as (Rint)minmeasured in the previous experiment. Continuing with two-electrode operation, various external resistors were placed in the counter electrode lead in order to vary the ratio (Ri,&in/Rext. In these experiments Rint had to be maintained constant because it was not possible to change RiDt without changing the composition of the solution and therewith the half-wave potential. Differences in the halfwave potentials for various ratios of (R&,in/Rext are given in Table 111. Values of CY were interpolated from tabulated data given by BrdiEka ( 6 ) for various ratios of (RinJmin/Rext.Although the values given in Table 111 were more difficult to measure than in the limiting cases, the agreement between the experimental and theoretical differences in half-wave potentials is satisfactory. The differences are seen to become more negative with increasing proportion of ReXtand also to pass through zero and to change sign as predicted. These results support the conclusions of our previous studies
that resistance compensation is not complete in three-electrode cells containing solutions of high specific resistance. In addition, they show that the magnitude of the uncompensated resistance in a three-electrode cell, as well as the apparent half-wave potential, varies significantly with the degree of damping employed in the recording of the wave. When corrections for iR losses in the cell approach or exceed 100 mV, it is advantageous to record undamped, maximum-current polarograms using a fast response recorder, regardless of the type of cell used. RECEIVED for review August 19, 1968. Accepted October 7, 1968. Work supported by the U.S. Atomic Energy Commission under Contract No. AT(l1-1)-256. The authors are also grateful to E. I. du Pont de Nemours and Co., and Lubrizol Corp., who sponsored fellowships held by one of us (W.E.T.). (A.E.C. document No. COO-256-89.)
Nonunity Electrode Reaction Orders and Stationary Electrode Polarography Mark S . Shuman Department of Chemistry, Texas Christian University, Fort Worth, Texas 76129 The theory of diffusion-controlled electrode processes, mO ne c? qR, where one or both of the stoichiometric numbers m and q are integers greater than unity, has been considered for the triangular-wave chronoamperometric experiment. Two cases were considered in detail, 2 0 ne e R and 3 0 ne R. Theoretical current-potential curves for these reactions are lower in height and broader than the curve for unity m and q. Satisfactory correlations have been made between the theory and the dissolution of mercury into cyanide solutions.
+
+
+
*
THEpolarographic oxidation of mercury into alkaline cyanide solutions ( I , 2) appears to be a diffusion-controlled process in which the overall electrode reaction order is higher than first order. The current is limited by diffusion of cyanide, and the electrode reaction can be written Hg mCN- + Hg(CN),+*”
+ 2e
with stationary electrode polarography and nonunity reaction orders corresponding to the reactions 21-
e Iz + 2e
(4)
+ 2e
(5)
and 31-
F?
Is-
have been proposed (5). However, theoretical current-potential curves for these reactions and this electroanalytical method have not been available for comparison with experiment. The work presented here outlines the general method for obtaining theoretical current-potential curves when the electrode mechanism is Reaction 3 and m and q have any integral values. Results are given for the important cases of a second-order reaction (m = 2, q = 1) and a third-order reaction (m = 3, q = 1). THEORY
This oxidation appears as an mth-order electrode reaction with polarographic current-potential characteristics given by the following equation [see for example, Ref. (3)].
Boundary Value Problem. For a reversible reduction and Reaction 3, the boundary value problem for diffusion to a planar electrode is (6, 7 ) :
- - - DO a2- v o at
The potential-time relationship for any nonunity reaction order represented by the general scheme
+
mO neeqR (3) is known for the constant-current experiment (4). Recently, the oxidation of iodide ion in acetonitrile has been studied (1) J. Revenda, Collect. Czech. Chem. Commun., 6, 453 (1934). (2) I. M. Kolthoff and C. S . Miller, J . Amer. Chem. Soc., 63, 1405 (1941). ( 3 ) J. Heyrovsky and J. Kuta, “Principles of Polarography,” Academic Press, New York, 1966, p 177. (4) W. H. Reinmuth, ANAL.CHEM., 32, 1514 (1960).
