freshly prepared solution of CrBr(H20)5+2the titration efficiency averages 99.1%. The titration efficiency drops slowly with time, reaching a value of 98.5yc after 6 hours. As the generating current is increased, the initial titration efficiency is lowered and the rate of decay of titration efficiency is increased. Thus with a generating current of 48.2 ma. the titration efficiency initially is 98.9% but begins to drop significantly after only 4 hours. When the generating current is increased to 193 ma. the initial titration efficiency has fallen to only 97%. Hence in practice 0.134 solutions of CrBr(HzO)a+2can be used for at least 4 hours with generating cur-
rents of 48.2 ma. and a t least 6 hours with generating currents of 19.3 ma. or less. Under these conditions the titration efficiency is sufficiently stable that titration of occasional standards interspersed with unknowns is sufficient to evaluate the titration efficiency and ensure accuracy. LITERATURE CITED
( I ) Bard, A. J., Petropoulos, A. G., Anal. Chim. Acta 27, 44 (1962). (2) Forbes, G. S., Richter, H., J . Am. Chem. SOC.39, 1140 (1917). (3) Gurney, R. W., "Ionic Processes in
Solution,'' pp. 132-7, RlcGraw-Hill,
New York, 1953.
14) James, G. S., Stephen, M.J., Analyst \
,
85,
n-
,.-,,A\
dD ( l Y b U j .
( 5 ) Reilley, C. N., in "Treatise on An-
alytical Chemistry," Part 1, 5'01. 4, pp. 2109-62, I. >I. Kolthoff and P. J. Elving, eds. Wiley, New York, 1963.
RECEIVED for review August 14, 1964. Accepted January 18, 1965. This work was supported in part by Research Corp. through a Grant in Aid from the Frederick Gardner Cottrell Fund and by the National Science Foundation through Grant No. GP 1996. Sr. M.C. acknowledges support from Sational Science Foundation for participation in a Summer Program in Instrumental Analysis held at Rensselaer Polytechnic Institute.
Standardization of Analytical Data Obtained with Silver-Silver Chloride Electrodes in M etha nol-Wa ter Solvents MAYA PAABO, ROGER G. BATES, and R. A. ROBINSON National Bureau of Standards, Washington,
D. C.
The standard potential of the silversilver chloride electrode has been determined in 10, 20, 45, and 70 wt. % methanol at 2 5 " C. These potentials, 0.2 1549, 0.20901, which are 0.1 94 14, and 0.1 6833 volt, respectively, provide a useful basis for the determination of data of analytical interest in methanol-water solvents. The activity coefficients of hydrochloric acid at concentrations from 0.01 to 0.1M in these mixed solvents have been calculated.
in water-methanol solvents and, thence, to calculate values of p,K for weak acids and values of pan* for buffer solutions in these solvents (3). The standard e.m.f. of this cell with 10 and 20% methanol as solvent has been determined by Harned and Thomas (9) from 0" to 40" C. and in 33.4% methanol by Bates and Rosenthal (4). Oiwa ( l a ) studied solvents containing 20, 40, 60, 80, and 90% methanol, and Paabo, Robinson, and Bates ( I S ) measured the standard e.m.f. in 50% methanol from 10" to 40" C. Austin, Hunt, Johnson, and Parton ( 2 ) measured ,Eo in solvents containing 43.3, 64, 84.2, 94.2, and 100% methanol (15" to 45" C,), while Nonhebel and Hartley (11) also made measurements using 1 0 0 ~ cmethanol as solvent and Koskikallio (10) studied the same cell in almost water-free methanol, Unless otherwise indicated, these measurements were made only at 25" C. I n spite of all this work, certain discrepancies remain. Thus, Harned and Thomas give E" = 0.20888 abs. volt for 20% methanol at 25" C., whereas Oiwa found 0.2094 volt. Again, for 43.12% methanol at 25" C., Feakins and Watson (6) calculated 0.1958 volt from the data of Schwabe and Ziegenbalg (Id),and this is not in good agreement with the value of 0.1939 volt found by Austin et al. for 43.3Yc methanol. Moreover, it is difficult to draw a smooth curve through the available points when ,Eo is plotted against the solvent composition. We have now re-
