Semiintegral electroanalysis. Theory and verification

monoxide and another portion of the sample is passed through the oxidation train.

The procedure is basically the same as for catalytic oxidation except the flow rate is decreased to 20 ml/min and the oxidation time increased to 40 minutes. Also, the computation of the mole fraction (Equation 4) must reflect the fact that only carbon monoxide and carbon dioxide are involved in the calculation. The radioactive concentration of the hydrocarbon 01-0001-324.

components is then determined by taking the difference between the catalytic and copper oxide results.

RECEIVED for review September 30,1971. Accepted February 16, 1972. This paper is based on work conducted by Bitu- minous Coal Research, Inc., for the Office of Coal Research, U.S. Department of the Interior, under Contract No. 14-

Se m i i n t eg r a I E I ec t r oa n a I y s i s :

Morten Grenness’ and Keith B. Oldham Trent University, Peterborough, Canada

Theory is presented which describes the properties of m, the semiintegral of‘ the faradaic current which flows when complete diffusion control follows a pre- existing equilibrium. According to the predictions of the theory, m will be proportional to the concentration of the electroactive species, to the square root of its diffusion coefficient, to the number of electrons involved, and to the area of the electrode. The theory has been verified using the reduction of Cd2+ and O2 at a mercury electrode. These studies demonstrate that the semiintegral electroanalysis method is valid for the measurement of concentrations and diffusion coefficients, and possesses a number of novel and potentially attractive features.

A PRELIMINARY ANNOUNCEMENT of the new analytical method “semiintegral electroanalysis” has already been made (I) ; the present article presents the theory in some detail together with an experimental verification.

THEORY

Consider an electroreducible species Ox to be dissolved in solution at a concentration C. Two or three electrodes are immersed in the solution, one of these being the working electrode at which the electrode reaction

(1)

is possible. Initially, however, this reaction does not occur, either because the working electrode is polarized at a potential sufficiently positive to inhibit reaction 1 or simply because the circuit is open. Commencing at time t = 0, a signal is applied to the working electrode as a result (though not neces- sarily as an immediate result) of which reaction 1 occurs. If transport of Ox to the working electrode is solely as a result of semiinfinite linear diffusion with D being the diffusion coefficient, then the equations

Ox + ne- -+ Rd

Theory and Verification

C( co ,t> = c (3)

Present address, School of Mathematics and Physics Sciences, The University of Sussex, Falmer, Brighton, England.

(1) K. B. Oldham, ANAL. CHEM., 44, 196 (1972).

and

C(r,O) = C (4)

apply where C(r,t) denotes the concentration of Ox at distance r from the electrode surface at time t . Moreover, the flux of Ox at the surface of the working electrode is given by

a D - C(0,t) = - br nAF

where A is the electrode area, F is Faraday’s constant and i( t) denotes the cathodic faradaic current. Now, it has been demonstrated rigorously (2) that the relationship

is a direct consequence of Equations 2 through 5 . denotes the semiintegral

Here m(t)

(7)

of the faradaic current i( t) with respect to time. The operation of semiintegration has been explained in the

electrochemical literature (2, 3) ; the result of this operation applied to i( t) is to produce a quantity m(t) intermediate between the current i( t) and the integral

of the current, i.e., the charge passed. As an alternative to regarding m(t) as the semiintegral of the current, as defined in Equation 7, it may, with equal validity, be regarded as the semiderivative (2 ) of the charge,

To exemplify the operations of semiintegration and semidifferentiation, Table I lists some particularly simple instances of i(t), together with the corresponding m(t) and q(t) functions. The interrelationship of these three quantities is summarized in the diagram

(2) K. B. Oldham and J. Spanier, J. Electroanal. Chem. Interfacial

(3) K. B. Oldham, ANAL, CHEM., 41, 1904, (1969). Electrochem., 26, 331 (1970).

ANALYTICAL CHEMISTRY, VOL. 44, NO. 7, JUNE 1972 1121

Table I. Examples of m(t) Functions and the Corresponding Current and Charge Functions~

abr

C z abtz

2 -

2u a .\/?rb d u w ( f i ) (1 - e-bt)

a u(0) denotes the unit step function at time t = 0. daw(y) denotes Dawson’s integral of y (M. Abramowitz and I. A. Stegun, “Handbook of Mathematical Functions, Applied Mathematics Series No. 55,” National Bureau of Standards, U.S. Government Printing Office, Washington, D.C., 1964, p 208). a, b, and c are constants with dimensions ampere, second-’ and amplomb, respectively.

integration i(t)

which clearly brings out the intermediacy of m(t) between current and charge. Another aspect of this intermediacy is that m(t) is measurable in the “amplomb” unit,

1 amplomb = 1 ampere sec1/2 = 1 coulomb sec-1/2

Relationship 6 is valid whatever signal is applied to the cell, but if the signal is such that after a time r the potential of the working electrode has reached such a negative value that the concentration of Ox at its surface is diminished to virtually zero, i.e. C(O,7) = 0, then

m(7) = n A F C ‘ d 5 (10) follows on rearrangement. This equation, demonstrating the proportionality between m(7) and concentration, constitutes the basis of semiintegral electroanalysis.

