1. Electro-Kinetics

2. Description of Electrochemical Techniques
The technique is named according to the parameters measured
E.g.
Voltammetry – measure current and voltage
Potentiometry – measure voltage
Chrono-potentiometry – measure voltage with time (under an applied current)
Chrono-amperometry – measure current with time (under an applied voltage)

3. Electro-Kinetics Movement of Ions Butler Volmer Equation
Rotating Disc Electrode
Rotating Cylinder Electrode
Voltammetry
Cyclic Voltammetry
Chrono-potentiometry
Chrono-amperometry

4. Movement of Ions in Solution
Diffusion – Movement under a concentration gradient. If an electrochemical reaction occurs the current due to this reaction is called, id , the diffusion current.
Migration or Transport – Movement of ions under an electric field due to coulombic forces. If an electrochemical reaction occurs the current due to this reaction is called, im , the migration current.
Convection – Movement due to changes in density at the electrode solution interface. This occurs due to depletion or addition of a species due to the electrochemical reaction.

5. The Capacitance Current
The charging or capacitance current, ic , is due to the presence of the electrical double layer and it is always present. This current, of course, is not related to any movement of ions.
Ic = Cdl x V
Where:
Cdl = the capacitance of the electrical double layer
V = voltage scan rate
The capacitance current makes its presence felt when measuring charge transfer (Faradaic) processes at concentrations of 10-5 M.

6. Diffusion
Molecular diffusion, often called simply diffusion, is a net transport of molecules from a region of higher concentration to one of lower concentration by random molecular motion.

7. Migration or Transport
Is the fraction of current carried by the ions.
For example in a solution of copper sulphate the transport number of Cu2+ is 0.4 and that of SO42- = 0.6.
t+ + t- = = 1
Since the migration current depends on the ionic strength of the solution it is usually eliminated by addition of excess of an inert supporting electrolyte (100 – 1000 fold excess in concentration)
The current is carried by the inert supporting electrolyte (e.g. NaCl , KNO3 etc) – because the ions produced do not undergo any electrochemical reaction the transport current is effectively removed.
In excess inert supporting electrolyte, the current measured due to the electro-active species of interest is due only to diffusion which can be related to mass transfer.

8. The supporting electrolyte
Voltammetry – the following example shows how the migration current is eliminated. Pb2+ + 2e → Pb0
The supporting electrolyte
Ensures diffusion control of limiting currents by eliminating migration currents
Table: Limiting currents observed for 9.5 x 10-4 M PbCl2 as a function of the concentration of KNO3 supporting electrolyte

9. Voltammetry
The example shown is for the reduction of Pb2+ at an inert mercury electrode.
Pb e → Pb(Hg)
At low inert electrolyte concentration a large fraction of the total current is due to the migration current, i.e. the currents due to the electrostatic attraction of ions to the electrode.
For solution 1:
i migration im 17.6 – 8.45 = 9.2 A
i diffusion id = 8.45 A

10. Fick’s First Law of Diffusion
Fick's first law relates the diffusive flux to the concentration field, by postulating that the flux goes from regions of high concentration to regions of low concentration, with a magnitude that is proportional to the concentration gradient (spatial derivative). In one (spatial) dimension, this is

11. Fick’s First Law of Diffusion
where
J is the diffusion flux in dimensions of [(concentration of substance) length−2 time-1], example mole (M) m-2 s-1.
J measures the amount of substance that will flow through a small area during a small time interval.
D is the diffusion coefficient or diffusivity in dimensions of [length2 time−1], example m2 s-1
 (for ideal mixtures) is the concentration in dimensions of [(concentration of substance) length−3], example M m-3
x is the position [length], example m

12. Fick’s First Law of Diffusion
D is proportional to the squared velocity of the diffusing particles, which depends on the temperature, viscosity of the fluid and the size of the particles according to the Stokes-Einstein relationship.
In dilute aqueous solutions the diffusion coefficients of most ions are similar and have values that at room temperature are in the range of 0.6x10-9 to 2x10-9 m2/s.
For biological molecules the diffusion coefficients normally range from to m2/s.

13. Ficks First Law of Diffusion
In two or more dimensions we must use, , the del or gradient operator, which generalises the first derivative, obtaining
J = -D  
The driving force for the one-dimensional diffusion is the quantity -/x
which for ideal mixtures is the concentration gradient. In chemical systems other than ideal solutions or mixtures, the driving force for diffusion of each species is the gradient of chemical potential of this species. Then Fick's first law (one-dimensional case) can be written as:

14. Fick’s First Law of Diffusion
where the index i denotes the ith species,
c is the concentration (mol/m3),
R is the universal gas constant (J/(K mol)),
T is the absolute temperature (K), and
μ is the chemical potential (J/mol).

