Multistep Charge Transfers in Stationary Electrode Polarography DANIEL S. POLCYN and IRVING SHAlN Chemistry Department, University of Wisconsin, Madison, Wis.
b
The theory of stationary electrode polarography for single scan and cyclic triangular wave experiments has been considered for uncomplicated multistep reversible or irreversible charge transfers. A numerical method was used to solve the integral equations obtained from the boundary value problem, and extensive data were calculated to characterize the systems quantitatively. The reduction of Cu(ll) in 1M ammonium chlorideammonium hydroxide was selected to test the theoretical calculations for an uncomplicated multistep reaction in which both charge transfers are reversible. Reasonable agreement between theory and experiment was obtained.
1
bCA/bt
0.6
5-
0.4-
3
0.2
k 0
T
+
+
A nle + B n2e + C (I) where substances A and B are electroactive, and n1and n2 are the number of electrons involved in the succesqive steps. If A and B react a t sufficiently separated potentials with A more easily reducible than B, the polarogram for the overall reduction of A to C consists of two separate waves. The first wave corresponds to the reduction of A to B with n1 electrons, and in this potential range, substance B diffuses into the solution. As the potential is scanned toward cathodic values, a second wave appears which is made up of two parts superimposed. The current related to substance A , which is still diffusing toward the electrode, increases since this species now is reduced directly to substance C by (nl n2) electrons. I n addition, substance B , which was the product of the first wave, can be reduced in this potential region, and a portion of this material diffuses back toward the electrode and reacts. The theory of stationary electrode polarography for this case has been given by
+
370
ANALYTICAL CHEMISTRY
(2)
bCc/bt = D c ( b ' C c / b ~ ~ ) 2 0: C A = C A * ;
(3)
t = 0,x
CB = CB*; CC = cC*
-
CB,
5
0.0 -
t
> 0, x
L1z U
t
- 0.4
W
I
120
I
0
I
:
A. Total current function: $ ( a t )
+
[+(at)
> 0, x
+
8. Current function corresponding to flux of substance A, $(at) C. Current function corresponding to Aux of substance 8, x(at)
Gokhshtein and Gokhshtein (8, 3) for the single scan method only. Their treatment was a very general approach to multistep charge transfers involving reversible, quasireversible, or irreversible systems in various combinations for any number of successive reactions. As a result, their work cannot be applied directly for the interpretation of experimental stationary electrode polarograms. Therefore a more straightforward treatment, limited to only two species, was derived in this work. This made it possible to investigate the effect of wave separation for cases in which they overlapJ and permitted extension of the method to the cyclic triangular wave method. Experimental work was carried out on the reduction of Cu(I1) in ammonium hydroxide-ammonium chloride solution as an example of a multistep charge transfer, BOUNDARY VALUE PROBLEM
For a two-step electron transfer (Equation I) the boundary value problem is:
CC
z5
(4)
0
C A --* C A * ; -f
0;
CC +
0
(5)
+
= 0: DA(~CA/~X) D B ( ~ C B / ~ X Dc(bCc/bx) ) = 0 (6)
+
CA, CB,
I
I
-120 - 2 4 0 -360
Figure 1 . Cyclic stationary electrode polarogram
x(0f)l
--*
CB
s -0.2 (E - E:)n, rnv.
theory of stationary electrode polarography for a single reversible or irreversible charge transfer has been studied by many workers, and their results have been reviewed previously (8). However, less attention has been directed toward electrode processes in which multi-electron consecutive charge transfers occur. A multistep charge transfer reaction can be represented by HE
bCB/bt = D B ( ~ ' C B / ~ X ' )
LL
-l
D A ( ~ ' ~ A / ~ X * )(1)
=
CC
=
f(E, t )
(7)
Here x is the distance from the electrode surface, t is the time, CA,C B ,and Cc are the concentrations of the substances A , B , and C, CA*, CB*, and CC*are the bulk concentrations, and DA, D E , and Dc are the respective diffusion coefficients. The function f ( E , t ) for stationary electrode polarography is a periodic triangular variation of the electrode potential. This in turn is related to the surface concentrations of the electroactive species through the Nernst equation for reversible charge transfers or the Eyring equation for irreversible charge transfers. The total current for the two step reaction is given by
it = nlFdfA(t) f d - 4 [ f A ( t ) f
fB(t)
1 (8)
I n the potential region of the first wave, the flux of substance B is the negative of the flux of substance -4; thus the entire second term does not appear and the current merely corresponds to the flux of substance A , I n the potential region of the second wave, however, where a portion of the substance B previously produced returns to the electrode for further reduction, the second term becomes important. The solution of this boundary value problem is very similar to that described previously for the case of the E C E reaction in stationary electrode polarography ( g ) , and is described in detail elsewhere (14). The explicit form of Equation 7 required to qefine the nature of the charge transfer reaction can be stated in a manner analogous to Equations 25-32 in Table I of Reference (9).