Few conditions were assumed in the derivation of Equation 10 and it is worthwhile emphasizing some of the assumptions that it is not necessary to make. Therefore note that

(i) The initial presence or absence of Rd is irrelevant. (ii) The nature of Ox (anion, cation, or molecule) is unimpor-

tant, the only requirement being its electroreducibility. The semiintegral electroanalysis of electrooxidizable species may be carried out by semiintegrating an anodic current in an exactly analogous way.

(iii) Whether Rd is soluble in the solution (or, in case the working electrode is of mercury and Rd is an amalgamable metal, in the electrode) or not is irrelevant.

(iv) The kinetics of the electrode reaction do not matter, the equations being applicable equally to reversible or irreversible processes.

(v) Whether or not the reduction process occurs in a single step is immaterial. For the oxygen reduction, for example, provided that at time the potential is sufficiently negative for complete reduction to OH-, Equation 10 applies with n = 4 even though at times less than r, hydrogen peroxide may have appeared as an intermediate product of the reaction.

(vi) The way in which the potential of the working electrode changes from its initial equilibrium value to the value corresponding to complete concentration polarization of Ox is irrelevant to the magnitude of m(7). At first encounter, this assertion may seem unacceptable to an electroanalyst accus- tomed to thinking in terms of currents. The situation becomes clearer if a parallel is drawn with the charging of a capacitor. The way which the potential across a capacitor changes from an uncharged condition to some value E(7) has no bearing on the magnitude of q(7), the integral of i ( t ) over the charging interval 7. Analogously, the way in which the potential across the working electrode changes from an initial equilibrium to some concentration-polarizing value has no bearing on the magnitude of m(r), the semiintegral of i(t) over the analysis interval r.

(vii) Any signal which achieves the desired change in the potential of the working electrode will validate Equation 10. The signal may involve control of potential, control of current, or some intermediate situation. The potential change may be mediated through the use of the three-electrode configuration which is now commonplace in electrochemical instrumenta- tion, but a two-electrode cell will suffice.

(viii) The time t = 7 need not correspond to a single instant. Usually there will exist a range of times over which concentration polarization is complete: r is any instant in this range. Indeed, this time range may even be infinite.

(ix) Where a plurality of t = r instants exists (Le., in the usual case where concentration polarization exists for a length of time), the potential of the working electrode may or may not be constant from one instant to the next. Provided only that its potentials all correspond to complete concentration polarization, the working electrode is free to vary in any way between successive t = r instants.

OELJECTIVES OF THE PRESENT STUDY

The experimental phase of this study was designed to verify different aspects of the theory of semiintegral electroanalysis, employing, for the most part, the electroreduction.

CdZ+(aq) + Ze-(Hg) -+ Cd(Hg) (1 1)

as the test system. The aim was to verify the following theo- retical predictions.

Prediction A. That, under a given set of experimental conditions, the same time-dependent function is produced by semiintegrating current as by semidifferentiating charge.

Prediction B. That, provided that the working mercury electrode acquires a final potential more negative than -670 millivolts us. SCE (corresponding to complete concentration polarization with respect to Cd2+), m(t) will achieve a final constant value m(r) which is independent of the way the potential changes from its initial value (-490 millivolts us. SCE at which Cd2+ ion is irreducible). The potential was changed in the following ways, which will be further clarified by reference to the diagrams in Figure 1.

(a) Simple ramp applied to the working electrode us. a

(6) Capped ramp applied to the working electrode us. a

(c) Sigmoid signal obtained by “degrading” a capped ramp

(4 Step applied to the working electrode us. a reference elec-

(e) Step applied to the series combination of a resistor and

reference electrode.

reference electrode.

by means of a capacitance.

trode.

the cell (working-reference electrode pair)

1122 ANALYTICAL CHEMISTRY, VOL. 44, NO. 7, JUNE 1972

m Ii I I 1 I I I I

. / O

0 t

Figure 1. Various signals applied to the cell in the study of the semiintegral electroanalysis of Cd2+ and O2

E,, denotes the potential of the working electrode us. a saturated calomel reference electrode in a three-electrode configuration and E,, the potential of the working electrode us. a mercury pool counter electrode in a two-electrode configuration. Ei, and E/ , , denote initial and fhal potentials, respectively

Rising exponential applied to the working electrode us. a reference electrode (by “degrading” a step by means of a capacitance). (g) Rising exponential applied to a two-electrode cell.