142
ANALYTICAL CHEMISTRY
ax
(7) For t
= 0, x
2 0,
(5) G . Dryhurst and P. J. Elving, ibid., 39,606 (1967). (6) P. Delahay, “New Instrumental Methods in Electrochemistry,” Interscience, New York, 1954, Chap. 3. (7) R. S. Nicholson and I. Shain, ANAL.CHEM., 36,706 (1964).
Forf>O,x-c
a,
c04cO*; C R + o
(9)
For the stationary electrode polarography experiment, the following boundary conditions give the concentration-potential relationship and the relation between the fluxes of substance 0 and substance R at the electrode surface when a triangular-wave potential function, S(t), is applied to the electrode (7). For t > 0, x = 0,
.5CA
=0
.3-
3
.2.I -
LL I-
z w
g
"rn
_ -- es(t) LO"'
'\\
I
I-
Y
;-
.4-
0-
-.I -
" -.23
cRq
\.,'
-.3-
and
i
q [Do
r3)]
= -m [ D R
f3)] = nFA
(11)
where
\
t
a
l
'
*
120 60 0 - 6 0 - 1 2 0
1 2 0 60 0 -60 -120
P O T E N T I A L , mV
Figure 1. Stationary electrode polarograms for second and third-order electrode reactions compared with a first-order reaction Potential axis is ( E - Ei!~)n. Curve A . Solid line is second order, dotted line is first order Curve B. Solid line is third order, dotted line is first order
or in terms of Eli2
and
nFv RT
a = --
Here Co and CR are the solution concentrations of 0 and R, Co* and CR* are initial bulk concentrations, DO and DR are diffusion coefficients, t is the time, y = d D o / D n , x is the distance from the electrode surface, u is the scan rate, Et is the initial potential, E" is the formal electrode potential, EIiz= E" - (RT/nF)[qln(yq/m) (q - m)ln CO*], X is the time the triangular potential scan reverses direction, m and q are stoichiometric numbers in Reaction 3, n is the number of electrons involved in the reaction, and R, T , and F have their usual significance. A closed solution to Equations 6-11 could not be obtained but the numerical approach developed by Nicholson and Shain (7) was used for calculating the current-potential relationships. The boundary value problem is first converted to a single dimensionless integral equation
+
where at) is a dimensionless parameter related to the current by i(t) = nFACo* d x a x(at) (17) Next, a numerical method is used to solve Equation 16 for x(at) in a chosen range of at and the desired current-potential data are obtained. The method of solution used here was nearly identical to the one detailed in Reference (7) but in addition included the modification of employing an iterative false-position interpolation procedure (8) to handle the non(8) F. B. Hildebrand, "Introduction to Numerical Analysis," McGraw Hill, New York, 1955.
linear equation properly. Calculations were made on a IBM 360 Model 50 digital computer. Correlations. To assist in the use of stationary electrode polarography for characterizing electrode mechanisms, diagnostic criteria have been developed. These correlations, as described in Reference (7), involve the width of the stationary electrode polarograms, the ratio of anodic to cathodic peak heights, and the variation of peak or half-peak potentials and peak currents with scan rate. The curve width is conveniently expressed as the difference between the peak potential and the potential preceding the peak at half-peak height (half-peak potential). Anodic peak heights are measured from extensions of the cathodic curves. With increasing scan rates, peak currents divided by the square root of the scan rate, i D / A may , increase, decrease, or remain constant and peak potentials may shift anodically or cathodically or may not shift at all depending on the overall electrolysis mechanism. The calculated current function, 4~ at), for selected values of potential, ( E - El&, is summarized in Table I. This table includes a single scan to a potential -36O/n mV of Eli2and a cyclic scan with the potential reversed at - 138/n mV of El/*. The calculations show that the reverse peak potential appears several millivolts anodic of that indicated in the table if the potential is switched at a value more negative than - 138/n mV. The numerical calculations reveal the following correlations. For the case of rn = 2 and q = 1, d . ? r ~ ( a t ) has a maximum value of 0.3533 at a potential -36.0/n mV cathodic of El/*. The half-prak potential precedes El,* by 45.4/n mV and the ratio of the anodic peak height to the cathodic peak height is 1.09 independent of switching potential. Results for rn = 3 and q = 1 show that x(at) has a maximum value of 0.3033 at -49.8/n mV cathodic of Ell2 and that the half-peak potential precedes E1/2by 54.7/nmV. The anodic to cathodic peak ratio is 1.16. In all cases the value of the peak current divided by the square root of the scan rate and the potential at which the peak appears is independent of scan rate. A comparison of cyclic polarograms for the two cases is made with the cyclic polarogram for rn = 1 and q = 1 [Reference (7)] in Figure 1. The second- and third-order curves appear lower and broader than the first-order curve.