15. Butler-Volmer Equation
The Butler-Volmer equation is one of the most fundamental relationships in electrochemistry. It describes how the electrical current on an electrode depends on the electrode potential, considering that both a cathodic and an anodic reaction occur on the same electrode:

16. Butler-Volmer Equation
where:
I = electrode current, Amps
Io= exchange current density, Amp/m2
E = electrode potential, V
Eeq= equilibrium potential, V
A = electrode active surface area, m2
T = absolute temperature, K
n = number of electrons involved in the electrode reaction
F = Faraday constant
R = universal gas constant
α = so-called symmetry factor or charge transfer coefficient dimensionless
The equation is named after chemists John Alfred Valentine Butler and Max Volmer

17. Butler-Volmer Equation
The equation describes two regions:
At high overpotential the Butler-Volmer equation simplifies to the Tafel equation
E − Eeq = a − blog(ic) for a cathodic reaction
E − Eeq = a + blog(ia) for an anodic reaction
Where:
a and b are constants (for a given reaction and temperature) and are called the Tafel equation constants
At low overpotential the Stern Geary equation applies

18. Current Voltage Curves for Electrode Reactions
Without concentration and therefore mass transport effects to complicate the electrolysis it is possible to establish the effects of voltage on the current flowing. In this situation the quantity E - Ee reflects the activation energy required to force current i to flow. Plotted below are three curves for differing values of io with α = 0.5.

19. Voltammetry
Although the Butler Volmer Equation predicts, that at high overpotential, the current will increase exponentially with applied voltage, this is often not the case as the current will be influenced by mass transfer control of the reactive species.
Take the following example of the reduction of ferric ions at a platinum rotating disc electrode (RDE).
Fe3+ + e = Fe2+
The rotation of the electrode establishes a well defined diffusion layer (Nernst diffusion layer)
The contribution of the capacitance current will also be demonstrated in this example.

20. Effect of the Capacitance Current in Voltammetry
Effect of the Capacitance Current in Voltammetry. The reduction of Ferric Chloride is carried out in the presence of 1 M NaCl to eliminate the migration current.
(a)
10-5 M Fe3+ Fe3+ + e → Fe2+
ild
Slope due to ic
Current
Applied Potential → -Ve
(b)
10-3 M Fe3+ Fe3+ + e → Fe2+
ild
Note that the iE curve in Fig. (a) is recorded at a much higher sensitivity than in Fig. (b).
Current
Applied Potential → -Ve

21. Charging Current or Capacitance Current
Note that due to the presence of the electrical double layer a charging or capacitance current is always present in voltammetric measurements.

22. Butler-Volmer Equation
where:
I = electrode current, Amps
Io= exchange current density, Amp/m2
E = electrode potential, V
Eeq= equilibrium potential, V
A = electrode active surface area, m2
T = absolute temperature, K
n = number of electrons involved in the electrode reaction
F = Faraday constant
R = universal gas constant
α = so-called symmetry factor or charge transfer coefficient dimensionless
The equation is named after chemists John Alfred Valentine Butler and Max Volmer

23. Butler Volmer Equation
While the Butler-Volmer equation is valid over the full potential range, simpler approximate solutions can be obtained over more restricted ranges of potential. As overpotentials, either positive or negative, become larger than about 0.05 V, the second or the first term of equation becomes negligible, respectively. Hence, simple exponential relationships between current (i.e., rate) and overpotential are obtained, or the overpotential can be considered as logarithmically dependent on the current density. This theoretical result is in agreement with the experimental findings of the German physical chemist Julius Tafel (1905), and the usual plots of overpotential versus log current density are known as Tafel lines.
The slope of a Tafel plot reveals the value of the transfer coefficient; for the given direction of the electrode reaction.

24. Butler-Volmer Equation
ia and ic are the exhange current densities for the anodic and cathodic reactions
These equations can be rearranged to give the Tafel equation which was obtained experimentally

25. Butler Volmer Equation - Tafel Equation

26. Tafel Equation
The Tafel slope is an intensive parameter and does not depend on the electrode surface area.
i0 is and extensive parameter and is influenced by the electrode surface area and the kinetics or speed of the reaction.
Notice that the Tafel slope is restricted to the number of electrons, n, involved in the charge transfer controlled reaction and the so called symmetry factor, .
n is often = 1 and although the symmetry factor can vary between 0 and 1 it is normally close to 0.5.
This means that the Tafel slope should be close to 120 mV if n = 1 and 60 mV if n = 2.