Moreover the values of m ( ~ ) should be independent of the signal parameters, Le., ramp rate, resistor value, rise time of the exponential, etc.

Prediction C. That, for a given value of C, the concentration of the electroactive species, m(7) is proportional to the area A of the working electrode.

Prediction D. That, for a given value of the area A , m ( ~ ) is proportional to the concentration C of the electroactive species.

Prediction E. That, other factors being maintained constant, m(7) is proportional to n, the number of electrons involved in the electroreduction.

Prediction F. The quantity [m(7)/nFACIz equals the diffusion coefficient of the electroactive species.

EXPERIMENTAL

Measurements of Transients. For the electroreduction of Cd2+, deoxygenated solutions of Cd(NO& of various concentrations in lOOmM KNO, were used (see Table 111). Also the electroreduction of 02 was studied, in which case mostly solutions of lOOmM KC1 saturated with either air or oxygen were used. The salts were generally of “laboratory grade” dissolved in singly distilled water, and no attempt was made to purify the stock solutions (other than by filtra-

/’ /’

t -200- a

I I tlmo (soconda)

0 0 4 2 3 4 5 6 7 s

Figure 2. Reduction of Cd2+. Application of a potential ramp to the working electrode Solution: 1.00mM Cd(NO& + lOOmM KNOa. Electrode area: 4.69 X 10- cmz

I. The potential ramp, E(?) II. The current-time transient, i(t) m. The current-time transient semiintegrated us. time

tion). All glassware was rinsed in alcoholic KOH solution followed by distilled water before use.

Throughout the study a Beckman No. 39016 Hanging Mercury Drop Electrode Assembly was used as the working electrode and by means of a micrometer with dial readout, calibrated in microliters, it was possible to meter the volume of the mercury drop sufficiently accurately. The cell was of a simple design used for polarography and the counter electrode was a pool of mercury in the bottom of the cell. Both “two-electrode’’ and, in most cases, “three-electrode” configurations were investigated. In the latter case, a filter plug type, saturated calomel reference electrode (Coleman type 3-712) was used and placed in a separate compartment connected to the main part via a Luggin capillary. Because of the low current ranges studied, the placing of the tip of the capillary was not critical. Throughout the experimental work, the cell was at a room temperature of 24 “C.

In order to apply the various potential-time profiles as illustrated in Figure 1, to the working electrode, a Princeton Applied Research “Model 170” (waveform generator and potentiostat) was used. The current-time transients were recorded on the X-Y recorder which is part of the instrument. The “Model 170” is provided with an analog integrating circuit which was used to record the charge-time transients in a similar way.

Calculation of m(t). From the recorded current-time transients, values i(O), i(A), QA),. . . .i(jA),. . . .i(t) were read off at evenly spaced time intervals. The algorithm

was then used in a computer program to determine values of m(t). This alogrithm, an example of an RLO algorithm, is derived from the Riemann-Liouville definition (2) of semiintegration and becomes exact as t / A approaches infinity.

In a similar way the algorithm

ANALYTICAL CHEMISTRY, VOL. 44, NO. 7, J U N E 1972 1123

Figure 3. Reduction of Cdz+. Application of a “capped” potential ramp to the working electrode

Solution: 1.00mM Cd(NO& + lOOmM KN03. Electrode area: 4.69 X 10- em2

I. Current-time transient, i(t) 11. Charge-time transient, q(r)

111. Current-time transient semiintegrated, m(r) IV. Charge-time transient semimerentiated, m(z)

an example of an RL1 algorithm, was used to semidifferentiate evenly-spaced q us. t data, obtained from the recorded charge-time transients.

The efficiency of these alogorithms in the present context was established by applying them to functions selected from Table I, upon which the operations of semiintegration and semidifferentiation act to yield functions of known values.

