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Mathematical model of catalytic processes at modified electrodes |

Femila Mercy Rani Joseph
Recebido em 12/11/2016 Endereço para correspondência
*e-mail: raj_sms@rediffmail.com RESUMO
A mathematical modeling of electrocatalytic processes taking place at modified electrodes is discussed. In this paper we obtained the approximate analytical solutions for the nonlinear equations under non steady state conditions using homotopy perturbation method. Simple and approximate polynomial expressions for the concentration of reactant, product and charge carrier were obtained in terms of diffusion coefficient and rate constant. In this work the numerical simulation of the problem is reported using Scilab program. In this manuscript analytical results are compared with simulation results and satisfactory agreement is noted. Palavras-chave: modeling-electrocatalysis; reaction-diffusion; modified electrodes; biosensor; homotopy perturbation method.
Electrocatalysis at modified electrodes has diverse applications, especially in sensors and biosensors and bio fuel cells. The model presented accounts for three processes, viz. (i) diffusion of reactant from solution into a layer of film, (ii) a chemical reaction between reactant and catalytically active centers of a film, and (iii) diffusion of charge carriers. Naujikas Recently, Naujikas
The diffusion of reactant into a modified electrode is described by the Fick's law: where D). Hence the nonlinear reaction diffusion equations for reactant (R), reaction product (P) and charge carrier (n) can be expressed as follows^{1}:where The schematic diagram of modified electrodes is given in Figure.1. Let
Figure 1. Schematic diagram of modified electrodes. Theoretical model of catalytic processes at modified electrodes are discussed. Mathematical solu tion of nonlinear reaction diffusion equations were obtained
Consequently, the boundary conditions are ( We introduce the following set of dimensionless parameters, The governing nonlinear reaction/diffusion Eqns. (2-4) are expressed in the following non - dimensional form: The dimensionless initial and boundary conditions becomes The dimensionless current density ψ(
Recently, many authors have applied the HPM to various problems and demonstrated the efficiency of the HPM for handling nonlinear structures and solving various physics and engineering problems. More recently, the system of coupled nonlinear reaction diffusionequations in an electroactive film deposited on aninlaid microdisc electrodes are solved using homotopy perturbation method. The current becomes
When Eqn. (22) - (24) represents new simple analytical expressions of concentration of reactant ( The steady state current becomes,
The non-steady state nonlinear differential equations (11)-(13)are also solved using numerical methods. The function pdex4 in Scilab software which is the function of solving the initial value problems for ordinary differential is used to solve this equation. Our theoretical results for the concentration of reactant (Eqn. (18)), reaction product (Eqn. (19)) and charge carrier (Eqn. (20)) are compared with simulation results (Scilab program 4.1) in Figures 2-4. Satisfactory agreement is found for all values of time
Figure 2. Plot of concentration of reactant versus distance from the electrode surface for various values of k_{0}, d and D, when time T=1. (___) denotes Eqn. (18) and (...) denotes the numerical simulation for the experimental values of the parameters (Appendix C)
Figure 3. Plot of concentration of reaction product versus distance from the electrode surface for various values of k_{0}, d and D, when time T=1. (... ) denotes Eqn. (19)
Figure 4. Plot of concentration of charge carrier versus distance from the electrode surface for various values of k_{0}, d and D, when time T=1. (___) denote Eqn. (20) and (...) denotes the numerical simulation for the experimental values of the parameters (Appendix C)
Eqns. (18)-(20) represents the concentration of the reactant ( The concentration of the reactant, reaction product and charge carrier on depends upon the reaction rate, thickness of the film and diffusion coefficient.The reaction rate for a given chemical reaction is the measure of the change in concentration of the reactant or product per unit time. The thickness of the film and diffusion coefficient are always critical parameters. Figures 2(a)-(c) represent the concentration of reactant versus distance from the electrode. Here the concentration is calculated for various values of the parameter Figure 3 (a)-(c) presents the dependence of the parameter Figure 4(a)-(c) represent the concentration of charge carrier versus the distance from the electrode for various values of parameters. In Figure 4 (a) and (c) the concentration decreases when the parameter Figure 5 represent the steady state concentration of reactant, reaction product and charge carrier respectively. From the Figure 5, it is observed that the concentration of reactant and reaction product are increasing function whereas charge carrier is a decreasing function from the film/solution interface.
Figure 5. Plot of steady state concentration of reactant, reaction product and charge carrier versus distance from the electrode surface
From Figure 6 (a), it is inferred that the current density increases when
Figure 6. Current versus the time for various values of k_{0} and D_{n}
Electrocatalysis of solute species at modified electrodes is discussed. The concentration of reactant, reaction product and charge carrier are obtained by solving the nonlinear reaction diffusion equation using homotopy perturbation method. Analytical results are compared with simulation result. A satisfactory agreement with the numerical results is noted. Based on the proposed model, optimization of reaction system parameters could be made for any particular case to get an optimum efficiency or reactant to product conversion. These analytical results are useful to predict and optimize the kinetic parameters in modified electrodes.
Appendix A, Appendix B, Appendix C and Appendix D can be found at http://qumicanova.sbq.org.br in pdf format with free access.
The authors are very much grateful to the referees for the valuable suggestions. The authors are thankful to Mr. S. Mohamed Jaleel, The Chairman, Dr. A. Senthilkumar, The Principal, Dr. P. G. Jansi Rani, Head of the Department of Mathematics, Sethu Institute of Technology, Kariapatti-626115, Tamilnadu, India for their encouragement.
1. Naujikas, R.; Malinauskas, A.; Ivanauskas, F.; 2. Puida, M.; Malinauskas, A.; Ivanauskas, F.; 3. Lyonsa, G.; Greera, C.; Fitzgerald, A.; Bannon, T.; Barlett, N.; 4. Lyons, G.; Bannon, T.; Hinds. G.; Rebouilat, S.; 5. Albery, J.; Hilman, R.; 6. Lyons, G.; Bannon, G.; Rebouilat, S.; 7. Maheswari, M.; Rajendran, L.; 8. Streeter, I.; Compton, G.; 9. Park, W.; Park, W.; Kim, Y.; Park, Y.; Oh, J.; 10. He, H.; 11. He, H.; 12. He, H.; 13. He, H.; 14. He, H.; 15. Golbabai, A.; Keramati, B.; 16. Ghasemi, M.; Kajani, M.; Babolian, E.; 17. Biazar, J.; Ghazvini, H.; 18. Odibat, Z.; Momani, S.; 19. Chowdhury, H.; Hashim, I.; |

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