of its stability, reproduciand applicability under a wide varietv of conditions, the silversilver chloride electrode is one of the most useful reference electrodes available for use in aqueous solutions. Its behavior in numerous nonaqueous and mixed solvents is likewise satisfactory. If its standard potential were known accurately, this electrode would provide a suitable basis for the determination of electrode potentials, redox potentials, equilibrium constants, and other data of analytical interest in these solvents. For example, the standard electromotive force (.E") of the cell P t ; Hz (g., 1 atm.), HC1 (m) in CHIOH (X tvt. 7 0 ) ) H2O [(loo - X ) iTt. 701, AgCl; Ag I must be known if cells of type I are to be used to measure values of p a ( u ~ y c l ) 462
ANALYTICAL CHEMISTRY
peated the measurement of .E" in 10 and 2070 methanol and have made new measurements in 45 and 70% methanol at 25" C. The new data for ,E" in 10 and 20% methanol agree well with the values found by Harned and Thomas (9). EXPERIMENTAL
The methanol was distilled from spectrograde material. Polarographic analysis showed the presence of only 10-BM formaldehyde. Hydrochloric acid was the middle fraction from constant boiling acid. The cells used were those described by Bates and Rosenthal (4), except that an additional presaturator was used for solvents containing 45 and 70yc methanol. The measured e.m.f. of the cell was corrected to 1 atm. pressure of hydrogen, using available vapor pressure data for watermethanol mixtures ( 5 ) . The corrected e.m.f. values are given in Table I. STANDARD POTENTIALS
The standard potential of the silversilver chloride electrode was calculated by the equation
.Eo
=
E
+ 2k log m + 2 k log ,y+
(1)
where m is molality, k is 2.3026RT/F, and
x
=
Table I. E.m.f. of the Cell Pt, Hz (g., 1 atm.), HC1 ( m )in CH9OH (X wt. %), AgCl; Ag x = 20 x = 45 m E m E m 0.004674 0.48898 0.01003 0.43838 0.05867 0.01124 0.44601 0.01117 0.43308 0.05968 0.01671 0.42673 0.01872 0.40833 0.06480 0.04223 0,38196 0.01994 0.07370 0.40555 0.05631 0.36825 0.02951 0.38703 0.08677 0.06577 0.36088 0.03147 0.38410 0.08934 0.07505 0.35455 0.03446 0.37978 0.09931 0.08455 0.34887 0.04640 0.36585 0.1051 0.1080 0.33707 0.04971 0.36247 0.1091 0.05784 0.35572
10 E 0.45322 0.41615 0.37560 0.36006 0.34260
m 0,01095 0.02348 0.05438 0,07520 0.1082
~
2c 3
02
0
03
0-
-Js Figure 1 . Molal activity coefficient of hydrochloric acid in 45 and 70% methanol Straight lines a r e Debye-HGckel limiting slopes
Here A and B are constants of the Debye-Huckel theory, dependent only on the temperature and dielectric constant of the solvent, and & is the density of the solvent. The symbol d represents the "ion-size" parameter and 5? is the mean molecular weight of the solvent defined b y . 1 a v
-
X MHtO
e
74.2
m
0.009875 0.01515 0.01991 0.02500 0.03507 0.04564 0.06012 0.07568 0.09147 0.1073
MeOK, volt Harned and Thomas, re- ,Eo = calculated 0.21849 Oiwa (12) ... This investigation 0.21549
Table V.
m
0.01 0.02 0.05 0.10 a
E 0.41583 0,39574 0.38299 0.37243 0,35706 0.34526 0.33257 0.32238 0.31384 0.30669
70.0
58.7 47.1
JG
A
B
a, A.
18.84 19.75 22.44 25.98
0.5545 0,6052 0.7881 1,0964
0.3381 0.3481 0.3801 0,4244
4.3 4.3 4.3 5.0
Table 111. Standard Potential of the Ag-ASCI Electrode at 25" C.