RESULTS AND DISCUSSION

Curve I1 on Figure 2 is a typical current-time transient recorded on application of the cathodic-going potential ramp, shown as I, to the working electrode (potential profile, case a, Figure 1). The solution was 1.00 mMCd(NO& + l00mM KNO,. As chronoamperometric theory predicts, virtually no current flows in the initial potential range, but at about -490 mV us. SCE there is a sharp rise in current, which then passes through a maximum, after which it decreases steadily. Curve I11 is the semiintegrated current-time transient, m(t), calculated by means of the computer program using the algorithm, Equation 12. Also here the increase starts at about -490 mV; m(t), however, does not go through a maximum but levels off at a plateau, almost independent of further negative increase of the potential.

Variation of Method for Determining m(r). Figure 3, curve I is another typical i-t transient as recorded, using the same Cd*+ solution. It is almost identical in shape to the previous one, even though a “capped” potential ramp (Figure 1, case b)

so - - 0 E

40-

f - E -

30 - E E

Expoud Electrode Area ( m d I I I I 3 6 9 I2 I5

Figure 4. Reduction of Cd2+. M ( T ) vs. exposed electrode area

m(7) is here defined as the value of m(t) at time of the end of ramp. Solution: 1.OOmM Cd(NO& + lOOmM KNOa. Ei, = -490 mV us. SCE, E/in = -670 mV us. SCE. Slope of line: 5.21 X ainplomb

was applied. The initial and final potentials were - 490 mV and -670 mV, respectively, us. SCE. Curve I1 on Figure 3 is a typical charge-time transient as recorded, applying the same capped ramp as above. The ramp speed was 50 mV/sec and the volume of the mercury drop 1.00 pl. Curves 111 and IV have been derived by semiintegrating curve I1 and semidifferentiating curve 11, respectively; the latter using the algorithm Equation 13. They clearly have the same shape but do not quite level off at the same height. Possible reasons for this include the inaccuracy of metering the volume of the mercury drop (the two transients were not recorded using the same drop), inaccuracy in the integrating circuit used for recording the charge-time transient, and discretization errors in the al- gorithms.

The plateaus on the m(t) curves on Figures 2 and 3 are not quite flat but slope upwards slightly-i.e., there is a small increase in m(T) with time. The increase is partly caused by the nonfaradaic component of the current (charging of the double layer), and partly due to the electrode-solution inter- face being curved and therefore the diffusion field not truly linear. The upward slope has been observed generally, but for the present investigation, the results are near enough to verify Prediction A.

Variation of the Signal. In order to verify Prediction B, a number of current-time and charge-time transients were recorded, where the applied potential of the working electrode in all cases initially was -490 mV and finally -670 mV us. SCE. The variation of the potential, going from the initial to the final value, was carried out in a number of different ways as shown on Figure 1. On one series, the ramp rate, dE/dt, was varied and in another, potential steps were applied


Table 11. Results Obtained for Cd2+ Reduction, Showing the Signal-Independence of m(7) m(r) at

Type of Ramp rate, Electrode Series resistor Potential of m(7) beginning transient mV/sec or step configuration or capacitor step, mV vs. SCE microamplomb of level

i(r) 10 three 0 . . . 24.22 . . . i ( r ) 10 three 0 . . . 24.42 . . . i(t) 20 three 0 . . . 23.74 . . . i( 1) 50 three 0 . . . 23.11 . . . X i ) 100 three 0 . . . 23.21 . . . i(r) 200 three 0 . . . 23.09 . . . i(t) 500 three 0 . . . 22.77 . . . d 1) step three 0.7 kohm - 680 24.1 23.9 dr) step three 1 . O kohm - 680 24.00 23.5 d r ) step three 1 . 5 kohm - 690 23.85 23.6 s(r) step three 2.2 kohm - 690 24.9 24.3 d r ) step three 3 . 3 kohm - 700 24.7 24.2 4(1) step three 4. I kohm -710 24.2 23.5 d r ) step three 6.8 kohm - 730 24.4 23.9 d t ) step three 10.0 kohm - 750 . . . 24.2

step three 15.0 kohm - 770 24.9 . . . 41) 100 three 32 p F . . . . . . 23.04 i(t) 200 three 32 p F . . . . . . 22.95 i( r ) 500 three 32 p F . . . . . . 23.16 i(r) step three 32 p F . . . . . . 23.00 i(t) step three 16 p F . . . . . . 23.00 i(r) 500 three 0 . 7 kohm . . . . . . 23.20 i( t ) 20 two 0 . . . 23.65 . . . i(t) 50 two 0 . . . 23.40 . . . i( 0 100 two 0 . . . 23.28 . . . $0 200 two 0 . . . 23.25 . . . i(t) 500 two 0 . . . 23.22 2 3 . 8

Table 111. Data Showing the Concentration-Dependence of m(7) Concentration Concentration m(r) (at end of ramp) m(r) corrected

