Analytical expression of non steady-state concentration profiles at planar electrode for the CE mechanism

The analytical solutions of the non-steady-state concentrations of species at a planar microelectrode are presented. These simple new approximate expressions of concentrations are valid for all values of time and possible values of rate constants. Analytical equations are given to describe the current when the homogeneous equilibrium position lies heavily in favour of the electroinactive species. Working surfaces are presented for the variation of limiting current with a homogeneous kinetic parameter and equilibrium constant. Moreover, in this work we employ the Homotopy perturbation method to solve the boundary value problem.


INTRODUCTION
One of the major achievements in electroanalytical chemistry in the 1980s was the introduction of microelectrodes, i.e., electrodes whose characteristic dimension is on the order of a few m  (the radius in the case of disc and hemispheres, band width in the case of bands, etc.).Microelectrodes have become more commonly used in electrochemistry to probe kinetics of fast chemical reactions [1].In this work, we are interested in finding the mass transport limiting current response for the CE mechanism at a microelectrode.For each mechanism, the electroinactive species A is in dynamic equilibrium with the electroactive species B via a homogeneous chemical step.The decay of species A is described by the first order forward rate constant f k and the reverse of this process is described by the rate constant b k , which is first order for the CE mechanism.All species are considered to have an diffusion coefficient D .
Oldham [2] made use of an analytical expression of CE mechanism at a hemispherical electrode.Lavagnini et al .[3] employed the hopscotch method and a conformal map to numerically simulate CE mechanism at a planar electrode.Values of limiting current were analysed for a range of equilibrium constants and rate constants.There have been many previous theoretical descriptions of the diffusion limiting current for the CE mechanism.In Reference [4] k using Homotopy perturbation method.

MATHEMATICAL FORMULATION OF THE BOUNDARY VALUE PROBLEM
As a representative example of the reaction-diffusion problems considered, the standard CE mechanism has been chosen, with initial and boundary conditions corresponding to the potential step for all planar electrodes.Under stead-state conditions, the local concentrations of the species do not change.Therefore the mass transport equations are set equal to zero.We consider the differential equations with diffusion described by the concentration of the two species leads to the following equations [5] where a and b denote the concentration of the species A and B .x and t stand for space and time, respectively.
where 0 a and 0 b are the bulk concentrations of the species A and B , s a and s b denote the concentrations at electrode surface .The flux j can be described as follows: The current density is defined as: Where n is the number of electrons and F is the Faraday constant.Using the following dimensionless parameters we obtained the dimensionless non-linear reaction diffusion equations for planar electrode as follows The initial and boundary conditions becomes: 0; 1; ; 1; The dimensionless current is as follows:

ANALYTICAL SOLUTION OF THE CONCENTRATIONS AND CURRENT USING HOMOTOPY PERTURBATION METHOD
Recently, many authors have applied the Homotopy perturbation method to various problems and demonstrated the efficiency of the Homotopy perturbation method for handling non-linear structures and solving various physics and engineering problems [6][7][8][9].This method is a combination of homotopy in topology and classic perturbation techniques.Ji-Huan He used the Homotopy perturbation method to solve the Lighthill equation [10], the Duffing equation [11] and the Blasius equation [12].The idea has been used to solve non-linear boundary value problems [13], integral equations [14][15][16], Klein-Gordon and Sine-Gordon equations [17], Emden -Flower type equations [18] and many other problems.This wide variety of applications shows the power of the Homotopy perturbation method to solve functional equations.The Homotopy perturbation method is unique in its applicability, accuracy and efficiency.The Homotopy perturbation method [19] uses the imbedding parameter p as a small parameter, and only a few iterations are needed to search for an asymptotic solution.By applying Laplace transformation to the partial differential Eqs.9 and 10 and using the condition Eq.11, the following diffential equations in Laplace space are obtained: Now the boundary conditions become where s is the Laplace variable and an overbar indicates a Laplace-transformed quantity.The set of expressions presented in Eqs.15-18 defines the initial and boundary value problem in Laplace space.The Homotopy perturbation method method has overcome the limitations of traditional perturbation techniques, so a considerable deal of research has been conducted to apply the homotopy technique to solve various strong non-linear equations.
The Homotopy perturbtion method [20][21][22] is used to give the approximate analytical solutions of coupled non-linear Eqs.15 and 16.Using this method (see Appendix -A and B) the approximate solutions of the Eqs.9 and 10 are The Eqs. 19 and 20 satisfies the boundary conditions Eq.11 to Eq.13.These equations represent the new approximate analytical expressions for the concentration profiles 1