(Molal Scale) Wt. 70 methanol
,E", volt 0.21549 0.20901 0.19414 0.16833
10
20 45 70
0
20% MeOH, volt
,E"
mV. 0.04 0.04 0.05 0.05
uC,
Table IV. Standard Potential of Ag-AgCI Electrode (Molal Scale) at Round Values of Weight Percentage of Methanol
Methanol, 70 10%
1-2
where x is the weight fraction of water in the solvent mixture. [Ext.] denotes the extended terms of the DebyeHdckel theory; these were taken from the tables of Gronwall, LaMer, and Sandved (8). The symbol b represents an adjustable parameter. The data necessary for the calculation of ,E" are given in Table 11, the dielectric constant values being those of Albright and Gosting (1) and the density data those of Griffiths (7) and Oiwa (12). With the aid of the parameters in Table 11, ,Eo for the silversilver chloride electrode was calculated from the experimental e.m.f. by the method of least squares. Satisfactory linear extrapolations of (dEo-2kbm) were obtained with d = 4.3 A, except for the 70y0 methanol mixture where
do
0.9799 0.9645 0.9223 0.8680
d = 5.0 A. gave a better extrapolation. The values of ,E" are given in Table 111, together with the standard deviation ( u i ) of the intercept. From a large-scale plot of all available values of ,E" us. weight percentage of methanol, ,Eo a t 25" C. was read a t round values of the weight composition; these values are summarized in Table IV. For comparison with the results of this investigation, the data of Harned and Thomas (9) have been recalculated by the method used in the present work together with the best recent values of the physical constants concerned. The following results were obtained:
+---
-@fOHaOE
X = 70 E 0.35470 0.35415 0.35023 0.34423 0.33663 0.33527 0.33051 0.32777 0.32604
Properties of Water-Methanol Solvents at 25" C.
Table II.
Wt. 70 methanol 10 20 45 70
at 25" C.
10 20 30 40 50 60 70
=
0.20905 0.2094
80
100
.E" at 25" C. 0,22234 0.2155 0.2091 0.2031 0.1968 0.1906 0.1818 0.1683 0.1492 -.0.0101
0.20901
Activity Coefficient yi of Hydrochloric Acid in Water-Methanol Solvents at 25' C.
x 0.896 0.865 0.820 0.783
=
10
(0.897)0 (0.866) (0.819) (0.780)
x = 20 0.888
0.855 0.805 0.771
x = 45
(0.888)a (0.856) (0.806) (0,762)
0.860 0.819 0.748 0.708
X
=
70
0.818 0.769 0,694 0.637
Values in parentheses are those of Harned and Thomas (9).
VOL 37, NO. 4, APRIL 1965
463
ACTIVITY COEFFICIENTS
Values of the molal activity coefficient of hydrochloric acid in these four methanol-water solvents have been calculated by Equation 2 and are given in Table V. The values of 6 were obtained from the slopes of the extrapolation lines whose intercepts were the standard potentials ,E”. The activity coefficients in 10 and 20% methanol are compared with those found by Harned and Thomas; the agreement is good except at O.lm in 20y0 methanol. Figure 1 is a plot of log -y& vs. m’l2 for 45 and 75y0methanol, the straight lines representing the limiting DebyeHuckel slopes. The activity coefficients
are much lower in 70 than in 45% methanol. Nevertheless, the experimental curves approach the theoretical straight lines from above in both instances. There is no evidence, therefore, of ion-pair formation even in 7oy0methanol.
Maclennan, W. H., J . Chem. SOC.1933,
p. 674.
(6) Feakins, D.. Watson. P.. Zbid.. 1963. (8) Gronwall, T. H., LaMer, V’ K.,Sandved, K., Physzk. 2. 29. 3.58 11428) (9) Harned, H. S., Th
LITERATURE CITED
(1) Albright, P. S.,Gosting, L. J., J . Am. Chem. SOC.68, 1061 (1946). (2) Austin, J. AI., Hunt, A. H., Johnson,
F. A., Parton, H. N., cited by Robinson, R. A., Stokes, R. H., “Electrolyte Solutions,” Butterworths, London, 1959. (3) Bates, R. G., Paabo, AI,, Robinson, R. A., J . Phys. Chem. 6 7 , 1833 (1963). (4)Bates, R. G., Rosenthal, D., Zbid., p. 1088. (5) Butler, J. A. V., Thomson, D. W.,
(1957).