Solution No. of KNOa, mM of Cd(NO&, mM microamplomb for non-faradaic currents 1 100 5.00 120.4 . . . 2 100 4.00 97.25 . . . 3 100 3 .OO 6 8 . 7 5 . . . 4 100 2.00 46.9 . . . 5 100 1 .oo 23.5 . . . 6 100 0.500 11.70 1 1 . 5 5 7 100 0.200 4.92 4.75 8 100 0.100 2.49 2.32 9 100 0.0500 1,371 1.20

10 100 0.0200 0.660 0.488 1 1 100 0.0100 0.412 0.252 12 100 0.0040 0.282 0.121

with resistors of different values connected in series with the working electrode, see Figure 1, case e. In a third series, capacitors of different values were connected internally in the potential programming unit so as to “degrade” the applied potential steps or ramps into exponential or sigmoid potential profiles as shown in Figure 1, case f o r c. Table I1 lists the results. The m(t) values listed in the table were read off at the time ( t = 7) at which the potential of the working electrode reached -670 mV cs. SCE.

The two-electrode configuration was also tried out as listed in Table 11. In this configuration the initial and final potentials were - 750 mV and - 1200 mV, respectively, us. the mercury pool counter electrode. The m(t) values listed were read off at the time ( t = 7) at which the potential reached -975 mV.

The values of m(7) summarized in Table I1 group around 23 to 24 microamplomb; however, there seems to be an increase of m(7) with the length of time required to reach the final potential. This increasing tendency is again explained in the charging of the double layer and the geometry of the electrode-solution system. For very long time spans, convec-

tion in the solution will also result in increased values of d 7 ) .

Variation of the Electrode Area. In this series, the area of the working electrode was varied by metering different volumes of the hanging mercury drop, from the smallest possible to the largest which surface tension would retain. The area was calculated from the metered volume, assuming the part of the drop exposed to the solution to be perfectly spherical. The cadmium ion concentration was throughout 1.00 mM, the potential profile applied was of the “capped” ramp type with initial and final potentials - 490 and - 670 mV os. SCE respectively and dE/dt = 100 mV/sec. Figure 4 shows the values of m(7) plotted L’S. the corresponding area: the points clearly lie along a straight line through the origin as would be expected according to Prediction C. The fairly small scatter is mainly due to the inaccuracy of metering the drop volume. This is proved by plotting the m(7) values against the peak height of the corresponding i-t transients which also should be proportional to the area of the electrode (4) . Con-

(4) R. S. Nicholson and I. Shah, ANAL. CHEM., 36,706 (1964).


I I I I

(25 c““1 d

8

5

(00-

t b c

75 -

50 -

25 -

Concentration of Cd“ (miromob.cmJ) I I I I \ 7 3 4 5

Figure 5. Reduction of Cd*. m(7) us. concentration of Cd(NOa)z

n i ( ~ ) is here defined as the value of m(r) at time of the end of ramp. Supporting electrolyte: 100mMKNOa. Electrode area: 4.69 X 10” cm*. Ei, = -490 mV us. SCE. ,!?/in = -670 mV US. SCE. Slope of line: 24.1 amplomb cma mole-*

sequently the inaccuracy of the volume has the same effect on the peak height as on the m(7) value and the quotient m(7)/ip, listed in Table IV is obtained with even less deviation than the quotient m(T)/A.

Variation of the Cadmium Ion Concentration. In order to verify Prediction D, the cadmium ion concentration of the solution was varied as reported in Table 111. The K N 0 3 concentration was lOOmM throughout, the volume of the mercury drop was metered to 1 .OO pl and the “capped” potential ramps were applied as before. The m(7) values are plotted us. the corresponding concentration as shown on Figure 5 . The points group along a straight line through the origin as according to Prediction D. Better to see the effect in the lower concentration region, the logarithm of the m(7) values are plotted against the logarithm of concentration in Figure 6. The proportionality between the m(7) and concentration is illustrated by the straight line of slope 1. With decreasing concentration, however, there is a persistent, increasing deviation from the line which is mainly due to the increasing pro- portion of the non-faradaic current component. A correction for this has been introduced on Figure 6 and the deviation is clearly reduced. At very low concentrations there is some residual deviation which possibly is due to the increasing significance of impurities in the solution.

Reduction of Oxygen. The reduction of oxygen on mercury is irreversible and takes place in two steps which are visible directly as two waves in the well known polarogram (5). In

(5) I. M. Kolthoff and J. J. Lingane, “Polarography,” Interscience Publishers, New York, N.Y., 1965.