As before, if quasireversible systems are not considered, four sets of conditions are possible depending on the reversibility of the charge transfer. For brevity, these systems are referred to as R-R, R-I, I-R, and I-I, which define the charge transfer reactions as being reversible or irreversible in sequence. The potential in thehe boundary condition equations can be written as a cyclic triangular function of time, and using the procedures and definitions described previously, two integral equations are obtained for each case (Table I). The dimensionless functions in the integral equations are related to the fluxes of substances A , B , and C, by:
f ~ ( t )=
C ~ * d * D a $(ad) l
Table
I.
Integral Equations for Multistep Charge Transfer Reactions Involving Two Electroactive Species
(17)
For Case R-R, the values of x(alt) and o(alt) can be solved for independently.
I n the other three ca mensionless function. $ , X , or @ can be solved for indel~entlently.and then these values can be suhtituted in the second integral equation to obtain values of the other unknown funcation. [-sing a numerical method siniilar to that described previously (9), each pair of integral equations was solved to obtain t'heoretical stationary electrode ~)olarograms, Calculations were carried out on an 11311 Type i o 4 digital coinputer u+ig integration hubintervals of 6 = 0.04 i ' ~ selected r valuw of potential beparation and n1 or n!. 13. comparing the valucs calculated liere for the first wave ivith thohe o b t a i n d previously ( S ) , where a mailer integration subinterval \vas used, the error in the peak current values vas less than 0.1%.
polarogram exhibits two waves. Each of the individual Iyaves has the shape and peak height of an uncomplicated reversible charge transfer, provided the descending branch of the first wave is uqed as the base line for the second wave (Figure 1). -1s discuqsed by Gokhshtein (Z), in order that the polarograms behave as independent reversible waves, a certain minimum potential separation, LE", of about 118,n mv. is required between the formal reduction potentials. -4theoietical polarogram calculated for a potential separation LEo of - 180 mv. is shown in Figure 2, Curve A. THEORETICAL CORRELATIONS X. the potential separation between The characteri4cs of the theoretical the successive reductions becomes less stationary electrode polarograms which than about 100/'n mV., the individual were obtained from the numerical soluwaves merge into one broad distorted tion of the integral equations depend on wave \Those peak height and shape are heveral factor., including the nature of no longer characteristic of a reversible the charge tranhfcr h t e p , the potential wave (Figure 2 , Curve B ) . The wave is wparation between the individual broadened similar to an irreversible cahai,getranhfer.;, the number of electrons wave, but can be distinguished from the in the specific bteps:, and whether the irreversible polarogram, in that the cxperimcnt is performed as a single distorted wive does not shift on the potential wan or a:' a cyclic potent,ial potential axis as a function of scan rate. scan. T o obtain useful information for For the particular case when both A rharacterizing experimental systems, and B are reduced a t the same potential, :ill the>c 1)3ralneters w r e varied in the AEo = 0 (and asbunling that nl and n2 cdculations. are both unity) the wave observed Reversible Charge Transfers (Figure 2 , Curve C) has a peak height (R-R). EFFECT O F POTESTIAL intermediate between a one-electron and SICPARATIOS.\\-hen both charge a two-electron reversible wave, and transfers are reversible and when E , - El,?is about 21 mv. As B bethe two electroactive species are comes easier to reduce than d , the lvave reduced a t sufficiently different potenheight increases and the peak narrows tials, the cathodic stationary electrode until it reaches the height and shape of
a two-electron wave (Figure 2 , Curve D ) . That is, the height of the twoelectron wave is (2)3/2t.imes the height of the corresponding single electron reversible wave, E , - Eli2is 14.23 mv., and the reaction behaves as a direct reduction of A to C. The effective E o for the composite t'wo electron wave is given by (Eo1 E o 2 ) / 2 . EFFECTO F S U U B E R OF E L E C T R O N S . The number of electrons involved in the charge transfer steps affects both the shape and position of t,he n-aves. The values of nl and n. increase the value of the current function since they occur as multiplying factors in Equation 8, and they also affect the width of the waves because the potential scale i q defined as ( E - E o ) n . This two-fold effect complicates the theoretical stationary electrode polarograms. I n general, the results of the calculations confirmed that for potential separations where one obtains two u v e s which can be considered a.; isolated reversible waves, the number of electrons affects the height and width of the waves individually, without interacting. Thus each wave is proportional to ( n ) 3 / 2 and the potential range covered by each wave is determined by the individual value of ( E - E")n. At potentials where distorted waves are obtained, the effect of varying nl and n? is complicated and extensive calculations are required to characterize the theoretical polarogram for any particular set of conditions.
