with EDTA [9] .

2. Liebig-Denigès Method of Cyanide Determination

In the system involved with Liebig-Denigès method, V mL of the solution composed of AgNO3 is added into mL of the solution composed of . At defined stage of the process, the following species are present in the system:

(N1), (N2), (N3, n3), (N4, n4), (N5, n5), (N6, n6), (N7, n7), (N8, n8), (N9, n9), NH3 (N11, n11), Ag+ (N12, n12), AgOH (N13, n13), (N14, n14), (N15, n15), (N16, n16), (N17, n17), (N18, n18), AgI (N19, n19), (N21, n21), (N22, n22), (N23, n23), (N24, n24), (N25, n25), (N26, n26), (N27, n27).

From the balances:

(1)

(2)

we have

(3)

Addition of (3) to charge balance (4) and balances: f(Na) (5), f(K) (6), 5 f(NO3) (7), f(CN) (8), 3f(NH3) (9), f(Ag) (10) and f(I) (11)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

gives the identity, 0 = 0, i.e., (3) is not an independent balance in the system. The identity is also valid before crossing the solubility product for AgI(s).

3. Complexometric Titration of Zinc with EDTA

Let us consider mL of titrand (D), containing (erio T, C0In) titrated with V mL of titrant (T) containing EDTA (C).

The titrand is composed of N01 molecules of ZnSO4・7H2O (goslarite), N02 molecules of NH3, N03 molecules of NH4Cl, N04 molecules of NaH2In = C20H12N3O7SNa, N05 molecules of H2O and the titrant is composed of N06 molecules of EDTA = Na2H2L・2H2O = C10H14N2O8Na2・2H2O and N07 molecules of H2O, at defined point of titration (V mL of T added). In the system in question, the following species are formed:

H2O (N1), H+ (N2, n2), OH (N3, n3), (N4, n4), (N5, n5), (N6, n6), Na+ (N7, n7), (N8, n8), (N9, n9), Zn2+ (N11, n11), ZnOH+ (N12, n12), soluble complex (N13, n13), (N14, n14), (N15, n15), (N16, n16), (N17, n17), (N18, n18), (N19, n19), ZnCl+ (N21, n21); ZnSO4 (N22, n22), C20H13N3O7S (N23, n23), C20H12N3O7S (N24, n24), C20H11N3O7S2− (N25, n25), C20H10N3O7S3− (N26, n26), C20H10N3O7SZn (N27, n27), (N28, n28), (N29, n29), (N31, n31), C10H16N2O8 (N32, n32), (N33, n33), (N34, n34), (N35, n35), (N36, n36), (N37, n37), (N38, n38), (N39, n39).

The complex is formed in reaction between and. The species can be arranged in the following balances:

(12)

(13)

The balance, obtained from Equations (12) and (13), is as follows

(14)

Addition of (14) to charge balance (15) and balances for (16), (17), (18),

(19), (20), (21), and (22):

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

gives the identity, 0 = 0, that testifies about linear dependence between the balances, i.e., Equation (14) is linearly dependent on other balances in the system. Note that the Equations (19)-(22) are specified separately, according to different “cores”: SO4, NH3, C20H10N3O7S, C10H12N2O8. Note that N enters the compounds and species in Equations (20)-(22); S enters the compounds and species in Equations (19) and (21); C enters the compounds and species in Equations (21) and (22). However, none transformations occur between the “cores” of the species that belong to different concentration balances. Generalizing, for any non-redox system, there are some numbers/multipliers for the relevant equations that reduce the sum received to the identity.

Referring again to Equation (21), related to the species involved with erio T, one can write the elemental balances:

(for C),

(for N), (for S),

and

(for O).

All the equations are identical and equivalent to Equation (21), because the “core” C20H10N3O7S is unchanged in reactions occurred during the titration. Similarly, the species involved with EDTA, see Equation (22), fulfill the relations:

(for C),

and

(for N).

Both equations are equivalent to.

4. Final Comments

Checking of linear dependence or independence of algebraic equations [10] is not a mathematical problem of the highest order. However, it requires an additional knowledge of the user, concerning the properties of the matrix (;;) of coefficients in the matrix equation, where

(23)

with vectors of variables, , and vector of constant terms,. For explaining these properties, some abstract terms such as (dimension of) vector space, (matrix, kolumn, row) ranks, are used.

It should be noted, however, that the coefficients and used in purely algebraic equations

(24)

do not have specific physical or chemical connotations. Assuming the charge balance as the second of the balances (24) considered for this purpose, we state that the coefficients in this balance are involved with external charges of the species in the system in question and in the vector. The coefficients in the elemental/core balances are involved with the number of elements/cores in the related species. The coefficients in depend on the form of equations for and.

