main peak. This does not prevent accurate measurements of carbonate carbon provided sufficient time is allowed between analyses. An unstable or drifting base line was observed with substances like malic acid, which decompose more slowly. Nitric and nitrous acids o r their salts were found to interfere at a temperature of 175’ C or higher. Larger concentrations than those indicated in Table I11 will interfere at 150’ C. Either of these compounds produces a sharp peak that coincides with the normal carbon dioxide peak. It is probably caused by nitrogen dioxide or nitric acid, both of which absorb infrared radiation in the carbon dioxide region. It is unlikely, however, that concentrations greater than those tested will be encountered in the analysis of natural waters. Those substances which gave no interference at the temperatures tested are included in Table I11 also. ~~
ACKNOWLEDGMENT
The authors thank William E. Hatton for his valuable technical assistance in the assembly of the apparatus used in this work. A patent application has been filed on certain features of the apparatus and procedure described herein. It is intended that an instrument incorporating these features will be made commercially available in the near future.
RECEIVED for review September 22,1966. Accepted February 1, 1967. Division of Analytical Chemistry, 152nd Meeting, ACS, New York, N. Y., September 1966. ~~
~
Transfer Coeff icielnts and Heterogeneous Rate Constants for Cadmium(l1:) in Various Supporting Electrolytes Evaluated by Phase-Angle Measurements with an A.C. Polarograph J. K. Frischmann and Andrew Timnick Departm -nt of Chemistry, Michigan State University, East Lansing, Mich.
Transfer coefficient!; evaluated for the reduction of l.OmM Cd(ll) at the dropping mercury electrode at 24O & lo C for the following supporting electrolytes, 0.5M Na2S04,1.0M Na,S04, 0.5M H2S04,1.OM H2SOI, 1.OM NaC104, 1.0mM HCIO.I, 1.OM KNOI, 1.OM HCI, 1.OM NaCI, 1.OM KCI were: 0.21, 0.20, 0.27, 0.24, 0.31, 0.26, 0.50, 0.42, 0.46,0.44, respectively and the values for the heterogeneous rate constants, kr, for the same order of the above supporting electrolytes were: 0.076, 0.063, 0.14, 0.12,0.34,0.25,0.63,0.94,1.2,1.2, cm per sec, respectively. Straight line cot Q plots, with intercepts cot Q = 1, extrapolated to zero frequency, were obtained through the frequency rangci 30-1100 cps for all supporting electrolytes containing Cd(ll). The heterogeneous rate constants decrease with double-layer capacitance of supporting electrolyte salts and this correlation arallels the decrease in adsorbability of the electroyte anions on the mercury electrode.
P
DURING RECENT
a number of methods have been developed for the evaluation of the heterogeneous rate constant, kh, and the transfer coefficient, cy, for an electrode reaction. The newer methods are usually tested by measuring these electrode kinetic parameters for the reaction of Cd(I1). Numerous studies of the reduction of Cd(I1) present in various supporting electrolytes by various methods and different investigators have yielded a range of values for a and k h (1-10). YEARS
(1) H. H. Bauer and P. J. :Eking, ANAL.CHEM.,30,341 (1958). (2) H. H. Bauer and P. ,J. Elving, J. Am. Chem. SOC.,82, 2094 (1960). ( 3 ) H. H. Bauer and P. J. Elving, ANAL.CHEM., 30, 334 (1958). (4) B. Breyer and H. H. E:auer, Australian J. Chem., 9,425 (1956). (5) T. Biegler and H. Laitinen, ANAL.CHEM.,37,572 (1965). (6) W. F. Head, Anal. Chr’m. Acra, 23, 294 (1960). (7) P. J. Lingane and J. H. Christie, J . Electroanal. Chem., 10, 284 (1965). (8) R. S. Nicholson, ANAL.CHEM.,37, 1351 (1965). (9) Y.Okinaka, T a h t a , 11, 203 (1964). (10) N. Tanaka and R. Tamamushi, Electrochim. Acta, 9, 963 (1964).