+
VOL. 38,
NO. 3,
MARCH 1966
371
'i z
0 I=
0
-0.5
z
C A
--"
t
3 LL i-
0.8
IA
-/
z
t
W
a
I
I
I
I
D
E 3 0
0
-200
U
( E - E:)n, rnv. I
Figure 2. Case R-R
Cyclic stationary electrode polarograrns for
Current function is +(at) A. A€' = -180 mv. B. AEo = -90 mv. C. AEo = 0 mv. D. A E O = 180 mv.
+ (m/nd[+(at) + x(at)]
CYCLIC TRIANGULAR RAVE VOLTAMFor the first cycle, the rathodic portion of the polarogram is the same as for a single scan, but the height and position of the anodic portion depend on the switching potential, EA. Extensive calculations were carried out using the numerical method, and provided the switching potential is not less than about 65/72 mv. past the most cathodic EO, all the waves can be considered isolated for cases where AEO is sufficiently large. Using the extension of the cathodic curve as a base line (as in Figure I), the anodic waves are the same height and shape as the cathodic waves, independent of the potential separation of the charge transfers (Figure 2, Curve A). For smaller values of AEO , the anodic portion of the cyclic polarogram still has the same shape as the cathodic portion. Thus, when a distorted, merged cathodic wave is obtained (Figure 2, Curve B ) , the anodic wave also is distorted in the same m y . The appearance of a distorted anodic wave in this manner differentiates this type of successive multistep charge transfer from an irreversible reaction. Irreversible Charge Transfers (&I, Z-R, Z-I). Provided the two electroactive species are reduced at METRY.
372
I
ANALYTICAL CHEMISTRY
I
I
0
I
I
- 200
0
I
-200
P O T E N T I A L , mv. nz/nl = 1.0
Figure 3. Cyclic stationary electrode polarograrns with irreversible charge transfers Case R-l; AED = -180 rnv. Case R-l; A€' = 0 mv. 180 mv. Case I-R; A€' = D. Case I-/; A€' = 1 8 0 rnv. Potential scale for curves A and B is (E Elo)n Potential scale for curves C and D is ( E - E l o ) a l n o l In each case, n l = n2 and a = 0.5 &X,/k8.