In this paper, the linear relationship between the balance and charge + elemental/core balances for elements/cores ¹ H, O was checked in extremely simple way (indicated in [4] ) and proved on examples of two electrolytic non-redox systems, of analytical importance, known from titrimetric analyses. Full complexity of these systems, known from preliminary physicochemical data, is involved in the related balances, expressed in terms of numbers of entities of particular components and species. The related balances can also be expressed in terms of molar concentrations: for the species (NA? Avogadro’s number) and analogous relationships for components forming the titrand (D) and titrant (T) in the related D + T system. It should be recalled and emphasized that a linear relationship between and charge + elemental/core balances for elements/cores ¹ H, O does not occur for redox systems, and the resulting balance is the basis of formulation of GEB for these systems, obtained according to Approach II to GEB [1] -[7] [11] .

References

  1. Michałowski, T. (2010) The Generalized Approach to Electrolytic Systems: I. Physicochemical and Analytical Impli- cations. Critical Reviews in Analytical Chemistry, 40, 2-16. http://dx.doi.org/10.1080/10408340903001292
  2. Michałowski, T., Toporek, M., Michałowska-Kaczmarczyk, A.M. and Asuero, A.G. (2013) New Trends in Studies on Electrolytic Redox Systems. Electrochimica Acta, 109, 519-531. http://www.sciencedirect.com/science/article/pii/S0013468613013947 http://dx.doi.org/10.1016/j.electacta.2013.07.125
  3. Michałowski, T., Michałowska-Kaczmarczyk, A.M. and Toporek, M. (2013) Formulation of General Criterion Distin- guishing between Non-Redox and Redox Systems. Electrochimica Acta, 112, 199-211 http://www.sciencedirect.com/science/article/pii/S0013468613016836 http://dx.doi.org/10.1016/j.electacta.2013.08.153
  4. Michałowska-Kaczmarczyk, A.M. and Michałowski, T. (2013) Comparative Balancing of Non-Redox and Redox Electro- lytic Systems and Its Consequences. American Journal of Analytical Chemistry, 4, 46-53. http://www.scirp.org/journal/PaperInformation.aspx?paperID=38569#.VGObL2dvHFw http://dx.doi.org/10.4236/ajac.2013.410A1006
  5. Michałowski, T., Pilarski, B., Asuero, A.G. and Michałowska-Kaczmarczyk, A.M. (2014) Chapter 9.4: Modeling of Acid- Base Properties in Binary-Solvent Systems. In: Wypych, G., Ed., Handbook of Solvents, Volume 1, Properties, 2nd Edi- tion, ChemTec Publishing, Toronto, 623-648. http://store.elsevier.com/Handbook-of-Solvents-Volume-1/isbn-9781895198645/
  6. Michałowska-Kaczmarczyk, A.M. and Michałowski, T. (2014) Compact Formulation of Redox Systems According to GATES/GEB Principles. Journal of Analytical Sciences, Methods and Instrumentation, 4, 39-45. http://www.scirp.org/journal/PaperInformation.aspx?PaperID=46335#.VGm1Tmfpt74
  7. Michałowska-Kaczmarczyk, A.M. and Michałowski, T. (2014) Generalized Electron Balance for Dynamic Redox Sys- tems in Mixed-Solvent Media. Journal of Analytical Sciences, Methods and Instrumentation, 4, 102-109. http://www.scirp.org/Journal/PaperInformation.aspx?PaperID=52018#.VIfpeMkdqUR
  8. Michałowski, T., Asuero, A.G., Ponikvar-Svet, M., Toporek, M., Pietrzyk, A. and Rymanowski, M. (2012) Principles of Computer Programming Applied to Simulated pH-Static Titration of Cyanide According to a MODIFIED Liebig- Denigès Method. Journal of Solution Chemistry, 41, 1224-1239. http://dx.doi.org/10.1007/s10953-012-9864-x
  9. Michałowski, T. (2001) Calculations in Analytical Chemistry with Elements of Computer Programming (in Polish), PK, Cracow. http://suw.biblos.pk.edu.pl/resourceDetails&rId=3974
  10. http://en.wikipedia.org/wiki/System_of_linear_equations
  11. Michałowski, T. (2011) Application of GATES and MATLAB for Resolution of Equilibrium, Metastable and Non- Equilibrium Electrolytic Systems, In: Michałowski, T., Ed., Applications of MATLAB in Science and Engineering, In- Tech―Open Access Publisher in the Fields of Science, Technology and Medicine, Rijeka, Chapter 1, 1-34. http://www.intechopen.com/books/show/title/applications-of-matlab-in-science-and-engineering

NOTES

*Corresponding author.