48823
Recently, Bauer (11) has presented an explanation for the range of values of a and kh reported for aqueous Cd(I1). The original equations for the a.c. wave were derived for diffusion to a plane electrode by Matsuda (12) and more recently Delmastro and Smith (13) derived equations for diffusion to a spherical electrode. From these equations it is apparent that the electrode kinetic parameters can be evaluated by alternating current measurements, but the work involved is tedious. A more direct approach is through the measurement of the phase angle between the faradaic alternating current and the applied a.c. potential. This study was undertaken to evaluate a and kh for the reduction at the DME of Cd(I1) present in various supporting electrolytes by phase-angle measurements. THEORY
On the basis of Matsuda’s theory (14)for a diffusion and charge transfer controlled (quasi-reversible) electrode process,
O+ne$R Tamamushi and Tanaka (15) showed that the phase angle, 4, can be expressed as
where D = DoaDRa
(11) (12) (13) (14)
H. H. Bauer, J. Electroanal. Chem., 12,64 (1966). H. Matsuda, 2.Elecktrochem., 61, 489 (1957). J. D. Delmastro and D. E. Smith, ANAL.CHEM.,38,169 (1966). H. Matsuda, Z. Elecktrochem., 62,977 (1958). (15) R. Tamamushi and N. Tanaka, 2.Phys. Chem., N.F., 11, 89 (1959). VOL. 39, NO. 4, APRIL 1967
507
Thus, one can write
(4)
p = 1 - a
(5)
when the amplitude of the applied alternating potential is less than 8/n millivolts. Delmastro and Smith (13) showed that these relations are unaltered by contributions of spherical diffusion regardless of whether the reduced form is soluble in the solution or the electrode. For the special case whenj = 0 ( E d . c .= El$), a plot of cot 4 against c o l i z should yield a straight line with a slope of
G (1)'Iz with the obvious intercept of 1. slope and D known, pression
kh
or
Taking the natural log of both sides of Equation 11 gives
Rearranging yields
With values for the
can be simply calculated by the ex-
By plotting cot 4 cs. u112for each of the applied potentials, the resulting slopes a n d j values are used to calculate a from Equation 13. EXPERIMENTAL
On plotting cot q5 us. Ed.c.,cot 4 will be at a maximum (IO) when
(7) With the value for Ed.c.when cot 6 is at a maximum, read off the plot of cot 6 us. Ed.c.,and the known value for E1izR,a can be calculated. a may also be calculated from the d.c. potential dependence of the cot 4 us. u1/2 plots. The variation of cot q5 with changing frequency is obtained for two applied potentials, E d . c . , l and From Equation 1, it is apparent that the values of the cot 4 - w112slopes at and Ed.c.,2may be written
(9)
Table I. Supporting electrolyte
E1/zR,
1M 1M KNO, 1M HClOr 1M NaClO4 1M KCI 1M HCl 1M NaCl 4
6
Determined at the log
AE/A log
cat
508
SCE, V"
-0.604 -0.606 -0.594 -0.597 -0.585 -0.607 -0.573 -0.642 -0.644 -0.635
0.5M NazS04 1M NaSO4 0.5M HzSO4
= 0 intercept
0.059 (iyi) __. =
- 0.750.V us. SCE ANALYTICAL CHEMISTRY
(16) D. E. Smith, ANAL.CHEM., 35, 610 (1963). (17) J. K. Frischmann, Ph.D. Thesis, Michigan State University, East Lansing, Mich., 1966.
id, and DOValues for l.OmM Cd(I1) in Various Supporting Electrolytes
Ellz R,
us.
