Larry B. Anderson1 and Charles N. Reillev University o f North Carolina Chapel Hill
I
I
Teaching Electroclnalytical Chemistry Diffusion-controlled processes
Electroana~ytical methods are conveniently divided into two groups, static and dynamic. The static or zero-current methods such as potentiometry and pH measurements are firmly rooted in the basic concepts of equilibrium and mass action, and the rigor of their presentation at the undergraduate level may be considered adequate. The same may not he said about the electrodynamic techniques, such as polarography and chronopotentiometr?r. The Illcovic equation and the Sand equation are usually presented to and accepted by the student without proof. Considering the intensive current research in these and other dynamic methods and because of their promising applications in chemical kinetics, analysis, reaction mechanisms, electron-transfer reactions, and many other areas, they merit more than a passing mention in the analytical curriculum. Most standard texts currently used in analytical course* at the advanced undergraduate level contain discussions of electroanalytical techniques. The depth and breadth of quantitative treat,ment of these met,hods varies greatly from text to text, but nearly all discuss polarography and cite the Ilkovic equation. I n the more recent books, attention is also given to chronopotentiometry (the Sand equation) and chronoamperometry (the Cottrell equation), but in no case are these equations derived. Step-by-step derivations must be sought in the original research papers or in advanced monographs (1,s). The reasons for this omission are understandable. Diffusion problems are usually formulated in differential form (using appropriate forms of Fick's second law), and their solution has customarily followed the integral transfom techniques employed by engineers and phyeicists t ~ solve ) problems in heat conduction (5-5). Although these well-documented procedures are a vesy powerful-sometimes even indispensable-tool for solving differential equations, the focus on the mathematical a,spects and the time required for their mastery by students at this level would only serve t,o divert attention from the microscopic nature of diffusional processes. I n the present discussion, we outline a general approach for the solution of diffusion problems requiring no more mathematical sophistication than elementary integral calculu~. Primary emphasis is placed on the fact that diffusion is a completely random process. First, an elementary model of the diffusion process is taken which consists uf a group of molecules, initially concentrated in a narrow spike, and thereafter spread 1 Present address: Department of Chemistry, Ohio State University, Culumbus, Ohio.
by random thermal motion into a Gaussian distribution. From this basis, other initial concentration conditions are constructed by simple summation of a number of such elementary spikes, and flux conditions at boundaries are treated as addition of a given number of spikes of matcrial at such boundaries per unit time. The simple addition of the contribution from earh expanding Gaussian describes the total concentration of the diffusing molecules as a function of distance and time. I n this way, formulation of the diffusion problem in the differentialform is circumvented. I n fact, Fick's laws of diffusion are themselves derived as a ('onsequence of the random nature of diffusion. Molecules in Motion
Diffusion of a solute molerule in a dilute solutiou tnay be considered a linear process; that is, it is not sig~~ificantly influenced by the presence of other solute molecules of the same type. For example, the solute molecule does not react chemically with other molecules of the same type, and collisions between solute mole~~ules of the same type are infrequent compared with t,hc number of collisions between the solute molecules and the solvent molecules. The movement of si~lutc molecules in the liquid is due solely to the thermal or Brownian mot,ion of the solvent. Figure 1 illustrates t,he alttion of the random thermd process on a group of solute molecules, originally locatcd at a position xo on the x-axis. The molecules are free to move an infinite distance t,o the right and an infinite distance to the left. After a time, At/2, half of the molecules will have diffused an average distance, Ax, from the center. Of course, as many move to the right as to the left. At successively longer times, Brownian motion will have caused a further net movement of the n~oleculesfrom the area of high concentration near the center, to the area of low conreut~ration near the edges. The histograms (bar graphs) in Figure 1 detail this exercisc in binomial mobahilities. From statistiral arguments (e.g., as outlined by Keeping (6) or Band (7), it may be proved that the histogram approximates more and more closely a normal or Gaussian distribution as t/At m when the number of molecules, No, is very large. A Gaussian is drawn as the smooth rurve in Figure 1and has the general formula
-
where No is the total number of moles in the orig~~lal spike of molecules, Ax cm wide, to is the time that the spike was initially formed, and D is the diffusion rocfficient. Equation (1) is converted into units of co~lVolume 44, Number I , Jonuory 1967
/
9
derived from statistical considerations in the following manner. Any concentration distribution may be easily described as a number of closely spaced spikes or impulse of molecules. For example, the concentration in Figure 2 has been schematically drawn as composed of impulses of molecules of varying height distributed a t intervals of Ax along the x-axis. The motion of the molecules depicted in Figure 2 is governed by the same selection rules as were applied to the molecules in .. Figure 1 . After some time, At, the molecules originally at X I , will have moved an average (root mean square) distance Ax. Because the driving force for the motion of these molecules is a random thermal process, exactly one half, N / 2 , of the molecules will have moved to the right and N / 2 to the left. A similar analysis applies to the moleculesoriginally located at the coordinate (xl Ax). Fick's First L a w : The net number of molecules moving between X I and X I Ax is the difference between the number of molecules moving to the right from X I and the number of molecules moving to the left from XI AX
+
+
+
Multiplying the rightrhand side of eqn. (5) b y ( A x ) % / ( A x ) = converts the number of moles into concentration flux per unit area
Figure 1. Schematic representation of the diffurian process. Left: Molecvier dispersed by random process Right: Hirtogramr show that molesulo count opprorimoter a Gourdan distribution.