4;
VOL. 41, NO. 1, JANUARY 1969
143
EXPERIMENTAL Table I. Current Functions l/&(at) for Second and Third-Order Electrode Reactions ( E - Elj2)nmV 130 100
Second order
Third order
l/nx(at)a
z/ix(atP
0.012 0,037 0.051 0.071 0.096 0.126 0.143 0.160 0.179 0.197 0.216 0.233 0.251 0.267 0.283 0.297 0.309 0.320 0.329 0.337 0.343 0.347 0.351 0.352 0.353 0.353 0.352 0.350 0.347 0.344 0.336 0.327 0.306 0.285 0.268 0.257 0.233 0.213 0.197 0.184 0.173 0.164 0.156
90 80 70 60 55 50 45
40 35 30 25 20 15 10 5 0 -5 10 - 15 -20 -25 - 30 - 35 -40 -45 - 50 - 55 - 60 - 70 - 80 - 100 - 120 - 138 - 150 - 180 -210 - 240 - 270 - 300 - 330 360
-
-
Potential scan reversed at
- 120 - 100 - 80 -70
- 60 - 50 -40 - 30 - 20 - 10 0 10 20 30 40 50 51.9b 60 64. 2b 70 90 120 150 a
b
0,208 0.166 0.122 0.098 0,072 0.053 0.012 -0.018 -0.054 -0.089 -0.123 -0.158 -0.183 -0.204 -0.218 -0.222 -0.223 -0,220 -0.211 -0.185 -0.141 -0.108
Accuracy 10.001. Reverse peak potential.
144
0
ANALYTICAL CHEMISTRY
0.018 0.048 0.066 0.087 0.111 0.138 0.151 0.164 0.178 0.191 0.204 0.216 0,228 0.239 0.249 0.258 0.266 0.273 0.280 0.285 0.289 0.293 0.298 0.299 0.301 0.302 0.303 0.303 0.303 0.302 0.300 0.297 0.287 0.275 0.264 0.257 0.238 0.221 0.206 0.193 0.181 0.171 0.163
-138 m V 0.186 0.139 0.095 0.073 0.052 0.028 0.003 -0.020 -0.044 -0.069 -0.094 -0.117 -0.138 -0.157 -0.171 -0.181 -0.186 -0.186 -0.185 -0.172 -0.138 -0.106
All chronoamperometric experiments were carried out with a three-electrode controlled-potential circuit using Philbrick Model 456, 656, and P25AH operational amplifiers in Q-3 modules (G. A. Philbrick Researchers, Inc., Boston, Mass.). The circuit employed was that of Figure 4, Reference (9), with a Hewlett Packard Model 330/3302A function generator (Hewlett Packard, Palo Alto, Calif.) used to obtain the triangular and square-wave forms. A Bolt, Bernak, and Newman Model 800A X-Y recorder (Data Equipment Co., Santa Ana, Calif.) was used as a readout device for slow scan experiments and to record constant-potential current-time curves. For scan rates corresponding to frequencies greater than 0.25 cps, a Tektronix Model 585 oscilloscope (Tektronix, Inc., Beaverton, Ore.) with a Type W plug-in unit was used. The cell design and electrodes were identical to those described previously (IO). The cell consisted of a 30-ml borosilicate glass beaker with a Teflon lid machined to fit. Holes were drilled in the lid to allow insertion of a deaerator, counter electrode, and working electrode. A 10/30 standard taper joint was sealed in the side of the cell to accept a Luggin capillary probe. This probe could be positioned very close to the working electrode to minimize iR drop. The counter electrode was a platinum wire helix (14-gauge). The reference electrode and salt bridge were separated into three sections by IO-mm fine-fritted-glass discs (Corning #39570F). The left compartment was a saturated calomel electrode, the middle section contained KCI, and the right compartment terminated with a 14/20 standard taper joint to provide the means of connecting the Luggin capillary probe to the reference electrode. The working electrode was a microburet type hanging mercury drop electrode (Metrohm Ltd., Switzerland) modified as described in (11) to reduce the resistance of the electrical contact to the mercury drop. Polarograms were obtained with a Sargent Model 15 Polarograph (E. H. Sargent and Co., Chicago, Ill.) using a two-electrode cell with a total resistance