A. B. C.
sufficiently different potentials, the waves can be considered as resulting from separate reactions even if one (or both) of the charge transfers is irreversible. Under these conditions, the irreversible wave will have the same characteristics described earlier (8) for an isolated irreversible charge transfer. The current corresponding to the irreversible wave is lower and the wave is more drawn out. The current function has a value of 0.496 a t the peak, E, - Epl2 = 47.70/ana, and since the peak potential is a function of scan rate, the wave shifts cathodically 30/an, mv. for each 10-fold increase in the rate of potential scan. For cyclic scans, no anodic wave is observed for the corresponding cathodic irreversible wave. Since irreversible waves depend on the charge transfer coefficient a which can have nonintegral values, i t is not feasible to construct many theoretical polarograms, but only to define general trends. As the potential separation between the successive reductions becomes less than about 100/an, mv., the
-
-
-
+ ( R T / F ) In
individual waves merge and a distorted wave is obseived. K h e n .i and B are reduced at the same potential, a single wave is observed n i t h the combined characteri3tics of the two charge transB becomes easier to fers involved. reduce than -1, the wave height increases and the peak becomes sharper. Under these conditions, the characteristics of the combined wave depend primarily on the first charge transfer in the sequence. Typical stationary electrode polarograms for selected systems involving an irreversible charge transfer are shown in Figure 3. Descending Branch of the Stationary Electrode Polarogram. When a stationary electrode polarogram involves two n-aves (either succe"'bbive cathodic peaks in single scan esperiments, or anodic peaks in cyclic experiments) investigation of the second wave requires knowledge of the descending branch of the preceding wave to establish a base line. Several forms of an equation for the descending branch of the polarogram were given by Gokhshtein and Gokh-
shtein ( 2 , 4 ) . -1more detailed basis for these expressions, and more precise values of certain adjustable parameters are presented here. .It potentials past the peak of t'he wave, the concentration of the reacting species a t the electrode surface approaches zero, and the current exhibits typical diffusion controlled l / d t d e c a y , independent of potential. The correqjondence between the l/d dependence and the descending branch of the polarographic wave is shown in Figure 4. The origin for the currenttime curve is b e h e e n Eo and the peak of the wave; its exact position depends on the potential region over which t'he two curves are fit. The form of the equation can be determined by comparing the equations describing the stationary electrode polarogram and the current-time curve for the potentiostatic case:
i
=
n ~ - 1 d R D a ~ y x(at) o*
(E, - E")
nF.ldDCo*/Z/rro
(21)
The term t' in Equation 21 is a constant, and is the displacement from .! = 0 along the time asis required for the current-time curve to coincide with the stationary electrode polarogram. For the currents given in Equations 20 and 21 to be equal a t any given time, it is necessary to select a value of t' such that
x(at)
=
l/dX=q
(22)
This value of t' can be determined from the numerical calculations. Once determined, that value of t' can be used with Equation 22 to determine values of x(at) for any value t o time. For convenience, Equation 22 can be rearranged so that calculations are made with respect to the potential axis rather than time. Thus,
4
I
-200
I
I
I
-400
-600
-800
(E - E') n, mv. Figure 4. Comparison of descending branch of stationary electrode polarogram with potentiostatic current-time curve Curves were shifted arbitrarily on the potential (time) axis to obtain the best flt
&,(at)
=
l / d r ( n F / R T )[(E' - E,)
- ( E -E,)] (26)
Since (E, - E") is equal to -28.5/n mv., the potential axis can be shifted between Equations 25 and 26 as convenient. It is not possible to determine a single value of (E' - E O ) for Equation 25 which would be suitable for calculation of d X x ( a t ) values over the entire time (potential) scale, because the concentration gradients generated by the currentpotential characteristics over the rising portion of the stationary electrode polarogram differ significantly from the potentiostatic case. Thus, depending on the accuracy desired and the range of values to be covered, several values of (E' - EO) must be provided. An equation which covers a much wider range can be obtained if a second arbitrary parameter p is included as a factor in the numerator of Equation 25 :
dlrx(at)
and Equation 22 becomes,
- [5.34/ana+
I
0.0
& x(at) = =
=
Thus, for an irreversible charge transfer, the expression used for calculation of the current function is
(20)
and
i
I n addition, the potential axis is almost always referred to the peak potential, which, in turn, can be related to the Eo for the system by the relation
=
~ / ~ / & F / R T ) ( E '- E )