The a x . polarograph employed was similar to one constructed by Smith ( I d ) . Modifications in the phase-angle detecting circuit and performance are described elsewhere (17). A Hewlett-Packard Model 202A signal generator, calibrated against a Hewlett-Packard Electronic Counter Model 521A, provided all a x . potentials. A Sargent S-30260 potentiometer was used to measure all d.c. potentials applied to the cell. A Tektronix Model 502 oscilloscope was employed to measure the amplitude of the alternating potential applied to the cell. The Sargent Model M.R. and S.R. recorders were employed to measure the d.c. currents, rectified a.c. currents, and signals proportional to the phase-angle. The polarographic cell was constructed so that nitrogen could be either bubbled through the solution by way of a coarse frit at the bottom of the cell or blown over the top of the solution. The cell was fitted with a 3-hole rubber stopper through which the electrodes (dropping mercury
Av. id, PA
sloe
5.31 4.87 5.98 5.81 6.40 6.54 6.33 6.85 6.69 6.59
0.0296 0.0292 0.0296 0.0295 0.0293 0.0296 0.0293 0.0295 0.0294 0.0298
from the plot of log
Do X IO6,
m,
cmz/sec
t, See
mg/secc
4.52 3.93 5.70 6.41 6.46 6.67 6.45 7.30 6.70 6.82
5.40 5.42 5.34 5.39 5.38 5.32 4.99 5.39 5.41 5.42
1.676 1.666 1.684 1.678 1.676 1.687 1.686 1.678 1.673 1.673
Table 11. Values of the Transfer Coefficient, a, Obtained from Slope Measurements for l.OmM Cd(I1) in Various Supporting Electrolytes Supporting Slope5 at Slope0 at Slope" at ffb electrolyte El14 Eilz E314 0.22 f 0.03 2.26 X 2.72 X 1.42 X lodz 0 . 5M Na2S04 3.05 X 10-* 0.22 & 0.03 2.63 X 1W2 1.67 x 1M NapS04 0.25 rt 0.03 1.42 X 10-* 1.58 X 0.911 X 0.5M 0.24 rt 0.03 1.57 X 1.77 X 1.01 x 10-2 1M 9.11 X lW3 0.29 f 0.04 8 . 3 3 X lW3 5.78 X 1k' HClO4 6.55 x 10-8 0.29 f 0.04 6.11 X lWa 4.11 x 10-3 1M NaClO4 A cot 4 a Determined from ~3 at a constant d.c. potential Obtained from ratio of slopes El14/E112, and
see Equation 13.
electrode, saturated calomel electrode, and a platinum wire auxiliary electrode) were introduced. The side arm of the reference electrode was fitted with a fine frit and was separated from the polarographic solution by an isolation compartment with a fine frit. The auxiliary electrode was also separated from the polarographic solution with a similar isolation compartment. The isolation compartments usually contained the supporting electrolyte. All polarograms were recorded manually and measurements were made in a constant temperature room which had an ambient temperature of 24" f 1" C. Measured phaseangles were corrected Yor the effects of double-layer capacitance and total uncompensated series resistance according to the method of Bauer artd Elvihg (18). Values of Cd.l. and R , necessary for the above corrections, were determined experimentally on supporting electrolyte solutions, and are tabulated in Table IV. An alternating potential amplitude of 5.00 mV was used and all measurements were made at maximum drop size. Calibration of the phase-angle was made before and after each measurement since the long-term phase stability of the oscillator was about 0.5". The uncertainty in the faradaic phase-angle measurements was at most 0.5" depending upon the magnitudes of the faradaic current, of the correction for total uncompensated series resistance and double-layer charging current. Solutions were prepared with reagent grade chemicals without further purification. A 0.0100M Cd(I1) stock solution was prepared from Cd(NO&. 4Hz0 and standardized by EDTA titration (IS'). None of the solutions contained maximum suppressor. All solutions were prepared from the laboratory distilled water supply. These solutions yielded no different results than when triply distilled water was used. Dissolved oxygen was removed from cell contents by bubbling with nitrogen, for a period of 20 minutes. Last traces of oxygen were removed from the nitrogen by bubbling it through an acid vanadous sulfaie solution, dilute NaOH, and HzO before passing it into the polarographic cell.
RESULTS AND DISCUSSIONS Direct current polarographic data obtained with 1.OmM Cd(I1) in various supporting electrolytes yielded linear log (idi)/i us. &. plots with a slope of 0.0295 0.004 indicating a d.c. polarographically reversible two-electron reduction. The halfwave potentials, determined from the log = 0 intercept of the above plot, and the diffusion coefficient, calculated by the Lingane-Loveridge equation (20), are tabulated in Table I for
*
(18) H. H. Bauer and P. J. Elving, J. Am. Chem. SOC.,82, 2091 (1960). (19) G. Schwarzenbach, "Complexometnc Titration," Interscience, New York, 1957, pp. 55, 83-86. (20) J. J. Lingane and B. A. Loveridge, J. Am. Chem. SOC.,72, 439 (1950).
Table 111. Values of the Transfer Coefficient, a, for l.OmM Cd(m in Various Supporting Electrolytes Supporting electrolyte
(&e
- EIItR).,
ab
a=
0.21 f 0.03 0.20 rt 0.03 0.27 f 0.03 0.24 f 0.03 0.50 i 0.05 0.26 f 0.04 0.31 f 0.04 0.44 f 0.05 0.42 f 0.05 0.46 f 0.05
0.21 rt 0.03 0.22 f 0.03 0.25 f 0.03 0.24 f 0.03
mV. -16.5 -17.5 -12.5 -14.5 0.0 -13.5 -10.0 -3.0 -4.0 -2.0
...
