KINETICS OF ELECTRODE REACTIONS_3

Lets continue to consider the equilibrium condition.........
At equilibrium, the net current density is zero, but the rates of anodic and cathodic reaction are not zero. The magnitude of current density at equilibrium condition is called as the exchange current density, io. The formulation of io is depicted in eqution (16).

                               (16)


if we apply logarithm of equation (16), then we will have equation (17)

                             (17)

arrangement of equation (17) may result equation (18), which is the similar term of Nersnt equation.

                             (18)

the explanation is

The deviation from the equilibrium potential represent as the overpotential as it described in equation (19)

                               (19)

substitution of equation (18) and (19) into equation (15) results equation (20)

         (20)


rearranging of equation (20) gives equation (21)

            (21)


io is the exchange current density. If we formulate the symmetri factor (bheta) into equation (22)

            (22)


therefore the substitution of equation (22) into equation (21) may result equation (23).

            (23)


equation (23) is usually called as the Butler-Volmer equation. It is a standard model to described the current-overpotential relationship for an electrode reaction at a specified temperature.
If overpotential value is high and positive, therefore, the second part of equation (23) can be eliminated due to exponential small value  will provide. Then equation (23) becomes equation (24)

                                    (24)



rearranging of equation (24) may result equation (25) and equation (26)

                                      (25)

                                      (26)


Equation (26) i ussually named as Tafel plot, a linear regression plot with B as slope and A as intercept.
In case of the overpotential is low, therefore after applying mc Laurin series expansion, equation (23) will transforms to equation (27).

                                     (27)


Now, how we can measure the overpotential value? Figure 3 describes the scheme of overpotential measurement.

Figure 3. Scheme of overpotential measurement

The calculation is described in equation (28) and equation (29)

                                                   (28)

                         (29)

Ewr,rev is the working potential in comparison to reference potential (volt) at equilibrium condition, Ew is working potential at equilibrium condition, Er,rev is the reference potential at equilibrium condition. Meanwhile Emeas is the measurement potential, the potential recorded by voltmeter, Ew is the working potential at definited time which is still in unequilibrium condition and thetha is the overpotential.

reference:
Prentice, G., 1997, Electrochemical Engineering Principles, Prentice Hall, Englewood Cliffs, New Jersey 07632.    


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KINETICS OF ELECTRODE REACTIONS_2

if we begin the reaction at the potential Q1 and reduce it to Q2 as shown in Figure 2. The activation energy for process 1 is Gc1 and Gc2 for process 2. The formula is in equation (8)

                     (8)


the subscript c indicates a cathodic process. The activation energy in the anodic direction is described in equation (9).

                     (9)


Bheta is the symmetri factor. This factor can vary between 0 and 1. It is the fraction of potential across the double layer to promote reaction. The form of kinetic expression for chemical reactions is (equation (10)).

                        (10)



G* is the free energy of activation and k' is a constant. The rate of an electrochemical reaction is proportional to the current density as described in equation (11).

                       (11)


r is reaction rate (mol/s cm2), c is the reactant concentration (mol/cm3). For anodic reaction involving a single electron transfer, the rate is expressed in equation (12).

                      (12)





therefore equation (12) transform to equation (13)

                                                  (13)


similarly, the rate equation for cathodic reaction is in equation (14)

                                                  (14)


The net reaction rate is the difference between the rate of anodic and cathodic reaction, as depicted in equation (15)

                   (15)

KINETICS OF ELECTRODE REACTION

Now we are discussing the kinetics of reaction at electrodes, whether anode or cathode. The electrode reaction occurs due to the potential difference of electrode surface. In electrochemical reaction, we can manipulate the additional driving force of reaction, i.e the electric potential, beside the conventional parameters such as temperature and catalyst.
Electrode kinetics are governed by the potential difference across the thin layer ( on the order of 10 A) at the electrode surface which is named as the electrical double layer. When we apply a potential to an electrode, the charges accumulate on the electrode surface and attract ions of the opposite charges from the electrolyte. The potential distribution in the double layer is complicated, however for the first approximation we can model the double layer as a simple parallel-plate capacitor (equation 1),

                     (1)



C is capacitance per unit area, D is the dielectric constant (or relative permittivity) and d is the separation between the two layer of charge. The model of double layer proposed by Helmholtz is described in in Figure 1.

Figure 1. Model of double layer proposed by Helmholtz. Two parallel layers of charge are separated by solvent molecules (unmarked cricle) at a distance d, representing the outer Helmholtz plane ( Prentice, 1991)

The ionic distribution at diffuse layer follow Boltzmann distribution,as described in equation (2).

             (2)



Ci is concentration of ion i in the bulk phase, zi is charge of ion and theta is the potential. 
The potential distribution in double layer follow Poisson Distribution as described in equation (3).

                                                            (3)

Charge density is a function of the concentration, Ci and the charge of ions, zi, as described in equation (4)

                  (4)



The combination of (2),(3) and (4) producing equation (5).

         (5)





In cartesian coordinate, equation (5) can be stated as equation (6),

             (6)



THE ELECTRODE KINETICS MODEL
 the reduction reaction of an ion at potential thetha_1 (Q1)is described in equation (7),

    
                     (7)




a more negative potential tends to promote reduction, but as in convension, more negative potential corresponds to a more positive energy. The energy versus reaction coordinate can be drawn as in Figure 2.

Figure 2. Energy along a reaction coordinate for an electrochemical reaction at three different potentials, Q1>Q2>Q3 (Prentice, 1991).

Reference:

Prentice, G., 1991, Electrochemical Engineering Principles, Prentice-Hall International (UK) Limited, London
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