-+2

.=. +J - -5 b 0 -

-+I

- 0

0

- 2 - 4 0 (micrornole-cmJ)

Figure 6. Reduction of Cd2+. lOg(m(T)) us. log[Cd(NO&]

Slope of line = 1, corresponding to a linear relationship o Not corrected for non-faradaic current component v Corrected for non-faradaic current component

neutral solution the reduction proceeds according to the scheme :

0 2 + H2O + 2e- + HOz- + OH-

H02- + H2O + 2e- -+ 30H-

and the potential ranges at which the two reduction steps are completely diffusion controlled are between - 300 and - 600 mV us. SCE for the first step and more negative than - 1300 mV for the overall process.

The fist reduction step was investigated using oxygen- saturated lOOmMKC1 solution. Table V summarizes a series of i-t transient recordings, where the rate of the applied “capped” potential ramps was varied. The initial and final potentials were f 3 0 mV and - 300 mV us. SCE, respectively. When this is compared to the similar measurements on the cadmium ion reduction (Table 11), there is a considerably larger variation of m ( ~ ) with the ramp rate; the m ( ~ ) value decreases with increasing ramp rate. The reason for this en- larged variation is believed to be due to the irreversibility of the oxygen reduction, combined with the final potential not being far enough inside the diffusion control region.

Figures 7 and 8 show m ( ~ ) values corresponding to the fkst reduction step, plotted us. electrode area and oxygen concentration, respectively, The initial and final potentials were the same as before but with a fixed ramp rate of 200 mV/sec. The values of m(7) lie along straight lines through the origin, thus verifying Predictions C and D for the first step of the oxygen reduction.

The Number of Electrons Involved in the Reduction. Curve I1 of Figure 9 is a current-time transient as recorded where the potential range of the “capped” ramp was extended to -1300 mV us. SCE, beyond the potential range of the second reduction step of oxygen. The current, in this case passes through two maxima, one for each reduction step, after which it steadily decreases, As before, the current data were read off at equal time intervals and used in the computer pro-

1126 * ANALYTICAL CHEMISTRY, VOL. 44, NO. 7, JUNE 1972

Volume of drop, mma 0.05 0.10 0.15 0.20 0.25 0.35 0.45 0.60 0.80 1 .oo 1.25 1.50 2.00 2.50 3.00 3.50

Exposed area, mmz

0.51 0.89 1.17 1.51 1.78 2.21 2.63 3.29 4.93 4.69 5.47 6.19 7.53 8.79

10.13 11.27

~~

Table IV. Data Showing the Area-Dependence of m(7)

m(s) (at end of ramp) Peak-height, pA microamplomb Ratio rn(r)/area

2.77 2.316 4.53 4.18 3.56 4.00 7.11 6.02 5.14 9.50 8.00 5.30

10.96 9.25 5.19 13.80 11.65 5.28 16.00 13.52 5.14 20.40 17.15 5.22 25.05 21.15 5.25 27.75 23.50 5.01 32.85 27.75 5.07 37.92 32.01 5.17 46.50 39.40 5.16 53.00 44.79 5.10 62.5 53.02 5.24 69 .O 58.62 5.19

Ratio rn(.r)/peak-height

0.835 0.850 0.848 0.844 0.844 0.846 0.845 0.841 0.846 0.847 0.846 0.845 0.847 0.847 0.849 0.849

Table V. Data Showing the Significant Ramp Rate-Dependence of m(7) in the Case of Oxygen Reduction

m(7) (at end of ramp) m(7) corrected for Type of transient Ramp rate mV/sec Electrode configuration microamplomb non-faradaic currents

20 50

100 200 500

three three three three three

57 .5 52.7 52.2 49.6 48.0

56.4 51.7 50.9 48.9 46.3

gram, yielding the semiintegrated current-time transient m(t) as shown on Figure 9, curve 111.

When the slopes on the plateaus are taken in account, the height of the first wave appears to be very nearly the half of the overall height. As two electrons are involved in each reaction step, it appears that the height of a wave is proportional to the number of electrons involved in the reaction. This was expected according to Prediction E. Compared to the cadmium ion reduction, the slope on a plateau appears to be steeper in the case of oxygen reduction. One reason for this could be due to the curvature of the electrode-solution inter- face, as before, and that the diffusion coefficient for oxygen is considerably larger than that for camdium ions.