centration by dividing both sides by the original volume, A A x
The iutermediate goal of mathematically describing the diffusion of molecules as a function of distance and time is achieved in eqn. ( 2 ) . Of course, we have assumed an intial distribution of molecules that approximat,es an impulse function (the narrow spike of molecules a t t = 0 in Figure 1 ) andunrestricted diffusion to the right and left of x = xo. I n subsequent discussion, we will consider how to use this equation to solve other prectical diffusion problems.
=
-
az
2At
(6)
where C = N / ( A A x ) . The first term in brackets on the right-hand side of eqn. (6) is a function of the average distance the molecules have moved, Ax, in a given time, At. It is a constant, characteristic of the particular system of solute and solvent molecules and is called the dlusion coefficient, D (9). By allowing Ax 0 and At -+ 0 in eqn. (6), we obtain the differential form, Fick's First Law.
-
Fick's Second Law: The second law may be derived from eqn. (7) by noting in Figure 2 that the change in concentration per unit time a t x2 will be equal to the difference between the net fluxes from the left and from the right of x2.
Derivation of Fick's Laws
Before continuing with development of solutions in an integrated form, it is instructive to see how the random motion of molecules may be expressed in a differential manner. The differential equations describing diffusional processes were first cited by Fick ( a ) , who recognized that Fourier's equations describing heat transfer ( 8 a ) could also be applied to diffusional mass transfer. A summary of Fick's contributions has been presented by Tyrrell (8b). Fick's Laws may be 10 /
Journal of Chemical Education
Figure 2. Schematic representation of an arbitrary mncentration proflle as closely spaced spikes (or impulrer) of molecule..
Chronoamperometry
when Ax Law.
+0
and At + 0, this becomes Fick's Second
or because D is assumed independent of x and t,
I n these derivations, the diffusion of molecules was treated as a probability or stochastic process (9). From such a model, two equivalent representations were derived for the change of concentration as a function of distance and time [eqns. (2) and (9)l. Equation (2), the Gaumian distribution, is usually obtained analytically as a solution of equ. (9), and the integro-differential relationship between them is easily shown by suitably differentiating eqn. (2)
The concentration a t the beginning of a chronoamperometric experiment is schematically drawn as line A in Figure 3. The concentration to the right of x = 0, the electrode surface, is a constant, Cn, and the concentration to the left is zero. At time, t > 0, the concentration of the electroactive species is held at zero a t the electrode surface. Of course, the electroactive material is free to diffuse toward the electrode from the right, but as soon as it arrives at the electrode surface, it is immediately electrolyzed. A short time after the electrolysis has begun, the concentration distribution to the right of the electrode will resemble line C in Figure 3. In order to derive the expression for this concentration and the current that flows a t the electrode, we must first consider what would happen if there were no electrode at so.
and
A more thorough treatment of stochastic processes may be found in Bailey (9). Similar techniques have also been used to describe the effects of diffusional processes in gas chromatography (10). In a practical diffusion experiment, the concentration profile a t time, t = 0, is not usually the impulse function shown in Figure 1, nor, in general, is the subsequent movement of molecules to the right and left of xo completely unrestricted. For instance, in an electrolysis experiment a t a planar electrode surface, the initial concentration is usually uniform throughout the solu-
Figure 4. Diffusion from a region of uniform concentration (right) to o region of rero concentration (left). A, initial concentration profile; B, concentration profile offer diffusion has occurred.