and values of y and ( E - E,)an, for use in Equation 30 are included in Table 111. Diagnostic Criteria. Reactions in which multistep charge transfers are present may appear similar to systems in which coupled chemical reactions are present, because under certain conditions distorted stationary electrode polarograms are observed. However, the same diagnostic procedures developed previously (8) for characterization of chemical reactions coupled to charge transfers can be used t o distinguish between the possible cases. T h a t is, over the entire range of scan rates, the quantity is a constant for these systems involving only multistep charge transfers. If AEo is sufficiently negative, the type of multistep charge transfer system
i,/A
Table II. Values of (E' - Eo)n and @ for Use in Equation 27 for Calculation X(d) of Current Function
4;
Potential range (from the ( E ' peak, mv.) E")n 20-830 50-830 100-830 20-300 50-300 100-300 100-200
3.28 15.11 17.30 2.68 16.31 20.84 22.95
p
Rel. error range, 70
0.987 0.986 0.985 0.998 0.977 0.970 0.961
0 . 1 -7.0 0 . 1 -1.4 0 . 1 -1.2 0 . 1 -7.6 0.054.1 0.05-0.1 0 . 0 -0.1
(24)
where E' is the potential (measured from the initial potential E,) eorresponding to at'. Since the potential axis is usually measured with respect to E" or E , rather than Ei, Equation 24 can be rewritten as
d&(at)
=
l / d ~ ( n F j R T[(E' ) - Eo)- (E -E0)] (25) or
Values of (E' - E o ) n and p , and the range over which they apply are presented in Table 11. If the charge transfer is irreversible, analogous equations can be derived for calculation of values of the current function 6 x(bt). The major difference is that the term ano appears in place of n, since
bt
=
(cun,F/RT)(vt) = (anaF/RT)(Ei - E ) (28)
Table 111. Values of (E' - €,)an, and y for Use in Equation 30 for Calculax(bt) tion of Current Function
6
Potential range (from the (E' peak, mv.) Ep)ane 36-845 55-845 36-309 55-305 100-300
15.0 12.2 14.3 10.7 11.5
y
Rel. error range, %
0.994 0.992 0.980 0.975 0.976
0.01-1.0 0.10-1.2 0.03-0.5 0.06-1.4 0.01-0.03
VOL. 38, NO. 3, MARCH 1966
373
30
d
i
20
cr
LT
3 V
VOLTS vs.
S.C.E
Figure 5. Single-scan stationary electrode polarogram of 2.5 X lO+M copper(l1) in 1 M ammonium chlorideammonium hydroxide Points, theory; line, experimental; dashed line, blank. Initial potential, -0.1 1 volt vs. S.C.E.; scan rate, 53 mv./second
can be determined froin the number of anodic waves. For decreasing values of LEO, the waves merge and become distorted. However, the invariance of the current function (inid? with scan rate would indicate that the total current function is made up of charge tran5fers not coupled to hoinogcneous chemical reactions. EXPERIMENTAL
T o verify the theory of stationary electrode polarography for uncomplicated multistep charge transfer systems and to evaluate the correlations, the reduction of copper(I1) in 1X ainmoniuin hydroxide-ammonium chloride solution was selected. The polaro-
Table IV.
Scan rate (volts/ second) 23.2 11.4 5.70 2.27 1.15 0.581 0.235 0.116 0.059
graphic behavior has been studied extensively using the dropping mercury electrode (6, 7 , 1 7 ) . Two \vel1 defined waves of equal height are obtained, corresponding to thc successive one electron reductions of Cu(I1) to Cu(1) and to the amalgam. The half wive potentials are -0.24 and -0.50 volt us. S.C.E. and the linear log (id - i)/i plots over the riqing portion of the polarograms indicate both charge transfers are reversible under these conditions. Herman and Bard ( 6 ) used this system to verify the theory of cyclic chronopotentiometr~ for multistep charge transfers. Peters and coworkers (la,I S ) studied the stepwise reduction of copper a t platinum and gold electrodes. XI1 experiments were carried out with a three electrode controlled potential circuit, using I'hilbrick Model K 2 - 5 , 122-P operational amplifiers (G. :I. Philbrick Researches, Inc., I3oston, Xass.). The circuit configuration, signal generators, and detector (recorder or oscilloscope, depending on the tiine scale of the esperiment) were similar to those described previously (1,10). The cell design also was the same as described previously ( I ) , except that the lid of Teflon was constructed in two parts. larger outer section (which fit the joint of the cell body), contained holes for the counter electrode, the reference electrode, and the nitrogen deaerator. A smaller plug which fit into an off-center machined hole in the main body of the cell lid contained holes drilled for the dropping mercury electrode, hanging mercury drop electrode, and the mercury drop tranjfer scoop. This made it possible to rotate the working electrode into position with respect to the reference electrode, and by this means the Luggin capillary of the reference electrode could be placcd within a millimeter of the n-orking electrode to minimize uncompensated iR drop. A11 espcriment: ivere carried out a t 25' C. Copper solutions were made by dissolving copper wire (99.99%) in nitric acid and diluting to volume. AU1 other chemicals were reagent grade and were used without further purification.
