levels below 10-5M (7). Titration of a blank of sulfolane with perchloric acid in sulfolane, using the glass electrode as indicator, shows no sign of a break in the titration curve. Over the entire range of perchloric acid added to lO-lM), the slope of the potential us. log CHCIO, curve is greater than 60 mV, and furthermore it increases smoothly during the titration. Also, even in sulfolane-water mixtures the AE values exceed 60 mV, as shown in Table I. However, for three buffer solutions in pure sulfolane consisting of 10-2Mtetraethylammonium picrate and and 10-3M picric acid, respectively, both A E values are equal to the Nernst coefficient of 60 mV. The anomalously high A E values are peculiar to highly acidic solutions. 2,4-Dinitrobenzenesulfonic acid, which approaches perchloric acid in strength in certain nonaqueous solvents ( I I ) , is very weakly dissociated in sulfolane. The same is true of fluorosulfonic acid, which has an equivalent conductivity of only 0.5 in a 0.1M solution. However, hydrogen fluoroborate in sulfolane-benzene mixtures has been shown to be an outstandingly effective acidic reagent for the isomerization
of olefins (12), and its analytical potentialities should be investigated further. We are proceeding with attempts to calibrate an acidity scale in sulfolane as solvent, but even if such attempts should be unsuccessful, there already exists sufficient evidence to recommend perchloric acid in sulfolane as a titrant for very weak bases, particularly in relatively inert solvents. Apart from the slight discoloration described, the solutions appear to be quite stable. Their total acidity, determined by titration in water, remained unchanged for a period of one year. Their hydrogen ion activity, as measured with the glass electrode, appears to be stable as long as they are well protected from the atmosphere. This is not the case for perchloric acid solutions in acetonitrile, which rapidly age with a decrease in hydrogen ion activity. RECEIVED for review December 12, 1968. Accepted February 17, 1969. We thank the National Science Foundation for financial aid under grant number GP-6478X. (12) J. W. Powell and M. C. Whiting, Proc. Chem. SOC.,1960, 412.
(11) D. J. Pietrzyk and J. Belisle, ANAL. CHEM., 38, 969 (1966).
II CORRESPONDENCE
1I
Cyclic Voltammetry Theory for the Disproportionation Reaction and Spherical Diffusion SIR: SAVEANT and coworkers recently proposed a probable relationship between ece mechanisms and the disproportionation reaction ( I ) : O+neeR
2R$O+Z
In addition to the general solution of the problem there are several limiting cases, which are discussed by Saveant and coworkers. The most important of these asymptotic solutions is the one for large kinetic parameter ($, vide infra) where a steady-state assumption for the concentration of R leads to the equation
(1)
These authors suggested that linear sweep voltammetry should provide a means for distinguishing between these two reaction paths. Thus, they were prompted to reconsider some of the theory of stationary electrode polarography for ece reactions, and also to present, for the first time, theory for Reaction I (Nernstian electron transfer). We independently derived the theory of stationary electrode polarography for Reaction I, but our treatment differs in two important respects from that of Saveant and coworkers: (1) our model is spherical rather than linear diffusion; and (2) our calculations are for cyclic rather than for single scan voltammetry. We believe that each of these differences represents an important extension of the data of Saveant and coworkers, and hence the purpose of this Correspondence is to present those results of our calculations which are complementary with respect to the data of Saveant and coworkers.
2
- c x ( y ) K ( y - z)dz
=
21/ax(y)2/3eusx(y)
(1)
Terms in Equation 1 have the following definitions: y = nFut/RT = at
x(y)
=
i ( y ) / n F A 6 CO*
(2)
(3)
There t is time, v is scan rate, CO*is initial bulk concentration, and ro is electrode radius. In addition, the quantities u and &(y) are defined by the equation u
+ In[sA(y)] = n F ( E - E") + In RT
(6)
where CALCULATIONS
The problem is stated by Saveant and coworkers with the two exceptions already mentioned : the electrode is assumed to be spherical and its potential varies as a triangular wave. In general, our calculation method was an integral equationfinite difference procedure, apparently similar to the method used by Saveant and coworkers. Our approach is described elsewhere in adequate detail (2). 862
ANALYTICAL CHEMISTRY
$ = k&*/a
(7)
Equation 1 is identical with Equation 116 of Reference ( I ) , except that the kernel function in the present case (Equa(1) M. Mastragostino, L. Nadjo, and J. M. Saveant, Electrochim. Acta., 13, 721 (1968). (2) M. L. Olmstead and R. S. Nicholson, ANAL.CHEM.,41, 260 (1969).
tion 4) corresponds to spherical diffusion (3). Solutions of Equation 1 were obtained by a method described previously by one of us ( 4 ) .