I n Figure 4, the initial concentration (line A) has been divided into a series of narrow spikes of concentrsr tion, Ax cm wide and C units high-that is, into a series of impulse functions. The first impulse is centered a t x = 0, the second a t x = Ax, the third a t x = 2Ax, etc. The molecules composing each impulse will be spread symmetrically about their center of origin by Brownian motion, and their Gaussian shape a t time, t, may be derived from eqn. (2). Because each molecule moves independently of the rest, the total concentration a t any distance, x, and time, t, will be simply the sum of the contrihitions to the concentrsr tion from each of the Gaussians: Figure 3. Diffusion into a region of rero concentration. A, initial concentration; B, concentration ot some later time; C, line B minvr the concentration represented by the hatched area.
tion, and the electrode surface acts as a barrier to the random motion of molecules. I n the remaining portion of this article, we will discuss the methods for using eqn. (2) to generate directly, in terms of x and t, mathematical expressions for the concentration in such practical diffusion experiments.
C(..e
=
2 m t [exp
z- 0 ) (- (4Dt ')
+ exp ( - ( x & t k ) 2 )
+ .. .
(10)
Or, in a more concise notation,
Making the substitution z = - (x - m ~ r ) / 2 6 and t Az = Ax/2&, we obtain Volume 44, Number
I, Jonuory 1967
/
11
As the interval Az approaches zero, the limit of Cc,,,)is an integral.
or by remembering the definition of erfc [ ( - x ) / ( " d 6 ) ] and noting the fact that e r f [ ( - x ) / ( 2 f i ) ] = - erfW ( 2 f i ) I,
This integral may he divided into two integrals
The first integral may be obtained from a table of standard integrals, and, by inverting the limits of the second integral, it is seen to he the negative of the error function of - x / 2 f i (see ref. (2), Chapter 3).
For convenience of expressing eqn. (15), (1 - erf( - x / 2 f i ) ) is defined as erfc(-x/2d&):
which is the well-known result of unrestricted diffusion into a region of zero concentration, and i t is graphed as line B in Figure 3. Since eqn. (16) was obtained from a linear combination of Gaussians, then it mnst also be a particular solution of Ficlc's second law, eqn. (9). This may be verified by taking the appropriate derive tives of eqn. (16).
This is the intuitively obvious result if all of the molecules are confined to the right-hand side of x = 0. The concentration does not change. I n the case of a chronoamperometric experiment,, xi-e are confronted by a different condition at the origin. An electrode, situated at x = 0, immediately electrolyzes any electroactive material that arrives at. its surface after time, 1 = 0. Of course, all of the electroactive material represented by the portion of line B to the left of the origin in Figure 3 would have been electrolyzed at the electrode. I n addition, because of the symmetry conditions discussed earlier, there would be an identical set of molecules to the right of the origin (the hatched area in Fig. 3) that had arrived at x = 0 and are now found on the right. These, too, mould have been removed from the solution upon contact,ing the electrode. The concentration in the chronoamperometric experiment will, therefore, be given by the difference between the mathematical expression for line B to the right of zero minus the mathematiral expression for line B to the left of zero.
