1. Introduction to Analytical Chemistry
CHAPTER A BRIEF LOOK AT SOME OTHER ELECTROANALYTICAL METHODS

2. 20A How Does Current Affect the Potential of Electrochemical Cells?
When a current develops in an electrochemical cell, the measured potential across the two electrodes is no longer simply the difference between the two electrode potentials (the thermodynamic cell potential). Two additional phenomena, IR drop and polarization, require application of potentials greater than the thermodynamic potential to operate an electrolytic cell.

3. 20A-1 Ohmic Potential; IR Drop
The product of the resistance R of a cell in ohms and the current I in amperes (A) is called the ohmic potential or the IR drop of the cell.
To generate a current of I amperes in this cell, we must apply a potential that is IR volts more negative than the thermodynamic cell potential, Ecell = Eright － Eleft . That is,
(20-2)

4. Figure 20-1
Figure 20-1 An electrolytic cell for determining Cd2+. (a) Current mA. (b) Schematic of cell in
(a) with the internal resistance of the cell represented by a 15.0-Ω resistor and Eapplied increased to give a
current of 2.00 mA.

5. 20A-1 Ohmic Potential; IR Drop
Usually, we try to minimize the IR drop in the cell by having a very small cell resistance (high ionic strength) or by using a special three-electrode cell (see Section 20C-2) in which the current passes between the working electrode and an auxiliary (or counter) electrode.

6. 20A-2 Polarization Effects
According to Equation 20-2, a plot of current in an electrolytic cell as a function of applied potential should be a straight line with a slope equal to the negative reciprocal of the resistance.

7. 20A-2 Polarization Effects
Cells that exhibit nonlinear behavior at higher currents are said to be polarized, and the degree of polarization is given by an overvoltage, or overpotential, symbolized by Π in Figure 20-2.
(20-3)

8. 20A-2 Polarization Effects
The degree of polarization can be so large that the current in the cell becomes independent of potential.
Polarization phenomena are conveniently divided into two categories: concentration polarization and kinetic polarization.

9. 20A-2 Polarization Effects
Concentration Polarization
Concentration polarization occurs when reactant species do not arrive at the surface of the electrode or product species do not leave the surface of the electrode fast enough to maintain the desired current. When this happens, the current is limited to values less than that predicted by Equation 20-2.

10. 20A-2 Polarization Effects
Reactants are transported to the surface of an electrode by three mechanisms: (1) diffusion, (2) migration, and (3) convection. Products are removed from electrode surfaces in the same ways.

11. 20A-2 Polarization Effects
Diffusion.
When there is a concentration difference between two regions of a solution, ions or molecules move from the more concentrated region to the more dilute. This process, called diffusion.

12. 20A-2 Polarization Effects
The rate of diffusion is given by
where [Cd²⁺] is the reactant concentration in the bulk of the solution, [Cd²⁺]₀ is its equilibrium concentration at the surface of the cathode.
The value of [Cd²⁺]₀ at any instant is fixed by the potential of the electrode.
(20-4)

13. 20A-2 Polarization Effects
As the applied potential becomes more and more negative, [Cd²⁺]₀ becomes smaller and smaller.
The rate of diffusion and the current become correspondingly larger until the surface concentration falls to zero, and the maximum or limiting current is reached.

14. 20A-2 Polarization Effects
Migration.
The process by which ions move under the influence of an electric field is called migration.
Migration of analyte species is undesirable in most types of electrochemistry, and migration can be minimized by having a high concentration of an inert electrolyte, called a supporting electrolyte, present in the cell.
The supporting electrolyte also serves to reduce the resistance of the cell, which decreases the IR drop.

15. 20A-2 Polarization Effects
Convection.
Forced convection, such as stirring or agitation, will tend to decrease the thickness of the diffusion layer at the surface of an electrode and thus decrease concentration polarization.
Natural convection resulting from temperature or density differences also contributes to the transport of molecules to and from an electrode.

16. 20A-2 Polarization Effects
Kinetic Polarization
In kinetic polarization, the magnitude of the current is limited by the rate of one or both of the electrode reactions, that is, by the rate of electron transfer between the reactants and the electrodes.
To offset kinetic polarization, an additional potential, or overvoltage, is required to overcome the activation energy of the halfreaction.

