during the program was poor. The packing was 10% DC200 silicone on C-22 firebrick (actually, 0.0116 gram of substrate per cm of column). Runs were made a t different temperature rise rates o n a mixture of three compounds which are listed in Table 11. Here the measured temperature rise rate is compared with the value calculated to be necessary to make the predicted (by Equation 2) retention temperature equal the actual retention temperature. Table I1 shows that discrepancies between the calculated and actual retention temperatures
are small, and the calculated rates are therefore close to the real ones. F o r the calculation of theta for all three compounds, the “calculated” temperature rise rate values for ethylbenzene were used. Two sets of theta values were computed: One was obtained by pairing the 0.42 and the 3.71 degrees per minute runs and the other by pairing the 0.42 and 1.89 degrees per minute runs. The results are shown in Table 111.
RECEIVED for review December 12,1966. Accepted May 17, 1967.
Current Transients at a Rotating Disk Electrode Produced by a Potential Step Stanley Bruckenstein and Stephen Prager
Department of Chemistry, Unicersity of Minnesota, Minneapolis, Minn. 55455
MANYPARTIAL DIFFERENTIAL EQUATIONS which occur in electrochemical problems are not amendable to closed-form solution. An example of such a problem is the calculation of the current transient accompanying a potential step which results on a rotating disk electrode a t hydrodynamic equilibrium. The electrochemical problem is, in itself, of fundamental interest for a t least two reasons. First, the “response time” of the disk electrode current to a potential change, and hence a t a n electrode surface concentration change, must be known in order t o evaluate the analytical utility of the electrode under certain conditions-e.g., in following the kinetics of an extremely rapid reaction. Second, film formation or adsorption often interferes with the interpretation of steady-state limiting currents, and the desired information can be obtained from the current transient before such complications interfere. The approximate mathematical solution t o this problem and to many others of electrochemical interest may be obtained by using the well-known “method of moments” ( I ) or the “Galerkin method” (2). A reviewer of this paper has pointed out that this latter method has been applied by Gelb (3)recently t o other electrochemical problems.
On applying a constant potential t o a disk electrode, the surface concentration of the electroactive material becomes constant c(0, r)
=
cs
t>O
(3)
A convenient, approximate, solution t o the above problem may be obtained by using the “method of moments” ( I ) or the “Galerkin” method (2). If Equation 1 is integrated with respect to y from 0 to 03, a gross mass balance is obtained for the entire diffusion layer
where
K
=
0.51 [
~ ~ / v ] l ’ ~ ,
Assuming a simple, time-dependent, Nernst diffusion layer of thickness 6(t) adjacent to the disk surface, we write our concentration distribution in the form
Therefore, from Equation 4
THEORY
The convective diffusion equation for a rotating disk electrode which has attained hydrodynamic steady-state is and In Equation 1 , D is the diffusion coefficient of the electroactive species whose bulk concentration is cb. The normal distance from the electrode surface is y , cy ( = 0.51 [ ~ ~ / v ] l ’ 2 y 2 ) is the velocity of the supporting electrolyte at a distance y , w is the angular velocity, and v is the kinematic viscosity of the supporting electrolyte. Initially there is a uniform concentration distribution throughout the solution c(y, 0 ) =
Cb
O> aDZD),deviations appear between the two methods. As has been mentioned ( 4 , Rp-’ approaches 0.94 under these conditions, almost exactly the value of the difference between the two methods of calculation, and suggests that Equation 10 is a better approximation when 0 is large. (4) V. Yu. Filinovskii and V. A. Kir’yanov, Dokl. Akad. Nauk., 156, 1412 (1964). (5) V. G. Levich, “Physicochemical Hydrodynamics,” Prentice-
Hall, 1963, p. 69.
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ANALYTICAL CHEMISTRY
0.040.080.12 016 0.20
e
Figure 2. Solid curve represents theory according to Equation 10; 0 - 1.96 X lO-3M Cu+Z in OSM KCI; -3.91 X lO-3M Cu+z in 0.5M KCI
+
Reduction process: Cu(I1) 2e -,Cu on gold disk electrode at -0.9 volt cs. the saturated calomel electrode. D = 6.8 X 10-6cmz/secdetermined from angular velocity studies of the steady-state disk current, and was used to calculate 8
Buck and Keller (6) have treated the chronopotentiometric problem a t a rotating disk electrode using a membrane electrode analogy. This can also be done for the potentiostatic problem under discussion, and yields the result (7) m
R-lA,* = 1
+2
~-m2r20“
m=l
(12)
where OAM = D t / P and it is assumed that the thickness of the membrane, I, equals 6. A plot of Rp/R.M is also given in Figure 1 and does not differ much from unity over a wide range of values of 0. EXPERIMENTAL
CUPRICCHLORIDE. Pure copper metal was dissolved in concentrated nitric acid. Hydrochloric acid was added t o the residue and the solution evaporated to dryness. The latter step was repeated and water was added t o the residue and the solution evaporated to dryness to remove any traces of hydrochloric acid. The residue was dissolved in doublydistilled water and diluted to volume t o prepare a stock solution. POTASSIUM CHLORIDE. The material used was crystallized two times. NITROGEN. Nitrogen was purified by passing over heated copper dispersed in Kieselgur. (6) R. P. Buck and H. E. Keller, ANAL.CHEM., 35, 400 (1963). (7) R. C. Bowers and A. M. Wilson, J . Am. Chem. SOC.,80, 2968 (1958).