Introduction of Boundaries
Brownian motion of the solvent is the driving force for the movement of the molecules represented in Figure 3 from their initial distribution, A, to their distribution, B, at time, t, if there is no barrier (or a totally permeable harrier) at x = 0. At any instant it is equally as likely that a given molecule will be moving to the right as the left. That is, the statist,ical probability that a given molecule will next move in the right-hand direction is 0.5 and the probability that it will move in the opposite direction is the same. For every molecule that arrives at x = 0 and goes to the left. there is another that arrives and gocs to the right. The shaded portion to the right of zero in Figure 3 represents molecules that have reached zero and are now found on the right. It is the exact mirror image of the distribution of molecules found on the left of zero at time, t. If there is a boundary at x = 0 (such as a wall or an active electrode), eqn. (16) mnst he modified so thnt the condition at the boundary is satisfied. For example consider the possibility of an inert glass barrier at the origin. Under this condition, the molecules represented by line B to the left of the origin in Fi,gure 3 would never have crossed the origin but would have been reflected back to the right whenever they arrived at x = 0. Thus, the correct concentration distribution will be obtained from eqn. (16) by adding the mathematical expression for the concentration to the left of the origin to the expression for the concentration to the right. 12
/
Journal of Chemical
Education
This result is identical with eqn. (3-4) of Delahay (2). A fundamental distinction may be made between the boundary conditions which define the concentration at some point in space, such as the conditions employed in the derivation of eqn. (19), and those boundary conditions that specify the Rux a t some point in space. I t may be shown (10a) that a concentration or Direchlet condition yields a unique, stable solution for a parabolic equation such as eqn. (9). On the other hand, a flux or Neumann condition yields an infinity of stable solutions, differing at most by a constant. It will be necessary to use Neumann conditions later in our derivation of the equations describing chronopotentiometry. The CoWrell Equation
The eurrenf; flowing at the electrode in the chronoamperometric experiment is related to the flux defined by eqn. (6)
and by taking t,he indicated derivative of eqn. (19), we obtain:
Table 1.
Recent Applications of Chronoomperometry
System \'ariow metal ions I? 3 3 21Cd(I1) -r- Cd(F1g)
Application Determination of diffusion eoeffidents Detection of iodine adsorbed on a Pt electrode using an integrated form of eqn. (16) Establish electrode kinetic parameters. k. and a
Table 2.
ence
(18)
\'?riu~w met,al IO~S
Reference
Determinstiuns of diffmion coefficienta
(18) and
*I
Ni(F1rO)Z'.
Kinetics of
HNF.
Mechanism of the reduction of difluoramine
@41
(19) d
. -a
2H +
I '?, ->
Dibenathiophene Co(en)? +2r
-
Application
System
-NH. .
P h ( I I ) ~ P h ( H g Adsorption ) stttdies in presence of I-, Br-, C1-, and SCN-
Recent Applications of Polarography
Refer-
(dl)
Olefins Peroxidase and cat,alase
(38
Correlations with LCAO-MO theory General investigation of electrochemistry of organometallics Electrolytic reductive coupling Electrochemistry of enzymes
+2e-
Al;ahew,ene
.C2HI
Hydrasobenzene +I%+
Bensidine Electrochemiluminescence
Study uf the rate of the benaidine rearrangement (Cottrell equation constitutes the starling point of the method) Modification of general technique outlined in eqn. (16) as applied to electrogenertllion of chemih~minescence
(%2)
(M 1
This may be rearranged to give the Cottrell equation
However, this equation predicts a diffusion current that is too low, and multiplication of this equation by a constant factor of (7/3)'12 is found to provide a closer fit to the instantaneous polarographic current. When the constants are gathered together, the result is the familiar Ilkovic equation. i = 708 nms/s t'/sD'/C" (27)
A very brief list of current research utilizing polarography as a tool is presented in Table 2. Chronopotentiometry
and some recent applications of chronoamperometry are presented in Table 1.
and then calcnlat,ingA:
Chronopotentiometry or voltammetry a t controlled current is carried out by applying a current to the indicator electrode and observing the change of the potential with time. When some electroactive species is present in the solution, the potential swings rapidly to that value where this species begins to be electrolyzed. Thereafter, the potential changes move slowly because the removal of electroactive species by the flow of current is partially balanced by the delivery of more material to the electrode by diffusion. Eventually, however, at some time, r , the rate of transfer of material to the electrode will become so small that the concentration of the electroactive species at the electrode will be reduced to zero, and the potential will swing rapidly to a value where some other electrode reaction can satisfy the current condition. Thus, it is necessary to develop a mathematical method for dealing with the effect of the constant current boundary condition on the concentration. The production (or removal) of the electroactive species a t the electrode corresponds to the addition (or subtraction) of a certain number of moles of that species per unit area per unit time at time, t,. I n terms of the Gaussian, eqn. (2), this may be treated as the addition of a Gaussian containing N(,,) moles in a volume AAx, at x = 0,per unit time, At.
