## Journal of the Chilean Chemical Society

##
*On-line version* ISSN 0717-9707

### J. Chil. Chem. Soc. vol.51 no.1 Concepción Mar. 2006

#### http://dx.doi.org/10.4067/S0717-97072006000100012

J. Chil. Chem. Soc., 51, Nº 1 (2006)

** POLYNOMIAL METHOD FOR CALCULATION OF ELECTROKINETIC PARAMETERS IN A CONTROLLED CHARGE-TRANSFER SYSTEM **

**M.GUZMÁN*, P.ORTEGA AND L.VERA**

Departamento de Ciencias Químicas y Farmacéuticas, Facultad de Ciencias, Universidad Católica del Norte, Casilla 1280, Antofagasta-Chile

*E-mail: mguzman@ucn.cl*

**ABSTRACT**

A method is presented for the calculation of electrokinetic parameters of a controlled electrochemical reaction employing only charge-transfer. The method includes development of a polynomial expression of the Butler-Volmer equation from which constants were obtained, which were then used to determine adequate relations for the calculation of electrokinetic parameters. This method can be used with polarization data obtained by potentiostatic or galvanostatic techniques. Calculations carried out with synthetic and experimental data showed that the calculation method was precise and uncomplicated to use. The method is limited in that it is unable to make use of the null data pair 0,0 .

**Keywords:** electrokinetic parameters, polynomial calculation, Butler-Volmer equation, Maclaurins series.

**INTRODUCTION**

The study of the mechanism of a charge-transfer reaction of an electrode can be made using the Tafel equation or the linear form of the Butler-Volmer relation at high values of overtension when the order of the reaction is known. Nevertheless, when the order of the reaction is unknown, an additional calculation must be carried out at low overpotentials. An additional complication appears when the reactions are very rapid. In this case, interferences attributable to mass transfer are produced in the Tafel region. Various algebraic and computational methods have been developed which can be used in any region of overtension which also serve for the determination of corrosion parameters, each one having its individual limitations.

One of these recently developed methods is termed the "polynomial method"^{1/2)} for calculation of corrosion parameters. Using the basic principle of this method, it is possible to evaluate relations which allow calculation of simple electrokinetic parameters for equations of charge-transfer applicable at any order level.

**METHOD**

The equation that represents the mechanism of charge transfer for any electrode reaction of the type (first order), the unitary stoichiometric number is represented by the Butler-Volmer equation of the following type:

For reactions of the type,

considered to be general for the upper order, class A^{3, 4)} and whose velocity is governed only by the charge transfer mechanism, representation is commonly given by ^{5, 7)} :

where ** i** is the current density ,

**is the current density of the interchange,**

*i*_{0}

**a****is the transfer coefficient,**

*h*overtension,

**number of electrons transferred and**

*n**n*is the stoichiometric number. The exponentials of equation (2) can be developed according to the Maclaurins series, following which, and ordering the terms, we obtain:

If we divide by *h*, the preceding equation is transformed to,

Equation (4) is a polynomial, and for a given system at a constant temperature, this becomes simplified to:

Where A, B, C...U, are constants.

Developing the respective differential, the three constants are obtained as:

Relating these last equations, we finally obtain:

**RESULTS AND DISCUSSION**

The methodology for finding the polynomial equation includes calculating the regression of ** i/n** on

**in a computational program such as Excel.**

*n*

In order to test equations (9, 10 and 11) equation (2) is used to calculate the data ** i** and

**with**

*n***=1.00 mA/cm**

*i*_{0}^{2};

**= 0.25;**

*a***=2,00;**

*n***=2.00;**

*n***=298,2 K;**

*T***=8.314 J/mol K y**

*R***=96485 C/mol, producing Table I.**

*F*

Running the data ** i/h **against

**in the Excel program, the polynomial equation obtained is with**

*h*

With the constants of this polynomial and equations (9, 10 and 11) we have:** i_{0 }**=1.00 mA/cm

^{2};

**= 0.25;**

*a**n*=2.00. These results show that equations (9, 10 and 11) are solutions for equation (2).If the polynomial equation were of an order higher than 4, the calculation would be precise, with more significant figures.