I I DISCUSSION AND CONCLUSIONS

The results of the experimental work described so far have verified Predictions A, B, C, D, and E. From Equation 10, it follows that it should be possible to calculate the diffusion coefficient

Exposed Electrode Area h m * ) 9 as Prediction F. In case of the electroreduction of CdZ+, the diffusion co-

efficient was calculated twice in this way, based on the slopes of the lines in Figure 4 and 5, i.e. m(7) /A = 5.21 x 10-4

spectively. By inserting the remaining constants, i.e., n = 2 equiv mole-', F = 96487 coulomb equiv-1 and C = 1.00 x

mole ern+ (or A = 4.69 X 10-2 cm* in the second case), two slightly different values were obtained:

amplomb cm-2, and m(.r)/C = 24.1 amplomb cma mole-', re- OO 3 6 9 12

Figure 7. Reduction of 0 2 . (First reduction step). ~ ( 7 ) us. exposed electrode area

Solution: Oxygen saturated lOOmM KCI. Ei,, = +30 mV _ _ US. SCE, ,!?fin = -300 mV VS. SCE. Slope of line: 1.062 X 10-8 amplomb cm* (Corrected for non-faradaic current com- D = 0.73 X cm2 sec-' (Variation of electrode area)

(Figure 4) ponent)


Table VI. A Summary of D Values for Cadmium Ion

Diffusion coefficient

< 106 cma/sec 0.73 0.71 0.6784 0.7254 0.7171 0.6793 0.690 0.716 0.717 0.770

concn Supporting electrolyte Cd2+, mM mM Salt

1 .00 . . . . . . . . . . . . . . .

2.50 2.50 2.50 0.50

100 100 100 100 100 100 100 100 100 100

Temp, “C 24 24 25 25 25 25 25 25 25 25

Method of determining D Comments m(7)iA m(7)lC Polarographic Same experi- Polarographic I mental data Polarographic Same experi- Polarographic 1 mental data Polarographic Polarographic Cottrell Polarographic

Figure 8. Reduction of 02. (First reduction step). m(r) us. partial pressure of oxygen

Supporting electrolyte: lOOmM KCI. Ei, = +30 mVus. SCE, Eji, = -300 mV us. SCE

0 Not corrected for non-faradaic current component V Corrected for non-faradaic current component

D = 0.71 X cm2 sec-1 (Variation of concentration) (Figure 5 )

As a check on the values obtained, the diffusion coefficient has also been calculated using the established theory of linear sweep chronoamperometry (6). This calculation i s based on the peak height of the recorded current-time transients and the corresponding ramp rate and gives :

D = 0.64 X cm2 sec-1

(6) P. Delahay, “New Instrumental Methods in Electrochemistry,” Interscience Publishers, New York, N.Y., 1954.

E

Figure 9. Reduction of 02. (Both reduction steps). Applica- tion of a “capped” potential ramp to the working electrode extending beyond the potential range of the second reduction step Solution: Oxygen saturated lOOmM KCI; Electrode area: 4.69 X 10- a*, Ein = +30 mV us. SCE Elzn = - 1300 mV us. SCE

I. The potential ramp E(r) II. The current-time transient. i(t)

III. The current-time transient semiintegrated us. time

which is significantly lower. The diffusion coefficient as calculated from Equation 14 compares well with the literature values for Cd2+ (7-9), see Table VI, although there is some scatter and different experimental conditions.

In the case of oxygen reduction, similarly, the diffusion coefficient was calculated from the slope of the line in Figure 7,

~~

(7) D. S. Turnham, J. Electroanal. Chem., 10,19 (1965). ( 8 ) M. v. Stackelberg, M. Pilgram, and V. Toome, 2. Elektrochem.,

(9) G. Faraone and M. Trozzi, Atti SOC. Pelorirana Sci. Fis. Mat., 57, 342 (1953).

8,249 (1962).


~ ( T ) / A = 1.062 x 1W8 amplomb cm-2, where correction for charging of the double layer was taken into account. With C = 1 .25 X mole cm-a, corresponding to PO, = 1 atm (IO), n = 2 equiv mole-I, corresponding to the first reduction step and A = 4 .69 X cm2, Equation 14 yields:

D = 2 . 0 X cm2 sec-’

which is to be compared to literature values (11): 2.6 X 10-5 cmzsec-l and (12): 2.08 X cm2 sec-’in HzO and 1.84 X 10-5 cm2 sec-1 in 7 . 4 z NaCl solution, both at 25 “C.

Curve I11 of Figure 2 and curve I11 of Figure 9 are reminis- cent of ordinary polarograms for the reduction of CdZ+ and 02, respectively. In fact, as we shall demonstrate in a future article, the m(t) us. E curves of semiintegral electroanalysis are identical in shape to the familiar i(tmax) US. E curves of polarography with a dropping electrode. In semiintegral electroanalysis, however, there is no need to employ the repetitive timing mechanism (natural or imposed) which characterizes polarography.