17. 20A-2 Polarization Effects
Kinetic polarization is most pronounced for electrode processes that yield gaseous products and is often negligible for reactions that involve the deposition or solution of a metal.
Kinetic polarization also causes the potential of a galvanic cell to be smaller than the value calculated from the Nernst equation and the IR drop (Equation 20-2).

18. Feature 20-1 Overvoltage and the Lead-Acid Battery
If it were not for the high overvoltage of hydrogen on lead and lead oxide electrodes, the lead-acid storage batteries found in automobiles and trucks would not operate because of hydrogen formation at the cathode both during charging and use. Certain trace metals in the system lower this overvoltage and eventually lead to gassing, or hydrogen formation, which limits the lifetime of the battery. The basic difference between a battery with a 48-month warranty and a 72-month warranty is the concentration of these trace metals in the system.

19. Figure 20F-1
Figure 20F-1 The lead-acid storage battery.

20. Example 20-2
Is a quantitative separation of Cu²⁺ and Pb²⁺ by electrolytic deposition feasible in principle? If so, what range of cathode potentials (versus SCE) can be used? Assume that the sample solution is initially M in each ion and that quantitative removal of an ion is realized when only 1 part in 10,000 remains undeposited.

21. Example 20-2
In Appendix 4, we find that

22. Example 20-2
It is apparent that copper will begin to deposit before lead. Let us first calculate the potential required to decrease the Cu²⁺ concentration to 10¯⁴ of its original concentration (that is, to 1.00 × 10¯⁵ M). Substituting into the Nernst equation, we obtain

23. Example 20-2
Similarly, we can derive the potential at which lead begins to deposit:

24. Example 20-2
Therefore, if the cathode potential is maintained between V and –0.156 V (versus SHE), a quantitative separation should in theory occur. To convert these potentials to potentials relative to a saturated calomel electrode, we must subtract the reference electrode potential and write or

25. Example 20-2
Therefore, the cathode potential should be kept between – and – V versus the SCE to deposit Cu without depositing Pb.

26. 20B Are Electrolytic Methods Selective?
Calculations such as those in Example 20-2 make it possible to compute the differences in standard electrode potentials theoretically needed to determine one ion without interference from another.
The practical way of achieving separation of species whose electrode potentials differ by a few tenths of a volt is to monitor the cathode potential continuously against a reference electrode whose potential is known.
An analysis performed in this way is called a controlled-potential electrolysis or a potentiostatic electrolysis.

27. Figure 20-5
Figure 20-5 Apparatus for electrodeposition of metals without cathode-potential control.

28. 20C-1 Electrogravimetry without Potential Control
Applying Electrogravimetric Methods
In practice, electrolysis at a constant cell potential is limited to the separation of easily reduced cations from those that are more difficult to reduce than hydrogen ion or nitrate ion. The reason for this limitation is illustrated in Figure 20-6.

29. Figure 20-6
Figure 20-6 (a) Current; (b) IR drop and cathode potential change during electrolytic deposition of copper at a constant applied cell potential.

30. 20C-1 Electrogravimetry without Potential Control
The decrease in current and the increase in cathode potential is slowed at point B by the reduction of hydrogen ions. Because the solution contains a large excess of acid, the current is now no longer limited by concentration polarization, and codeposition of copper and hydrogen occurs simultaneously until the remainder of the copper ions is deposited. Under these conditions, the cathode is said to be depolarized by hydrogen ions.

31. 20C-1 Electrogravimetry without Potential Control
Codeposition of hydrogen during electrolysis often leads to formation of nonadherent deposits, which are unsatisfactory for analytical purposes.

32. 20C-2 Potentiostatic Gravimetry
A large negative drift in the cathode potential can be avoided by employing a three-electrode system, such as that shown in Figure 20-7.

33. Figure 20-7
Figure 20-7 Apparatus for controlled-potential, or potentiostatic, electrolysis. Contact C is adjusted as necessary to maintain the working electrode (cathode in this example) at a constant
potential. The current in the reference electrode is essentially zero at all times. Modern potentiostats are fully automatic and frequently computer controlled.

