## Electrolysis And Voltammetry

For chemists, the second important application of electrochemistry (beyond potentimetry) is the measurement of species-specific [e.g., iron(III) and iron(II)] concentrations. This is accomplished by an experiment whereby the electrolysis current for a specific species is independent of applied potential (within narrow limits) and controlled by mass transfer across a concentration gradient, such that it is directly proportional to concentration (i = kC). Although the contemporary methodology of choice is cyclic voltammetry, the foundation for all voltammetric techniques is polarography (discovered in 1922 by Professor Jaroslov Heyrovsky; awarded the Nobel Prize for Chemistry in 1959). Hence, a historical approach is used with a recognition that cyclic voltammetry will be the primary methodology for most chemists.

A. Principles and Fundamental Relations

### 1. Diffusion to a Planar Electrode

The basic approach in controlled-potential methods of electrochemistry is to control in some manner the potential of the working electrode while measuring the resultant current, usually as a function of time. When a potential sufficient to electrolyze the electroactive species completely is applied to the electrode at (t = 0), the concentration at the electrode surface is reduced to zero and an electrode process occurs, for example, ox + ne~ red, (45)

where ox and red represent an oxidized and reduced form of an electroactive species. Passage of current requires material to be transported to the electrode surface as well as away from it. Thus, relationships must be developed which involve the flux and diffusion of materials; this is appropriately accomplished by starting with Fick's second law of diffusion,

where D represents the diffusion coefficient, C represents the concentration of the electroactive species at a distance x from the electrode surface, and t represents the amount of time that the concentration gradient has existed. Through the use of Laplace transforms with initial and boundary conditions;

for t = 0 and x > 0 C = Cb, for t > 0 and x ^ 0 C ^ Cb, for t > 0 and x = 0 C = 0.

Equation (46) can be solved to give a relationship for concentration in terms of parameters x and t, bx

where Cb is the bulk concentration of the electroactive species.

By taking the derivative of Eq. (47) for the proper boundary condition, namely at the electrode surface (x = 0), the diffusion gradient at the electrode surface is expressed by the relation

This flux of material crossing the electrode boundary can be converted to current by the expression d C(0,t)

where n is the number of electrons involved in the electrode reaction, F is the faraday, and A is the area of the electrode. When Eq. (48) is substituted into this relation a complete expression for the current that results from semiinfinite linear diffusion is obtained (the cottrell equation for a planar electrode), nFACbD1/2

This relationship holds for any electrochemical process that involves semi-infinite linear diffusion and is the basis for a variety of electrochemical methods (e.g., polarogra-phy, voltammetry, and controlled-potential electrolysis). Equation (50) is the basic relationship used for solid-electrode voltammetry with a preset initial potential on a plateau region of the current-voltage curve. Its application requires that the electrode configuration be such that semi-infinite linear diffusion is the controlling condition for the mass-transfer process.

B. Voltammetry

### 1. Polarography

The most extensively studied form of voltammetry has been polarography (first described by Heyrovsky in 1922, with the quantitative relationships of current, potential, and time completed by the early 1930s with the assistance of associates such as Ilkovic). The potential-time dependence that is used for polarographic measurements is presented in Fig. 2 (solid line). The potential is scanned from Ei to E2 to obtain a current response that qualitatively and quantitatively characterizes the electroactive species present. The vast body of data from polarographic measurements can be adopted by other electroanalytical methods. Moreover, pulse polarographic methods and anodic stripping analysis, which are still used for determination of trace amounts of metal ions, are closely related to po-larography. The unique characteristic of polarography is its use of a dropping mercury electrode, such that the electrode surface is continuously renewed in a well-defined and regulated manner to give reproducible effective electrode areas as a function of time. The diffusion current equation [Eq. (50)] can be extended to include a dropping mercury electrode by appropriate substitution for the area of the electrode. Thus, the volume of the drop for a dropping mercury electrode is given by the relationship

where r is the radius of the drop of mercury, m is the mass flow rate of mercury from the orifice of the capillary, t is the life of the drop, and d is the density of mercury under the experimental conditions. When this equation is solved for r and the latter is substituted into the equation for the area of a sphere, an expression for the area of the dropping mercury electrode drop as a function of the experimental parameters is obtained, i =

FIGURE 2 Potential-time profile that is used for polarography and linear-sweep voltammetry (solid line) and cyclic voltammetry (both solid and dashed lines).

