1. Voltammetry and Polarography
Lecture 4

2. Diffusion Controlled Processes
The flux of material to and from the electrode surface is a complex function of all three modes of mass transport.
In the limit in which diffusion is the only significant means for the mass transport of the reactants and products, the current in a voltammetric cell is given by:

3. where n is the number of electrons transferred in the redox reaction, F is Faraday's constant, A is the area of the electrode, D is the diffusion coefficient for the reactant, CbuIk and Cx=o are the concentration of the analyte in bulk solution and at the electrode surface, and  is the thickness of the diffusion layer.

4. For the above equation to be valid, migration and convection must not interfere with formation of diffusion layer around the electrode surface.
Migration is eliminated by adding a high concentration of an inert supporting electrolyte to the analytical solution.
The large excess of inert ions, ensures that very few reactant and product ions will move as a result of migration.

5. Although convection may be easily eliminated by not physically agitating the solution, in some situations it is desirable either to stir the solution or to push the solution through an electrochemical flow cell. Fortunately, the dynamics of a fluid moving past an electrode results in a small diffusion layer (stagnant solution), typically of ‑ 0.01 cm thickness, in which the rate of mass transport by convection is close to zero.

6. Voltammograms
For the reduction of an analyte A to give a product P at a mercury film microelectrode (for example), the microelectrode is connected to the negative terminal of a linear potential scan generator, cathodic currents are positive (by convention), while anodic currents are negative. The figure below represents the linear scan voltammogram under hydrodynamic conditions:

8. As seen from the figure, linear scan voltammograms assume a sigmoidal curve called a voltammetric wave. The constant current beyond the steep rise is called the limiting current il, because it arises from the limitation at which reactants are brought to electrode surface. Limiting currents are proportional to concentration, and thus are used in quantitative analysis. il = kCA Where CA is the analyte concentration, and k is a constant. The potential at half the limiting current is called the half wave potential (E1/2), a characteristic property of a component.

9. Obtaining a stable limiting current
Reproducible limiting currents can be obtained rapidly when:
The solution (or the microelectrode) be in continuous and reproducible motion.
Or a dropping mercury electrode is used
Or the solution is forced through a flow cell comprising the three electrodes (as in HPLC).
Linear scan voltammetry in which the solution is stirred or the electrode is rotating is called hydrodynamic voltammetry.

10. Hydrodynamic Voltammetry
In hydrodynamic voltammetry the solution is stirred by rotating the electrode, or using a stirrer.
Current is measured as a function of the potential applied to a solid working electrode.
The same potential profiles used for polarography are used in hydrodynamic voltammetry.
The resulting voltammograms are identical to those for polarography, except for the lack of current oscillations resulting from the growth of the mercury drops.
Because hydrodynamic voltammetry is not limited to DME electrodes, it is useful for the analysis of analytes that are reduced or oxidized at more positive potentials.

11. Hydrodynamic Voltammetry

12. What Happens During a Voltammetric Experiment
The reactant is carried to the surface of the electrode by one or more of the following:
Diffusion
Migration
Convection
2. Migration is minimized using a supporting electrolyte ( times more concentrated than analyte), therefore the fraction of current carried by the analyte through migration approaches zero. That means that the rate of migration of analyte to the electrode of opposite charge becomes independent of applied potential.

13. Concentration Profiles at Microelectrode Surface
Assume an electrode reaction: A + ne D P Taking place at a Hg coated microelectrode in a solution of A containing an excess of a supporting electrolyte. Also, assume the following: CA : initial concentration of A in the bulk soln CP : concentration of P in the bulk solution CAo and CPo are concentrations of A and P at the thin layer adjacent to electrode surface In addition, P is insoluble in the mercury film. The reduction is rapid and reversible (this means that the concentrations at electrode surface can be calculated from Nernst equation), therefore:

14. Where the Eappl is the potential between the microelectrode and the reference electrode. It can also be assumed (since the microelectrode is very small) that electrolysis will not change the bulk concentration. Therefore, the bulk concentration of A is unchanged and is equal to CA while the concentration of P (CP) in the bulk solution is practically zero.

15. Profiles for Planar electrodes in Unstirred Solutions
In this case, mass transport of solute to electrode surface occurs by diffusion. Assume that a square wave potential is applied (Eappl) to the working microelectrode for a period of time (t). Also, assume that Eappl is large enough that CPo /CAo is very large (>1000). Under this condition, the concentration of CAo at the electrode surface is zero, for all practical purposes, A is immediately reduced as soon as it approaches electrode surface.

17. In the previous figure, the current rises to a peak value, resulting in conversion of all A to P at the electrode surface layer of the solution. Diffusion of A from the bulk into this surface occurs which results in more reduction of A. The current required to satisfy the concentration of A in Nernst equation:
Decreases rapidly with time as A is required to travel longer distances to reach the surface. That is i a dCA/dx

20. It is then clear that it is not practical to obtain limiting currents with planar electrodes in unstirred solutions, because the current will continually decrease with time as the slopes of the concentration profiles become lower:

21. Profiles of Microelectrodes in Stirred Solutions
At any time, it can be assumed that the concentrations at the stagnant solution, region adjacent to electrode surface and at low potential, x, obey the equation: Cpo = CA – CAo At the half wave potential, the concentration of P is half that of A (at the electrode surface), which means that: Cpo = C A/2 However, applying a negative potential z or larger will make the reduction of A complete, that no A will be present at the electrode surface, therefore: Cpo = CA, and will remain constant, resulting in a limited current, regardless of applied potential.