(11) Nonhebel, G., Hartley, H., Phil.
Mag. 50, 729 (1925). (12) Oiwa, I. T., Sei. R e p k . Tohoku L’niv., Ser. 1 41, 47 (1957). (13) Paabo, AI., Robinson, R. A , , Bates, R. G., J . Chem. Eng. Data 9,374 (1964). (14) Schwabe, K., Ziegenbalg, S., 2. Elektrochem. 6 2 , 172 (1958). RECEIVED for review October 29, 1964.
Accepted January 27, 1965.
General Equation for Current-Potential Relationships at Rotating Disk Electrode ILANA FRIED and PHILIP J. ELVING Department o f Chemistry, The University of Michigan, Ann Arbor, Mich.
b The rotating disk electrode has been extensively investigated. The present paper discusses some of the existing theoretical treatment of the rotating disk electrode, evaluating its usefulness for actual electrochemical investigation, and presents the derivation of an equation which describes the complete current-potential curve obtained under standard voltammetric conditions: potential varying linearly with time and current varying as a function of potential. The treatment used assumes that the electrode reaction is reversible and controlled by the rate of mass transport and that an excess of nonelectroactive electrolyte is present. The treatment does not presuppose any particular mechanism for the electrode reaction.
T
HE rotating disk electrode has been the subject of many theoretical and experimental investigations. There are three reasons for the increasing current interest in this electrode type: the relative ease with which the electrode and the means of rotation are obtained; the availability of a rigorous hydrodynamical theory for this configuration; and the desirability of a well understood solid electrode for voltammetry and related techniques, especially for studying the electrochemical oxidation of organic compounds. The voltammetric theory of the rotated disk electrode essentially began with Levich (6), who proposed the following equation for the limiting current
464
ANALYTICAL CHEMISTRY
ili, = 0.62 nFAD1/I v - ’ / ~ w ~ C” / ~ (1) where ili, is the observed limiting current, n the number of electrons involved per molecule in the faradaic process, F the Faraday, A the area of the electrode, D the diffusion coefficient of the electroactive species, Y the kinematic viscosity, w the angular velocity of the disk, and C” the concentration of the electroactive species. The most important concept introduced by Levich is that of the hydrodynamic diffusion layer, 6. To solve the equation of diffusive convective mass transport
bt
(x,y, and z are the system’s coordinates; and vz, vy, and v,, the components of velocity of the solution in the respective directions) for the hydrodynamic conditions of the rotated disk, assuming that (bC/bt) = 0, Levich proved that the mathematical description of the rotating disk leads to results which are formally the same as if a layer of thickness 6 existed adjacent to the disk. The concentration gradient is limited to this layer. Outside this layer, the concentration is maintained constant by convection; inside the layer, diffusion gradually predominates over convection as the means of mass transport of the electroactive species as the surface of the disk is approached.
6 = 1.61 (D/V)’”( Y / U ) ~ ’ ~
(3)
6, as is evident from Equation 3, is independent of the concentration of the electroactive species in the solution and a t the electrode surface. Consequently, 6 does not change even if the concentration a t the electrode surface changes with time. By assuming (bC/bt) = 0 in Equation 2, Levich (6) found the current resulting from a faradaic process a t effectively constant current and constant potential conditions. The next step in the development of the theory of a rotated disk electrode is to let the current-or the potential-vary during the time of electrolysis. Siver (10) proposed equations for the current-time curve at constant potential for both reversible and irreversible electrode processes, Siver (11) also derived an equation for the potential of a rotating disk electrode as a function of time when the current is held constant. Bowers et al. (1) and Rosebrugh and Miller (9) solved the same boundary value problem, although they did not consider rotating disk electrodes. The resulting equation derived by Rosebrugh and Miller (9) is identical with the one derived by Siver (11). Bowers et al. (1) considered diffusion through a membrane, with the concentration on the side facing the solution held constant by stirring. The latter approach is applicable to the study of the rotating disk electrode as indicated by Buck and Keller ( 2 ) . -4 more detailed discussion of the equa-
