3. 4.2.2. Linear Diffusion at a Planar Electrode
The diffusive event involves two aspects:
The variation of the concentration of the active species along the approaching distance to the electrode surface (concentration gradient with space)
The variation of the concentration of the active species with time (concentration gradient with time)

4. Diffusion coefficient
According to Fick’s first law, the flux of Ox is proportional to the concentration gradient of Ox along the direction of propagation:
Diffusion coefficient
flux of Red

5. Fick’s second law just determines how the concentration of species Ox changes with time
COx(x,t) : the concentration of Ox in the infinitesimal volume of solution between the planes x and x – dx :
Substituting such a relationship in Fick’s first law:
The same holds for the species Red moving away from the electrode:

6. 4.2.3. Spherical Diffusion Example : a hanging drop mercury electrode
Changes in Fick’s second law:

7. 4.2.4. Concentration Profiles; Cottrell Equation:
Concentration profiles : The graphs that show the dependence of the concentration of a species on distance from the electrode surface and how it evolves with time
To obtain such diagrams, solve Fick's second law:
erf (error function) and erfc [error function complement (erfc = 1 - erf)] are transcendental functions of exponential type
Cottrell equation

8. The thickness of the diffusion layer is 6(Dox* t)1/2
At t = 0 only Ox is present in solution, then the electrode potential is suddenly changed from a value more positive than the formal potential of the couple Ox/Red to a value much more negative, so that the reduction immediately takes place.
The slope of each concentration profile expresses the concentration gradient of Ox at various times
δCox(x,t)/δx = Cox(x,t) - C*ox
The point at which the concentration gradient becomes zero (COx(x,t)= C*ox, or Cox(x,t)/C*ox = 1) identifies the thickness of the diffusion layer
The thickness of the diffusion layer is 6(Dox* t)1/2
a series of concentration profiles at different times
Ox + ne- → Red

9. 4. 3. Influence of Mass Transport on Charge Transfer
4.3. Influence of Mass Transport on Charge Transfer. Electrochemically ‘Reversible’ and ‘Irreversible’ Processes
This concept is relative
there are two fundamental types of behavior:
electrochemically reversible : if the rate of the electron transfer is higher than the rate of the mass transport (or, if both kRed and kox are large, and greater than the rate constant of the mass transport
electrochemically irreversible: if the rate of the electron transfer is lower than the rate of the mass transport (or, if kRed and kox are not both large, and lower than the rate constant of the mass transport)

10. 5. NON-FARADAIC PROCESSES. CAPACITIVE CURRENTS

11. For:
Ox + ne Red
A charge distribution of the double layer is completely analogous to that of a capacitor.
Double layer :
Capacitor :

12. capacitive current Then :
A difference of potential is applied to the two plates, an excess q of electrons accumulates on one of them. The current flows through the circuit until the capacitor is charged is called the capacitive current
q = the charge on the capacitor
ΔE = the difference of potential applied between the two plates
C = the capacitance of the capacitor
This is just what happens in an electrochemical cell
Evaluation of the magnitude of capacitive currents in an electrochemical experiment: consider the equivalent circuit of an electrochemical cell
Then :

13. to minimize the capacitive currents:
The use of high concentrations of supporting electrolyte
Considering of time dependence of capacitive (non-faradic) currents

14. 6. THE ELECTRICAL DOUBLE LAYER.
A DEEPER EXAMINATION

15. internal Helmholtz
plane
outer Helmholtz
plane
The charge on the metallic electrode, qM, which depending on the potential applied, can be negative or positive compared to the charge of the solution, qS