27. Tafel Equation
We can write:

28. Current Voltage Curves for Electrode Reactions
Without concentration and therefore mass transport effects to complicate the electrolysis it is possible to establish the effects of voltage on the current flowing. In this situation the quantity E - Ee reflects the activation energy required to force current i to flow. Plotted below are three curves for differing values of io with α = 0.5.

29. Tafel Equation The Tafel equation can be also written as: where
the plus sign under the exponent refers to an anodic reaction, and a minus sign to a cathodic reaction, n is the number of electrons involved in the electrode reaction k is the rate constant for the electrode reaction, R is the universal gas constant, F is the Faraday constant. k is Boltzmann's constant, T is the absolute temperature, e is the electron charge, and α is the so called "charge transfer coefficient", the value of which must be between 0 and 1.

30. Tafel Equation The following equation was obtained experimentally
Where:
 = the over-potential
i = the current density
a and b = Tafel constants

31. Tafel Equation Applicability
Where an electrochemical reaction occurs in two half reactions on separate electrodes, the Tafel equation is applied to each electrode separately.
The Tafel equation assumes that the reverse reaction rate is negligible compared to the forward reaction rate.
The Tafel equation is applicable to the region where the values of polarization are high. At low values of polarization, the dependence of current on polarization is usually linear (not logarithmic):
This linear region is called "polarization resistance" due to its formal similarity to Ohm’s law

32. Stern Geary Equation
Applicable in the linear region of the Butler Volmer Equation at low over-potentials

33. Tafel Equation Overview of the terms
The exchange current is the current at equilibrium, i.e. the rate at which oxidized and reduced species transfer electrons with the electrode. In other words, the exchange current density is the rate of reaction at the reversible potential (when the overpotential is zero by definition). At the reversible potential, the reaction is in equilibrium meaning that the forward and reverse reactions progress at the same rates. This rate is the exchange current density.

34. Tafel Equation
The Tafel slope is measured experimentally; however, it can be shown theoretically when the dominant reaction mechanism involves the transfer of a single electron that
T is the absolute temperature,
R is the gas constant
α is the so called "charge transfer coefficient", the value of which must be between 0 and 1.

35. Levich Equation
The Levich Equation models the diffusion and solution flow conditions around a rotating disc electrode (RDE). It is named after Veniamin Grigorievich Levich who first developed an RDE as a tool for electrochemical research. It can be used to predict the current observed at an RDE, in particular, the Levich equation gives the height of the sigmoidal wave observed in rotating disk voltammetry. The sigmoidal wave height is often called the Levich current.
In work at a RDE the electrode is usually rotated quite fast (1000 rpm) in order to establish a well defined diffusion layer.
The scan rate is relatively slow – typically 2-5 mV s-1

36. Current Voltage Curve at a RDE
It is important to remember that in order to determine the diffusion current and the mass transfer coefficient using volatmmetry, excess inert supporting electrolyte must be present to eliminate the migration current

37. Levich Equation The Levich Equation is written as: where
iL is the Levich current
n is the number of electrons transferred in the half reaction
F is the Faraday constant
A is the electrode area
D is the diffusion coefficient (see Fick's law of diffusion)
w is the angular rotation rate of the electrode
v is the kinematic viscosity
C is the analyte concentration
While the Levich equation suffices for many purposes, improved forms based on derivations utilising more terms in the velocity expression are available.[1][2]

38. Rotating Disc Electrode

39. Levich Equation

40. Levich Equation

41. Levich Equation
It is important to note that the layer of solution immediately adjacent to the surface of the electrode behaves as if it were stuck to the electrode. While the bulk of the solution is being stirred vigorously by the rotating electrode, this thin layer of solution manages to cling to the surface of the electrode and appears (from the perspective of the rotating electrode) to be motionless.
This layer is called the stagnant layer in order to distinguish it from the remaining bulk of the solution. The act of rotation drags material to the electrode surface where it can react. Providing the rotation speed is kept within the limits that laminar flow is maintained then the mass transport equation is given by the Levich equation.