The direct substitution of this arca into eqn. (22) gives the following approximate expression for the diffusion current in polarography :
This Gaussian will spread in the manner described in eqn. (2) (where C(,.,-h, = N(t-h)/AAx) and shown inFignre5:
The Ilkovic Equation
The many practical advantages of polarography ( I ) make it desirable to develop an expression for the current at an expanding spherical electrode when the surface concentration is held at zero. Several approximations to the solution of this formidable problem have been discussed by Kolthoff and Lingane (1). One of these solutions may he readily derived from the Cottrell eqn. (22). The volume of an expanding mercury drop is related to the Hg flow rat,e, m, in g per see and the time, t, since the drop began to form a t the capillary tip, by the following relation:
where ro is the radius of the drop at time, t, and d is the density of mercury. The surface area of the drop, A, is easily related to the volume by first solving for ro,
Volume 44, Number I , January 1967
/
13
w Ck.crm,= N -X Aaz
1
2 d U D ( t - t,) eXP [ a;? t d l
Or, replacing the number of moles by the current expression from eqn. (28) :
This result is identical with the general expression presented by Reinmuth (11), usually derived by Laplace transform methods. By substitution of appropriate current functions, i(z), and performing the indicated integration, solutions for many of the common electrochemical techniques may he obtained. Constant Current-The
where i ( k ) is the value of the current a t the time of addition of the Gaussian, t, and At is the time interval during which i(L) flows. Because any current function may he thought of as a composite of many such Gaussians, the concentration of the substance a t any distance from the origin and any time, C(x, t ) , will he the sum of the magnitude of the concentration, a t distance z, contributed by each G a u s sian added up to time, t, (Fig. 6). Of course, the time elapsed since addition will determine the amount that any given Gaussian has spread.
Sand Equation
When the current in eqn. (33) is a constant, the concentration a t the electrode surface, x = 0,is
Integrating this expression gives, for the concentration of the product of the electrode reaction Zit'/. n5AcD
CIO.II =
At time, 7 , the concentration of the reactant species becomes zero a t the electrode surface and the product concentration equals CO, resulting in the Sand equation:
Chronopotentiometry and the Sand equation have been
Making the substitution z = kAt and allowing Az to approach zero, eqn. (31) becomes
In order for this solution to apply to the physically realizable generating electrode (which is also a reflecting harrier), eqn. (32) must be reflected a t x = 0 and the two halves added together as was done to obtain eqn. (18).
-x Figure 5. tion.
14
/
Spreading of o spike
" d
molecvler at various timer after oddi-
Journol of Chemicol Education
Figure 6. Generation of material at a constant rote at on electrode surface. An impulse of molecules i, added each At second. Broken line represents total concentration.
Table 3.
Recent Applications of Chronopotentiometry
System II(Pd) Hz0
+ OR-+
Reference
Application -a-
Determination of OH-concentration and diffusion coefficient of OHPb in PhCll and Zn in Determination of metals disZnC1. solved in their molten salts
($1) (St)
-4e-
H -N=N-H Products
-Be-
HONIT* +Praducts
Determination of hydrazine and hydroxylamine in mixtwes mism of the re.cid Ion c* I and
Ferrocene oxidation
Electrochemistry of organic compounds dissolved in carbon paste electrodes
(3s)
(34) 1.96) \-,
(58)
+ze-
On.
HnO?
Studies of surface effects and ageing on oxygen reduction at platinum Adsorption studies
Phz+ in C1; Br-, I-, and SCN- media p-Aminophenal rednc- Characterization of chemical tion kinetics Iron-probporphyrins Electrochemioal studies of F e proto-porphyrin complexes with CN-. ovridine. ete.
($7)
($1 (38) ($9)
employed to characterize electrochemically a number of different chemical reactions, several of which are presented in Table 3. First-Order Coupled Chemical Reaction
When the concentration of the product of an electrochemical reaction is a function of time, a modified form of eqn. (2) must be used to describe properly the change in the total area of the Gaussian with time. For example, a first-order following chemical reaction, 0 neR 2 Products, results in R Gaussian whose total area is an exponential function of time.