POTENTIOMETRIC APPARATUS AND ELECTROLYSIS CELL. A conventional three-electrode apparatus was used. ROTATINGDISK ELECTRODE. A 5.00-mm gold disk electrode insulated with Teflon was used. The gold was polished to a mirror-like surface, and cleaned in a warm mixture of sulfuric and nitric acids immediately before use. Experiments were performed at 20’ =t 1 O C in nitrogenbubbled solutions. RESULTS AND DISCUSSION
Cupric ion in 0.5 Mpotassium chloride is reduced in a stepwise fashion Cu(I1) E Cu(1)
+ Cu(1) +
-
Current-time curves were obtained a t w1I2 = 5.86, 10.4, and 16.7 (radians/sec)*’2 for the reduction of Cu(I1) to copper metal in 0.5 M potassium chloride. Experimental values of cm2/ R are plotted us. 0 (calculated using D = 6.8 X sec) in Figure 2. The solid curve represents the predicted relationship according to Equation 10. Agreement is excellent. In conclusion, three different mathematical methods yield solutions differing only slightly from one another for the time-dependent current through a disk electrode on application of a potential-step, and are in satisfactory agreement with experiment.
--+
E
cu
ACKNOWLEDGMENT
at a gold electrode. The results obtained for the second wave closely resemble those reported by Nekrasov and Berezina, (8) using an amalgamated gold disk electrode. The diffusion coefficient of Cu(I1) calculated from the limiting current for the process Cu(I1) 2e + Cu was 6.8 X 10-6 cm*/second.
Experimental work reported in this paper was performed at the Chair of Electrochemistry of the M. V. Lomonosov Moscow State University, during an exchange jointly sponsored by the National Academy of Sciences and the Academy of Sciences of the U.S.S.R. during the winter of 1964/65. S. B. acknowledges helpful discussions with Academician A. N. Frumkin, and with L. N. Nekrasov.
(8) L. N. Nekrasov and N. P. Berezina, Dokl. Akad. Nauk., 142, 855 (1962).
RECEIVED for review April 13, 1967. Accepted May 11, 1967.
+
Aqueous Zinc Chloride as a Stationary Phase for Liquid-Liquid Chromatography of Organic Sulfides Wilson L. Orr Field Research Laboratory, Mobil Oil Corp., Dallas, Texas 75221
A PREVIOUS REPORT ( I ) described a liquid-liquid chromatographic (LLC) method by which saturated organic sulfides could be separated from most other classes of organic compounds. The stationary phases consisted of mercuric acetate in aqueous acetic acid. n-Hexane was used as the mobile phase. Hydrocarbons and most other compound types which occur in petroleum were eluted quickly with essentially no retention by the stationary phase. Saturated sulfides were delayed according to distribution constants which were shown to be proportional to the molecular weight of the sulfide and to the acetic acid content of the stationary phase. Separations on mercuric acetate phases were shown to be optimum for alkyl sulfides with carbon numbers between 10 and 18. Sulfides with more than 18 carbon atoms per molecule were not resolved sufficiently from hydrocarbons, and sulfides with less than 10 carbons were strongly adsorbed and had excessively large retention volumes. Although the strongly adsorbed sulfides could be recovered from the mercuric acetate phases by displacement, they were displaced as a group without resolution. A stationary phase which would allow direct elution, and hence resolution, of lower molecular weight sulfides would be a useful supplement to the LLC technique. The suggestion that aqueous zinc chloride might be a suitable stationary phase for low molecular weight sulfides was indicated by the batchwise extraction experiments reported by C.J. Thompson et al. (2). The distribution and chromato(1) W. L. Orr, ANAL.CHEM., 38,1558 (1966). (2) C. J. Thompson, H. J. Coleman, R. L. Hopkins, and H. T. Rall, Chem. Eng., Dafa, 9, 473 (1964).
graphic behavior of some saturated sulfides (chain and cyclic) with an aqueous stationary phase consisting of concentrated zinc chloride is the subject of this report. EXPERIMENTAL
Experimental methods were essentially identical with those used previously ( I ) . The same chromatographic column, solid support, flow rate, and general techniques were used. The stationary phase consisted of 66.6 zinc chloride and 33.3x water by weight. This phase is designated phase D to distinguish it from mercuric acetate phases A, B, and
c (1).
Redistilled n-pentane or n-hexane (Phillips Pure Grade) was used for the mobile phase. Alkyl sulfides and thiacyclopentane were commercial chemicals. The other cyclic sulfides were kindly supplied by H. T. Rall of the Bureau of Mines, Bartlesville, Okla. RESULTS AND DISCUSSION
Zinc chloride (phase D) is very weak compared to mercuric acetate (phases A, B, and C) in its interaction with sulfides. Measurable interactions were found only for lower molecular weight sulfides. There was no indication of decomposition or alteration of the sulfides by zinc chloride. Recoveries were good and GLC retention times were identical for all components in the sample introduced and in the fractions eluted. Distribution constants, K, and equivalent retention volumes, R, for the sulfides studied are listed in Table I. Values are averages of two or more determinations. The relative error is estimated to be about i8 % for the higher values of K VOL. 39, NO. 10, AUGUST 1967
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