Stationary Electrode Polarographic Behavior of 2.5 X 1 O-3M Cu(l1) in 1M Ammonium Hydroxide-Ammonium Chloride Solution Initial Potential -0.11 volt VS. S.C.E. Peak current
(pa.).
I
I1
I11
IT'
497 360 264 172 125 86.3 55.7 39.7 28.7
525 376 262 172 125 83.0 54.6 38.9 28.1
526 373 261 174 124 89.8 5S.9 43.1 30.5
559 386 260 175 127 SS.9 57.7 41.8 31.7
Peak potentials (volts zs. S.C.E.) I I1 I11 IT' -0.248 -0.250 -0.230 -0,244 -0.256 -0.253 -0.255 -0.253 -0.235
-0.612 -0,512 -0.510 -0.XI6 -0,518 -0.507 -0.510 -0.513 -0.501
-0.440
-0.465 --0.437 -0.4*55
-0.449 -0.46i
-0.481 -0.465 -0.462
ANALYTICAL CHEMISTRY
/-
I
I
,
-0.20
-0.40
-0.60
-Ieo/
VOLTS vs. S.CE.
Figure 6. Cyclic stationary electrode polarogram of 2.5 X 1 O+M copper(l1) in 1 M ammonium chloride-ammonium hydroxide Waves numbered in order of appearance during cyclic scan. Points, theory; line, experimental. Initial potential, -0.1 1 volt VI. S.C.E.; scan rote, 1.1 6 volts/second
I n ammonium hydroxide-aminoniuin chloride solution, the foot of the first wave i i not well defined because it overlaps the dissolution wave of the mercury electrode. Other experiments were carried out in 1-11 ammonium hydroside-ammonium nitrate solution, and although the anodic range is extended, a large nonreproducible masimuin was observed on the second wave (11). This could be eliminated by the addition of surface active materials to the solution, but because of possihle interference with the charge transfer reaction, the chloride system, which could be used without deliberate addition of surface active material3, was considered more suitable. Vsing this system, stationary electrode polarograms were obtained for
Table V. Variation of Peak Current Function with Scan Rate for 2.5 X 10-3M Cu(ll) in 1M Ammonium Chloride-Ammonium Hydroxide Solution
Ei
= -0.11
volt
1's.
S.C.E.
Peak current function"
dG
Scan rate (volts/second)
I
I1
29.4 11.9 5.72 2.98 1.21 0.596 0.296 0.118 @ ,060
109 116 117 119 119 119 119 119 118
151 160 157 1-59 161 157 159 15s 160
-0.194 -0.220 -0.177 -0.197 -0.192 - 0.208 -0 230 -0,205 -0.201
a Measured ivith respect to a base line determined by a blank for Wave I, and by the descending branch of the preceding wave for Waves 11,111, and I T
374
240
i p l
Currents ( p a . ) for both waves measured to the blank 0
both single scan and cyclic triangular wave experiments. When applicable, currents were measured by using a blank to determine the base line which was distorted by the interfering mercury dissolution reaction. Varying the initial potential over a range of 50 mv. anodic and cathodic of the value Et = -0.11 volt us. S.C.E. did not affect the results. RESULTS AND DISCUSSION
The experimental results are summarized in Tables IV and V. n’umerous correlations were made (IC),but only a few are included here. As expected for reversible charge transfers, the peak cuirent function i p / d i f o r Waves I and 11 were independent of the scan rate. l l o r e scatter than expected was observed in the experimental points, caused by the difficulty in establishing a suitable base line. Similar correlations for the anodic waves also of &/doshowed uncomplicated reversible behavior, except that the increased difficulty in establishing the anodic base lines resulted in greater scatter in the experimental points. The separation in the peak potentials of Waves I and I1 is fairly constant with scan rate. The average value is 258 f 5 mv., in good agreement with the difference in polarographic Eliz (260 mv.) given by Lingane (7’). The potential sepalation of the anodic waves is also constant with scan rate and is 254 =t5 mv . The ratio of the peak currents for the various m v e s also can be used to characterize the system, since the waves are sufficiently separated on the potential axis. If the peak height of each successive wave is measured to a base line determined from the descending branch of the preceding wave, the peak height ratio should be independent of the potential separation of the individual reductions, and should depend only on the nature of the charge