34. 20C-2 Potentiostatic Gravimetry
The electrolysis circuit consists of a dc source, a potentiometer (ACB) that permits continuous variation in the potential applied across the working electrode and a counter electrode, and a current meter.
The control circuit is made up of a reference electrode (often a SCE), a high-resistance digital voltmeter, and the working electrode.

35. 20C-2 Potentiostatic Gravimetry
The purpose of the control circuit is to monitor continuously the potential between the working electrode and the reference electrode and to maintain it at a constant value.
The current and the cell potential changes that occur in a typical constant-cathode-potential electrolysis are depicted in Figure 20-8.
The applied cell potential has to be decreased continuously throughout the electrolysis.
Controlled-potential electrolyses are generally performed with automated instruments called potentiostats.

36. 20D Coulometric Methods Of Analysis
Coulometric methods are performed by measuring the quantity of electrical charge (electrons) required to convert a sample of an analyte quantitatively to a different oxidation state.
Coulometric methods are as accurate as conventional gravimetric and volumetric procedures.

37. 20D-1 Determining the Quantity of Electrical Charge
number of coulombs (Q) resulting from a constant current of I amperes operated for t seconds is
For a variable current i,
The faraday is the quantity of charge that corresponds to one mole or × 10²³ electrons. The faraday also equals 96,485 C.
(20-5)
(20-6)

38. Example 20-3
A constant current of A is used to deposit copper at the cathode and oxygen at the anode of an electrolytic cell. Calculate the number of grams of each product formed in 15.2 min, assuming no other redox reaction.
The two half-reactions are

39. Example 20-3
Thus 1 mol of copper is equivalent to 2 mol of electrons, and 1 mol of oxygen corresponds to 4 mol of electrons.
Substituting into Equation 20-5 yields

40. Example 20-3
The masses of Cu and O₂ are given by

41. 20D-2 Characterizing Coulometric Methods
Two methods have been developed potentiostatic coulometry and amperostatic coulometry, or coulometric titrimetry. Potentiostatic methods are performed in much the same way as controlled-potential gravimetric methods.
However, the electrolysis current is recorded as a function of time to give a curve similar to curve B in Figure The analysis is then completed by integrating the current-time curve to obtain the number of coulombs.

42. 20D-2 Characterizing Coulometric Methods
In a coulometric procedure, the “reagent” is composed of electrons, and the “standard solution” is a constant current of known magnitude. Electrons are added to the analyte that immediately reacts with the analyte until the point of chemical equivalence is indicated.

43. Figure 20-9
Figure 20-9 Electrolysis cells for potentiostatic coulometry. Working electrode: (a) platinum gauze, (b) mercury pool. (Reprinted with permission from J. E. Harrar and C. L. Pomernacki, Anal. Chem., 1973, 45, 57. Copyright 1973 American Chemical Society.)

44. Figure 20-11
Figure Conceptual diagram of a coulometric titration apparatus. Commercial coulometric titrators are totally electronic and usually computer controlled.

45. Figure 20-12
Figure A typical coulometric titration cell.

46. 20D-5 Coulometric Titrations
Comparing Coulometric and Conventional Titrations
A coulometric titration offers several significant advantages over a conventional volumetric procedure. Principal among these is the elimination of the problems associated with the preparation, standardization, and storage of standard solutions.

47. 20D-5 Coulometric Titrations
Comparing Coulometric and Conventional Titrations
Coulometric methods also excel when small amounts of analyte.
A further advantage of the coulometric procedure is that a single constantcurrent source provides reagents for precipitation, complex formation, neutralization, or oxidation/reduction titrations.

48. 20E Voltammetry
Voltammetry consists of a group of electroanalytical methods in which information about the analyte is derived from measurement of current as a function of applied potential under conditions that encourage polarization of the indica-tor or working electrode.

