## Journal of the Chilean Chemical Society

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

### J. Chil. Chem. Soc. vol.50 no.4 Concepción Dec. 2005

#### http://dx.doi.org/10.4067/S0717-97072005000400016

J. Chil. Chem. Soc., 50, N° 4 (2005), págs: 739-743
This study proposes the use of a numerical calculation method for optimizing non-linear physico-chemical models based on the Gauss-Newton algorithm. This method can be applicable to the Wagner-Traud model used to determine the parameters of corrosion and to adsorption isotherm models, like those of Langmuir and Langmuir-Freundlich, to determine the free energy of adsorption. This method in present work was applicated to experimental data of polarization of the iron electrode in medium of sulfuric acid 0.5 M and experimental values of adsorption of 3-Mercaptopropyltrimethoxisilane on copper surface. For the first case (polarization) the results showed with the proposed method was obtained smaller relative errors than the relative errors obtained by the Polynomial method and the computational method of Betacrunch. In the second case, the results showed that the adsorption parameters agree with the model of isotherm of Langmuir-Freundlich, obtaining a free energy of adsorption of 40 kJ/mol.
The efficiency of interpretation of experimental data requires good determination of the parameters which regulate the physico-chemical model, a condition which is not generally met when the model is non-linear. One example is the use of the non-linear Wagner-Traud
We considered that the function which represents the non-linear physico-chemical model to be optimized is given in its general form by equation 1:
where:
_{j} = parameters of the model. n = number of parameters in the mode Thus, if there is a set of experimental values available [y
where:
i = 1, m. m = Number of experimental data points. If we define the residual according to Eq. 3:
Then the optimal values for a
Given that the function i.e.:there exist a larger number of data points than equations), to find the solution we use the Gauss-Newton algorithm following Eq. 5:
Equation 5 can be written in vectorial form according to Eq. 6:
Where vector represents the parameters which lead to the solution, and vector represents the variation of the parameters in each iteration and which must converge on values which produce minimal residuals.
Theoretical data were generated using the Wagner-Traud model to test the validity of the model for corrosion currents according to Eq. 9:
where:
ba = Slope of anodic Tafel. bc = Slope of cathodic Tafel. h = Overpotential.The data generated by this equation are included in Table 1, together with the information generated by the Gauss-Newton method. According to the data of Table 1, complete coincidence is observed between the theoretical model and the optimized model (residual value=0), validating the proposed method. Experimental polarization data were selected in order to determine the error of the proposed method. Table 2 and Figure 1 show the optimization of the experimental data of polarization of the Fe electrode in 0.5 M H2SO4 at 25 ºC as obtained by J. Jankowski and R. Jchniewczs
Based on the data from Table 3, it is observed that the Gauss-Newton method showed a lower percentage of relative error, that is, 0.7 %, compared with the polynomial method2 and the Betacrunch10 computational method, demonstrating that the proposed method was stable and reliable in quantifying the corrosion parameters and that it also permitted working over the entire range of overpotential. This cannot be done with the polynomial method, which does not allow the use of overpotential values near zero. In the case of the Betacrunch computational method, work was done within an overpotential range of ± 100 mV and with an odd number of data points.
The adsorption equilibrium in the electrode / electrolyte interface is described for non-linear models, termed adsorption isotherms, which in their general form are given by the function in Eq.10 as:
where:
The constant K is related to the free energy of adsorption by Eq 11:
where:
The isotherm most used because of its simplicity is that of Langmuir
This isotherm has the following restrictions regarding the adsorbed molecules:
As can be observed from equations 12 and 13 the Langmuir adsorption model is non-linear, however, it is common to transform it to a linear form using logarithmic relations or reciprocal values for the data, which implies the use of transformed values in the optimization of the system. The Langmuir isotherm can be improved by introducing the heterogeneity parameter
The heterogeneity parameter may assume a range of values between 0< h <1 and it is considered that it is a measure of the distribution of energy of adsorption at the different active sites over the surface
The data from Table 3 (Figures 2 and 3) show that the results of the free energy of adsorption are similar in the three cases. Nevertheless, the application of Gauss-Newton method to the model of adsorption of Langmuir- Freundlich is obtained the greater coefficient of correlation (0.99). From this result can conclude that the Langmuir-Freundlich isotherm is the best that represent the experimental
Application of the Gauss-Newton model for optimizing non-linear physico-chemical models produces optimal results. The proposed model has the advantage of being applicable to a complete range of experimental data. The non-linear model is a good alternative for determining physico-chemical parameters in problems concerning both corrosion and adsorption. The proposed model can be applied to any physico-chemical system which has non-linear behavior
The calculation programs developed in this paper were programmed in Matlab 6.5.
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