This then can be substituted into Eq. (50) to give a calculated diffusion current for the dropping mercury electrode,

Actually, experimental results indicate that the constant in Eq. (53) is too small by a factor of V7/3, due to the growth of the mercury drop into the solution away from the capillary orifice. Thus, the correct diffusion current expression for a dropping mercury electrode is i = 706nCbD1/2 m 2/311/6,

which gives the current at any time up to the lifetime of the drop. If the drop time, td, is substituted, the well-known Ilkovic equation results, id = 706nCbD1/2 m2/3 tj/6, (55)

where id is in A if D is in cm2 s-1, Cb is in mol cm-3, m in mg s-1, and t is in s. Alternatively, id is in ¡iA when Cb is in mM.

The polarographic current-potential wave that is illustrated by Fig. 3 conforms to the Nernst equation for reversible electrochemical processes. However, it is more convenient to express the concentrations at the electrode surface in terms of the current, i, and diffusion current, id. Because id is directly proportional to the concentration of the electroactive species in the bulk and i at any point on the curve is proportional to the amount of mate-

rial produced by the electrolysis reaction, these quantities can be directly related to the concentration of the species at the electrode surface. For a generic reduction process [Eq. (45)], the potential of the electrode is given by the Nernst equation

Initially, the solution contains only ox (concentration, Cob*). When a potenital is applied and reductive current flows, the oxidized form of the electroactive species diffuses toward the electrode. From Eq. (63) it follows that i = 706n[COx - Cox(0,t>]DOX2m2/31, = id - 706nCox(0,t)DO/2m2/3td/6.

Hence,

706nDOX2m2/311/6 '

After reduction of ox, its reduced form (red) diffuses either into the bulk of solution or into the mercury to form an amalgam. In either case, i = 706n [Cred(0,t) - Cbed] D1/m2/3td/6. (59)

FIGURE 3 Polarograms for (a) 0.5 mM CdN(OH2)2+ in 1 M HCl and (b) 1 M HCl

But the reduced form of the electroactive species was not present in the solution before electrolysis, therefore,

706nD1ed2m2/311/6'

Substitution of Eqs. (58) and (60) into Eq. (56) gives n RT (Dred E — E ° + — ln — nF \ Dox

nF i

At the half-height of a polarographic wave (i — id/2), the corresponding potential is defined as the half-wave potential (E1/2). Therefore, Eq. (61) takes the form

For the reduction of a simple solvated metal ion to its amalgam, E1/2 is given by

where yion is the activity coefficient for the ion and ya is the activity coefficient for the amalgamated species. The diffusion coefficients for the amalgam and ionic species also are a part of this expression. The standard reduction potential is for reduction of the ion to the amalgamated species. These expressions also hold for the reduction of an ion to a lower oxidation state, but require that the appropriate value be used.

### 2. Linear Sweep and Cyclic Voltammetry

The potential-time relation for voltammetric measurements is presented in Fig. 2. With linear-sweep voltamme-try, the potential is linearly increased between potentials E1 and E2. Cyclic voltammentry is an extension of linear-sweep voltammetry with the voltage scan reversed after the current maximum (peak) of the reduction process has been passed. The voltage is scanned negatively beyond the peak and then reversed in a linear positive sweep. Such a technique provides even more information about the properties and characteristics of the electrochemical process and also gives insight into any complicating side processes such as pre- and post-electron-transfer reactions as well as kinetic considerations. Whereas in classical polarog-raphy the voltage-scan rate is about 1 V min-1, linear-sweep voltammetry uses scan rates up to 100 V s-1 for conventional microelectrodes (and up to 10,000 V s-1 for ultra-microelectrodes; 10-6 m diameter).

Figure 4 illustrates the shape of a cyclic voltammo-gram with an electrode of fixed area. The voltammogram is characterized by a peak potential, Ep, at which the current reaches a maximum value, and by value of the peak current, ip. When the reduction process is reversible the peak current is given by the relation ip — 0.4463nFA( Da)1/2 Cb with a nFv ~RT

FIGURE 4 Linear voltage-sweep voltammogram with reversal of sweep direction to give a cyclic voltammogram. The initial sweep direction is to more negative potential.

where v is the scan rate in volts per second. This relation results from the set of differential equations for Fick's second law of diffusion (with the appropriate initial and boundary conditions for ox and rei ). Thus, in terms of the adjustable parameters the peak current is given by the Randles-Sevcik equation ip — 2.69 x 105n3/2ADl/2Cbv1/2

where ip is in A, A is in cm2, D is in cm2 s , Cb is in mol cm-3, and v is in V s-1.

Nicholson and Shain revolutionized the voltammetric experiment with their elegant development and demonstration of linear-sweep and cyclic voltammetry. In their approach, the current-potential curve is presented as i — nFACb (nDa)1/2 x (at ).

From tabulations for the relation between n1/2x (at) and n(E - E1/2) and converting the term n1/2x (at) to xrev, Eq. (67) takes the form i — nFACb ( Da)1/2 x™-

For a given potential (E) the value of xrev is obtained from tabulations.