CONCLUSIONS
In this work, the coupled time dependent linear dif-       sponds to changes in electrode size.Further, based on the outcome of this work it is possible to calculate the concentration and current at cylindrical and hemispherical electrode for CE mechanism. : Solving the Eqs.A7 to A10, and using the boundary conditions (A3) and (A4), we can find the following results

f
 and b are described the forward and backward rate constants respectively.The boundary conditions k k and 4 k .The third term in the Eqs.19 and 20 are in opposite sign when dimensionless concentration u and v are equal when 1 2 m m  (ratio of concentration at electrode surface for the bulk concentration) and 20 are the new and simple approximate analytical expressions of concentrations of the isomers calculated using Homotopy perturbation method for the initial and boundary conditions Eqs.11-13.The closed analytical expression of current is represented by the Eq.21.The dimensionless concentration profiles of u and v versus dimensionless distance X are given in Figures 1-4 and Figures 5-8 respectively.From these figures, we can see that the value of the concentration u and v decreases when T increases and attains the steady-state value at 40 X  .When the rate constants are small (less than 1) and 1 2 m m the concentration decreases slowly and reaches the minimum value and then increases in Figures1 and 5. From Figures 2, it is inferred that the concentration u attains the steady-state value at 5 X  .Also, when all the parameters are small and 1 T  , the concentration attains maximum value at 4 X  in Figures3 and 4. For large value of parameters 34 1   , , k k m and 2 m , the concentration v decreases when T increases in Figures 6.For the small values of parameter and time T ( 1 T  ), there is no significance different in the concentration.(Refer Figures 7(a,b), Figures 8(a,b).The dimensionless current  versus T for various values of 3 k and 4 k is given in Figures 9 and 10.From these figures the value of current decreases as the time T and 4 k increases.But the value of current increases when 3 k increases.

Figure 1 .
Figure 1.Normalized concentration u at microelectrode.The concentrations were computed using Eq.19 for some value of 1 1 m  ,

Figure 2 .Figure 3 .
Figure 2. Normalized concentration u at microelectrode.The concentrations were computed using Eq.19 for some value of 1 0.5 m  ,

Figure 4 .
Figure 4. Normalized concentration u at microelectrode.The concentrations were com puted using Eq.19 for some fixed value o f 1 2 0.001, 0.005 m m   a n d t h e r e a ction/diffusion parameter 1 0.005 k  and 2 0.001 k  for various values of (a) 0.1 T  ; (b) 1 T  ; (c) 10 T  ; (d) 100 T  .

Figure 5 .
Figure 5. Normalized concentration v at mi- croelectrode.The concentrations were computed using Eq.20 for some value of 1 2 1, 1 m m   and for various values of T and the reaction/diffusion parameter 3 0.01 k  and

Figure 6 .Figure 7 .Figure 8 .Figure 9 .
Figure 6.Normalized concentration v at microelectrode.The concentrations were computed using Eq.20 for some value of 1 2 1, 0.5 m m   and for various values of T and the reaction/diffusion parameter 1 5 k  and 2 10 k  .ferentialequations at planar electrode have been solved analytically.In the first part of the paper, we have derived the analytical expressions of the concentrations of the species for all values of rate constants for planar electrode.In the second part of the paper we have presented approximate analytical expressions corresponding to the species A and B in terms of the kinetic parameters 1 k , 2 k , 3 k and 4 k based on the Homotopy perturbation method.In addition, we have also presented an analytical expression for the non-steady state current.The kinetics of this homogeneous step can in principle be studied by observing how the limiting current re-

Figure 10 .
Figure 10.Plot of the dimensionless current,  versus time.The current were calculated using Eq.21 for the fixed value of 4 1 k  and for various values of the reaction/diffusion parameter 3 k .
After putting Eqs.A11 and A12 into Eq.A15 and Eqs.A13 and A14 into Eq.A16.Using inverse Laplace transform, the final results can be described in Eqs.19and 20 in the text.The remaining components of