0.29 f 0.04 0.29 f 0.04
... ... ...
Potential at which cot 9 is maximum. Calculated when cot 9 was maximum by Equation 7. 9 average value determined from AAcot at two different d.c. wl/* potentials, see Equation 13. 0
~
the Cd(I1) reduction in the various media. It should be noted that the half-wave potential for Cd(1I) in 1 M HCIOI shifts with the type and concentration of solution in the reference isolation compartment due to a junction potential at the A log (id-i)/i frits. However the value of the slope is indeAE pendent of the concentration and type of electrolyte in the isolation compartment. For the experiments involving C104-, the reference isolation compartment contained 1M N a N 0 3 to prevent precipitation of Kclo4 in the frit of the reference electrode, and the half-wave potentials are reported on this basis. Phase-angle measurements were made on l.OmM Cd(I1) in various supporting electrolytes at fifteen different frequencies between 30 to 1100 cps and at three d.c. potentials, Ell4,EIIZR, E3,a. At 30 and 820 cps, phase angle measurements were made at 10-mV intervals through the d.c. potential range from 50 mV to 50 mV beyond it. before EllZR The transfer coefficients for the reduction of Cd(I1) in various media are tabulated in Table I1 and were calculated by Equation 7 employing the d.c. potential when cot 4 is a maximum in the plot of cot 4 us. Ed.c.at 820 cps and the El$ values in Table I. The values of the transfer coefficient were also calculated by Equation 13 using the slopes A ~
A u1I2
at two different d.c.
VOL 39, NO. 4, APRIL 1967
509
""
1.8
-
-26
2.6
2.2-
m W 7
-
2.2
1.85
0
cg
tJ.2
I- 1.8
1.4
0
0 0
U
Lot -
I .4
I
I
I
I
I
I
I
I
30
20
IO
0
-10
-20
-30
-40
Ed.c:
1.0
I
0
10
I
20
I
30
wi?,
I
I
40 50 RAOIANS~
I
60
I
SEC-t
70
I
80
J 90
Figure 1. Variation of cot $ with changing a x . frequency for 1mM Cd(I1) in various supporting electroIytes Curve A B
C D
Supporting electrolyte
EY,J, M V
0
Experimental points
0 Calculated points; k = 0.14 cm/sec, a = 0.27
Curve
LCPS
34.4 828
A B
A cot 4 slope -at Ear.c. = E1lZRand values of D. D was calA w1'2
culated with the aid of Equation 2 using the values of D o listed in Table I, a value of DR = 1.61 X 10+ (4, and CY obtained by Equation 7. The plots of cot 4 us. w1/2 at j = 0 for 1.OmM Cd(I1) in various media yielded straight lines up to frequencies of 1100 cps with intercepts at cot 4 = 1 when extrapolated to zero frequency, which is expected for an electrode process which is
1
Figure 2. Variation of cot $ with changing Edc for 1mM Cd(II) in 0.5M H2.904. Uncertainties of * O S 0 in phase angle indicated on experimental points
1M HCI 1M NaCIOp 1M HBOr 0.5M HzSOa
potentials and are tabulated in Table 111. The transfer coefficients for the reduction of Cd(1I) in the chloride and the nitrate media were not calculated in this manner since the ratio of the uncertainties in the slopes becomes competitive with the values of the ratio of the slopes (Table 111, Column 3, last 3 entries). The heterogeneous rate constants (Table IV) were calculated with the aid of Equation 6 employing the values of the
41.0
I
40
diffusion and charge transfer controlled. Previous investigators (I, 2) were not successful in attaining linearity over such a wide frequency range. Typical plots of cot $I us. d/* are illustrated in Figure 1 for 1mM Cd(I1) in some of the supporting media. From the experimental values of kh, CY,and D,a theoretical plot of cot 4 us. Ed.c.was calculated by Equation 1 for frequencies of 30 and 820 cps. The calculated and the experimental points are all in excellent agreement if the uncertainty of the faradaic phase-angle is assumed to be =k 0.5 Figure 2 illustrates the agreement between the calculated points and experimental points for the reduction of Cd(1I) in 0.5M HzS04at low and high frequency. The gross apparent differential capacity of the doublelayer of the supporting electrolytes at various d.c. potentials is illustrated in Figures 3 and 4. The decreasing order of the double-layer capacitance for 1 M salts is: KC1 > NaCl > KNOI > NaCIOa > Na2S04; and for the 1M acids is: HC1 > HCIOl > HzS04. The values of kh for both the salt and acid decrease with decreasing double-layer capacitance.