Table I. Current Function for a Planar Electrode and the Spherical Correction Function
RESULTS
Potential" 120 100
Saveant and coworkers present data for single scan voltammetry and linear diffusion for the entire range of $. For very large $, no anodic peak is observed for the cyclic experiment, and hence the only difference between our results for large $ and those of Saveant and coworkers is the fact that we include spherical diffusion. Although the influence of sphericity in Equation 1 cannot be expressed analytically, we were able to develop a simple empirical relationship. Thus, we represent the current function (Equation 3) for spherical diffusion by xs(y), the function for linear diffusion by x&), and then write
.\/Lm= .\/, X p l w + 4e(Y)
0.004 0.010 0.018 0.032 0.056
0.0
40
0.100 0.170 0.280
30 25
0.536
0.004 0.014 0.042 0.110 0.172
0.636 0.740 0.834 0.918 0.982
0.254 0.368 0.504 0.672 0.848
1.026 1.048 1.052 1.016
1.028 1.212 1.376 1.528 1.650
0.986 0.952 0.912 0.840 0.774
1.756 1.836 1.902 1.988 2.034
90
80 70 60 50
10
There is defined by Equation 5 and O(y) is an empirical and e(y) are precorrection function ( 5 ) . Values of x&) sented in Table I. As pointed out by Saveant and coworkers, Equation 1 is formally identical to the equation for a rapid dimerization following electron transfer, and therefore the spherical correction data of Table I also are applicable to this case ( 4 ) . For values of less than about 2, an anodic wave is ob-
0
(3) M. L. Olmstead and R. S. Nicholson, J . Elecrroanal. Chem. Interfacial Electrockem., 14, 133 (1967). (4) R. S. Nicholson, ANAL.CHEM., 37, 667 (1965). (5) R. S. Nicholson and I. Shain, ibid., 36,706 (1964).
0.440
20 15
(8)
+
d&l(Y)
5 -5 10
15 20 25
1.040
- 30
35 40
50 60 a
Potential is ( E - E")n
+ (RT/3F)ln(6n/$)at 25 "C.
~~~~
Table 11. Current Function for a Planar Electrode n(E - E"),
mV 120 100 80 60 40 20 10 0 - 10 - 20 - 30 -40
- 50
-60 - 80 -100 - 102 - 100 - 80 -60 -40 - 20 0 10
20 30 40 50 60 70 80 100 120 140
0.005 0.009 0.020 0.042 0.085 0.161 0.270 0.329 0.380 0.420 0,442 0.447 0.439 0.422 0.401 0.355 0.314 0.311 0.294 0.232 0.165 0.069 -0.052 -0.173 -0,217 -0,244 -0.254 -0.250 -0.237 -0.219 -0.199 -0.180 -0.146 -0.119 -0.099
0.0625 0.009 0.020 0.042 0.085 0.161 0.272 0.331 0.384 0.425 0.449 0.456 0.449 0.434 0,414 0.371 0.333 0.330 0.313 0.255 0.191 0.101 -0.017 -0.135 -0.179 -0.207 -0.218 -0.217 -0.206 -0.191 -0.174 -0.156 -0.126 -0.102 -0.084
0.150 0.009 0.020 0.042 0.085 0.162 0.274 0.335 0.390 0.432 0.458 0.468 0.464 0.450 0.432 0.392 0.356 0.353 0.338 0.282 0.221 0.135 0.021 -0.094 -0.137 -0.166 -0.180 -0.181 -0.174 -0.161 -0.147 -0.132 -0.106 -0.085 -0.070
0.350 0.009 0.020 0.042 0.085 0.164 0.279 0.343 0.402 0.449 0.479 0.493 0.493 0.482 0.466 0.429 0.395 0,392 0.378 0.324 0.265 0.184 0.077 -0.034 -0.078 -0.109 -0.126 -0.132 -0.129 -0.121 -0.111 -0. 100
-0.079 -0.063 -0.051
*
0.675 0.009 0.020 0.042 0.086 0.167 0.287 0.356 0,420 0.473 0.509 0.527 0.530 0.521 0.507 0.470 0.434 0.431 0.417 0.363 0.305 0.227 0.125 0.069 -0.028 -0.061 -0.081 -0.091 -0.092 -0.088 -0.082 -0.074 -0.059 -0.046 -0.037
1.5 0.009 0.020 0.042 0.088 0.173 0.307 0.385 0.461 0.524 0.568 0.590 0.594 0.584 0.566 0.522 0.480 0,476 0.463 0.407 0.349 0.264 0.178 0.074 0,028 -0.007 -0,032 -0.046 -0.053 -0.054 -0.051 -0.047 -0.038 -0.030 -0.023
3.5 0.009 0.020 0.043 0.093 0.189 0.349 0.442 0.536 0.612 0.656 0.674 0.669 0.649 0.622 0.565 0.513
10.0
0.009 0.021 0.046 0.106 0.231 0.445 0.562 0.669 0.739 0.766 0.761 0.736 0.700 0.661 0.589 0.531
35.0 0.010 0.023 0.057 0.145 0.336 0.621 0.743 0.827 0.855 0.843 0.805 0.757 0.710 0.665 0.588 0.530
VOL. 41, NO. 6, MAY 1969
863
Table 111. Peak Current Function for a Spherical Electrode p X 1040 0.0 6.25 25.0 56.25 0.005 0.447 (28.6) 0.449 (29.2) 0.450 (28.8) 0.451 (28.9) 0.010 0.448 0.450 0.452 0.454 0.015 0.449 0.451 0.453 0.456 0.025 0.450 0.453 0.456 0.459 0.040 0.452 0.456 0.460 0.464 0.0625 0,456 0.460 0.465 0.470 0.100 0.461 (30.6) 0.467 0.473 0.479 0.150 0.468 0.476 (32.0) 0.483 (32.5) 0.490 (33.0) 0.225 0.478 0.487 0.497 0.506 0.325 0.491 (34.2) 0.520 0.514 0.525 0.475 0.509 0.523 0.537 0.551 0.675 0.531 0.547 (38.0) 0.564 (39.0) 0.581 (40.1) 1.ooO 0.560(37.6) 0.580 0.601 0.622 1.500 0.594 0.620 0.646 0.672 (41.8) 2.70 0.650 (34.2) 0.685 (36.3) 0.722 (38.4) 6.00 0.724 (27.9) 24.00 0.832 (14.8) Values in parentheses are peak potentials, (E" - E&, mV.