It is this decoupling of the voltammetry from any dependence on time or frequency which constitutes the novel feature of semiintegral electroanalysis, and from which the ad- vantages of the method accrue. Thus it leads to the independence of m(7) on the form of the applied signal, permitting,

(10) “Handbook of Chemistry and Physics,” 50th Edition, The Chemical Rubber Co., Cleveland, Ohio, 1969-70.

(11) I. M. Kolthoff and C. S . Miller, J. Amer. Chem. Soc., 63, 1063 (1941).

(12) R. E. Collington, P. L. Blackshear, and E. R. G. Eckert, (Univ. Minnesota), Chem. Eng. Progr., Symp. Ser., 66(102), 141 (1970).

as we have demonstrated in this article, a large series resistor to be present in the circuit without impairment of the analysis. Equally, the large resistance may be present in the solution, so that semiintegral electroanalysis can tolerate very low supporting electrolyte concentrations, as will be reported in a later publication.

A disadvantage of semiintegral electroanalysis, as here described, is the data-reduction stage which requires tedious transcription of current-time or charge-time data from a chart recording into a computer. We are presently perfecting methods of replacing this stage by an analog technique, permitting m(t) to be measured directly. With this added feature, semiintegral electroanalysis possesses a marked advantage over such voltammetric methods as chronopotentiometry, linear sweep voltammetry, or pulse polarography, in that the shape of the measured response is unimportant, all that is needed being the constant m(T) value. Hence no expensive two-dimensional recording device is required for semiintegral electroanalysis, the need for a readout instrument being adequately met by a simple meter, which may even be directly calibrated in concentration units.

Studies are continuing in these laboratories on the following aspects of semiintegral electroanalysis : extension to other systems, the shapes of m(t) us. Ecurves, the role of non-faradaic current, the replacement of digital semiintegration by analog techniques, exact corrections for non-planarity of the working electrode, the development of inexpensive and robust instru- mentation, and the applicability of the method in the absence of excess supporting electrolyte.

RECEIVED for review October 28, 1971. Accepted February 16, 1972. The financial support of the National Research Council of Canada is gratefully acknowledged.

Voltammetry in Methanol, Ethanol, and Sulfolane as Solvents

J. F. Coetzee’ and J. M. Simon Department of Chemistry, University of Pittsburgh, Pittsburgh, Pa. 15213

The Polarographic half-wave Potentials of the alkali metal and barium ions have been measured in methanol and ethanol as solvents. For methanol, reliable standard potentials are available, and the sets of half- wave and standard potentials are reasonably consis- tent. Voltammetric data also are reported for a variety of electroactive species in sulfolane as solvent, and Some comparisons with corresponding values in acetonitrile are presented.

sulfolane a valuable medium for acid-base reactions in partic- ular, as has been demonstrated by the work of several groups, Some of which are referred to elsewhere (2) . In general, however, much less is known about solute properties in sulfolane than in other dipolar aprotic solvents, such as dimethyl- sulfoxide or acetonitrile. We report here some results of voltammetric measurements in sulfolane as a supplement to those presented earlier by Headridge et al. (3) and by us (4).

SULFOLANE IS ONE of the more recently introduced members of the class of dipolar aprotic solvents, but it already has attracted considerable attention as a medium for several types of investigation. Its major asset is that it couples high polarity (dipole moment = 4.8D, dielectric constant = 43 at 30 “C) with relatively low chemical reactivity. For example, while it is a typical dipolar aprotic solvent in being an extremely

One point of interest will be to determine to what extent the relative donor properties of sulfolane, acetonitrile, and water toward metal ions parallel the proton acceptor properties of these solvents. We also report here the results of polarographic measurements on the alkali metal and barium ions in methanol and ethanol as solvents, for reasons that will be presented in the discussion section.

weak proton donor, it is also a very weak proton acceptor: the pK, value of its conjugate acid is - 13 (I). This makes

Please address all correspondence to this author.

(2) J. F. Coetzee and R. J. Bertozzi, ANAL. CHEM., 43,961 (1971). (3) J. B. Headridge, D. Pletcher, and M. Callingham, J. Chem. SOC.

(4) J. F. Coetzee, J. M. Simon, and R. J. Bertozzi, ANAL. CHEM., A , 1967,684.

41, 766 (1969). (1) S. K. Hall and E. A. Robinson, Can. J. Chem., 42, 1113 (1964).


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Semiintegral electroanalysis. Theory and verification