of 150 ohms. The pH measurements were made with a Beckman Model 76 Expandomatic (Beckman, Fullerton, Calif.) pH meter. All chemicals were reagent grade and used without further purification. Experiments were carried out at 25 "C. RESULTS AND DISCUSSION The dissolution of mercury into dilute cyanide solutions was selected as the experimental system with which to verify the theory for Reaction 3 and for rn = 2 and 3, and q = 1. This reaction has been studied elsewhere (1, 2, 12-15) with classical polarography and chronopotentiometry. Solution conditions of cyanide concentration and pH can be controlled so that the overall electrode process is either 2CN-
2 Hg(CN)2 + 2e
3CN-
2 Hg(CNI3- + 2e
or (19)
The results of Tanaka and Murayama indicate that Reaction 18 describes the overall reaction mechanism for the polaro(9) W. L. Underkofler and I. Shain, ANAL.CHEM., 35, 1778 (1963). (10) M. S. Shuman, Ph.D. Thesis, University of Wisconsin, Madison, Wis., 1966. (11) R. H. Wopschall and I. Shain, ANAL.CHEM., 39, 1527 (1967). (12) T. Murayama, Science Reports of the Tohoku University, Series I, Vol. XLV, No. 2, July 1961. (13) N. Tanaka and T. Murayama, 2. Phys. Chem. (Frankfurt), 14, 370 (1958). (14) N. Tanaka and T. Murayarna, ibid., 21, 146 (1959). (15) N. Tanaka and T. Murayama, ibid., 11, 365 (1957).
16-
2
g
12-
w n
n 3
8-
0
.2
VOLTS vs. SCE
Figure 2. Comparison of polarographic data with third-order behavior
Figure 3.
Points are experimental for a solution of pH = 7.0 and 6mM cyanide in 0.2M Na2HPO4-NaHZPO4 buffer; the line is the theoretical slope of 30 m V
graphic oxidation of mercury into solutions around pH = 7.6 and cyanide concentrations of less than 1.0 mM. The conditions under which Reaction 19 describes the overall mechanism are not readily apparent nor can they be accurately calculated since large discrepancies exist in available data for the formation constants of mercury cyanide complexes (16). Therefore, Reaction 19 was considered operative for a set of solution conditions that met the polarographic criterion of fitting Equation 2 for m = 3. These conditions were found with a solution of pH = 7.0 and an analytical concentration of cyanide approximately 6 mM. Figure 2 shows that a polarogram of this solution fits the theoretical slope of 30 mV when log [i/(id - i)a] is plotted against potential. Correlations were made between experiment and theory by doing both linear-scan experiments and constant-potential experiments in succession on the same solution. The con- the necessity of destant-potential experiment eliminated termining each quantity in A C o * d D o , a term that appears in Equation 17. Because the equation for the current at a spherical electrode for a diffusion controlled process at constant potential is (6)
-
i = nFADoCo* [(l/dTDOt)
+ l/roI
.4 p 2
.5
.6
Analysis of current-time curves
Line A . Solid line through experimental points has a slope of 31.8pA secl’*. Solution is pH = 7.0,6mMcyanide, O.2M NazHPOa-NaH2P04buffer and the potential is -0.100 V us. SCE Line B. Solid line through experimental points has a slope of 3.OOpA sec1’*. Solution is pH = 7.5, 0.4mM cyanide, 0.2M NazHPOa-NaHzP04 buffer, -0.100 V us. SCE
I
0.2
z
Q +
0.1
0
Z
3 LL
0.0
c
z w [r
(20)
where ro is the radius of the electrode and other symbols have their usual -meaning, a plot of i us. l/d;yields the slope nFA C o * qDola. Substitution of the value of this slope into Equation 17 results in elimination of ACo*