For a reversible process the peak potential can be related to the polarographic half-wave potential, E1/2, by the expression

RT 0.0285

Another useful parameter of the voltammetric curves is the half-peak potential, Ep/2, which is the potential at which the registered current reaches half its maximum value and is used to characterize a voltammogram. For a reversible process, E1/2 is located halfway in between Ep and Ep/2.

The ratio of the peak current for the cathodic process relative to the peak current for the anodic process is equal to unity (ip,c/ ip,a = 1) for a reversible electrode process. To measure the peak current for the anodic process the extrapolated baseline going from the foot of the cathodic wave to the extension of this cathodic current beyond the peak must be used as a reference, as illustrated by Fig. 4.

For the condition

0.141

where Ex is the extent of the voltage sweep, the difference in the peak potentials between the anodic and ca-thodic processes of the reversible reaction is given by the relationship

0.059

which provides a rapid and convenient means to determine the number of electrons involved in the electrochemical reaction. For a reversible system, ip is a linear function of yV, and Ep is independent of v.

### C. Controlled-Potential Bulk Electrolysis

Because of the extensive amount of data available from the polarographic and voltammetric literature, the optimum conditions for macroscopic electrolyses often are established. In particular, controlled-potential electrolysis at a mercury pool can be approached with predictable success on the basis of available polarographic information for the system of interest. An electrolysis can be accelerated by maximizing the electrode surface area and minimizing the thickness of the diffusion layer. However, the same electrode material must be used as in polarography. Thus, a conventional approach in controlled-potential electrolysis is the use of a mercury pool stirred as vigorously as possible with a magnetic stirring bar to minimize the concentration gradient. Under such conditions the decay of n n the current as well as the decay of the concentration of the electroactive species is given by the relation it it=0

Ct=c

where V is the volume of the solution to be electrolyzed and Ax is the thickness of the concentration gradient. Thus, the current and concentration decay exponentially. Under idealized conditions, 90% of the electroactive species will be electrolyzed in approximately 20 min. Increases in the temperature as well as in the electrode area relative to the solution volume will accelerate the rate of electrolysis. The fundamentals of the controlled-potential bulk electrolysis are discussed in recent monographs (see Bibliography).

### D. Applications of Controlled-Potential Methods

To date, the most extensive application of electrochemical methods with controlled potential has been in the area of qualitative and quantitative analysis. Because a number of monographs have more than adequately reviewed the literature and outlined the conditions for specific applications, this material is not covered here. In particular, inorganic applications of polarography and voltammetry have been discussed in great detail in the classic treatise by Kolthoff and Lingane.

An important specialized type of voltammetric system is a self-contained cell for the determination of -O2- in the gas or solution phases. This is the so-called Clark electrode, which consists of a platinum or gold electrode in the end of a support rod that is covered by an -O2-permeable membrane (polyethylene or Teflon) such that a thin film of electrolyte is contained between the electrode surface and the membrane. A concentric tube provides the support for the membrane and the means to contain an electrolyte solution in contact with a silver-silver chloride reference electrode. The Clark device has found extensive application to monitor -O2- partial pressure in blood, in the atmosphere, and in sewage plants. By appropriate adjustment of the applied potential, it gives a voltammetric current plateau that is directly proportional to the O2 ■ partial pressure. The membrane material prevents interference from electroactive ions as well as from surface-contaminating biological materials.

In addition to the analytical applications discussed above, controlled-potential methods are used for the evaluation of thermodynamic data and diffusion coefficients in both aqueous and nonaqueous solvents. Polarographic and voltammetric methods provide a convenient and straightforward means to the evaluation of the diffusion coefficients in a variety of media. The requirements are that the current be diffusion controlled, the number of electrons in the electrode reaction be known, and the concentration of the electroactive species and the area of electrodes be known. With these conditions satisfied, diffusion coefficients can be evaluated rapidly over a range of temperatures and solution conditions.

Voltammetric methods also provide a convenient approach to establish the thermodynamic reversibility of an electrode reaction and for the evaluation of the electron stoichiometry for the electrode reaction. As outlined in earlier sections, the standard electrode potential, the dissociation constants of weak acids and bases, solubility products, and the formation constants of complex ions can be evaluated from polarographic half-wave potentials, if the electrode process is reversible. Furthermore, studies of half-wave potentials as a function of ligand concentration provide the means to determine the formula of a metal complex.

The techniques of voltage sweep and cyclic voltamme-try provide the analytical and physical chemical capabilities of classical voltammetry and, in addition, provide the means to perform these measurements much more rapidly for a broader range of conditions. Cyclic voltammetry is particularly useful for the rapid assessment of thermody-namic reversibility and for the evaluation of the stoichiom-etry for the electrode reaction.

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