+
-
The appropriate form of the concentration function corresponding to eqn. (33) is
-" exp [-k(t
- z)ldz
(36)
This addition, obviously, complicates the process of integration, but for many specific current functions, solution is still possible. Extension of the Method
Application of the linear transformation technique to the solution of more complicated boundary value problems is feasible in many cases. However, there are limitations to the method, and some suggestions may be helpful to the person interested in continuing further. The concentration response to various forms of steppotential and step-current programs (1B,13, dB) may be derived from modest extensions of the techniques de-
scribed here (such as substitution into eqn. (33) and integration). Treatment of finite boundary problems by folding techniques has proved to be a relatively simple and direct method of solution (14, 15). The solutions for these thin-layer electrochemical techniques are most conveniently expressed in terms of an infinite series, which follows directly from reflection and superposition, but is not obtained nearly as directly if the problem is solved by Laplace transform or operational calculus methods. The linear transform methods may not be used if a nonlinearity such as the changing potential in stripping voltammetry, is introduced as a boundary condition (16). Under some circumstances, e.g., A.C. polarography, the amplitude of the potential excursion may be sufficiently small that a linear approximation is valid. In these cases, the current excitation may be substituted directly into eqn. (33) (11). The mathematical validity of summation of Gaussiaus and of reflection and superposition is retained if they are applied to the diffusion equations for cylindrical or spherical geometries. However, the mechanics of solution rapidly increase in complexity, and any insight into the behavior of the diffusing molecule becomes more difficult. The elegant simplicity of the pictorial representations of Figures 2 and 3 is lost. Perhaps other geometries are more expeditiously handled by other methods (4) if it is experimentally necessary to deviate from conditions of planar diffusion. A generalization of eqn. (33) to include electron transfer between or within a multi-component chemical kinetic sub-system would be possible (17). On the other hand, it is not possible, at present, to obtain solutions in closed form for coupled chemical reactions of second or higher order, and we have stated at the outset that linear transformations may be validly applied, in general, only to linear equations. From the preceding examples, it is evident that a considerable number of laborious derivations could be greatly shortened by judicious choice of a linear transformation of a known solution to a related boundary value problem. Nonetheless, the principal application of the techniques outlined here would appear to be as a simple, graphic, and instrnctive tool for teaching M u sion-controlled methods in electroanalytical chemistry. Acknowledgment
The authors wish to thank J. W. Ashley, Jr., for his helpful and clarifying discussions of many of the subjects discussed in this paper. Literature Cited
(1) KOLTHOFF, I. M., AND LINGANE, J. J., " P d ~ r ~ g r ~ ~ p 2ud hy,'' Ed., Interscience Puhlkhers, (divkion of John Wiley & Sons, Inc.), New York, 1952. (2) DELAHAY,P., "New Instrumental Methods in Eleet,ruchemistry," Interscience Poblishem, (division of John Wiley & Sons, Inc.), New York, 1954. of that in (3) CARSLAW, H. S., AND JAGER,J. C., "Condu~ti~u Solids," 2nd Ed., Oxford, Clsrendon Prws, 1959. (4) Jam, W., "Diffusion," Academic Press, New York, 1960, 1). 16H. (5) CRANK, J., "The Mathematics of l ) i f i ~ ~ ~ i ~Oxfwd, n , ' ' The liniversity Press, London and New York, 1956. (6) KEEPING,E. S., "Introduction to Statist,id Infeve~~m," D. Van Nostrand Co., Inc., Princeton, N. J., 1962, 1,. 6411. Volume 44, Number 1 , January 1967
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15
(7) RAND,W., "Introduction to Mathematical Physics," D. Van Nostrand Co., Inc., Princet,on, N. J., 1959, p. 109ff, 163ff. (8) Rcn, A., Annaln der P k p i k (Pogg.) 94, 59 (18.55). (a) FOURIER, J., "The Analytical Theory of Heat,"A. FREEMAN, translator, Ihver Puhlicatiuns, New York, 1955. ( b ) TYRELL, H.J., J. CHEM.EDUC.41, 397 (1964). (!I) 9-~ILEY, N. T., The Elements of Stochastic Processe~," John Wiley & Sons, h e . , New York, 1964, p. 198R. ( 1 0 ) GIDDINGS, J. C.. "1)ynarnics of Chromatography," Vol. I, Part I, Marcel Dekker, Ine., New York, N. Y., 1965, p. 227ff. (a) HILDEBRIND,P. B., "Xlethod~of Applied
Mathematics," Prentice-Hall, N. Y., 1952. (11) KEINMUTH, W. H., Anal. Chem., 34, 1446 (1962). (12) I ~ E R M AH. N , B., .ANDB.ARD,A. J., Anal. C h a . , 35, 1121 119631. , (I.?) REILLEY, C. N., AND MUHIIIY,R. W., "Treatise on Analytical Chemistry," Vol. 4, I. M. KOLTROFF and P. J. ELYING, Eds., Interscience Publishers, (a diviiion of John Wiley & Sons, Inc.), New Ywk, 1963, pp. 2225-8. (14) OGLESBY, D. M., OMANG, S. H., .NU REILLEY,C . N., Amd. Ckem., 37, 1312 (196.5).