transfer. For a system in which the charge transfers are reversible, the ratio of peak currents of any two of the four waves (anodic and/or cathodic) should be unity. For the copper system, peak current ratios were obtained for various combinations of the currents given in Table IV. The base line for Wave I was taken as the blank. For the other waves, the base line was measured by stopping the scan and allowing the current t o decay at constant potential (15). Over the range of scan rates from 0.059 to 2 3 . 2 volts per second, the average and range of selected ratios of peak currents were: 1/11, 1.00 i 0 . 0 4 ; III/IV, 0.99 f.0.04; I/III, 0.96 f 0.05; and II/IV, 0.96 f 0.05. I n spite of the uncertainties introduced by the method of obtaining a base line, the experimental values are in reasonable agreement with theory. The ratios I/III and II/IV, which involve the ratio of a cathodic current to an anodic current probably are low because of the difficulty in correcting for the charging current. The excellent agreement between experiment and theory over the entire polarogram is shown in Figure 5 for a single-scan experiment using a rather low scan rate, and in Figure 6 for a cyclic experiment a t a n intermediate scan rate. For the low scan rate (53 mv. per second), a spherical correction as given by Reinmuth (16) m s applied to the theoretical values. At this scan rate, the spherical correction amounted to about 6% a t the peak of K a v e I and about 10% a t the peak of Wave 11. The theoretical curve was located on the potential axis to provide the best fit with the experimental points. Data obtained a t the foot of the w,ve were not included because of the distortion resulting from the reaction of the mercury electrode. For the cyclic experiment at 1.16 volts per second, the spherical correction is negligible. These quantitative correlations between experiment and theory indicate
that the theory of multistep charge transfers in stationary electrode polarography provides a valid and useful approach to the study of uncomplicated reactions of this class. For the copper system, both charge transfers exhibit reversible characteristics over a time scale considerably wider than previously reported, and from these data, the heterogeneous rate constant, k,, for both charge transfers must be greater than about 1.5 cm./sec. LITERATURE CITED
( 1 ) Alberts, G. S., Shain, I., ANAL.CHEM. 35, 1859 (1963). ( 2 ) Gokhshtein, Y. P., Gokhshtein, A. Y.,
“Advances in Polarography,” I. S. Longmuir, ed., Vol. 11, p. 465, Pergamon Press, New York, 1960. 131 Gokhshtein. Y. P.. Gokhshtein. A. Y..
(4 ( 5 ) Herman, H CHEM. 36,971 (1964). ( 6 ) Kolthoff, I. ,,hf., Lingane, J,. J., “Polarography, 2nd ed., Interscience, New York. 1952. (7) Lingane,’J. J., ANAL. CHEM.15, 583 (1943). (8) Nicholson, R. S., Shain, I., Zbid., 3 6 , 7 0 6 (1964). ( 9 ) Sicholson, R. S., Shain, I., I b i d . , 37. 178 (1965). (10) ’Ibid.,’p. 190. (11) Olver, J. W., ROSS,J. W.. Ibid.. 34. 791 (1962). (12) Peters, D. G., Cruser, S. A., J . EZectrounaZ. Chem. 9 , 2 7 (1965). .nklin, L. A., Zbid..
1965. (15) Reinmuth, W. H., Columbia Univeraity, New York, N. Y., unpublished work, 1964. (16) Reinmuth, W. H., J . Am. Chem. SOC.79,6358 (1957). (17) Stackelberg, hI. von, Freyhold, H., 2. Elektrochem. 46, 120 (1940).
RECEIVEDfor review October 25, 1965. Accepted January 10, 1966. Work supported in part by funds from the U. s. Atomic Energy Commission, Contract NO.AT(11-1)-1083.
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