Table IV. Values for the Heterogeneous Rate Constant, kh, for l.OmM Cd(I1) in Various Supporting Electrolytes. Values for Cd+ and R ,for Electrode Used Are Included Slopea Supporting electrolyte 0.5M Na2S04 1M NazS04 0.5M HzSO4 1M HzS04 1M KN03 1M HClOi 1M NaC104 1M KCl 1M HC1 1M NaCl
510
ANALYTICAL CHEMISTRY
A cot $ Awl/z
2.26 2.63 1.42 1.57 0.358 0.833 0.611 0.189 0.233 0.189
x
102
D X lo6,
Cdli,'
R I,
cmP/sec9
Ff
ohms
crnlsec
5.89 5.51 1.54 7.03 10.20 8.38 8.57 10.34 9.68 10.36
0.816 0.833 0.857 0.952 1.14 1.09 1.11 1.11 1.12 1.08
87.3
0.076 0.063 0.14 0.12 0.63 0.25 0.34 1.2 0.94 1.2
11.2
64.4 58.9 76.4 60.6 81.3 72.6 60.2 78.0
kh,
1-35
I .25
IC
1.15 r
a
-- 1.05
d
0
0.95 0.85
0.52
0.54
0.56 0.58 Edc,
VS.
0.60 0.62
0.64
0.66
Figure 3. Variation o:f double-layer capacitance with changing applied d.c. potential for various supporting electrolytes Curve A B C D E
Supporting electrolytes 0.5M HzS04 1M Na2SO4
1M NaC104 1M KNOI 1M HCl
Curve
0.63
D
0.94
cesses. Delahay (21) predicts that the parameter, kh, should be increased by specific adsorption of an anion for the reduction of a nonadsorbed cation and decreased by specific adsorption of a cation. Experimental results from this study bear out the anion effect prediction. Delahay (22) lists the amounts of specifically adsorbed anion on a mercury electrode at a constant d.c. potential. The degree of adsorption is
> sod-’
while Grahame (23) in earlier work determined the order to be
For sodium or potassium as the cation, the decrease in doublelayer capacitance parallels the decrease in adsorbability of the (21) P. Delahay, “Double Layer and Electrode Kinetics,” Interscience. New York, 1965.. P. _ 206. (22) Zbid:,p. 61. . (23) D, C. Graharne, Cherrl. Rev., 41,441 (1947).
0.54
0.56
058
0.60
0,62 0.64
0.66
Figure 4. Variation of double-layer capacitance with changing applied d.c. potential for various supporting electrolytes
k , cm/sec 0.14 0.063 0.34
It is not too unexpected that changes in the double-layer structure or electrode surface would affect the charge transfer rate because of the heterogeneous nature of the electrode pro-
c1- > N03-
0.52
S.C.E., v
A B
C
Supporting electrolyte 0.5M Na2S04 1M HzSO4 1M HClO4
1M NaCl or KCl
k, cm/sec 0.076
0.12 0.25
1.2
anions. On comparing the values of the heterogeneous rate constants for NaC1, NaC104, or NazSOa media to those of the corresponding acid and making the same comparison for double-layer capacitance, no direct correlation is possible. APPENDIX a,p = Transfer coefficients Di = diffusion coefficient of species i. Ed.c.= Applied d.c. potential. E l i Z R = d.c. reversible half-wave potential. E” = standard redox potential. F = Faraday’s constant. k,, = apparent heterogeneous rate constant uncorrected for activities. = number of electrons transferred. n R = the gas constant. T = absolute temperature. = angular frequency of applied alternating potential. w = phase-angle between the applied alternating potential $t and the fundamental faradaic current.
RECEIVED for review August 4, 1966. Accepted February 13, 1967. Fellowship support was provided by the Socony Mobil Co. and the National Science Foundation.
VOL. 39, NO. 4, APRIL 1967
51 1