*
0
Table IV. Ratio of Anodic to Cathodic Peak Current for Planar and Spherical Electrodes x 104 Wa 0.0 6.25 25.0 56.25 100.0 0.998 0.998 0.998 0.998 0.02 0.998 0.996 0.995 0.995 0.995 0.04 0.995 0.992 0.992 0.992 0.992 0.992 0.06 0.986 0.986 0.986 0.987 0.10 0.985 0.977 0.978 0.978 0.976 0.977 0.16 0.965 0.966 0.967 0.964 0.25 0.962 0.943 0.945 0.947 0.949 0.40 0.940 0.917 0.921 0.924 0.928 0.60 0.913 0.889 0.895 0.900 0.877 0.883 0.90 0.845 0.854 0.862 0.870 1.30 0.835 0.835 0.812 0.824 0.798 1.90 0.784 0.752 0.770 0.786 0.801 2.70 0.732 0.724 0.746 0.766 0.672 0.700 4.00 0.681 0.710 0.736 0.613 0.649 6.00 a For p > 0, w = k&*r, and ar = 4.
tained if a cyclic potential scan is employed. In this case, observation of the entire cyclic polarogram adds a valuable diagnostic test to those tests already discussed by Saveant and coworkers. Moreover, the anodic current provides a sensitive measure of the rate constant, kz. From a diagnostic point of view the least ambiguous evaluation of experimental data is a complete comparison with theory of entire cyclic polarograms recorded at a series of scan rates. We have found that such comparisons provide considerably more mechanistic insight than simply measuring the variation with scan rate of a single observable. Data from which theoretical cyclic polarograms can be constructed are presented in Table 11. Data of Table I1 are for linear diffusion, because no relationship could be found for conveniently including effects of sphericity; To provide some estimate of the influence of sphericity, however, maximum values of x(x,) and peak potentials are listed in Table I11 for several values of 9 and a parameter, p , defined by Reference (3): p = D/ro2Co*kz = qV/1c.
(9)
Anodic peak currents provide a measure of kz, because the anodic peak of a cyclic polarogram is quite sensitive to the 864
0
ANALYTICAL CHEMISTRY
100.0 0.453 (29.0) 0.456 0.458 0.462 0.468 0.475 0.485 0.498 (33.5) 0.515 0.536 0.564 0.598(41.1) 0.643 0.699 (43.4)
rate of the disproportionation. In an effort to obtain a quantitative correlation between peak current and kz, polarograms were calculated for several values of 1c. and three values of the switching potential, Ex. For each case the ratio, i& of anodic to cathodic peak currents [the baseline for calculation of i, is the extension of the cathodic waves (5)] was evaluated. For each switching potential the ratios i,/ic were plotted against the quantity log (+UT) [= log (kzCo*r)],where ar is UT
=
nF -
RT
(E"
- Ex)
and r is the time from E" to Ex [r = (E" - Ex)/v]. These plots for UT = 3.5, 4.0, and 5.0 differed only by small shifts along the log (kzCo*r)coordinate proportional to UT. Thus, it was possible to incorporate empirically effects of switching potential in a variable, w, defined by
Values of i& as a function of w are given in Table IV. Equation 11 and the data of Table IV can be used to construct working curves of i& us. kzCo*r from which kz can be determined. For spherical electrodes, no convenient method of incorporating simultaneously effects of sphericity and switching potential could be found. Thus, values of i& were determined for fixed UT and four values of p. These data, which are rigorously applicable for ar = 4, are included in Table IV. RECEIVED for review January 22, 1969. Accepted March 3, 1969. This research was supported by the National Science Foundation and United States Army Research OfficeDurham (Contract No. DA-31-124-ARO-D-308). MICHAEL L. OHMSTEAD' RICHARD S. NICHOLSON Chemistry Department Michigan State University East Lansing, Mich. 48823 1 Present address, Bell Telephone Laboratories, Murray Hill, N. J. 07971