23. Voltammetric Currents
In the stirred solution (or using a dropping mercury electrode), the current at any point will depend on:
The rate of mass transport of A to the diffusion layer (stagnant phase) by convection
The rate of transport of A from the outer edge of the diffusion layer to the electrode surface
Since the concentration of P formed in the diffusion layer is continuously swept away by convection, a continuous current will be necessary to maintain the surface concentrations of A and P to satisfy Nernst equation.

24. Convection maintains a constant supply of A at the outer edge of the diffusion layer, therefore a steady state current results that is determined by the applied potential:
Where i is the current in A, n is the number of moles, A is the electrode surface area in cm2, DA is the diffusion coefficient of A in cm2s-1, and F is the Faraday constant.

26. dCA/dx is the slope of the initial part of the concentration profile and can be approximated to:

27. CAo becomes negligible at high negative potentials, therefore it can be concluded that:
This derivation is based on a simplified assumption that the interface between the moving and stationary layers is well defined, where as transport of A ceases, its diffusion begins!!!

28. Polarographic Wave Equation
We have the two previous equations where: Subtraction of the later from the first and rearrangement gives:
Where CAO is the surface concentration of A

29. The surface concentration of P can also be expressed in a similar way: Or: And thus:

30. Since we have: Substitution for Cpo tio and CAo we get: When i = il/2, we have Eappl = E1/2
And E1/2 = EAo –Eref since usually kA = kp, both log terms will equal zero

31. Finding E and Number of Electrons involved
Id/2 Id
The polarographic wave
The linear relationship

32. Pulsed Voltammetric Techniques
There are three main pulsed polarographic techniques based on the excitation waveform and the current sampling regime:
- Normal-Pulse Polarography
- Differential-Pulse polarography
- Square-Wave polarography

33. Why use pulse techniques?
The basis of all pulse techniques is the difference in the rate of the decay of the charging and the faradaic currents following a potential step (or "pulse"). The charging current decays exponentially, whereas the faradaic current (for a diffusion-controlled current) decays as a function of 1/(time)½; that is, the rate of decay of the charging current is considerably faster than the decay of the faradaic current. The charging current is negligible at the end of the potential pulse. Therefore, at the end of the potential pulse, the measured current consists solely of the faradaic current; that is, measuring the current at the end of a potential pulse allows discrimination between the faradaic and charging currents.

34. Reduction of the capacitive current during the pulse time iC Capacitive current ∆EA Pulse amplitude R Discharge resistance t Time after pulse application CD Double layer capacitance This means that the charging current decays exponentially (ic = k”e-kt)

35. Reduction of the capacitive current during the pulse time
iC Capacitive current
∆EA Pulse amplitude
R Discharge resistance
t Time after pulse application
CD Double layer capacity of working electrode

36. Cottrell Equation
For diffusion controlled processes at planar electrodes:
i = current, in unit A, n = number of electrons (to reduce/oxidize one molecule of analyte), F = Faraday constant, 96,485 C/mol, A = area of the (planar) electrode in cm2 , C = initial concentration of the reducible analyte in mol/cm3; D = diffusion coefficient for analyte in cm2/s, and tm = time in s. OR: The limiting Faradaic current is proportional to t-1/2, which is a slow decay as compared to exponential decay of the charging current.

37. In addition, because of the short pulse duration, the diffusion layer is thinner than that of DC polarography (i.e., greater flux of analyte) and hence the faradaic current is increased. The resulting polarogram has a sigmoidal shape, with a limiting current given by Cottrell equation:
where tm is the time after application of the pulse where the current is sampled

39. The important parameters for pulse techniques are as follows:
Pulse amplitude is the height of the potential pulse. This may or may not be constant depending upon the technique.
Pulse width is the duration of the potential pulse.
Sample period is the time at the end of the pulse during which the current is measured.
Note that the end of the drop time coincides with the end of the pulse width).

40. Normal Pulse Polarography (NPP)
Normal-pulse polarography consists of a series of pulses of increasing amplitude applied to successive drops at a preselected time near the end of each drop lifetime. Between the pulses, the electrode is kept at a constant (base) potential at which no reaction of the analyte occurs. The amplitude of the pulse increases linearly with each drop.The current is measured about 40 ms after the pulse is applied, at which time the contribution of the charging current is nearly zero.

43. In a reduction, if the Initial Potential is well positive of the redox potential, the application of small amplitude pulses does not cause any faradaic reactions, hence there is no current response. When the pulse amplitude is sufficiently large that the pulse potential is close to the redox potential, there is a faradaic reaction in response to the potential pulse (assuming moderately fast electron transfer kinetics), and the magnitude of this current may depend on both the rate of diffusion and the rate of electron transfer. When the pulsed potentials are sufficiently negative of the redox potential that the electron transfer reaction occurs rapidly, the faradaic current depends only on the rate of diffusion; that is, a limiting current has been attained.