49. 20E-1 Linear-Scan Voltammetry
Voltammetric Instrumentation
A three-electrode potentiostat, an all-electronic version of the one shown in Figure 20-7, is employed in linear-scan voltammetry. The potential sweep is provided by an electronic linear-sweep generator.
The cell is made up of the three electrodes immersed in a solution containing the analyte and an excess of a supporting electrolyte.
The potential of the microelectrode or working electrode is varied linearly with time.
The second electrode is a reference electrode whose potential remains invariant throughout the experiment.
The third electrode is the auxiliary or counter electrode.

50. 20E-1 Linear-Scan Voltammetry
What Microelectrodes Are Used?
The microelectrodes used in voltammetry take a variety of shapes and forms.

51. Figure 20-13

52. Figure 20-13(cont.)
Figure Some common types of microelectrodes: (a) a disk electrode, (b) a hanging mercury drop electrode, (c) a dropping mercury electrode, (d) a static mercury dropping electrode.

53. 20E-1 Linear-Scan Voltammetry
Describing the Voltammogram
The shape of a sigmoidal curve called a voltammetric wave. The constant current beyond the steep rise (point Z on Figure 20-15) is called the limiting current il.
Limiting currents are generally directly proportional to reactant concentration.

54. Figure 20-15
Figure Linear-scan voltammogram for the reduction of a hypothetical species A to give a product P. The limiting current il is proportional to the analyte concentration and is used for quantitative analysis. The half-wave potential E1/2 is related to the standard potential for the half-reaction and is often used for qualitative identification of species. The half-wave potential is the applied potential at which the current i is il/2.

55. 20E-1 Linear-Scan Voltammetry
To obtain reproducible limiting currents rapidly, (1)the solution or the microelectrode be in continuous and reproducible motion or (2) a dropping mercury electrode be used. Linear-scan voltammetry in which the solution is stirred or the electrode is rotated is called hydrodynamic voltammetry. Voltammetry with the dropping mercury electrode is called polarography.

56. 20E-1 Linear-Scan Voltammetry
Cyclic voltammetry is an example in which forward and reverse linear scans are applied. With cyclic voltammetry, products formed on the forward scan can be detected on the reverse scan if they have not moved away from the electrode or have not been altered by a chemical reaction.

57. 20E-2 Applying Voltammetric Methods
Voltammetric Detectors
Voltammetric detectors are widely used for detection and determination of oxidizable or reducible compounds or ions in flowing streams. Compounds that have been separated by liquid chromatography or are present in flow injection analyzers are typical examples.

58. Figure 20-16
Figure The Clark voltammetric oxygen sensor. Cathode reaction: O2 + 4H+ + 4e– H2O. Anodic reaction: Ag + Cl–AgCl(s) + e–.

59. 20E-2 Applying Voltammetric Methods
Voltammetric Sensors
When the oxygen sensor is immersed in a flowing or stirred solution of the analyte, oxygen diffuses through the membrane into the thin layer of electrolyte immediately adjacent to the disk cathode, where it diffuses to the electrode and is immediately reduced to water.

60. 20E-2 Applying Voltammetric Methods
Voltammetric Sensors
A number of enzyme-based voltammetric sensors are available commercially. An example is a glucose sensor that is widely used in clinical laboratories.

61. 20E-2 Applying Voltammetric Methods
Amperometric Titrations
Hydrodynamic voltammetry can be used to estimate the equivalence point of titrations, provided at least one of the participants or products of the reaction involved is electroactive.

62. Figure 20-17
Figure Typical amperometric titration curves: (a) analyte is reduced,
reagent is not; (b) reagent is reduced, analyte is not; (c) both reagent and analyte are reduced.

63. 20F Some Additional Electroanalytical Methods
Stripping methods
The analyte is first deposited on a microelectrode, usually from a stirred solution. After an accurately measured deposition period, the electrolysis is discontinued, the stirring is stopped, and the deposited analyte is determined by one of the voltammetric procedures described in the preceding section.

64. 20F Some Additional Electroanalytical Methods
In anodic stripping methods, the microelectrode behaves as a cathode during the electrodeposition step and as an anode during the stripping step. In cathodic stripping methods, the working electrode is an anode during deposition and a cathode during stripping.

65. THE END