In order to test the method with real data we selected the Fe^{2+/}Fe^{3+} system, which has been well studied using various concentrations, media, and methods^{8-23)}.Data obtained from the literature^{22)} were used with equimolar Fe^{2+/}Fe^{3+ }(0.08 M) in NaHSO_{4} 0.912 M between a bright platinum electrode and a platinum-platinum plated reference electrode at 25 ºC, using the pulsed potentiostatic technique, Table II.

Ordering the ** i/h **data

**against**

*h*, Figure 1 is obtained,

Using the Excel program, the following polynomial equation is obtained:

**Figure 1.** Plot of ** i/h**_ against

**_ for the Fe**

*h*^{2+}/Fe

^{3+ }system in NaHSO

_{4}at 25ºC

Using the constants from this equation and the relations from (9, 10 and 11) we have:

The Fe^{2+}/Fe^{3+} first order electrochemical system (*n *=1), (Barnartt

^{23)}for the same system under the same conditions, shows, using the polarization technique and two-point methods, that the average of the current density was 35,8 mA/cm

^{2}and the transfer coefficient was 0.48, also showing that the concentration range studied was 0.020-0.096 M with

*a*=0.50; other authors

^{22)}using the same pairs of values as in Table II and using the "

*six point calculation method"*obtained mean values of

**=32,6) mA/cm**

*i*_{0 }^{2 };

_{ }

*a*=0.52 y

*n*=1.00 . The previous calculation methods used two or six pairs of i,h values with a constant range of potential.

The proposed method can use data obtained by potentiostatic of galvanostatic techniques with any numbers of pairs of i,h values without the need for having a range for each measurement. The proposed calculation method can employ values for any polarization region. The limitation of the proposed method is that it is unable to employ the null data pair 0, 0.

**ACKNOWLEDGEMENTS**

The authors thank the General Research and Postgraduate Office of the Universidad Católica del Norte for promoting this research.

**REFERENCES**

1. M.Guzmán,P.Ortega and L.Vera, *Bol.Soc.Chil.**Quím*.,**43**,461(1998).

2. M.Guzmán,P.Ortega and L.Vera, *Bol.Soc.Chil. Quím*.,**45**,191 (2000).

3. S.Barnartt, *Electrochim.Acta*,**11**,1531(1966).

4. S.Barnartt, *Electrochim.Acta*,**13**,901(1968).

5. R.Parsons, *Trans.Faraday Soc*.**47**,1332(1951).

6. S.Barnartt, *J.phys.Chem.*** 70**,412(1966).

7. K.J.Vetter, *Electrochemical Kinetics*.Academic Press,New York (1967).

8. H.Gerischer, *Z.Elektrochem*.**54**,366(1950).

9. J.V.Petrocelli and A.A.Paolucci,*J.Electrochem.Soc*.**98**,291(1951).

10. J.E.B.Randles and K.W.Somerton,*Trans.Faraday Soc*.**48**,937(1952).

11. R.Parsons,*Handbook of Electrochemical Constants*. Butterworths, London (1959).

12. M.D.Wijnen and W.M.Smith, *Rec.Trav.Chim*.**79**, 289(1960).

13. T.Hurlen,*Acta Chem.Scand*.**14**,1533(1960).

14. F.C.Arson, *Analyt.Chem*.**33**, 939(1961).

15. K.J.Vetter,Elektrocheische Kinetik,Springer(1961).

16. T.Hurlen,*Acta Chem.Scand*.**15**,621(1961).

17. D.Jahn and W.Vielstich,* J.Electrochem.Soc.109, 649*(1962).

18. J.Jordan and R.A.Javick,*Electrochim.Acta*,**6**,22(1962).

19. H.P.Agarwal,*J.Electroanalyt.Chem*.**5,**236(1963).

20. Z.Galus and R.N.Adams,*J.Phys.Chem*.**67**,866(1963).

21. S.Barnartt,Canadian J.of Chemistry,47,1661(1969).

22. M.Guillen, M.Guzmán and R.Lara, *Bol.Soc.Chil.**Quím*.,**40**,427(1995)

23. S.Barnartt, *Electrochim.Acta*,**15**,1313 (1970).