42. Levich Equation RDE
The Levich equation takes into account both the rate of diffusion across the stagnant layer and the complex solution flow pattern. In particular, the Levich equation gives the height of the sigmoidal wave observed in rotated disk voltammetry. The sigmoid wave height is often called the Levich current, iL, and it is directly proportional to the analyte concentration, C. The Levich equation is written as:
iL = (0.620) n F A D2/3 w1/2 v–1/6 C
where w is the angular rotation rate of the electrode (radians/sec) and v is the kinematic viscosity of the solution (cm2/sec). The kinematic viscosity is the ratio of the solution's viscosity to its density.

43. Current Voltage Curve at a RDE

44. Ilimiting vs ½ (electrode rotational velocity)

45. Levich Equation - RDE
The linear relationship between Levich current and the square root of the rotation rate is obvious from the Levich plot. A linear least squares fit of the data produces an equation for the best straight line passing through the data. The specific experiment shown, the electrode area, A, was cm2, the analyte concentration, C, was 2.55x10–6 mol/cm3, and the solution had a kinematic viscosity, v, equal to cm2/sec. After careful substitution and unit analysis, you can solve for the diffusion coefficient, D, and obtain a value equal to 4.75x10–6 cm2/s. This result is a little low, probably due to the poor shape of the sigmoidal signal observed in this particular experiment.
The kinematic viscosity is the ratio of the absolute viscosity of a solution to its density. Absolute viscosity is measured in poises (1 poise = gram cm–1 sec–1). Kinematic viscosity is measured in stokes (1 stoke = cm2 sec–1). Extensive tables of solution viscosity and more information about viscosity units can be found in the CRC Handbook of Chemistry and Physics.

46. Cyclic Voltammetry
Cyclic Voltammetry is carried out at a stationary electrode.
This normally involves the use of an inert disc electrode made from platinum, gold or glassy carbon. Nickel has also been used.
The potential is continuously changed as a linear function of time. The rate of change of potential with time is referred to as the scan rate (v). Compared to a RDE the scan rates in cyclic voltammetry are usually much higher, typically 50 mV s-1

47. Cyclic Voltammetry
Cyclic voltammetry, in which the direction of the potential is reversed at the end of the first scan. Thus, the waveform is usually of the form of an isosceles triangle.
The advantage using a stationary electrode is that the product of the electron transfer reaction that occurred in the forward scan can be probed again in the reverse scan.
CV is a powerful tool for the determination of formal redox potentials, detection of chemical reactions that precede or follow the electrochemical reaction and evaluation of electron transfer kinetics.

48. Cyclic Voltammetry

49. Cyclic Voltammetry
For a reversible process
Epc – Epa = 0.059V/n

50. The Randles-Sevcik equation Reversible systems

51. The Randles-Sevcik equation Reversible systems
n = the number of electrons in the redox reaction
v = the scan rate in V s-1
F = the Faraday’s constant 96,485 coulombs mole-1
A = the electrode area cm2
R = the gas constant J mole-1 K-1
T = the temperature K
D = the analyte diffusion coefficient cm2 s-1

52. The Randles-Sevcik equation Reversible systems
As expected a plot of peak height vs the square root of the scan rate produces a linear plot, in which the diffusion coefficient can be obtained from the slope of the plot.

53. Cyclic Voltammetry

54. Cyclic Voltammetry

55. Cyclic Voltammetry

56. Cyclic Voltammetry – Stationary Electrode
Peak positions are related to formal potential of redox process
E0 = (Epa + Epc ) /2
Separation of peaks for a reversible couple is 0.059/n volts
A one electron fast electron transfer reaction thus gives 59mV separation
Peak potentials are then independent of scan rate
Half-peak potential Ep/2 = E1/2  0.028/n
Sign is + for a reduction

57. Cyclic Voltammetry – Stationary Electrode
The shape of the voltammogram depends on the transfer coefficient 
When   deviates from 0.5 the voltammograms become asymmetric -cathodic peak sharper as expected from Butler Volmer eqn.

58. Web Sites http://calctool.org/CALC/chem/electrochem/levich

59. Tafel Equation
The Tafel slope is an intensive parameter and does not depend on the electrode surface area.
i0 is and extensive parameter and is influenced by the electrode surface area and the kinetics or speed of the reaction.
Notice that the Tafel slope is restricted to the number of electrons, n, involved in the charge transfer controlled reaction and the so called symmetry factor, .
n is often = 1 and although the symmetry factor can vary between 0 and 1 it is normally close to 0.5.
This means that the Tafel slope should be close to 120 mV if n = 1 and 60 mV if n = 2.

60. Tafel Equation
We can write:

61. Evans Diagrams

62. Evans Diagrams

63. Evans Diagrams