.
(15)
ANDERSON, L. B.,
AND
~IEILLEY, C. N., J.Eleclroana1. Chem.,
10., 538~,~ 1196Ri. ,
116) I l ~ V n m sW. , T., A N D V.\N DLEN, E., J.Electroanal. Ckem., 8, 366 (1964). (17) ASHLEY,J. W., JR., AND REILLEY,C. N., J . Electroanal. Ckem., 7, 253 (1964).
Applications of Chronoomperometry 118) MACERO, D. J., AND RULFS,C. L., J . Am. Ckem. Soc., 81, 2942 119591. ~, (I!)) CHRISTIE, J . H., L ~ E HG., , :AND OSTERYOUNG, 12. A,, J . Ekctroanal. Chem., 7, 60 (1964). ('20) ANSON,F. C., Anal. Ckem., 38,M : (1966). i l l ) MURRAY, R. W., A N D GROSS,11. J., Anal. Chm., 38, 392 (1966).
16
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Journal of Chemical Education
( 2 2 ) SCHWARZ, W. M., AND SHAIN,I., J . Pkys. Ckem., 69, 30 (1965). (23) FELDBERG, S. W., J . Am. Chem. Soc., 88, 390 (1966). Applicotions of Polorography (24) TURNHAM, D. S., J . Ekdrvanal. Chem., 10, 19 (1965). (25) HUSA,M. S., AND SCARROTT, J . W., J. Eleelroanal. Chem., 7, 26 (1964). (26) WARD,G. A,, WRIGHT,C. M., I N D CRAIG,A. D., J . Am. Chem. Soe., 88, 213 (1966). (27) GERDEL,R., AND LUCXEN, El A, C., J. Am. Ckm.Soe., 88, 733 (1966). (28) I ~ S S Y ,R. E., ET AL.,J. Am. Chem. Soe., 88, 453, 460,467, 471 1Q6RI .. .(*-..--,.
(29) OILT,M. R.,
AND
BAIZER,M. M., J . Org. C k a . , 31, 1646
llORRI
(30) GUILRAULT, G. G., Anal. Riockm., 14, 61 (1966).
Applicotions of Chronopotentiometry (31) BLACKBURN, T. R., Anal. Ckem., 38, 619 (1966). (32) VIN NORMAN, J . D., BOOKLESS,J. S., AND EGIN, J. J., J . Pkys. Cham., 70, 1276 (1966). (33) MORRIS,M. D., A N D LINGANE, J . J.. J. Eledmanal. Ckem.. 8. 85 (1964). (34) DESIDERI,P. G., J. Eketroanal. Chem., 9, 218 (1965). (35) SCEULTZ, F. A., AND KI'WANA,T., J. Ekdroanal. Chon., 10, 95 (19651 , , (36) PETERS,D. G., AND FRANKLIN, L. A., J . Ekct~oanal.Ckcm., 9, 385 (1965). (37) PETERS,D. G., AND MITCHELL,R. A,, J . Electroanal. Chem., 10, 306 (1965). (38) HERMAN, H. B., AND B.\RD, A. J., J. Phys Chem., 70,396 (1965). (39) DAVIS, D. G., AND MARTIN,R. F., J . Am. Ckem. Sac., 88, I365 (1966).
